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Visual PEST-ASP

1. 0 000 5 16 10 1 10 6 12 6 Parameter variation file c cccccecssccsseceeseeeesseeees 11 3 PataMetersye csi non r a AR ieee 1 1 1 5 PARCHG IM iiss iaa aaa 2 21 4 18 Parent parameter 1 5 2 19 4 17 4 19 PARGP nsina ia NR E a 4 19 PARGPNME eenean nerit 4 14 PARLBND 5 fecsesciscesviccnscvecssvesspesevntcbsecsicecvevens 2 19 4 18 PARNMEB ee ris Se ted eres ee get Sat 4 17 4 25 PARREP 1 12 5 39 5 40 10 8 PARTIED innra desiccation eee 4 19 PARTRANS e EREE TOATA 2 19 4 17 PARUBND ssnecrciisicinieviiiiiiariikii 2 19 4 19 PARVAL lyeevennenun aran aE Ou 4 18 PATH 4 21 5 23 PRO PANE ESES AE AN OES EEE 6 12 PO linan a E RR 6 13 PDZ a svecisvcessetevec descent ostvenve orale E 6 13 PEST control file eccccesscesssceeseceeseeeeseeeesseeees 6 10 PEST SUP innnan 5 23 PESTCHEK 0 0 000 1 12 3 26 4 1 9 12 10 4 12 9 PESTGEN c cccccecesecesseeessseeeneeees 1 12 4 1 10 5 12 8 PESTMODE sivesecisteceteccveies cooseisvacdivectvesdsscsthecventeasened 6 12 PEST to model cccccccscccssscessceseseceesseeesseeees 2 32 8 4 PHIMLIM PHIRATSUF PHIREDLAM PHIREDST Presisccssiscvcceovecsncctvsnavecs POSt PplOCESSIN Gs inier ai 5 36 PPAUSE eroen e 1 11 5 23 9 13 PPES Tnt r E EAR 5 1 9 13 PRECIS pera ee 3 6 4 6 5 16 10 2 11 6 Precision 2 31 3 4 3 6 4 6 4 22 5 9 Predictive Analysis nacosoreni ite S 6 1 Primary Marker soussessetecamaeosesciwe
2. scf control data noverbose 5 19 2 3 single point sensan files parvar dat outl dat out2 dat out3 dat model command line model gt nul model input output ves tpl vesl inp ves tp2 ves2 inp vesl ins vesl out ves2 ins ves2 out ves3 ins ves3 out Example 11 2 A SENSAN control file SENSAN 11 5 scf control data SCREENDISP NPAR NOBS NTPLFLE NINSFLE PRECIS DPOINT sensan files VARF LE ABSFLE RELE SENSFL model command line write the command which SENSAN must use to run the model model input output TEMPFLE INFLE I one such line for NTPLFLE template files SFLE OUTFL one such line for NINSFLE instruction files Example 11 3 Structure of the SENSAN control file The role of each SENSAN input variable is now discussed 11 2 6 Control Data SCREENDISP SCREENDISP is a character variable which can take either one of two possible values These values are noverbose and verbose In the former case when SENSAN runs TEMPCHEK and INSCHEK it redirects all of the screen output from these programs to the nul file hence the user is not aware that they are running In the latter case TEMPCHEK and INSCHEK output is directed to the screen in the usual fashion Once you have set up a SENSAN run and ensured that everything is working correctly a nicer screen display is o
3. 2 2 PEST s Implementation of the Method So far this chapter has discussed the theory behind PEST viz the method of weighted least squares and its application to nonlinear parameter estimation and predictive analysis The remainder of this chapter discusses the ways in which the least squares method has been implemented in PEST to provide a general robust parameter estimation and predictive analysis package that is useable across a wide range of model types 2 2 1 Parameter Transformation The PEST Algorithm 2 19 PEST allows for the logarithmic transformation of some or all parameters It often happens that if PEST or any other parameter estimation program is asked to estimate the log of a parameter rather than the parameter itself the process is much faster and more stable than it would otherwise be however sometimes the opposite can occur PEST requires that each parameter be designated in the PEST control file as untransformed log transformed fixed or tied the latter two options will be discussed shortly If a parameter is log transformed any prior information pertaining to that parameter must pertain to the log to base 10 of that parameter Also elements of the covariance correlation coefficient and eigenvector matrices calculated by PEST pertaining to that parameter refer to the log of the parameter rather than to the parameter itself However PEST parameter estimates and confidence intervals listed in the run record f
4. Example 10 2 Running PESTCHEK and PEST as a batch process 10 4 PESTGEN Program PESTGEN generates a PEST control file In most cases this file will need to be modified before PEST is run as PESTGEN generates default values for many of the PEST input variables supplied on this file it is probable that not all of these default values will be appropriate for your particular problem PESTGEN is run using the command pestgen case parfile obsfile where PEST Utilities 10 6 case is the case name No filename extension should be supplied PESTGEN automatically adds the extension pst to case in order to form the filename of the PEST control file which it writes parfile is a parameter value file and obsfile is an observation value file A parameter value file is shown in Example 5 2 Example 10 1 shows an observation value file The former file must include all parameters used in the current case these parameters may be cited in one or a number of template files Similarly the observation value file must provide the name and value for all observations used in the current problem the observations too may be cited on one or a number of instruction files The observation values provided in this file may be field laboratory measurements or if PEST is being run on theoretical data model generated observation values In the latter case program INSCHEK may be used to generate the file if there are multiple model output files
5. Each line within the second part of the parameter data section of the PEST control file consists of two entries The first is PARNME the parameter name This must be the name of a parameter already cited in the first part of the parameter data section and for which the PARTRANS variable was assigned the value tied The second entry on the line the character variable PARTIED must hold the name of the parameter to which the first mentioned parameter is tied ie the parent parameter of the first mentioned parameter The parent parameter must not be a tied or fixed parameter itself Note that PEST allows you to link as many tied parameters as you wish to a single parent parameter However a tied parameter can naturally be linked to only one parent parameter The PEST Control File 4 20 4 2 6 Observation Groups In the observation groups section of the PEST control file a name is supplied for every observation group Like all other names used by PEST observation group names must be of twelve characters or less in length and are case insensitive A name assigned to one observation group must not be assigned to any other observation group Observation group names are written one to a line NOBSGP such names must be provided where NOBSGP is listed on the fourth line of the PEST control file If PEST is running in predictive analysis mode one of these group names must be predict If it is running in regularisati
6. del or it can be the name of a user supplied executable program or batch file The system command to be run after any particular model run should be written to the parameter variation file following the parameter values pertinent to that model run In many cases the command will simply be the copy command ensuring that model output files are stored under different names before they are overwritten during subsequent model runs Example 11 5 shows such a case SENSAN 11 8 dep1 dep2 resl res2 res3 1 0 10 0 Bia 2 0 10 0 copy model out modell out 2 0 10 0 5 0 2 0 10 0 copy model out model2 out 1 0 TELO 5 0 2 0 10 0 copy model out model3 out 1 0 10 0 6 0 2 0 10 0 copy model out model4 out 1 0 TOSO 5 0 340 10 0 copy model out model5 out 1 0 10 0 53 0 2 0 11 0 copy model out model6 out Example 11 5 A parameter variation file for a SENSAN run in which system commands are run after the model In Example 11 5 the model run by SENSAN produces a file called model out After each model run this file is copied to a different file whose name is associated with that run These files can later be inspected or processed in a fitting manner SENSAN assumes that any characters following the NPAR numbers representing the NPAR parameter values for a particular run constitute a system command which it duly delivers to the operating system after the model has run SENSAN takes no responsibility for incorrect commands nor does it
7. or switch If it is supplied as always_2 derivatives calculation is through forward differences for all parameters within the group throughout the estimation process if it is always_3 central ie three point derivatives will be used for the entirety of the estimation process However if it is switch PEST will commence the optimisation process using forward differences for all members of the group and switch to using central differences on the first occasion that the relative reduction in the objective function between optimisation iterations is less than the value contained in the PEST input variable PHIREDSWH Two group input variables pertain specifically to the calculation of derivatives using the central method viz variables DERINCMUL and DERMTHD The latter variable must be one of outside_pts parabolic or best_fit this determines the method of central derivatives calculation to be used by PEST the three options having already been discussed The variable DERINCMUL contains the increment multiplier this is the value by which DERINC is multiplied when it is used to evaluate increments for any of the three central derivatives methods Sometimes it is useful to employ larger increments for central derivatives calculation than for forward derivatives calculation especially where the model The PEST Algorithm 2 30 output dependence on parameter values is bumpy see the next section Use of
8. parameter data infiltl 0 3456 infilt2 infiltl Sinfiltrat2 infilt3 infilt2 Sinfiltrat3 template and model input files model tpl model in Example 10 8 A template for the PAR2PAR input file of Example 10 7 It is apparent that when using PAR2PAR as part of a composite model run by PEST there are two sets of template files involved in the inversion process viz that used by PEST to write a PAR2PAR input file and those used by PAR2PAR to write model input files These should not be confused PEST should never be instructed to use a template file to write a model input file that is also cited in the template and model input files section of a PAR2PAR input file If this happens the model input file generated by PEST will be overwritten by that generated by PAR2PAR It often happens that only a few parameters required by a model need to be calculated by an expression cited in a PAR2PAR input file other model parameters can be estimated directly by PEST These latter parameters can simply be passed through PAR2PAR by assigning them numerical values in the pertinent expressions in the PAR2PAR input file Example 10 9 shows a PAR2PAR input file in which only parameter par is calculated through manipulation of other parameters Example 10 10 shows the corresponding template file of the PAR2PAR input file Parameters par to par5 are passed directly to the model through the template file model tp of the model input file
9. partial parallelisation thanks to the fact that lambda values used by PEST in this search are all related to each other by multiples of LAMFAC Thus if PEST is run as Parallel PEST and if it has access to a number of processors it can undertake simultaneous model runs across these different processors using parameters calculated on the basis of a series of lambda values related to each other by factors of LAMFAC To some extent PEST must guess which lambdas to use for these parallelised model runs If it turns out that some of these lambdas are actually not required then nothing will have been lost because the respective processors contributions to the lambda search procedure would have also been zero if they were undertaking no model runs at all While Parallel PEST allows such a partial parallelisation of the lambda search to be undertaken parallel lambda runs will not be undertaken e if at least one parameter is frozen at its upper or lower bound e if PEST is running in predictive analysis mode and a line search is undertaken as part of the predictive analysis process e if the model run time for the fastest processor involved in the parallelisation process is less than 1 8 times the model run time for the second fastest processor see below e if only one slave is currently available for the undertaking of model runs this may happen if all but one slave is currently completing redundant model runs arising out
10. section of a PEST control file The regularisation section of the PEST control file must begin with a single line containing the character string regularisation Then follow three lines each of which contains a number of variables which control the way in which PEST operates when working in this mode The role of each of these variables is now discussed PHIMLIM This is m of equation 2 31 That is it is the upper limit of the measurement objective function ie the upper level of model to measurement misfit that is tolerable when trying to minimise the regularisation objective function In some cases a PEST regularisation run will postdate a normal parameter estimation run If the latter run was successful it will have informed the user of how low the measurement objective function can be if all parameters are adjusted without reference to any regularisation conditions Pm should be set somewhat above this for the imposition of regularisation constraints will mostly result in a slight diminution of PEST s ability to fit the field data exactly The user informs PEST of the extent to which he she will tolerate a less than optimal fit between model outputs and field data for the sake of adhering to the reality check imposed by the regularisation constraints through the variable PHIMLIM PHIMACCEPT During each optimisation iteration just after it has linearised the problem through calculating the Jacobian matrix and
11. xxx Parallel PEST 9 19 9 3 12 The Model The model can be any executable or batch program which can be run from the command line 9 4 An Example Once PEST has been installed a subdirectory of the main PEST directory called ppestex will contain all the files needed to undertake a Parallel PEST run on a single machine Before running this example make sure that the PEST directory is cited in the PATH environment variable Change the working directory to the ppestex subdirectory and create two subdirectories to this directory called test and test2 Now open three command line windows you probably have one open already In one of these windows transfer to subdirectory test and type the command pslave When prompted for the command to run the model type a_model Do the same in another command line window for subdirectory test2 In the third command line window transfer to the directory holding the example files ie the parent directory of test and test2 and type the command ppest test Parallel PEST should commence execution and after verifying that it can communicate with each of its slaves undertake parameter optimisation for the a_model model If you wish you can also carry out the parameter estimation process using the single window version of PEST While situated in the ppestex subdirectory type pest test For this particular case the single window version of PEST will run faster than Parallel
12. Example 3 9 demonstrates the use of semi fixed observations Semi fixed observations are similar to fixed observations in that two numbers are provided in the pertinent instruction item the purpose of these numbers being to locate the observation s position by column number on the model output file However in contrast to fixed observations these numbers do not locate the observation exactly When PEST encounters a semi fixed observation instruction it proceeds to the first of the two nominated column numbers and then if this column is not occupied by a non blank character it searches the output file line from left to right beginning at this column number until it reaches either the second identified column or a non blank character If it reaches the second column before finding a non blank character an error condition arises However if it finds a non blank character it then locates the nearest whitespace on either side of the character in this way it identifies one or a number of non The Model PEST Interface 3 21 blank characters sandwiched between whitespace whitespace includes the beginning and or the end of the model output file line It tries to read these characters as a number this number being the value of the observation named in the semi fixed observation instruction Obviously the width of this number can be greater than the difference between the column numbers cited in the semi fixed observation instruction Like a
13. and 10 However if there are many adjustable parameters and PEST consumes a large amount of time in determining the optimal weight factor a tolerance of somewhat higher than 10 may prove suitable 7 3 4 The Control Variable FRACPHIM As was mentioned above a non zero value can be supplied for FRACPHIM if you would like to use PEST in regularisation mode but you are unsure of what value to use for PHIMLIM If FRACPHIM is provided with a value of zero or less or if this variable is absent from the PEST control file then PEST s action when working in regularisation mode will be exactly the same as that already described However if FRACPHIM is provided with a value of between 0 0 and 1 0 values of 1 0 or greater are illegal then PEST will calculate a new value for PHIMLIM at the beginning of each optimisation iteration This value will be calculated as the current value of the measurement objective function times FRACPHIM Thus PEST will always aim for a measurement objective function that is lower than the current one This allows it to lower the measurement objective function as the parameter estimation process progresses while at the same time making use of the numerically Regularisation 7 9 stabilising effects of regularisation The following aspects of the use of the FRACPHIM variable should be carefully noted 1 If FRACPHIM is supplied as 0 0 or as less than 0 0 PEST will assume that no value has been supplied for
14. button Q Other options are specific to each graph and will be discussed in the following sub sections WinPEST includes two text viewing windows one for the PEST log and one for the PEST run record file as well as a single window for general text editing Options available in this latter window are as follows Close Close the file that is currently being edited Go to Go to a specified line number Font Choose the font used for displaying text Print Print the file being currently edited Preparing for a PEST Run 6 3 Preparing fora PEST Run 3 1 Reading a Case Prior to starting WinPEST all PEST input files should have been prepared viz the PEST control file and all template and instruction files cited therein Once this has been done WinPEST execution can be started in either of the following ways 1 selection through the Windows Start menu 2 by typing WinPEST at the command line prompt In the latter case a desired PEST control file can be read on startup by typing the command winpest f case where case is the filename base of the PEST control file for the current case you can include the pst extension if you wish Once WinPEST has been started the WinPEST window appears This window can be maximised or minimised by clicking on the appropriate button in its top right corner it can be re sized using the mouse in the normal Windows fashion A PEST control file can be opened
15. r Switch 0 ee eeeesceseceseceeeeeneeeseeeseeesseeeseeeaes 5 24 5 4 3 Restarting PEST with the j Switch sssescssrsssssessesssssseesennenscsssvonsoaes 5 25 22D H PESE Wont Optimise eneo irnn cease ccaduseutecentgns sed EEEE O RE ETOO ERER 5 25 Dia General oein E aE EE E EAEE EE aK A ASSE a iS 5 25 5 5 2 Derivatives are not Calculated with Sufficient Precision ccesseeeseeees 5 26 5 5 3 High Parameter Correlation ccecceecceseceseeseeeceseceaceesceesaeesaeeseeesaeesaeeeseeeues 5 26 5 5 4 Inappropriate Parameter Transformation ceeeeceseceseeceeeceeeceeeeeeeeeeeeees 5 27 5 5 5 Highly Nonlinear Problems 00 eeeeeceecceseeceeeceeecesceesceesaeesaeeeeeesaeeeaeeeseeeues 5 28 5 5 6 Discontinuous Problems ccccssssesseccccccccesssceccecsccscessscescescesseuasecescess 5 28 Table of Contents 5 5 7 Parameter Change Limits Set Too Large or Too Small eee eeeeeseeeeeeeees 5 28 5 5 8 Poor Choice of Initial Parameter Values cecceesececeseecesteeeeeeeeesteeeeeeeeens 5 29 5 5 9 Poor Choice of Initial Marquardt Lambda tees eesessceesceesceeeeeeeeeeeeeeees 5 29 5 5 10 Observations are Insensitive to Initial Parameter Values eecceeeeeeeees 5 30 5 5 11 Parameter Cannot be Written with Sufficient Precision cceeseeeeeeeees 5 31 5 5 12 Incorrect Instructions eneen oiea e e ea e e a 5 31 5 5 13 Upgrade Vector Dominated by Insensitive Parame
16. 4 9 5 10 5 28 5 33 Factor limited parameters 2 22 4 9 5 10 5 29 Fixed observations ccccccccssscesssceeseeeesseeeesseeesseeees 3 19 Fixed parameters 2 sc csceicsoscendeonescaereoxenseesd devestasvuzewes 4 17 FORCEN 2 29 4 15 5 11 Formatted input osere rea E a a 3 5 FORTRAN areeiro tiraa AE 200s 3 5 3 7 3 20 Forward derivatives 1 8 2 27 2 29 4 10 4 15 FRACPHIM reregan renina n oiia a 7 7 7 9 Frozen parameters cece eee eee eee sees 2 20 4 8 5 11 Gradient vectori no n a 2 8 2 20 Hemstitching T OR i riye paiana ia EERE E IEO D AEETI T A I AT JEI G rniii iena i TENET DA EEEE E OE INCTYP INFLE Initial parameter values INITSCHFA C nrinn ioiei 6 14 INSCHEK ccccccsceeeseeees INSFLE Instability Installation Instruction files 0ceeeee 1 7 4 6 4 24 10 3 12 7 Tastiictions 3s es sisi A ARATE 1 7 10 3 Interpretation s secese iaa 1 2 TGC 8I AEEA A EE E E E 1 11 JACFILE 4 6 8 7 Jacobi sreo a a Noted 1 13 Jacobian matrix cccceeeeee 2 6 2 27 4 15 5 9 5 35 JACWR ITA ai EVA ARa 1 13 10 9 LAMBDA anane a A EEA 5 33 Line advance 0 cccccccessscessscceseeeeeseceesseeeseeesseeeeeeee 3 15 Lineat modelSirsnoriiissosiniiisiis iiaii 1 7 List directed input 00 0 ee eee eee eseeteeseeeeeeeeeee 3 5 12 5 Log least squares 0 0 cece nas 4 21 Logarithmic transformation 2 1
17. However defining the parameter spaces in this way would achieve nothing as there would be no advantage in using less than the full 10 characters allowed by the model A B C D E Example 3 5 Fragment of a template file corresponding to Example 3 4 Parameters D and E are treated very differently to parameters A B and C As Example 3 3 shows the model simply expects two numbers in succession If the spaces for parameters D The Model PEST Interface 3 6 and E appearing in Example 3 5 are replaced by two numbers each will be 13 characters long the model s requirement for two numbers in succession separated by whitespace or a comma will have been satisfied as will PEST s preference for maximum precision Comparing Examples 3 4 and 3 5 it is obvious that the spaces for parameters D and E on the template file are greater than the spaces occupied by the corresponding numbers on the model input file from which the template file was constructed the same applies for the parameter spaces defined in Example 3 2 pertaining to the model input file of Example 3 1 In most cases of template file construction a model input file will be used as the starting point In such a file numbers read using free field input will often be written with trailing zeros omitted In constructing the template file you should recognise which numbers are read using free field input and expand the parameter space to the right accordingly be
18. It uses an identical model interface protocol to PEST writing model input files on the basis of user supplied templates and reading output files with the aid of a user prepared instruction set In fact SENSAN communicates only indirectly with a model using the PEST utilities TEMPCHEK and INSCHEK to write and read model files these programs are run by SENSAN as system calls Like PEST SENSAN runs a model through a command supplied by the user There is no reason why a model cannot be a batch file housing a number of commands Thus a model can consist of a series of executables the outputs of one constituting the inputs to another or simply a number of executables which read different input files and generate different output files SENSAN can write parameter values to many input files and read model outputs from many output files SENSAN is limited in the number of parameters and observations that it can handle both through the internal dimensioning of its own arrays and those belonging to TEMPCHEK and INSCHEK which it runs This will rarely pose a problem for it is in the nature of sensitivity analysis that adjustable parameters do not number in the hundreds nor selected model outcomes in the thousands Nevertheless if either SENSAN TEMPCHEK or INSCHEK reports that it cannot allocate sufficient memory to commence or continue execution or that the maximum number of parameters or observations has been exceeded contact Waterma
19. The fact that model input files are written and model output files are read by PEST across a network underlines the point made above that the potential reduction in overall PEST run time that can be achieved by undertaking model runs in parallel will only be realised if model run times are large compared with the delays that may be incurred in writing and reading these files across a network 9 2 2 The PEST Slave Program While Parallel PEST is able to achieve access to model input and output files residing on other machines through the use of shared subdirectories it cannot actually run the model on another machine only a program running on the other machine can do that Hence before PEST commences execution a slave program must be started on each machine on which the model will run Whenever PEST wishes to run a model on a particular machine it signals the slave running on that machine to start the model Once the model has finished execution the slave signals PEST that the model run is complete PEST can then read the output file s generated by the model The slave program named PSLAVE must be started before Parallel PEST on each machine on which model runs are to be undertaken It detects the commencement of PEST execution through reading a signal sent by PEST as the latter starts up It then awaits an order by PEST to commence a model run upon the arrival of which it sends a command to its local system to start the model It is possi
20. The result may be that convergence takes place intolerably slowly or not at all with a huge wastage of model runs This phenomenon can be avoided by temporarily holding troublesome parameters at their current value for an iteration or two Such parameters are then not involved in the calculation of the parameter upgrade vector and hence do not get the chance to have an adverse impact on it Offending parameters can be identified as those undergoing the maximum relative or factor limited changes during a particular optimisation iteration where this maximum change is equal to RELPARMAX or FACPARMAX or as those parameters whose current sensitivity is very low PEST recording information by which to make this assessment every time it calculates the Jacobian matrix PEST records the composite sensitivity of each parameter ie the magnitude of the column of the Jacobian matrix pertaining to that parameter modulated by the weight attached to each observation divided by the number of observations or V m where Vj is the inverse of S defined in equation 2 22 and m is the number of observations see equation 5 1 to a parameter sensitivity file this file being updated during every optimisation iteration Those parameters with the lowest sensitivities are the most likely to cause trouble and hence the most likely candidates for being temporarily held if the parameter estimation process proceeds too slowly as a result of one or more parameter
21. These latter two parameters are defined by the relationships infiltrat2 infilt2 infiltl 10 1a PEST Utilities 10 11 and infiltrat3 infilt3 infilt2 10 1b Desired infiltration parameter ordering relationships will be maintained if each of infiltrat2 and infiltrat3 is provided with a lower bound of 1 0 in the parameter estimation process implemented by PEST In using this device to ensure that correct infiltration parameter ordering relationships are maintained PEST must see parameters infilt infiltrat2 and infiltrat3 while the model must see parameters infiltl infilt2 and infilt3 The necessary parameter transformation process can be accomplished by running the utility program PAR2PAR as a model preprocessor contained in a composite model run by PEST as a batch file PAR2PAR receives the current PEST calculated values of infiltl infiltrat2 and infiltrat3 it then transforms these into values for infilt infilt2 and infilt3 Then it writes one or more model input files based on appropriate template files containing the current values of these native model parameters Based on equations 10 1a and 10 1b PAR2PAR must be programmed to calculate infilt2 and infilt3 using the relationships infilt2 infilt infiltrat2 infilt3 infilt2 infiltrat3 10 7 1 2 Seasonal Parameter Variations Some model parameters show seasonal variation For environmental models which simulate water
22. WFFAC WFTOL When PEST calculates the appropriate regularisation weight factor to use during any optimisation iteration it uses an iterative procedure which begins at the value of the regularisation weight factor calculated for the previous optimisation iteration for the first optimisation iteration it uses WFINIT to start the procedure In the process of finding the weight factor which under the linearity assumption used in its calculation will result in a measurement objective function ie m of PHIMLIM ie m PEST first travels along a path of progressively increasing or decreasing weight factor it decides which one of these alternatives to explore on the basis of the value of the current measurement objective function with respect to PHIMLIM In undertaking this exploration it either multiplies or divides the weight factor by WFFAC it continues to do this until it has found two successive weight factors which lie on either side of the optimal weight factor for that optimisation iteration Once it has done this it uses Newton s method to calculate the optimal weight factor through a series of successive approximations When two subsequent weight factors calculated in this way differ from each other by no more than a relative amount of WFTOL the optimal weight factor is deemed to have been calculated Experience has shown that a suitable value for WFFAC is about 1 3 it must be greater than 1 WETOL is best set at somewhere between 10
23. bo Bu 2 26 2 1 9 Predictive Analysis Let X represent the action of a linear model under calibration conditions Let b represent the parameter vector for this model while is a vector of field or laboratory observations for which there are model generated counterparts As is explained in Section 2 1 2 when calibrating a model it is PEST s task to minimise an objective function defined as c Xb Q c Xb where Q is the cofactor matrix a diagonal matrix whose elements are the squares of observation weights Let Z represent the same linear model under predictive conditions Thus the action of the model when used in predictive mode can be represented by the equation d d Zb 2 27 where d is a 1 x 1 vector ie a scalar d representing a single model outcome ie prediction Naturally when run in predictive mode the model operates on the same parameter vector as that on which it operates in calibration mode ie b The PEST Algorithm 2 12 The aim of predictive analysis is to maximise minimise d while keeping the model calibrated d will be maximised minimised when the objective function associated with b lies on the min contour Pmin is the lowest achievable value of the objective function while 6 is an acceptable increment to the objective function minimum such that the model can be considered calibrated as long as the objective function is less than min see the discussion in Chapter 6 f
24. henceforth as the Marquardt lambda Then the largest diagonal element of the scaled normal matrix JS QJS oS S of equation 2 23 will be 1 2 1 7 Determining the Marquardt Lambda PEST requires that the user supply an initial value for During the first optimisation iteration PEST solves equation 2 23 for the parameter upgrade vector u using that user supplied It then upgrades the parameters substitutes them into the model and evaluates the resulting objective function PEST then tries another lower by a user supplied factor than the initial If is lowered is lowered yet again However if was raised by reducing below the initial then is raised above the initial lambda by the same user supplied factor a new set of parameters is obtained through solution of equation 2 23 and a new is calculated If was lowered A is raised again PEST uses a number of different criteria to determine when to stop testing new s and proceed to the next optimisation iteration see Section 4 2 2 Normally between one and four s need to be tested in this manner per optimisation iteration At the next iteration PEST repeats the procedure using as its starting A either the from the previous iteration that provided the lowest if needed to be raised from its initial value to achieve this or the previous iteration s best reduced by the user supplied factor In most cases this process results in an overa
25. if a decimal point is absent from an input number Hence if you are unsure what to do assign the string point to the control variable DPOINT this will ensure that all numbers written to model input files will include a decimal point thus overriding point location or scaling conventions implicit in some FORTRAN format specifiers Note that if a parameter space is 13 characters wide or greater and PRECIS is set to single PEST will include the decimal point regardless of the setting of DPOINT for there are no gains to be made in precision through leaving it out Similarly if PRECIS is set to double no attempt is made to omit a decimal point if the parameter space is 23 characters wide or more Table 3 1 shows how the setting of DPOINT affects the representation of the number 12345 67 In examining this table remember that PEST writes a number in such a way that the maximum possible precision is squeezed into each parameter space The Model PEST Interface 3 8 Table 3 1 Representations of the number 12345 67 parameter space width DPOINT characters point nopoint 8 12345 67 12345 67 7 12345 7 12345 7 6 12346 12346 5 1 2e4 12346 4 1 e4 12e3 j TAN le4 2 EF As explained below a template file may contain multiple spaces for the same parameter In such a case PEST will write the parameter value to all those spaces using the minimum parameter space width specified
26. observation O is defined as A 11 2 P P oe where Op and pp are model outcome and parameter base values and O and p are the model outcome and parameter values pertaining to a particular model run Hence if NPAR 1 parameter sets are provided to SENSAN where the first set contains parameter base values and the subsequent NPAR sets contain parameter values identical to the base values except that each parameter in turn is varied from the base value by an incremental amount then the last NPAR rows and NOBS columns on the SENSAN sensitivity output file SENSFLE approximates the transpose of the Jacobian matrix Note that if p p in equation 11 2 is equal to zero then SENSAN writes the corresponding sensitivity as 10 except for the first data line assumed to be the base value line where all sensitivities are provided as 0 0 Note also that the L2 norm can only be positive However when only a single parameter is varied the sign of that variation is taken into account resulting in a negative denominator for equation 11 2 if p lt p 11 5 2 Other Files used by SENSAN As has already been discussed SENSAN uses programs TEMPCHEK and INSCHEK to prepare model input files and read model output files SENSAN writes a parameter value file SENSAN 11 11 for the use of TEMPCHEK naming this file t par This filename should be avoided when naming other files INSCHEK writes the values of the observations which it reads from a model out
27. observation value files generated on successive INSCHEK runs could be concatenated to form an appropriate observation value file to provide to PESTGEN PESTGEN commences execution by reading the information contained in files parfile and obsfile see above checking them for correctness and consistency If there are any errors in either of these files PESTGEN lists these errors to the screen and terminates execution Alternatively if these files are error free PESTGEN then generates a PEST control file Files parfile and obsfile provide PESTGEN with the names of all parameters and observations which need to be listed in the PEST control file They also provides PEST with initial parameter values these must be provided in the second column of the parameter value file the scale and offset for each parameter in the third and fourth columns of the parameter value file the laboratory or field measurement set in the second column of the observation value file and values for the variables PRECIS and DPOINT on the first line of the parameter value file For all other variables listed in the PEST control file PESTGEN uses default values For the parameter and observation value files shown in Examples 5 2 and 10 1 the PESTGEN generated PEST control file is shown in Example 10 3 PEST Utilities 10 7 pct control data restart estimation fo 19 bo 0 1 1 1 single point 1 0 0 0 2 0 Uaa 003 10 3 0 3 0 0 001 al 0 D 0 01 3
28. parameter set saves PEST from having to carry out another set of model runs at the end of the optimisation process in order to re calculate the Jacobian matrix on the basis of best fit parameters Normally this is quite acceptable However if you want to be absolutely sure that the covariance matrix and its derived statistics are as right as they can be notwithstanding the fact that they are based on an often poorly met linearity assumption then you way wish to undertake a special PEST run simply to calculate these statistics on the basis of optimal parameter values You can do this by following these steps 1 Use program PARREP see Section 10 5 to build a new PEST control file in which Running PEST 5 40 initial parameter values are actually optimised parameter values determined on a previous PEST run 2 Set the control variable NOPTMAX see Section 4 2 2 to 1 in the new PEST control file 3 Run PEST Greater accuracy in derivatives calculation and hence greater accuracy in calculation of the parameter covariance matrix can be achieved with the FORCEN variable see Section 4 2 3 for all parameter groups set to always_3 in the new PEST control file 5 7 7 Model Outputs based on Optimal Parameter Values Whether or not you have undertaken the steps outlined in the previous section for obtaining statistics based on optimal parameters the last model run undertaken by PEST prior to termination of execution will not normally
29. then FACORIG times its initial value is substituted into equation 2 44 and the denominator of equation 2 45 for the parameter s current value bo A suitable value for FACORIG varies from case to case though 0 001 is often appropriate Note however that FACORIG is not used to adjust change limits for log transformed parameters For more information on FACORIG see Section 4 2 2 The PEST Algorithm 2 23 It should be noted that problems such as those described above incurred by parameters with low absolute values can also be prevented from occurring by providing such parameters with a suitable OFFSET value accompanied by appropriate lower upper bounds that prevent them from being assigned such troublesome values 2 2 6 Damping of Parameter Changes Parameter over adjustment and any resulting oscillatory behaviour of the parameter estimation process is further mitigated by the damping of potentially oscillatory parameter changes The method used by PEST is based on a technique described by Cooley 1983 and used by Hill 1992 To see how it works suppose that a parameter upgrade vector Bu has just been determined using equations 2 23 2 24 and 2 26 Suppose further that this upgrade vector causes no parameter values to exceed their bounds and that all parameter changes are within factor and relative limits For relative limited parameters let the parameter undergoing the proposed relative change of greatest magnitude be parameter i
30. then become large in comparison with model run times 9 1 5 Installing Parallel PEST The command line version of the Parallel PEST executable ppest exe is automatically installed when you install PEST on your machine As is explained below for Parallel PEST to run a model on another machine it must signal a slave named PSLAVE residing on the other machine to generate the command to run the model Thus pslave exe must be installed on each machine engaged in the Parallel PEST optimisation process To do this copy pslave exe also provided with PEST to an appropriate subdirectory on each such machine This subdirectory can be the model working directory on that machine if desired if not it should be a directory whose name is cited in the PATH environment variable on that machine Alternatively if each slave has access to the PEST directory on the master machine PSLAVE can be loaded from that directory each time it is run on each slave machine This is most easily accomplished if the PEST directory on the master computer as seen by each slave computer is cited in the latter s PATH variable 9 2 How Parallel PEST Works 9 2 1 Model Input and Output Files The manner in which Parallel PEST carries out model runs on different machines is just a simple extension of the manner in which PEST carries out model runs on a single machine Before running a model on any machine Parallel PEST writes one or more input files for that model th
31. 0 log parl 1 0 log par2 1 60206 2 0 group_pr To the right of the sign of each article of prior information are two real variables and a character variable viz PIVAL WEIGHT and OBGNME The first of these is the value of the right side of the prior information equation The second is the weight assigned to the article of prior information in the parameter estimation process As for observation weights the prior information weight should ideally be inversely proportional to the standard deviation of the prior information value PIVAL it can be zero if you wish but must not be negative in practice the weights should be chosen such that the prior information equation neither dominates the objective function or is dwarfed by other components of the objective function In choosing observation and prior information weights remember that the weight is multiplied by its respective residual and then squared before being assimilated into the objective function The final item on each line of prior information must be the observation group to which the prior information belongs Recall that each observation and each element of prior information cited in a PEST control file must be assigned to an observation group In the course of carrying out the parameter estimation process PEST calculates the contribution made to the objective function by each such observation group The name of any observation group to which an item of prior informatio
32. 0 0 Note that if INCTYP is absolute DERINCLB is ignored FORCEN The character variable FORCEN an abbreviation of FORward CENtral determines whether derivatives for group members are calculated using forward differences one of the variants of the central difference method of whether both alternatives are used in the course of an optimisation run It must assume one of the values always_2 always_3 or switch If FORCEN for a particular group is always_2 derivatives for all parameters belonging to that group will always be calculated using the forward difference method as explained in Section 2 3 filling of the columns of the Jacobian matrix corresponding to members of the group will require as many model runs as there are adjustable parameters in the group If FORCEN is provided as always_3 the filling of these same columns will require twice as many model runs as there are parameters within the group however the derivatives will be calculated with greater accuracy and this will probably have a beneficial effect on PEST s performance If FORCEN is set to switch derivatives calculation for all adjustable group members will begin using the forward difference method switching to the central method for the remainder of the estimation process on the iteration after the relative objective function reduction between successive optimisation iterations is less than PHIREDSWH a value for which is s
33. 10 SSTRAINS 11 SSTRAIN str1 11 STRAIN str2 Example 3 13 Extract from an instruction file Whitespace The whitespace instruction is similar to the secondary marker in that it allows the user to navigate through a model output file line prior to reading a non fixed observation see below It directs PEST to move its cursor forwards from its current position until it encounters the next blank character PEST then moves the cursor forward again until it finds a nonblank character finally placing the cursor on the blank character preceding this nonblank character ie on the last blank character in a sequence of blank characters ready for the next instruction The whitespace instruction is a simple w separated from its neighbouring instructions by at least one blank space Consider the model output file line represented below MODEL OUTPUTS 2 89988 4 487892 4 59098 8 394843 The following instruction line directs PEST to read the fourth number on the above line MODEL OUTPUTS w w w w obsl The instruction line begins with a primary marker allowing PEST to locate the above line on the model output file After this marker is processed the PEST cursor rests on the character of OUTPUTS ie on the last character of the marker string In response to the first whitespace instruction PEST finds the next whitespace and then moves its cursor to the end of this whitespace ie just before the
34. 80 02 ro3 ro2 observation groups group_l group_2 group_3 group_4 observation data arl 1 21038 1 0 group_l ar2 1 51208 1 0 group_l ar3 2 07204 1 0 group_1l ar4 2 94056 1 0 group_1l ar5 4 15787 1 0 group_l ar6 5 77620 1 0 group_l ar7 7 78940 1 0 group_2 ar8 9 99743 1 0 group_2 arg 11 8307 1 0 group_2 ar10 12 3194 1 0 group_2 arll 10 6003 1 0 group_2 ar12 7 00419 1 0 group_2 ar13 3 44391 1 0 group 2 ar14 1 58279 1 0 group 2 ar15 1 10380 1 0 group_3 arl6 1 03086 1 0 group_3 ar17 1 01318 1 0 group_3 ar18 1 00593 1 0 group_3 ar19 1 00272 1 0 group_3 model command line ves model input output ves tpl ves inp ves ins ves out prior information pil 1 0 hl 2 0 3 0 group_4 pi2 1 0 log ro2 1 0 log h2 2 6026 2 0 group_4 Example 4 2 A PEST control file A PEST control file must begin with the letters pef for PEST control file Scattered through the file are a number of section headers These headers always follow the same format viz an asterisk followed by a space followed by text When preparing a PEST control _The PEST Control File file these headers must be written exactly as set out in Examples 4 1 and 4 2 however if there is no prior information the prior information header can be omitted On each line of the PEST control file variables must be separated from each other by at least one space Real numbers can be supplied with the minimum precision neces
35. 9 of the PEST manual for full details of the operation of Parallel PEST As is explained in the PEST manual Parallel PEST does not actually run the model itself rather it sends signals to its various slaves to run the model These slaves can be running on the same machine as that on which PEST is running or on other machines linked to the user s machine across a local area network Parallel WinPEST 28 PEST Model Independent Parameter Estimation Watermark Numerical Computing Acknowledgements and Disclaimer Acknowledgments Some of the improvements made to PEST documented in this manual were carried out while I was employed as a Research Scientist at the University of Idaho Idaho Falls Also while I occupied that position I was able to devote a considerable amount of time to writing software to expedite PEST s usage in the calibration and predictive analysis of surface water models Much of this software is now part of the PEST Surface Water Utilities Suite I wish to publicly acknowledge my gratitude to the University of Idaho for providing me with the time and resources necessary to carry out this important work Disclaimer The user of this software accepts and uses it at his her own risk The author does not make any expressed of implied warranty of any kind with regard to this software Nor shall the author be liable for incidental or consequential damages with or arising out of the furnishing use o
36. E EEEN 8 1 8 2 Externally Supplied Derivatives eeeeecescceseecseeceseceaceeceeseeesaeeseeesaeesaeeeseeeseees 8 1 8 2 1 The External Derivatives File oo eee ee eseesceeseceeeceeeeceeeceseceacecseecaeenaceeseeeaeeses 8 1 8 2 2 File Management osete erns ee eei dea e e eea E ROEE E EESE 8 2 8 2 3 File Formatoa ene iiia iea E EEEa E eeo a aa aiar eA 8 2 8 2 4 Derivatives Types eieren eeina eaa E E Ene AET aeaa EE SE Deea EKER EE EELER 8 3 8 2 5 Use of Derivatives Information cee eeseeseceeeceeeecececeseeeacecseecnaeesacenseeeaeeees 8 3 8 2 6 Tied Bata meter Seoca cise tes osov ee e e R snc E See AE RE EE 8 4 8 2 7 Name of the Derivatives File 0 eee eesesscceseceeeeeeeeceeeceseceaeecseecaeesaeeeseeeaeeses 8 4 8 2 8 Predictive Analysis Mode ieee inerente ie a ae o atai E E e EEEa 8 4 8 2 9 Parallel PES Tornia p inrer aeaeaei ne ernea ai r Saa t e e TE A aa E A 8 4 8 3 Sending a Message to the Model eects escesccescecseeceeeceseeeceeseeeseeseessaeesaeeeseseeeees 8 4 8 4 Multiple Command Lines cece eeceeecceeecesecescecscecaeceaceesceesaeesaeeeseeesaeeeaeeeseeesaees 8 6 8 5 External Derivatives and the PEST Control File ce eeeeeeceeeceeeeceeeceneeeeseeeneees 8 6 8 5 1 General 0essieu eit casi testis eee eine E Ea a ae ete teas 8 6 8 5 2 Control Data Section 0 0 o e a e e EE E EE E ER 8 6 8 5 3 Parameter Data Section ssssssesesssessseeseeesseesssesssesssessseesssesssesssessseesseesseessee 8 7 8 5 4 Der
37. Example Instruction File ccc ceccccsscccsseceesseceeseecsneeeseecssaeeeeseeeeseeeeaaes 3 11 3 3 4 The Marker Delimiter 0 ccccccccccccsccceessccceeseeseeeceeeeeeeeeeceeeeesesseeesenseeeeeeees 3 12 3 35 ObservationsNamessi3 4 ciccai antes iis ieee este eee 3 13 3 3 6 The Instruction Set ccccccccccccccccssscceeeesscceceessceeeeessaeeccesseeccessseeeesssaeeeessneeeees 3 13 3 3 7 Making an Instruction File cee eeeeeseceseceeeceeeecececeeeceaeecseeceeeenaeeeaeeeseeeenees 3 25 4 The PEST Control Bile aana aaae A AEA ATEA 4 1 4 1 The Role of the PEST Control Files nsrnrrarsiiniinain akena A 4 4 2 Consttuction Details aia Eae NA AEAEE DA EAA S ALE sage Meets 4 1 4 2 1 The Structure of the PEST Control File cccccccccccssssccccccceesenssseceeeeeeeeeenees 4 1 A 22 Control E D L1 r EEEE O EEE ENE S 4 4 4 2 3 Parameter Groups csccesceesecesecsereceeeceeeeescecseecsaccaceesecesaeesaeesseeesaeesaeesseeeues 4 13 4 2 4 Parameter Data First Part ccccccccccccccssssssssecececcceecesssssceeecceeeeensssseeeeceeeeeees 4 16 4 2 5 Parameter Data Second Part cccccccccccesesssssecececcceesenseseceeeccceesenseseceeeceeeeeens 4 19 A 2 6 ObServ ation Groups wecveresccssvstezeehetivssacesizcaedubatgesccsacetera bbacgavectadebunsebnre iaae uani 4 20 A D 7 Observation Daliani iaria sasageseacs E ead hod a dda aasaeaea See eee 4 20 4 2 8 Model Command Tine nannan ann a e a a a diat 4 21 Table of Contents
38. FRACPHIM at all The value of PHIMLIM used in every optimisation iteration will thus be that supplied in the PEST control file 2 If FRACPHIM is supplied with a value of 1 0 or greater PEST will cease execution with an error message 3 For other values of FRACPHIM PEST will adjust the value of PHIMLIM at each optimisation iteration by multiplying the current value of the measurement objective function by FRACPHIM However it will lower PHIMLIM no further than the value for this variable supplied in the PEST control file 4 Optimal values for FRACPHIM are normally in the range 0 1 to 0 3 5 As well as adjusting the value of PHIMLIM during every optimisation iteration PEST also adjusts the value of PHIMACCEPT This adjustment is made such that during every optimisation iteration the ratio of PHIMACCEPT to PHIMLIM is the same as that supplied in the PEST control file Normally this ratio should be no greater than 1 1 7 4 Working with PEST in Regularisation Mode 7 4 1 Run Time Information As it runs PEST records information to both the screen and to its run record file When PEST is run in parameter estimation mode the principal items of interest during each optimisation iteration are the value of the objective function at the beginning of the iteration and new values of the objective function which are calculated as PEST tests a series of parameter upgrade vectors calculated on the basis of a number of different Marquardt lambdas
39. Frequently Asked Questions 13 1 PEST When I run PEST with certain older models a message pops up asking me if I would like to run the model in MS DOS mode How can I prevent this from happening Click with your right mouse button on the bar at the top of the DOS window in which you are running PEST and choose properties from the pop up menu In the program advanced section of the pop up dialogue box uncheck the Suggest MS DOS mode as necessary box To prevent this from ever happening again click on the model icon in Windows Explorer with the right mouse button and alter its properties in similar fashion When I run PEST with a certain model a message pops up telling me that the computer is about to enter MS DOS mode and that I must close all other programs How can I prevent this from happening Click on the model icon in Windows Explorer as described above This time uncheck the MS DOS mode box I am having trouble running PEST in WINDOWS NT and or WINDOWS 2000 Sometimes a command line window just disappears See Section 13 3 below 13 2 Parallel PEST Can a slave be introduced part of the way through a Parallel PEST run As presently programmed no All incidences of PSLA VE must begin execution before PEST A slave can drop out of the Parallel PEST optimisation process or its execution can be terminated eg by hitting lt Ctl C gt and PEST will simply carry on without it as long as there are other slave
40. However if you set it too low the model solution procedure may not converge worse still it may converge for some parameter sets and not for others To overcome this you may need to make a small change to the model such that it prints out its solution vector even if it has not converged according to your very stringent convergence setting alternatively you could employ a batch process such as was demonstrated in Example 4 3 5 5 3 High Parameter Correlation There is often a temptation in fitting models to data to improve the fit between modelled and measured observations by increasing the number of adjustable parameters While it is true that this can result in a lowered objective function it is not always true that such an improvement increases a model s ability to make reliable predictions or that a high number of parameters represents a valid interpretation of the dataset to which the model s outcomes are matched Furthermore as the number of parameters requiring estimation is increased there will come a stage where PEST s ability to lower the objective function by adjusting the values of these parameters is diminished due to the effects of roundoff error particularly for highly nonlinear models this applies not just to PEST but to any parameter estimation Running PEST 5 27 package The trouble with increasing the number of parameters without limit is that sooner or later some parameters become highly correlated This results
41. PEST s user intervention functionality this allowing the user to hold troublesome parameters normally insensitive and or highly correlated parameters at their current values so that the parameter estimation process could proceed without the damaging effects that these parameters have on that process The second was the incorporation of PEST s nonlinear predictive analysis functionality Preface this denoting a recognition of the fact that increased parameterisation normally results in increased parameter nonuniqueness at the same time as any semblance of model linearity rapidly fades from view Now with PEST ASP comes the advent of advanced regularisation functionality At the time of writing this preface PEST s new regularisation functionality has already proven itself enormously useful in the parameterisation of heterogeneous two and three dimensional spatial model domains especially when accompanied by the use of flexible methods of spatial parameter definition such as pilot points see the PEST Ground Water Data Utility Suite Use of PEST in regularisation mode allows the modeller to estimate many more parameters than would otherwise be possible Thus when working with spatial models PEST is able to find for itself regions of anomalous physical or hydraulic properties rather than requiring that such areas be delineated in advance by the modeller using zones of piecewise parameter constancy Furthermore the proc
42. PEST This is because the model run time is too small to justify Parallel PEST s run management overheads Furthermore unless you are using a multi processor machine there is nothing to be gained by undertaking parallel model runs on a single machine anyway Note however that Parallel PEST s speed can be increased somewhat by reducing the value of the WAIT variable from that provided on the test rmf file 9 5 Frequently Asked Questions For more information on Parallel PEST see the frequently asked questions listed in Chapter 13 PEST Utilities 10 1 10 PEST Utilities PEST is accompanied by five utility programs whose role is to assist in PEST input file preparation These are programs TEMPCHEK INSCHEK PESTCHEK PESTGEN and PARREP The first three of these programs are used to check template files instruction files and the PEST control file respectively for a particular PEST case prior to actually running PEST on that case In this way you can be sure that the input dataset which you supply to PEST is correct and consistent The PESTGEN utility creates a PEST control file using default values for many of the PEST input variables PARREP builds a new PEST control file based on an existing PEST control file and a set of values cited in a parameter value file A sixth utility program named JACWRIT acts in a PEST postprocessing capacity It allows a user to generate and ASCII file in which the Jacobian matrix for a particular paramete
43. PESTCHEK using the command pestchek twofit If all is correct you can now run PEST using the command pest twofit An Example 12 10 A run record file twofit rec will be written by PEST in the pestex subdirectory so too will file twofit par containing the optimised parameter set Figure 12 3 shows the lines of best fit superimposed on the laboratory data Specific volume cu m Mg 0 0 0 2 0 4 0 6 Water content cu m Mg Figure 12 3 Soil clod shrinkage data with lines of best fit superimposed 12 2 Predictive Analysis 12 2 1 Obtaining the Model Prediction of Maximum Likelihood Files for this example can be found in the papestex directory of the PEST directory after installation This example builds on the soil clod shrinkage example discussed in the previous section Based on the dataset supplied with the example PEST lowers the objective function to a value of 6 71E 4 when estimating values for the model parameters Best fit parameter values are listed in Table 12 2 An Example 12 11 Parameter PEST estimated value sl 0 238 s2 0 963 yl 0 497 XC 0 174 Table 12 2 Optimised parameter values for soil clod shrinkage example After PEST is run in parameter estimation mode best fit model parameters can be found in file twofit par Insert these into the model input file in dat by running TEMPCHEK as follows tempchek in tpl in dat twofit par If the value for specific volume at a water c
44. a PEST control file by another set of values the latter being supplied in a PEST parameter value file Recall from Section 5 3 1 that in the course of the parameter estimation process PEST writes a parameter value file every time it improves its parameter estimates After a PEST run has finished either of its own accord or if it was manually halted optimised parameter values can be found in the parameter value file The parameter value file possesses the same filename base as the current case but has an extension of par Because it has such a simple PEST Utilities 10 9 structure a parameter value file can also be easily built by the user with the help of a text editor PARREP is useful when commencing a new PEST run where an old run finished An updated PEST control file can be produced by replacing parameter values in the old file with the best parameter values determined during the previous PEST run as recorded in the parameter value file written during that run Recommencing a PEST run in this way rather than through use of the r or j switches allows a user to alter certain PEST control variables fix or tie certain parameters or adjust PEST s management of the parameter estimation process in other ways prior to re commencement of the run PARREP is also useful when undertaking a single model run on the basis of a certain set of parameters in order to calculate the objective function Simply modify an existing PE
45. adjustable parameters in order to fill the Jacobian matrix during each optimisation iteration The second reason is based on the possible near singular condition of the normal matrix and the way in which PEST adjusts the Marquardt lambda upwards in response to this In general while high lambda values can lead to a rapid lowering of the objective function at the early stages of the parameter estimation process when parameter values are far from optimal it is normally far better to decrease lambda as the objective function minimum is approached As discussed in Section 2 1 5 use of a high Marquardt lambda is equivalent to the use of the steepest descent optimisation method however this method is notoriously slow when parameters are highly correlated due to the phenomenon of hemstitching as the parameter upgrade vector oscillates across narrow objective function valleys in parameter space But if lambda cannot be lowered because the normal matrix would then become singular or at best ill conditioned due to the excessive number of parameters requiring estimation there will be no way to prevent this The incorporation of prior information into the estimation process can often add stability to an over parameterised system Likewise removing a number of parameters from the process by holding them fixed at strategic values may yield dramatic improvements in PEST s performance In many environmental modelling contexts a spectacular increase i
46. along the boundaries of the parameter domain for the smallest to which it has access when the global minimum of lies outside of the parameter domain beyond PEST s reach At the beginning of each new optimisation iteration all temporarily frozen parameters are freed to allow them to move back inside the allowed parameter domain if solution of equation 2 23 deems this necessary The stepwise temporary freezing of parameters is then repeated as described above 2 2 4 Scale and Offset For every parameter you must supply a scale and offset variables SCALE and OFFSET in the PEST control file Before writing a parameter value to a model input file PEST multiplies this value by the scale and adds the offset The scale and offset variables can be very convenient in some situations For example in a particular model a certain parameter may have a non zero base level you may wish to redefine the parameter as the actual parameter minus this base level Elevation may be such a parameter If a reference elevation is subtracted from the true model required elevation the result may be thickness this may be a more natural parameter for PEST to optimise than elevation In particular it may make more sense to express a derivative increment see Section The PEST Algorithm 2 21 2 3 as a fraction of thickness than as a fraction of elevation to which an arbitrary datum has been added Also the optimisation process may be better behaved if t
47. an ability to quantify the uncertainty associated with such predictions is as important as the ability to make such predictions in the first place The concept of a prediction can be broadened to describe PEST s use in those fields where parameter estimation is an end in itself This is especially the case in the geophysical context where PEST is used to infer earth properties from measurements made at the surface and or down a number of boreholes In this case it would appear that model parameters as determined through the nonlinear parameter estimation process are of overriding importance and indeed that the concept of a model prediction is inapplicable However this is not the case in fact PEST s predictive analysis capabilities have proved an extremely useful addition to the exploration geophysicist s arsenal Use of PEST in predictive analysis mode allows the geophysical data interpreter to ask and have answered such questions as is it possible that a Preface hole drilled at a certain location will not intersect any conductive material or what is the maximum possible depth extent of the conductor giving rise to anomalous surficial measurements The term predictive analysis as used in this manual describes the task of calculating the effect of parameter uncertainty as estimated through the calibration process on predictive uncertainty A number of methods have been documented for undertaking s
48. and or the parameter covariance matrix and other quantities derived from this matrix on the basis of the measurement dataset alone you should undertake the following steps 1 After the regularisation run create a new PEST control file with optimised parameter values substituted for initial parameter values using the PARREP utility program supplied with PEST 2 Set all regularisation weights to zero Regularisation 7 13 3 Alter PESTMODE on the new PEST control file to estimation 4 Adjust NOPTMAX to 1 As explained in Section 4 2 2 of this manual when PEST is run with NOPTMAX set to 1 it undertakes enough model runs to calculate the Jacobian matrix and then terminates execution with a full statistical printout at the end of its run record file 7 5 2 Initial Parameter Values When running PEST in regularisation mode you can supply initial parameter values that are far from optimal or you can supply optimised parameter values from a previous unregularised PEST run if you wish In the latter case the initial measurement objective l function m may actually be less than m In most cases it makes no difference to PEST what the starting parameter values are if they are a long way from optimal PEST will reduce them to near optimal before the regularisation mechanism begins to have a strong effect on the optimisation process However in difficult cases it has been found from experience that it is sometimes better to sta
49. any parameter It also records the names of the parameters undergoing these maximum changes This in combination with the contents of the parameter sensitivity file may assist the user in deciding which if any parameters to temporarily hold at their present values using directives supplied in the parameter hold file 5 7 PEST Postprocessing 5 7 1 General After the parameter estimation process is complete a thorough examination must be undertaken of the results of this process before the estimated parameters can be used in a calibrated model which is run for predictive purposes This section presents a guide to some of this analysis procedure The discussion necessarily pertains to general parameter estimation post processing extra case specific analysis will be required for each particular instance of PEST deployment The information upon which to base post processing analyses such as those outlined in this section is contained in the PEST run record file and in a number of other files that are written by PEST often in a format that facilitates plotting using commercial graphing and spreadsheet packages See Sections 5 2 and 5 3 for details 5 7 2 Parameter Values Parameter values should be inspected for reasonableness It is often a good idea to undertake initial PEST runs with parameter bounds set very wide In this way PEST is free to assign unrealistic values to certain parameters if this is required in order to achieve goodness of
50. are near or at their upper or lower bounds as defined by the parameter variables PARLBND and PARUBND discussed below In this case it is possible that the magnitude of the parameter upgrade vector may be curtailed over one or a number of optimisation iterations to ensure that no parameter value overshoots its bound The result may be smaller reductions in the objective function than would otherwise occur It would be a shame if these reduced reductions were mistaken for the onset of parameter convergence to the optimal set NPHINORED If PEST has failed to lower the objective function over NPHINORED successive iterations it will terminate execution NPHINORED is an integer variable a value of 3 or 4 is often suitable RELPARSTP and NRELPAR If the magnitude of the maximum relative parameter change between optimisation iterations is less than RELPARSTP over NRELPAR successive iterations PEST will cease execution The relative parameter change between optimisation iterations for any parameter is calculated using equation 4 3 PEST evaluates this change for all adjustable parameters at the end of each optimisation iteration and determines the relative parameter change with the highest magnitude If this maximum relative change is less than RELPARSTP a counter is advanced by one if it is greater than RELPARSTP the counter is zeroed All adjustable parameters whether they are relative limited or factor limited are involved in the calculation
51. because most models generate a wealth of data for which we may have only a handful of corresponding field measurements on which to base our estimates of the system properties Hence as we include in the vector c only those model outcomes for which there are complementary laboratory or field measurements it is appropriate to distinguish them from the remainder of the model outcomes by referring to them as the model generated observations The complementary set of field or laboratory data is referred to as measurements or as experimental observations in the following discussion Let it be assumed that the elements of X are all known For most models these elements will include the effects of such things as the system dimensions physical chemical or other constants which are considered immutable independent variables such as time and distance etc For example equation 2 1 may represent the response of the system at different times where the response at time p is calculated using the equation Xpl bi Xp2 b2 Fse Xpn bn Cp 2 2 where xpi is the element of X found at its p th row and 7 th column As X has m rows there are m such equations one for each of m different times Hence for any p at least one of the xpi depends on time The PEST Algorithm 2 2 Suppose that m is greater than n ie we are capable of observing the system response and hence providing elements for the vector c at more times than there are para
52. by two parameter delimiters the name of the parameter to which the space belongs must be written between the two delimiters If a model input file is such that the space available for writing a certain parameter is limited the parameter name may need to be considerably less than twelve characters long in order that both the name and the left and right delimiters can be written within the limited space available The minimum allowable parameter space width is thus three characters one character for each of the left and right delimiters and one for the parameter name 3 2 5 Setting the Parameter Space Width In general the wider is a parameter space up to a certain limit see below the better PEST likes it for numbers can be represented with greater precision in wider spaces than they can be in narrower spaces However unlike the case of model generated observations where maximum precision is crucial to obtaining useable derivatives PEST can adjust to limited precision in the representation of parameters on model input files as long as enough precision is employed such that a parameter value can be distinguished from the value of that same parameter incremented for derivatives calculation Hence beyond a certain number of characters the exact number depending on the parameter value and the size and type of parameter increment employed extra precision is not critical Nevertheless it is good practice to endow parameter values with as much p
53. can recommence a lengthy execution using more less and or different slaves than those that were used for the initial part of the Parallel PEST run as long as the new run management file is prepared in the correct fashion and PSLAVE execution is recommenced on each of the slave machines identified in that file before PEST execution is recommenced 9 3 5 Parallel PEST Errors As in normal PEST operation Parallel PEST run time error messages are written to both the screen and the run record file 9 3 6 Losing Slaves If during the course of a Parallel PEST run a slave machine drops out of the network under many circumstances PEST will continue execution If communications are lost during the course of a model run then PSLAVE executing on the lost machine will not be able to inform PEST of the completion of that model run PEST will soon grow tired of waiting and allocate that run to another slave It will thus continue execution with one less slave at its disposal However if the slave machine drops out just while PEST is trying to read a model output file PEST will terminate execution with an error message Even complete network failure may not result in the termination of a Parallel PEST run for if one slave is running on the same machine as PEST PEST will be able to continue execution Parallel PEST 9 15 using just that single slave as long as the time of network failure did not coincide with the time at which PEST was reading a mod
54. chosen wisely As three point derivative calculations are normally carried out by first adding an increment to a current parameter value and then subtracting an increment the method is referred to herein as the central method of derivatives calculation Note that if a parameter value is at its upper bound the parameter increment is subtracted once and then twice the model being run each time if it is at its lower bound the increment is added once and then twice PEST uses one of three methods to calculate central derivatives In the first or outside points method the two outer parameter values ie that for which an increment has been added and that for which an increment has been subtracted are used in the same finite difference type of calculation as is used in the forward difference method This method yields more accurate derivative values than the forward difference method because the unused current parameter value is at the centre of the finite difference interval except where the parameter is at its upper or lower bound The second method is to define a parabola through the three parameter observation pairs and to calculate the derivative of this parabola with respect to the incrementally varied parameter at the current value of that parameter This method referred to as the parabolic method can yield very accurate derivatives if model calculated observation values can be read from the model output file with sufficient
55. command line for slave slave 2 ves Slave slave JT rnme gt Model input files on slave slave 3 m model ves inl m model ves in2 Model output files on slave slave 3 m model ves ot1l Parallel PEST 9 16 m model ves ot2 m model ves ot3 Model command line for slave slave 3 ves AVERAGE WAIT INTERVAL 50 hundredths of a second RUN MANAGEMENT RECORD RUNNING MODEL FOR FIRST TIME gt 21 50 00 19 slave slave 1 commencing model run 21 55 05 00 slave slave 1 finished execution reading results OPTIMISATION ITERATION NO 1 gt Calculating Jacobian matrix running model 5 times 21 55 23 65 slave slave 1 commencing model run 21 55 33 92 slave slave 3 commencing model run 21 55 44 20 slave slave 2 commencing model run 22 00 05 79 slave slave 2 finished execution reading results 22 00 17 77 slave slave 3 finished execution reading results 22 00 29 47 slave slave 2 commencing model run 22 00 39 58 slave slave 3 commencing model run 22 00 50 07 slave Slave 1 finished execution reading results 22 05 11 83 slave slave 2 finished execution reading results 22 05 43 70 slave slave 3 finished execution reading results Testing parameter upgrades 22 06 01 69 slave slave 2 commencing model run 22 11 16 45 slave slave 2 finished execution reading results 22 11 27 49 slave slave 2 commen
56. detail When creating your own instruction files the syntax provided for each instruction must be followed exactly If a number of instruction items appear on a single line of an instruction file these items must be separated from each other by at least one space Instructions pertaining to a single line on a model output file are written on a single line of a PEST instruction file Thus the start of a new instruction line signifies that PEST must read at least one new model output file line just how many lines it needs to read depends on the first instruction on the new instruction line The Model PEST Interface 3 14 Note however that if the first instruction on the new line is the character amp the new instruction line is simply a continuation of the old one Like all other instruction items the amp character used in this context must be separated from its following instruction item by at least one space PEST reads a model output file in the forward top to bottom direction looking to the instructions in the instruction file to tell it what to do next Instructions should be written with this in mind an instruction cannot direct PEST to backtrack to a previous line on the model output file Also because PEST processes model output file lines from left to right an instruction cannot direct PEST backwards to an earlier part of a model output file line than the part of the line to which its attention is currently focus
57. does not start after one minute use Windows Explorer to navigate to the CD ROM drive and run the program SETUP32 EXE After the installation program is run the following programs will be installed onto your machine s hard disk e the command line version of PEST e the command line version of Parallel PEST e WinPEST the Windows version of PEST with real time graphics e Parallel WinPEST the Windows version of Parallel PEST e a number of utility programs which are supplied with PEST to carry out PEST data checking and pre post run data manipulation and e the complete manual for PEST and Visual PEST in pdf format The Visual PEST ASP manual provides complete documentation for PEST and all the supporting software provided with it It also contains a number of working examples where you can gain experience in using PEST and its supporting software Familiarity with the contents of this manual is a necessary precursor to using PEST and WinPEST 1 2 Other Software The Visual PEST ASP CD also contains the following software you may wish to install e the PEST MODFLOW MTS3D Utilities e the PEST Groundwater Data Utilities e the PEST Surface Water Modeling Utilities e MODFLOW ASP The MODFLOW MT3D Utilities have now been largely superseded by programs of the Groundwater Data Utilities Programs of the Groundwater Data Utilities and Surface Water Utilities can provide great assistance in file preparation for a PEST ru
58. e aa aaia 11 8 11 3 2 Ru nijge SENS CHEK 5 cn tecrantecaassenaninssn oei ee e aa E analy 11 8 11 4 Running SENSAN seiceanna a i i a a iia 11 9 11 4 1 SENSAN Command Line iee aeoiee ei ie enie osori denat ieia ea ERROSE 11 9 11 4 2 Interrupting SENSAN Execution esesssseeeeereesessrerrerresersresresrrerresessesseesreses 11 9 11 5 Files Written by SENSAIN 4 5 iccestesveside onthe bivedesnenncsachsoedsyvaswatcateobinowesousmetoactebees 11 9 11 5 1 SENSAN Output Files eee csseesseessecesecseeeeeeeceeececeaeesaeeeeeesaeesaeesseeenes 11 9 11 5 2 Other Files used by SENSAN cccssssssssscnrsrsseosenrcsrsteesoaserereossarerersnses 11 10 11 6 Sensitivity of the Objective Function ole eee eseceseceeeeeeeecececeaeecaeecaecaeesseesaes 11 11 11 7 SENSAN Error Checking and Run Time Problems ecceeeceeeeeeneceneeeeeenees 11 12 LL B An Exam ples c2cceiccuseiti ese tebeten anenai na a ia a a i iiae 11 13 AE EXAM o A E E E A E EO 12 1 12 1 Parameter Estimation ineco eeen oaa TE i e RTEA RE aE E h 12 1 12 1 Laboratory Data reaot ieee a o Ea i Eiei Tena eei 12 1 12 12 The Model vices csicccissoctes ietiie ats eea a aa i i a aai 12 2 12 1 3 Preparing the Template File ccc cecccccsssceesseeceeeceseeeeeeceeseeeeseceeeeeeeas 12 5 12 1 4 Preparing the Instruction File ccc ceccceessceessseeceeeceseeeeeeeeeseeeeseeeeeseeenaas 12 7 12 1 5 Preparing the PEST Control File ccc ceeccecsseeeeseecesneecesneeeeeeeeeseeeeeaeeeens 12 8 12 2
59. e ae R e iE 10 5 10S PARR EP sess A esa d i EAT ea A EE EE dag ate nb va reac E aia A 10 8 1 06 TA COW a O E A E EE E EE T EE 10 9 10 7 PARZPAR ona ar Te et ke See ea eee eee 10 10 LOG General isccsuseetei clei E etek ternational 10 10 10 752 Using PARQPAR iasence rancareda aait 10 12 10 7 3 Using PAR2PAR With PES Terenten eieiei ee o aeei 10 16 TISSENSAN pme anai a aa e aaa ea awsome Ee 11 1 11 1 troduction 2 fi vecscscaeerctese i iecadocteys eg oletndide ha esehon Saeed E A AE aa Ea 11 1 11 2 SENSAN File Requirement 0 cc cesceeccesscesseesscessceesseesseesseceseeeeeseeeceeeceaeeeaeeess 11 2 TD 2A General E EEEE 11 2 1152 2 Template Files eseria e AR E R 11 2 11 2 3 Instruction Filesi ni e eaei a a tales e a avis 11 2 11 2 4 The Parameter Variation File sseeesseesseessessesssessssresseesseesseesseessresssesssees 11 3 11 25 SENSAN Control File vy secsceteiessscesstcessecguhes heescstanedunabian gavsctaaeeunrabucs EKA EnS 11 4 11 2 6 Control Datassist EEE E EEEE EE E AK i ii 11 5 T T SENSA N FE S e a E E E R Eei 11 6 Table of Contents 11 2 8 Model Command Line 0 ccecceccccesseceeseceececceceeeeecceseeceeaeeceeeeeeseeeesaeeesas 11 6 11 2 9 Model Input Output 0 0 cece ceecceececeeeceeecsceceeceaecsseeeseeeeaeesaeeeseeesseesseeeaeeeaes 11 7 11 2 10 Issuing a System Command from within SENSAN oseere 11 7 11 3 SENSCHEK gt iina ians iaeia aaa anus aa a eaaa e i aE A aA 11 8 13 About SENSGHEK geile i iae areetan
60. expressions by which parameter values are calculated sin cos tan asin acos atan sinh cosh tanh exp log log10 abs and sqrt Note the following rules governing use of these functions e As is the FORTRAN convention the arguments of the trigonometric functions sin cos and tan and the values returned by their inverse functions asin acos and atan are assumed to be in radians There are 27 radians in a circle thus 27 radians are equal to 360 degrees e The log function is to base e for logarithms to base 10 use the og 0 function e For some of the functions listed above arguments must lie within a specific numerical range for example the argument of the log function must always be greater than zero If a function argument is provided which is outside of its legal range PAR2PAR will often trap the error and cease execution with an appropriate error message However in some rare instances the argument may slip through and a compiler generated error message will be supplied upon termination of PAR2PAR execution PEST Utilities 10 15 The following rules apply when formulating mathematical expressions to calculate parameter values e Expressions may contain both numbers and parameters However where a parameter is used its value must have been calculated or supplied in a previous expression e Asis the normal PEST convention parameter names must be 12 characters or less in length e Spaces can be placed next
61. extremely small numbers can be represented If a model s input data fields are small and there is nothing you can do about it every effort must be made to squeeze as much precision as possible into the limited parameter spaces available PEST will do this anyway but it may be able to gain one or more extra significant figures if it does not need to include a decimal point in a number if the decimal point is redundant Thus if a parameter space is 5 characters wide and the current value of the parameter to which this field pertains is 10234 345 PEST will write the number as 1 0e4 or as 10234 depending on whether it must include the decimal point or not Similarly if the parameter space is 6 characters wide the number 106857 34 can be represented as either 1 07e5 or 1069e2 depending on whether the decimal point must be included or not By assigning the string nopoint to the PEST control variable DPOINT you can instruct PEST to omit the decimal point in the representation of a number if it can However this should be done with great caution If the model is written in FORTRAN and numbers are read using free field input or using a field width specifier such as F6 0 or E8 0 the decimal point is not necessary However in other cases the format specifier will insert its own decimal point eg for specifiers such as F6 2 or enforce power of 10 scaling eg for specifiers such as E8 2
62. factor limited and RELPARMAX must be at least unity Suitable values for RELPARMAX and FACPARMAX can vary enormously between cases For highly non linear problems these values are best set low If they are set too low however the estimation process can be very slow An inspection of the PEST run record will often reveal whether you have set these values too low for PEST records the maximum parameter factor and relative changes on this file at the end of each optimisation iteration If these changes are always at their upper limits and the estimation process is showing no signs of instability it is quite possible that RELPARMAX and or FACPARMAX could be increased or that an insensitive parameter should be held at its current value see Section 5 6 If you are unsure of how to set these parameters a value of 5 for each of them is often suitable In cases of extreme nonlinearity be prepared to set them much lower Note however that FACPARMAX can never be less than 1 RELPARMAX can be less than 1 as long as no parameter s upper and lower bounds are of opposite sign Values assigned to RELPARMAX and FACPARMAX can be adjusted in the course of the optimisation process through the user intervention functionality discussed in Section 5 6 FACORIG If in the course of the estimation process a parameter becomes very small the relative or factor limit to subsequent adjustment of this parameter may severely hamper its growth back to higher val
63. field or laboratory observations After calculating the mismatch between the two sets of numbers and evaluating how best to correct that mismatch it adjusts model input data and runs the model again For PEST to take control of an existing model in this fashion in order to optimise its Introduction 1 4 parameters and or excitations certain conditions must be met These are as follows e While a model may read many input files some of which may be binary and some of which may be ASCII the file or files containing those excitations and or parameters which PEST is required to adjust must be ASCII ie text files e While a model may write many output files some of which may be binary and some of which may be ASCII the file or files containing those model outcomes which complement field or laboratory measurements must be ASCII ie text files e The model must be capable of being run using a system command and of requiring no user intervention to run to completion see below for further details e PEST uses a nonlinear estimation technique known as the Gauss Marquardt Levenberg method The strength of this method lies in the fact that it can generally estimate parameters using fewer model runs than any other estimation method a definite bonus for large models whose run times may be considerable However the method requires that the dependence of model generated observation counterparts on adjustable parameters and or excitations be c
64. fit between model outcomes and corresponding field data When reasonable parameter bounds are then imposed it will often be possible to obtain just as good a fit this being an outcome of the parameter correlation that is a feature of most real world model calibration However if it is not possible to achieve just as good a fit with parameters constrained to reasonable values then PEST is providing valuable information on model adequacy for this state of affairs indicates that the model may not be simulating all aspects of the system that it is intended to portray Whether this is a worrying matter or not depends on the particular modelling application It is sometimes valid modelling practice to allow unsimulated subprocesses to be represented by out of range surrogate parameter values In other applications this is completely unacceptable The making of the appropriate decision in each case is part of the art of modelling 5 7 3 Parameter Statistics This manual discusses at length the role of the various statistics that are produced as an outcome of the PEST parameter estimation process These include the parameter covariance matrix correlation coefficient matrix covariance matrix eigenvalues and eigenvectors and parameter confidence intervals Though all of these quantities are only approximate because of the fact that their calculation is based on a linearity assumption that is often violated there is nevertheless much to be learned about the m
65. fixed observation the instruction to read a semi fixed observation consists of two parts viz the observation name followed by two column numbers the latter being separated by a colon the column numbers must be in ascending order However for semi fixed observations the observation name is enclosed in round brackets rather than square brackets Again there must be no space separating the two parts of the semi fixed observation instruction Reading a number as a semi fixed observation is useful if you are unsure how large the representation of the number could stretch on a model output file as its magnitude grows and or diminishes in PEST controlled model runs it is also useful if you do not know whether the number is left or right justified However you must be sure that at least part of the number will always fall between and including the two nominated columns and that whenever the number is written and whatever its size it will always be surrounded either by whitespace or by the beginning or end of the model output file line If when reading the model output file PEST encounters only whitespace between and including the two nominated column numbers or if it encounters non numeric characters or two number fragments separated by whitespace an error condition will occur and PEST will terminate execution with an appropriate error message As for fixed observations it is normally not necessary to have secondary markers whitespace and ta
66. flood peak may be sensitive to parameters which are highly correlated with each other under the more benign conditions under which the model was calibrated the possibility of predictive uncertainty is very high so too is the imperative for predictive analysis 6 1 4 Dual Calibration There are two means whereby PEST can be used to establish the degree of uncertainty associated with a particular model prediction The first is approximate and can be done using PEST under normal parameter estimation conditions The second involves determination of the actual critical point The first of these is discussed in the present section while the second is discussed in the next section Dual Calibration gets its name from the fact that PEST is asked to calibrate a model that is actually comprised of two models Recall that the model run by PEST can actually be a batch file comprised of many executables A batch file can easily be written such that it first runs the model under calibration conditions and then runs the model under predictive conditions The dataset for calibration of this dual model should be the normal calibration dataset for which corresponding model generated numbers are produced by the first component of the dual model ie the model run under calibration conditions plus a single extra observation which corresponds to a best or worst case model outcome of a certain type for which the corresponding model generated num
67. for temporary storage This will be done using yet another model comprised of a batch or script file which includes the pertinent copy commands 8 5 External Derivatives and the PEST Control File 8 5 1 General Formatting of the PEST control file must be slightly expanded from that discussed in Section 4 2 to accommodate the use of external derivatives and to control PEST to model messaging Pertinent variables in the PEST control file which govern this aspect of PEST s functionality are now discussed 8 5 2 Control Data Section As is explained in Section 4 2 the fifth line of the PEST control file begins with the PEST control variables NTPLFLE NINSFLE PRECIS and DPOINT The three variables situated to the end of this line for which values of 1 O and 0 were suggested in Section 4 2 are named NUMCOM JACFILE and MESSFILE in that order For the sake of backwards Model Calculated Derivatives 8 7 compatibility with older versions of PEST these variables may be omitted from this line However it is important to note that either these three variables must be completely absent from the fifth line of the PEST control file or all of them must be present The roles of these variables are now discussed NUMCOM This is the number of different command lines which can be used to run the model The actual commands themselves are listed in the model command line section of the PEST control file see below For pr
68. for that particular parameter for the wider spaces the number will be right justified with spaces padded on the left In this way a consistent parameter value is written to all spaces pertaining to the one parameter 3 2 7 Multiple Occurrences of the Same Parameter Large numerical models which calculate the variation of some scalar or vector over two or three dimensional space may require on their input files large amounts of system property data written in the form of two or three dimensional arrays For example a finite difference ground water model may read arrays representing the distribution of hydraulic conductivity storage coefficient and other aquifer properties over the modelled area each element within each array pertaining to one rectangular finite difference cell A finite element model used in simulating geophysical traverse results over an orebody may require an array containing conductivity values for the various elements into which the orebody and surrounding earth are subdivided For large grids or networks used to spatially discretise two or three dimensional inhomogeneous systems of this kind hundreds perhaps thousands of numbers may be required to represent the distributed system properties If it is required that field measurements be used to infer system properties using models such as these to link these properties to system response certain assumptions regarding the variation in space of the distributed param
69. from the fact that the measurement set upon which the parameter estimation process is based may not have the ability to discriminate between different combinations of parameter values each combination giving rise to an equally low objective function As has already been discussed the extent to which parameter pairs and or groups are correlated can be gleaned from an inspection of the correlation coefficient and or eigenvector matrices If parameters are too highly correlated the matrix J QJ of equation 2 18 becomes singular However because PEST adds the Marquardt parameter to the diagonal elements of this matrix before solving for the parameter upgrade vector see equation 2 20 rendering it singular no longer an upgrade vector can nevertheless be obtained Eventually unless circumvented by roundoff errors an objective function minimum will be obtained through the normal iterative optimisation process However the parameter set determined on this basis may not be unique Hence if you are running a theoretical case PEST may determine a parameter set which is entirely different from the one which you used to generate the artificial measurement set In spite of this the objective function may be very small In addition to the nonuniqueness problem the optimisation process may become very slow if there are many parameters in need of estimation There are two reasons for this The first is that PEST requires at least as many model runs as there are
70. geotechnical mechanical aeronautical and chemical engineering ground and surface water hydrology and other fields Through the use of PEST in model calibration and data interpretation many PEST users have been able to use their models to much greater advantage than was possible when such tasks were attempted manually by trial and error methods This second edition of the PEST manual coincides with the release of version 3 5 of PEST Some of the enhancements that were included in this new PEST have arisen out of my own experience Preface in the application of PEST to the calibration of large and complex models Others have been included at the suggestion of various PEST users some of whom are applying PEST in unique and interesting situations For those already familiar with PEST a brief summary of new features follows A version of PEST called Parallel PEST has been created This allows PEST to run a model on different machines across a PC network thereby reducing overall optimisation time enormously By popular demand parameter observation parameter group and prior information names can now be up to 8 characters in length The previous limit of 4 characters per name was set as a memory conservation strategy a matter of diminishing concern as computing hardware continues to improve Observations can now be collected into groups and the contribution made to the objective function by each group reported through the optimisation
71. have been undertaken on the basis of optimal parameters If the instructions outlined in the previous section are followed the last model run will in fact have been undertaken with one parameter value slightly incremented or decremented from optimality for the purpose of derivatives calculation Hence model output files available at the termination of PEST execution will not contain best fit model outputs These can only be obtained if a model run is explicitly undertaken on the basis of optimised parameter values This can be achieved in either of two ways The first is to use TEMPCHEK to generate a set of model input files based on template files for the current case together with the parameter value file produced by PEST in the course of its previous parameter estimation run see Section 10 1 for details The second option is to use PARREP in the manner described in the previous section but with NOPTMAX set to 0 in the new PEST control file so that PEST undertakes only one model run this being for the purpose of objective function calculation Predictive Analysis 6 1 6 Predictive Analysis 6 1 The Concept 6 1 1 What Predictive Analysis Means In the discussion that follows reference will be made to PEST s usage in the role of model calibration However PEST is often used in the role of data interpretation as well particularly geophysical data interpretation in which earth properties are inferred from a number of discrete surf
72. ie using parameters corresponding to min Predictive Analysis 6 5 direction of increasing prediction value critical point Figure 6 4 The critical point in parameter space The situation is only slightly more complicated when parameter bounds are imposed Figure 6 5 shows objective function contours model prediction contours and the critical point in this case The prediction corresponding to parameter values at the critical point is now the worst or best model prediction that it is possible to make after both knowledge and calibration constraints have been imposed Predictive Analysis 6 6 critical point Figure 6 5 The critical point with parameter bounds taken into account The importance of predictive analysis in model deployment cannot be understated In most instances of model usage only a single prediction is made However where a range of predictions is possible it is poor modelling practice not to provide at least some indication of the extent of this range The model prediction corresponding to the critical point defines the extent of this predictive range in one direction For example if a rainfall runoff model has been calibrated in order to make predictions of flood height following future high rainfall events then the model s prediction of the maximum possible flood height may be of critical importance to formulation of a flood management strategy Due to the fact that the height of the
73. in conductors of strange shape the growth of plants the population dynamics of ants the distribution of stress in the hulls of ships and on and on Modelling programs generally require data of four main types These are e Fixed data These data define the system for example in a ground water model the shape of the aquifer is fixed as are the whereabouts of any extraction and injection bores e Parameters These are the properties of the system parameters for a ground water model include the hydraulic conductivity and storage capacity of the rocks through which the water flows while for a stress model parameters include the elastic constants of the component materials A model may have many parameters each pertaining to one particular attribute of the system which affects its response to an input or excitation In spatial models a particular system property may vary from place to place hence the parameter data needed by the model may include either individual instances of that property for certain model subregions or some numbers which describe the manner in which the property is spatially distributed e Excitations These are the quantities which drive the system for example climatic data in a plant growth model and the source and location of electric current in electromagnetic boundary value problems Like parameters excitations may have spatial dependence e Control data These data provide settings for the numerical solution
74. in parameter estimation mode by assigning the weight as zero the prediction observation will contribute nothing to the objective function if that occurs Alter the model command line to model bat in the model command line section of the new PEST control file The two model input files are named inI dat and in2 dat the first is used for the calibration component of the composite model the second is used for the predictive component of the composite model A PEST template file already exists for the first model input file ie in tpl We will introduce a new template file for the second model input file shortly it will be called in2 tpl So alter the model input filename to inl dat on the first line of the model command line section of file twofitl pst then add another line underneath this comprised of the entries in2 tpl and in2 dat The model output files are named outl dat and out2 dat the first is produced by the calibration component of the composite model while the second is produced by the predictive component of the composite model A PEST instruction file already exists An Example 12 14 for the first model output file ie out ins We will introduce a new instruction file for the second one shortly it will be called out2 ins So alter the model output filename to outl dat on the third line of the model command line section of file twofit pst then add another line underneath
75. in that if r is 1 or greater b may fall to a minute fraction of bo or even to zero without transgressing the parameter change limit For some types of parameters in some models this will be fine in other cases a parameter factor change of this magnitude may significantly transgress model linearity limits In implementing the conditions set by equations 2 44 and 2 45 PEST limits the magnitude of the parameter upgrade vector Bu such that neither of these equations is violated Naturally if only one type of parameter change limit is featured in a current PEST run ie parameters are all factor limited or are all relative limited only the pertinent one of these equations will need to be obeyed If in the course of an optimisation run PEST assigns to a parameter a value which is very small in comparison with its initial value then either of equations 2 44 or 2 45 may place an undue restriction on subsequent parameter adjustments Thus if bo for one parameter is very small the changes to all parameters may be set intolerably small so that equation 2 44 or equation 2 45 is obeyed for this one parameter To circumvent this problem PEST provides another input variable FACORIG which allows the user to limit the effect that an unduly low parameter value can have in this regard If the absolute value of a parameter is less than FACORIG times its initial absolute value and PEST wishes to adjust that parameter such that its absolute value will increase
76. information on parameter sensitivity and correlation 6 2 The Prediction Window As is documented in Chapter 6 of the PEST manual under some circumstances special significance is attached to the observation group named predict When PEST is run in predictive analysis mode this observation group must have only one member this member being the prediction which PEST is trying to maximise or minimise while keeping the model calibrated In this case WinPEST makes a special window available for inspection this being the Prediction window Through this window the user can monitor the value of the prediction as it changes through the predictive analysis process Special significance can also be attached to the observation group predict when PEST is run under dual calibration conditions see Chapter 6 of the PEST manual again In this case evolution of the value of the normally single member of this observation group as the optimisation process progresses is often of great interest PEST and thus WinPEST is actually unaware of whether you are attempting a dual calibration run or not because there is no special mode for this PEST simply works in its traditional estimation mode when undertaking a dual calibration run However if there is an observation group named predict and if this group has only one member WinPEST plots the evolving value of the pertinent observation in its prediction window It sh
77. is subject to user alteration midway through the optimisation process See Section 5 6 for further details RLAMFAC RLAMFAC a real variable is the factor by which the Marquardt lambda is adjusted see Section 2 1 7 RLAMFAC must be greater than 1 0 When PEST reduces lambda it divides by RLAMFAC when it increases lambda it multiplies by RLAMFAC PEST reduces lambda if it can However if the normal matrix is not positive definite or if a reduction in lambda does not lower the objective function PEST has no choice but to increase lambda PHIRATSUF During any one optimisation iteration PEST may calculate a parameter upgrade vector using a number of different Marquardt lambdas First it lowers lambda and if this is unsuccessful in lowering the objective function it then raises lambda If at any stage it calculates an objective function which is a fraction PHIRATSUF or less of the starting objective function for that iteration PEST considers that the goal of the current iteration has been achieved and moves on to the next optimisation iteration Thus PEST will commence iteration i 1 if at any stage during iteration i il i lt PHIRATSUF 4 1 where is the lowest objective function calculated for optimisation iteration i 1 and hence the starting value for optimisation iteration i and q is the objective function corresponding to a parameter set calculated using the j th Marquardt lambda tested during optimisation iteration i
78. is used with such systems To enhance its use in conjunction with a complex modelling system PEST offers the user increased flexibility in derivatives calculation over that which is available through finite differences alone 8 2 Externally Supplied Derivatives 8 2 1 The External Derivatives File Some models are able to calculated derivatives of their outputs with respect to their parameters themselves If so it is often better for PEST to use these derivatives instead of the derivatives that it calculates itself using finite parameter differences There are two reasons for this 1 Code included within the model itself for the purpose of derivatives calculation can often exploit certain aspects of the mathematics underlying the numerical simulation process to calculate derivatives far more quickly than they can be calculated using finite differences 2 Derivatives calculated directly by the model are often numerically more precise than those calculated by taking differences between model outputs calculated on the basis of incrementally varied parameter values Hence if a model can calculate derivatives itself PEST should use these derivatives PEST can read derivatives of model calculated observations with respect to adjustable parameters from a special file written by the model whenever derivatives are requested by PEST Because this file must satisfy special formatting requirements it will normally be required that the user add a few
79. it can be log transformed by PEST for greater optimisation efficiency 1 4 2 Observation Definition and Recognition From this point onwards those numbers on a model output file for which there are corresponding real world values to which they must be matched will be referred to simply as observations Of the masses of data produced by a model only a handful of numbers may actually be observations For example a population dynamics model may calculate population figures on a daily basis yet measurements may only have been taken every week In this case most of the model s output data will be redundant from the point of view of model calibration Similarly a model for the stress field surrounding an excavation may calculate stress figures for each of the thousands of nodes of a finite element mesh through the use of which the system differential equations are solved however stress measurements may be available at only a handful of points viz at the locations of specially installed stress sensors in the rocks surrounding the excavation Again PEST must be able to identify a handful of numbers viz stress values calculated at those points for which stress measurements are available out of the thousands that may be written to the model s output file In order to peruse a model output file and read the observation values calculated by the Introduction 1 7 model PEST must be provided with a set of instructions Unfortuna
80. just before it begins calculation of the parameter upgrade vector PEST calculates the optimal value of the regularisation weight factor for that iteration This is the value which under the linearity assumption encapsulated in the Jacobian matrix results in a parameter upgrade vector for which the measurement component of the objective function is equal to PHIMLIM However due to the approximate nature of the linearity assumption PEST may not be able to lower the measurement component of the objective function to PHIMLIM on that iteration in spite of the fact that it uses a number of different values for the Marquardt lambda in attempting to do so If it cannot lower the measurement objective function to an acceptable level it simply accepts the upgraded parameters proceeds to the next optimisation iteration and tries again However if it does succeed in lowering the measurement objective function to an acceptable level or if it has succeeded in doing this on previous iterations then PEST slightly alters its philosophy of choosing new Marquardt lambdas in that it now attempts to lower the regularisation component of the objective function while maintaining the measurement component of the objective function m below this acceptable level This acceptable level is PHIMACCEPT it should be set slightly higher than PHIMLIM ie m in order to give PEST some room to move in its attempts to lower while keeping m below or close to Om
81. lines of code to the model to endow it with the ability to write this file to PEST s specifications The external derivatives file as it is called herein produced by the model must contain a derivatives matrix This is slightly different from the Jacobian matrix in that the latter matrix takes account of whether parameters are log transformed or not during the inversion process The derivatives matrix like any other matrix is comprised of rows and columns Each column contains the derivative of every model outcome for which there is a complementary observation with respect to a particular parameter Each row contains the derivatives of a single observation with respect to all parameters Example 8 1 shows an example of an external derivatives file expected by PEST Model Calculated Derivatives 8 2 4 9 5 00000 1707 60 34 4932 42 1234 5 25066 8 79458 93 2321 23 5921 1 04819 1 16448 5 34642 19 3235 1 52323 0 11418 0 59235 75 2354 3 21342 0 48392 9 49293 95 3459 2 49321 5 39230 0 49332 9 22934 19 4492 9 93024 0 49304 5 39234 36 3444 10 4933 0 59439 6 49345 95 4592 86 4234 47 4232 324 434 Example 8 1 An external derivatives file The first line of an external derivatives file contains two integers listing the number of parameters and number of observations represented in the derivatives file These correspond to the number of columns and the number of rows respectively in the derivatives matrix They must als
82. linked to more than one model input file This may occur if the same model is being run over more than one historical time period and parameter data for the model resides in a different file from excitation data A separate line must be provided for each such pair of files in the model input output section of the PEST control file A model input file cannot be linked to more than one template file As explained in Chapter 3 a model may have many input files However PEST only needs to know about those that contain parameters The second part of the model input output section of the PEST control file contains instruction file model output file pairs There should be one line for each of NINSFLE such pairs the value of NINSFLE having been provided to PEST in the control data section of the PEST control file Pathnames must be provided for both instruction files and model output files if they do not reside in the current directory Construction details for instruction files are provided in Chapter 3 of this manual A single model output file may be read by more than one instruction file perhaps you wish to extract the values for observations of different types from the model output file using different instruction files However any particular observation can only ever be referenced once hence a particular instruction file cannot be matched to more than one model output file 4 2 10 Prior Information If the value of NPRIOR
83. method by which the system equations are solved Examples are the specifications of a finite element mesh the convergence criteria for a preconditioned conjugate gradient matrix equation solver and so on The distinction between these different data types may not always be clear in a particular case Introduction 1 2 The purpose of a mathematical model is to produce numbers These numbers are the model s predictions of what a natural or man made system will do under a certain excitation regime It is for the sake of these numbers that the model was built be it a ten line program involving a few additions and subtractions or a complex numerical procedure for the solution of coupled sets of nonlinear partial differential equations Where a model simulates reality it often happens that the model user does not know what reality is in fact models are often used to infer reality by comparing the numbers that they produce with numbers obtained from some kind of measurement Thus if a model s parameter and or excitation data are tweaked or adjusted until the model produces numbers that compare well with those yielded by measurement then perhaps it can be assumed that the excitations or parameters so obtained have actually told us something which we could not obtain by direct observation Thus if a ground water model is able to reproduce the variations in borehole water levels over time a quantity which can be obtained by direct observati
84. model domain is complex and or the hydraulic conductivity of the porous medium through which the water flows is nonuniform this equation must be solved numerically using a technique such as the finite difference finite element or analytical element method In most ground water modelling applications the hydraulic conductivity at different locations within the model domain is only poorly known However if the strengths of the various sources and sinks of water and the characteristics of the pertinent boundary conditions affecting the system are known then hydraulic conductivities in different parts of the domain can be inferred from borehole water level measurements using PEST assuming that the coverage of observation bores is good enough However before doing this the user must decide on an appropriate parameterisation strategy for the model domain A common strategy is to subdivide the model domain into zones of assumed parameter constancy based on geological or other information Unfortunately such information is often absent or unreliable Furthermore there can be a considerable degree of variation of hydraulic conductivity within each geological unit as a result of many factors including lithological heterogeneity differential weathering structural features created during tectonic events etc Thus in many instances it is necessary to consider a more complex parameterisation scheme A simple but effective scheme is to divide the model do
85. model in The template file model tpl thus cites all of parameters parl to par5 as well as parameter parS It may also cite par6 and par7 PEST Utilities 10 18 parameter data parl 1 583745e 4 par2 395832e 1 par3 583924e 2 par4 389028e 5 par5 389428e 2 par6 559313e 1 par7 395355e 2 par8 par6 exp par7 template and model input files model tpl model in Example 10 9 A PAR2PAR input file ptf parameter data parl Sparl par2 Spar2 par3 Spar3 par4 Spar4 par5 Spar5 par6 Spar6 par7 Spar7 par8 par6 exp par7 template and model input files model tpl model in Example 10 10 A template file for the PAR2PAR input file of Example 10 9 10 7 3 2 Numerical Precision As is explained elsewhere in this manual when PEST writes a number to a model input file on the basis of a template file it alters its internal representation of that number to account for the fact that the number may be written to the model input file with less than the maximum number of significant figures with which that number can be represented internally within the computer Thus when PEST calculates derivatives of model outputs with respect to parameters using finite differences the differences between incrementally varied parameter values will be exactly correct because both PEST and the model use exactly the same parameter values The ability for PEST to compensate for limited parameter space widths on model in
86. multiplied by the optimised regularisation weight factor To calculate a parameter covariance matrix which takes no account of members of the observation group regul follow the instructions provided in Section 7 5 1 of the PEST manual 4 12 Correlation Conditions under which the correlation coefficient matrix will be displayed are similar to those under which the covariance matrix will be displayed however the pertinent control variable is ICOR rather than ICOV WinPEST Graphical Windows 18 Elements of the correlation coefficient matrix are displayed in similar fashion to the elements of matrices represented in other WinPEST windows ie positive valued elements are represented by red circles while negative valued elements are represented by blue squares with the size of the symbol representing the magnitude of the element Click on any such symbol to see the element indices and the numerical value of the element Note that all diagonal elements of the correlation coefficient matrix are equal to one Non adjustable parameters ie tied and fixed parameters are not represented in the matrix 4 13 Eigenvectors Eigenvectors of the covariance matrix will be displayed only if calculation of the covariance matrix is not prevented through singularity of the normal matrix As for other plots of this type eigenvectors can be displayed and updated during the course of the parameter estimation process if the IEIG variable in the PEST contr
87. na none ro2 log factor 5 00000 0 100000 10 0000 ro ro3 tied to ro2 na 0 500000 na na ro h1 none factor 2 00000 5 000000E 02 100 000 h h2 log factor 5 00000 5 000000E 02 100 000 h Name Scale Offset rol 1 00000 0 000000 ro2 1 00000 0 000000 ro3 1 00000 0 000000 h1 1 00000 0 000000 h2 1 00000 0 000000 Prior information Prior info Factor Parameter Prior Weight name information pil 1 00000 h1 2 00000 3 000 pi2 1 00000 log ro2 1 00000 log h2 2 60260 2 000 Prior Info Name Observation Group pil group_4 pi2 group_4 Observations Observation name Observation Weight Group arl 1 21038 1 000 group_1l ar2 1 51208 1 000 group_1 Running PEST 5 4 ar3 2 07204 000 group_1 ar4 2 94056 000 group_1 ars 4 15787 1 000 group_1 ar 5 77620 1 000 group_1 ar7 7 78940 1 000 group_2 ar8 9 99743 1 000 group_2 arg 11 8307 000 group_2 ar10 12 3194 1 000 group_2 ar11 0 6003 000 group_2 ar12 7 00419 1 000 group_2 ar13 3 44391 000 group_2 ar14 1 58279 000 group_2 ar15 1 10380 1 000 group_3 ar16 1 03086 000 group_3 arl7 1 01318 000 group_3 ar18 1 00593 1 000 group_3 ar19 1 00272 000 group_3 Inversion control settings Initial lambda 5 0000 Lambda adjustment factor 2 0000 Sufficient new old phi ratio per iteration 0 40000 Limiting relative phi reduction between lambdas 3 00000E 02 Maximum trial lambdas per iteration 10 Maximum factor parameter change factor limited changes 3 0000 Maximum relative p
88. now be expressed in terms of covariance matrices rather than simply in terms of weights Preface If derivatives of model outputs with respect to adjustable parameters can be calculated by the model rather than by PEST through the use of finite differences then PEST can use these derivatives if they are supplied to it through a file written by the model Different commands can be used to run the model for different purposes for which the model is used by PEST viz testing parameter upgrades calculating derivatives with respect to different parameters etc PEST can now send messages to a model allowing the model to adjust certain aspects of its behaviour depending on the purpose for which it is run by PEST PEST stores the Jacobian matrix corresponding to the best set of parameters achieved up to any stage of the parameter estimation process in a special binary file which is updated as the parameter estimation process proceeds A new utility program named READJAC re writes the Jacobian matrix in text format for user inspection PEST prints out a more comprehensive suite of information on composite parameter sensitivities than was available in previous versions A new utility named PAR2PAR has been added to the PEST suite This is a parameter preprocessor which allows the user to manipulate parameters according to mathematical equations of arbitrary complexity before these parameters are supplied to the model John Dohe
89. number you intend if you get an instruction wrong If PEST tries to read an observation but does not find a number where it expects to find one a run time error will occur PEST will inform you of where it encountered the error and of the instruction it was implementing when the error occurred this should allow you to find the problem However if PEST actually reads the wrong number from the model output file this may only become apparent if an unusually high objective function results or if PEST is unable to lower the objective function on successive optimisation iterations In this case you should interrupt PEST execution asking PEST to terminate execution with a statistics printout see Chapter 5 Included in this printout are the current model generated observation values if PEST is reading the wrong number it will then become apparent Included in the PEST suite are two programs which can be used to verify that instruction files The Model PEST Interface 3 26 have been built correctly Program PESTCHEK when checking all PEST input data for errors and inconsistencies prior to a PEST run reads all the instruction files cited in a PEST control file see the next chapter ensuring that no syntax errors are present in any of these files Program INSCHEK on the other hand checks a single PEST instruction file for syntax errors If an instruction file is error free INSCHEK can then use that instruction file to read a model output file printin
90. numbers recorded on any of its output files SENSCHEK should be run immediately if it has not already been run If SENSCHEK has not been used to verify an input dataset and SENSAN finds an error in a parameter variation file such as an unreadable parameter value it will not terminate execution Instead SENSAN reports the error to the screen and moves on to the next parameter set However it writes the offending line of the parameter variation file to its three output files If a trailing system command is present on this line this too will be written to the SENSAN output files however the command is not executed Naturally model outcome values are not written to the SENSAN output files because they cannot be calculated in these circumstances It is possible that model execution will fail for some parameter value sets supplied by the user SENSAN ensures that old model output files are deleted before the model is run so that should this occur out of date model outcome values are not read as current values If after the model has been run a certain model output file is not found SENSAN reports this condition to the screen records the current set of parameter values to its output files and moves on to the next parameter set If a model run terminates prematurely for a particular parameter set and all model outcomes cannot be read INSCHECK run by SENSAN will fail to produce an observation value file which SENSAN reads to ascertain model outco
91. objective function viz the regularisation component and m the measurement component can be compared from iteration to iteration Regularisation 7 5 Because when operating in regularisation mode PEST s intention during each optimisation iteration remains the same as in parameter estimation mode ie to lower an objective function all of the input variables which control its operation in parameter estimation mode are still required for its operation in regularisation mode Furthermore they still have the same roles However as will be discussed below a number of new variables are required in the PEST control file to control PEST s operation in regularisation mode 7 3 Preparing for a PEST Run in Regularisation Mode 7 3 1 The PEST Control File Control Data Section As was discussed in Section 4 2 2 of this manual the variable PESTMODE on the third line of the PEST control file must be provided as estimation prediction or regularisation the last of these options must be supplied for this variable for PEST to run in regularisation mode If so there must be a regularisation section at the end of the PEST control file following either the model input output section or if present the prior information section of the PEST control file 7 3 2 The PEST Control File Observation Groups As has already been discussed when working in regularisation mode observations and or prior in
92. of the previous parallel Jacobian calculation process e the user indicates to PEST that only the Jacobian matrix calculation process is to be parallelised see below The user should take particular note of the first of the above exceptions to PEST s ability to undertake a partial parallelisation of the lambda search The reason for this exception is that while PEST may be able to guess the values of future Marquardt lambdas to be used in the lambda search procedure with a high probability of success it has far more difficulty in Parallel PEST 9 3 predicting whether a parameter is to be frozen at its upper or lower bound and the order in which parameters are to be frozen if more than one of them are at their limits The fact that a parameter is at its upper or lower bound is no guarantee that it will be frozen for it may need to move back into adjustable parameter space from that bound as different lambdas are tested A continuation of the discussion on how Parallel PEST undertakes partial parallelisation of the lambda testing procedure will be presented in Section 9 2 6 below after a discussion of how Parallel PEST actually works 9 1 4 A Warning If model run times are short gains in computational efficiency that are achievable using Parallel PEST will not be as great as when model times are large for the time taken in writing and reading possibly lengthy model input and output files across a local area network may
93. of the PEST nonlinear parameter estimation algorithm and the means by which this theory has been implemented in the construction of the powerful parameter optimiser which is PEST However the discussion is brief and no proofs are presented The reader is referred to the limited bibliography at the end of the chapter for a number of books which treat the subject in much greater detail 2 1 The Mathematics of PEST 2 1 1 Parameter Estimation for Linear Models Let us assume that a natural or man made system can be described by the linear equation Xb c 2 1 In equation 2 1 X is a m x n matrix ie it is a matrix with m rows and n columns The elements of X are constant and hence independent of the elements of b a vector of order n which we assume holds the system parameters is a vector of order m containing numbers which describe the system s response to a set of excitations embodied in the matrix X and for which we can obtain corresponding field or laboratory measurements by which to infer the system parameters comprising b Note that for many problems to which PEST is amenable the system parameters may be contained in X and the excitations may comprise the elements of b Nevertheless in the discussion which follows it will be assumed for the sake of simplicity that b holds the system parameters The word observations will be used to describe the elements of the vector c even though c is in fact generated by the model This is
94. on these outcomes over other parts of its domain If the optimised value lies within the insensitive area a large degree of uncertainty will surround its estimate However if the optimal value lies in the sensitive part of the parameter s domain it is likely that the parameter will be well determined unless of course it is highly correlated with some other parameter In either case you should take care to ensure that the number which you supply as Running PEST 5 31 the parameter s initial value is within the sensitive part of its domain 5 5 11 Parameter Cannot be Written with Sufficient Precision In certain unusual cases an optimal parameter value may not be capable of representation in a field of limited width on the model input file The obvious solution to this problem is to increase the width of the parameter field in the corresponding template file However this may not be possible if model input format requirements are too rigid The only other remedial action that can be taken is to set the DPOINT variable to nopoint thus allowing a gain of one extra significant figure in some circumstances Before you do this however make sure that the model can still read the parameter value correctly with its decimal point omitted Note that PEST omits the decimal point only if it is redundant However because some models may use format specifiers which make certain assumptions about where an absent decimal point should be it is possib
95. only 1 as described in Section 2 2 5 it may then take many iterations to re adjust upwards this causing serious inefficiencies in the parameter estimation process If you wish to limit the extent of its downward movement during any one iteration to less than this you may wish to set RELPARMAX to for example 0 5 however this may unduly restrict its upward movement It may be better to declare the parameter as factor limited If so a FACPARMAX value of say 5 0 would limit its downward movement on any one iteration to 0 2 of its value at the start of the iteration and its upward movement to 5 times its starting value This may be a more sensible approach for many parameters Alternatively provide the parameter with a non zero OFFSET value and adjust its upper and lower bounds such that it never becomes zero or nearly zero It is important to note that a factor limit will not allow a parameter to change sign Hence if a parameter must be free to change sign in the course of the optimisation process it must be relative limited furthermore RELPARMAX must be set at greater than unity or the change of sign will be impossible Thus the utility program PESTCHEK see Chapter 10 will not allow The PEST Control File 4 10 you to declare a parameter as factor limited or as relative limited with the relative limit of less than 1 if its upper and lower bounds are of opposite sign Similarly if a parameter s upper or lower bound is zero it cannot be
96. parameter estimation process is better served by terminating the current lambda search procedure and moving on to the next optimisation iteration The criteria by which this decision is made are supplied through the variables appearing on the sixth line of the PEST control file Hence the size of the packet of parallel model runs ordered by PEST is determined solely on the basis of the number and speed of available slaves and not on the basis of foreknowledge of the number of parallel runs required for completion of the lambda search procedure The decision making process involved in the lambda search procedure is activated only after each packet of model runs is complete a process that may result in some of these runs being ignored The fact that the lambda adjustment procedure then becomes a combination of parallelisation with intermittent decision making is the basis for its classification as a partial parallelisation procedure During any optimisation iteration upon commencement of the lambda search procedure for that optimisation iteration PEST s first packet of model runs is based on Marquardt lambdas which are generally lower than the optimal lambda determined during the previous optimisation iteration However if there are enough slaves at its disposal PEST also carries out model runs based on one or a number of higher Marquardt lambda values as well On subsequent occasions during the same lambda search procedure on which PEST o
97. parameter is provided as 1 0E10 while the lower bound is 1 0E10 It is strongly suggested that you modify these bounds to suit each parameter It is also recommended that you consider log transforming some parameters for greater optimisation efficiency see Section 2 2 1 Note however that the lower bound of a log transformed parameter must be positive and that its changes must be factor limited e All observations are provided with a weight of 1 0 e PESTGEN assumes that the model is run using the command model It also assumes that the model requires one input file viz model inp for which a template file model tpl is provided It further assumes that all model generated observations can be read from one output file viz model out using the instructions provided in the instruction file model ins You will almost certainly need to alter these names If there are in fact multiple model input and or output files don t forget to alter the variables NTPLFLE and NINSFLE in the control data section of the PEST control file e The default values for all other variables can be read from Example 10 3 Once you have made all the changes necessary to the PESTGEN generated PEST control file you should check that your input dataset is complete and consistent using program PESTCHEK If PESTCHEK informs you that all is correct then you are ready to run PEST 10 5 PARREP Program PARREP replaces initial parameter values as provided in
98. parameters named in the template file To run TEMPCHEK in this fashion enter the command tempchek tempfile modfile parfile The name of the parameter value file is optional If you don t supply a name TEMPCHEK generates the name itself by replacing the extension used in the template filename with the extension par if tempfile has no extension par is simply appended Hence the naming convention of the parameter value file is in accordance with that used by PEST which generates such a file at the end of every optimisation iteration see Section 5 3 1 A PEST parameter value file is shown in Example 5 2 The first line of a parameter value file must contain values for the character variables PRECIS and DPOINT the role of these variables is discussed in Section 4 2 2 These variables must be supplied to TEMPCHEK so that it knows what protocol to use when writing parameter values to the model input file which it generates The second and subsequent lines of a parameter value file each contain a parameter name a value for the named parameter and the scale and offset to be used when writing the parameter value to the model input file Because TEMPCHEK is supplied with a scale and offset for each parameter it is able to generate model input files in exactly the same way that PEST does see Section 4 2 4 If TEMPCHEK finds a parameter in a template file which is not listed in the parameter value file it terminates execution with
99. place of individual observation weights if the cofactor matrix Q is calculated as the inverse of a user supplied observation covariance matrix Note however that as was mentioned above a user supplied observation covariance matrix can only be considered as proportional to the true observation covariance matrix the latter can be determined after the inversion process is complete by multiplying the user supplied covariance matrix by the reference variance determined through equation 2 5 At any stage of the optimisation process the objective function is computed using equation 2 8a this is equivalent to equation 2 8b when Q is a diagonal matrix whose elements are the squares of observation weights Use of PEST s regularisation functionality does not preclude use of a covariance matrix to characterise either or both of measurement and regularisation observations However when PEST is used in this mode the covariance matrix supplied for the regularisation observations must be separate from that supplied for the measurement observations If a covariance matrix is provided for the regularisation observations PEST will calculate a weight factor by which to multiply the regularisation cofactor matrix Q calculated by PEST as the inverse of the user supplied regularisation covariance matrix in order to satisfy the goal of the regularisation process ie that the measurement component of the objective function be no greater than the user
100. precision The third method is to use the least squares principal to define a straight line of best fit through the three parameter observation pairs and to take the derivative as the slope of this line This method may work best where model calculated observations cannot be read from the model output file with great precision because of either deficiencies in the model s numerical solution method or because the model writes numbers to its output file using a limited number of significant figures If the central method of derivatives calculation is used for all parameters each optimisation iteration requires that at least twice as many model runs be carried out than there are adjustable parameters If the central method is used for some parameters and the forward method for others the number of model runs will lie somewhere between the number of adjustable parameters and twice the number of adjustable parameters 2 3 2 Parameter Increments for Derivatives Calculation Because of the importance of reliable derivatives calculation PEST provides considerable flexibility in the way parameter increments are chosen Mathematically a parameter increment should be as small as possible so that the finite difference method or one of its three point variants provides a good approximation to the derivative in a theoretical sense remember that the derivative is defined as the limit of the finite difference as the increment approaches zero However if th
101. run the model as many times as it needs to without any human intervention PEST can be used with models written in any programming language they can be home made or bought You do not need to have the source code or know much about the internal workings of the model Models can be small and fast finishing execution in the blink of an eye or they can be large and slow taking minutes or even hours to run it does not matter to PEST 1 4 An Overview of PEST PEST can be subdivided into three functionally separate components whose roles are Introduction 1 5 e parameter and or excitation definition and recognition e observation definition and recognition and e the nonlinear estimation and predictive analysis algorithm Though the workings of PEST will be described in detail in later chapters these three components are discussed briefly so that you can become acquainted with PEST s capabilities 1 4 1 Parameter Definition and Recognition From this point on the single word parameter is used to describe what has hitherto been referred to as parameters and or excitations Of the masses of data of all types that may reside on a model s input files those numbers must be identified which PEST is free to alter and optimise Fortunately this is a simple process which can be carried out using input file templates If a model requires for example five input files and two of these contain parameters which PEST is fre
102. runs as part of its lambda search procedure is similar to that used for parallelisation of the Jacobian matrix calculation process However there are some important differences One such difference is that if any slave carries out model runs with a run time which is greater than 1 8 times that of the fastest slave then that slave is not used in the partial parallelisation process This is because PEST sends model runs to its slaves in packets of 1 or more runs and will not resume its normal execution until all of those runs are completed The size of this packet of runs at any stage of the lambda search procedure is equal to the number of available slaves whose execution speed is roughly equivalent to that of the fastest slave The packet is limited to this size because if one particular slave can complete two model runs in the same or less time than that required for another slave to complete only one model run then it would be more efficient to undertake these model runs in serial with the proper decision making process taking place after each such run Another difference between the procedure by which Jacobian runs are carried out in parallel and that by which lambda search runs are carried out in parallel is that in the former case PEST knows the number of runs that must be carried out before the Jacobian calculation process is complete However the lambda search procedure is deemed to be complete when PEST judges that the overall
103. section of the PEST control file 4 2 8 Model Command Line This section of the PEST control file supplies the command which PEST must use to run the model The command line may be simply the name of an executable file or it may be the name of a batch file containing a complex sequence of steps Note that you may include the path name in the model command line which you provide to PEST if you wish If PEST is to be successful in running the model then either the model must be in the current directory its full path must be provided or the PATH environment variable must include the directory in which the executable or batch file is situated Consider the case of a finite difference model for the stress field surrounding a tunnel The input file may be very complicated involving one or a number of large two or three dimensional arrays While parameters can be written to such files using appropriate templates you may prefer a different approach Perhaps you wish to estimate rock properties within a small number of zones whose boundaries are known these zones collectively covering the entire model domain Furthermore as is often the case you may have some The PEST Control File 4 22 preprocessing software which is able to construct the large model arrays from the handful of parameters of interest viz the elastic properties of the zones into which the model domain has been subdivided In this case it may be wise to run the preprocessor prior t
104. such parameter space with the corresponding current parameter value hence it will reconstruct an array containing hundreds maybe thousands of elements but in which only n different numbers are represented The occurrence of multiple incidences of the same parameter is not restricted to the one file If a model has multiple input files and if a particular parameter which you would like to optimise appears on more than one of these files then at least one space for this parameter will appear on more than one template file PEST passes no judgement on the occurrence of parameters within template files or across template files However it does require that each parameter cited in the PEST control file see Chapter 4 occur at least once on at least one template file and that each parameter cited in a template file be provided with bounds and an initial value in the PEST control file 3 2 8 Preparing a Template File Preparation of a template file is a simple procedure For most models it can be done in a matter of moments using a text editor to replace parameter values on a typical model input file by their respective parameter space identifiers Once a template file has been prepared it can be checked for correctness using the utility program TEMPCHEK see Chapter 10 TEMPCHEK also has the ability to write a model input file on the basis of a template file and a user supplied list of parameter values If you then run your model providing it wi
105. surface of a layered half space for different surface electrode configurations Suppose that we wish to use this program ie model to estimate the properties for each of three half space layers from apparent resistivity data collected on the surface of the half space The parameters for which we want estimates are the resistivity and thickness of the upper two layers and the resistivity of the third its thickness is infinite A suitable template file appears in Example 3 2 The Model PEST Interface 3 3 z O is EL INPUT FILE 19 no of layers no of spacings 0 1 0 resistivity thickness layer 1 0 0 20 0 resistivity thickness layer 2 0 resistivity layer 3 0 electrode spacings 47 TES 16 64 6 81 IROSO BwONRFP OPP W w eao 68 100 149 219 316 464 681 1000 Example 3 1 A model input file ptf MODEL INPUT FIL Sipe Jey no of layers no of spacings resl t1 resistivity thickness layer 1 res2 t2 resistivity thickness layer 2 res3 resistivity layer 3 1 0 electrode spacings 1 47 225 3 16 4 64 6 81 10 0 Fl Se oe w eeo 68 100 149 215 316 464 681 1000 Example 3 2 A template file 3 2 3 The Parameter Delimiter The Model PEST Interface 3 4 As Example 3 2 shows the first line of a template file must contain the letters ptf followed by a space followed by a single character ptf stands for
106. that any variables governing the numerical solution procedure be set in favour of precision over time Although the model run time may be much greater as a result it would be false economy to give reduced computation time precedence over output precision Accurate derivatives calculation depends on accurate calculation of model outcomes If PEST is trying to estimate model parameters on the basis of imprecise model generated observations derivatives calculation will suffer and with it PEST s chances of finding the optimal parameter set Even if PEST is still able to find the optimal parameter set which it often will it may require more optimisation iterations to do so resulting in a greater overall number of model runs removing any advantages gained in reducing the time required for a single model run Even after you have instructed the model to write to its output file with as much precision as possible and you have adjusted the model s solution settings for greatest precision model generated observations may still be granular in that the relationship between these observations and the model parameters may be bumpy rather than continuous In this case it may be wise to set parameter increments larger than you normally would If a parameter increment is set too small PEST may calculate a local erroneous bump derivative rather than a derivative that reflects an observation s true dependence on a parameter s value While
107. that there are often many different sets of parameter values for which the objective function is at its minimum or almost at its minimum Thus there are many different sets of parameters which could be considered to calibrate a model A question that then arises is if I use different sets of parameter values when using the model to make predictions all of these sets being considered to calibrate the model will I get different values for key model outcomes This question can be answered by running PEST in predictive analysis mode To run PEST in predictive analysis mode the user informs PEST of the objective function value below which the model can be considered to be calibrated this value is normally just slightly above the minimum objective function value as determined in a previous PEST calibration run A key model prediction is then identified on one of the model output files this may involve setting up a dual model run by PEST through a batch file consisting of the model run under both calibration and predictive conditions PEST is then asked to find that parameter set which results in the maximum or minimum model prediction while still calibrating the model In doing this PEST uses an iterative solution procedure similar in many ways to that used for solution of the parameter estimation problem The key model prediction made with a parameter set calculated in this way defines the upper or lower bound of the uncertainty interva
108. the PEST control file contains values for the variables PARNME PARTRANS PARCHGLIM PARVALI PARLBND PARUBND PARGP SCALE OFFSET and DERCOM in that order Only the variable DERCOM is used in implementing PEST s external derivatives functionality As was discussed above the various commands which can be used to run the model for purposes other than external derivatives calculation are listed in the model command line section of the PEST control file The value of DERCOM pertaining to each parameter denotes which of these commands will be used to run the model when PEST calculates Model Calculated Derivatives 8 8 derivatives with respect to that parameter using finite differences commands within the model command line section are numbered from first to last beginning at 1 Alternatively if the derivatives of all observations with respect to a particular parameter are to be supplied externally then the DERCOM value for that parameter should be supplied as zero If this is the case PEST will not undertake any model runs to calculate derivatives with respect to this parameter using the finite difference method For a particular parameter derivatives for some observations may be calculated using finite differences while derivatives for others may be supplied externally by the model In this case a non zero value should be provided for DERCOM thus ensuring that PEST calculates derivatives using finite differences for this para
109. the decision of whether to test more lambdas or to move on to the next iteration If the objective function is below PD1 and successive Marquardt lambdas have not succeeded in raising lowering the model prediction by a relative value of more than RELPREDLAM or by an absolute value of more than ABSPREDLAM PEST will move on to the next optimisation iteration Due to the fact that the approach to the critical point is often slow these values may need to be set low A value of 0 005 for RELPREDLAM is often suitable the value for ABSPREDLAM depends on the context If you would like one of these variables to have no effect on the predictive analysis process which is mostly the case for ABSPREDLAM use a value of 0 0 INITSCHFAC MULSCHFAC and NSEARCH When undertaking predictive analysis PEST calculates a parameter upgrade vector in accordance with the theory presented in Section 2 1 9 of this manual However it has been found from experience that the critical point of Figures 6 4 and 6 5 can be found more efficiently if PEST undertakes a line search along the direction of its calculated parameter upgrade vector each time it calculates such a vector in order to find the exact point of intersection of this vector with the min 6 contour However this search will not be undertaken unless the objective function has fallen below min at least once during any previous optimisation iteration A line search is undertaken for each trial value o
110. the higher order interpolation scheme provided by the parabolic method may allow you to place parameter values and hence model generated observation values farther apart for the calculation of derivatives this may have the effect of increasing the degree of significance of the resulting differences involved in the derivative calculation However if the increment is raised too high derivative precision must ultimately fall Note that through DERINCMUL you can also reduce the increment used for central derivatives calculation if you wish For increments calculated using the relative and rel_to_max methods the variable DERINCLB has the same role in central derivatives calculation as it does in forward derivatives calculation viz to place a lower limit on the increment absolute value Note however that DERINCLB is not multiplied by DERINCMUL when derivatives are calculated using the central method If a parameter is log transformed then it is wise that its increment be calculated using the relative method though PEST does not insist on this As PEST reads the data contained in its input control file it will object if a parameter increment either read directly as absolute or calculated from initial parameter values as relative or rel_to_max exceeds the range of values allowed for that parameter as defined by the parameter s upper and lower bounds divided by 3 2 as the increment is then too large compar
111. the parameter initial value PARVAL 1 and its upper and lower bounds PARUBND and PARLBND these must pertain to the parameter itself likewise at the end of the parameter estimation process PEST provides the optimised parameter value itself rather than the log of its value Experience has shown repeatedly that log transformation of at least some parameters can make the difference between a successful parameter estimation run and an unsuccessful one This is because in many cases the linearity approximation on which each PEST optimisation iteration is based holds better when certain parameters are log transformed However caution must be exercised when designating parameters as log transformed A parameter which can become zero or negative in the course of the parameter estimation process must not be log transformed hence if a parameter s lower bound is zero or less PEST will disallow logarithmic transformation for that parameter Note however that by using an appropriate scale and offset you can ensure that parameters never become negative Thus if you are estimating the value for a parameter whose domain as far as the model is concerned is the interval 9 99 10 you can shift this domain to 0 01 20 for PEST by designating a scale of 1 0 and an offset of 10 0 Similarly if a parameter s model domain is entirely negative you can make this domain entirely positive for PEST by supplying a scale of 1 0 and an offset of 0 0 See Sect
112. this observation it should be the sole member of an observation group named predict It is important to note that PEST takes no notice of either the observed value of this observation or of the weight assigned to this observation PEST s job is simply to raise or lower the model output corresponding to this observation while maintaining the objective function at or below min Before using PEST to undertake predictive analysis you should have already calibrated the model Because the model has been calibrated you will have determined a parameter set corresponding to the objective function minimum you will also have determined the objective function minimum itself ie min When run in predictive analysis mode PEST must be supplied with the value of min henceforth referred to as o PEST then maximises or minimises the model prediction you tell it which while ensuring that the objective function calculated on the basis of calibration observations alone is as close as possible to o See Section 2 1 9 for further details PEST s operation in predictive analysis mode has much in common with its operation in parameter estimation mode Like parameter estimation the process required to determine the critical point is an iterative one beginning at some user supplied initial parameter set Initial parameters can be either inside the po contour or outside of it in fact they can be the same initial values that were used for
113. those for which there are template files As explained later a single template file may under some circumstances be used to write more than one model input file In such a case you must treat each template file model input file pair separately in determining NTPLFLE NINSFLE This is the number of instruction files There must be one instruction file for each model output file containing model generated observations which PEST uses in the determination of the objective function In some circumstances a single model output file may be read by more than one instruction file however each instruction file model output file pair is treated separately in determining NINSFLE PRECIS PRECIS is a character variable which must be either single or double If it is supplied to PEST as single PEST writes parameters to model input files using single precision protocol ie parameter values will never be greater than 13 characters in length even if the parameter space allows for a greater length and the exponentiation character is e If PRECIS is supplied as double parameter values are written to model input files using double precision protocol the maximum parameter value length is 23 characters and the exponentiation symbol is d See Section 3 2 6 DPOINT This character variable must be either point or nopoint If DPOINT is provided with the value nopoint PEST will omit the decimal point
114. through which PEST is provided with an appropriate set of optimisation control variables and in which the laboratory measurements of specific volume are provided First copy file out obf to file measure obf Then replace the value of each model generated observation with the corresponding value from Table 12 1 ie with the appropriate laboratory measurement see Example 12 9 Then run PESTGEN using the command pestgen twofit in par measure obf ol 0 501 o2 OSZI 03 0 520 o4 0531 55 0 534 o6 0 548 o7 0 601 o8 0 626 o9 0 684 o10 0 696 oll 0 706 O12 10 57 83 61 3 1 0 832 Example 12 9 File measure obf PESTGEN generates a PEST control file named twofit pst see Example 12 10 File twofit pst should now be edited as some of the default values used by PESTGEN in writing this file are not appropriate to our problem In particular our model is run using the command twoline not model the filenames listed in the model input output section of twofit pst need to be altered as well Once you have made these changes Example 12 11 lists that part of twofit pst to which the alterations have been made preparation for the PEST run is complete It would be a very good idea to make some other adjustments to twofit pst as well such as providing more appropriate upper and lower bounds for each of the parameters However at the risk of leading you into bad habits this will not be done An Example 12 9 pet con
115. tune it to a specific model and you need to know what these adjustments are often it is only by trial and error that you can determine what are its best settings for a particular case The fact that PEST s operation can be tuned in this manner is one of its strengths however this strength can be properly harnessed only if you are aware of what options are available to you So I urge you to take the time to understand the contents of this manual before using PEST to interpret real world data In this way you will maximise your chances of using PEST successfully Experience has shown that for some difficult or messy models the setting of a single control variable can make the difference between PEST working for that model or not Once the correct settings have been determined PEST can then be used with that model forevermore maybe saving you days perhaps weeks of model calibration time for each new problem to which that model is applied Hence a small time investment in understanding the contents of this manual could yield excellent returns So good luck in your use of PEST I hope that it provides a quantum leap in your ability to calibrate models and interpret field and laboratory data John Doherty February 1994 Preface to Second Edition Since the first version of PEST was released in early 1994 it has been used all over the world by scientists and engineers working in many different fields including biology geophysics
116. use of a large increment incurs penalties due to poor representation of the derivative by a The PEST Algorithm 2 32 finite difference especially for highly nonlinear models this can be mitigated by the use of one of the central methods of derivatives calculation available in PEST particularly the parabolic and best fit methods Due to its second order representation of the observation parameter relationship the parabolic method can generate reliable derivatives even for large parameter increments However if model outcomes are really bumpy the best fit method may be more accurate Trial and error will determine the best method for the occasion 2 3 4 Model Calculated Derivatives As has been discussed above the calculation of derivatives by finite differences is both time consuming and numerically intensive If a model can calculate derivatives of its outputs with respect to its adjustable parameters itself use of these derivatives is normally extremely beneficial to the parameter estimation process This is a result of the greater accuracy with which a model can normally calculate its own derivatives especially if these are calculated using analytical equations and the likelihood that a model can undertake such calculations comparatively quickly through modification of the calculations that it undertakes in the normal simulation process Hence if they are available model calculated derivatives should be used by PEST in preference to f
117. used to write a PEST control file 4 2 Construction Details 4 2 1 The Structure of the PEST Control File The PEST control file consists of integer real and character variables Its construction details are set out in Example 4 1 where variables are referenced by name A sample PEST control file is provided in Example 4 2 Note that the PEST control file demonstrated in these two The PEST Control File 4 2 examples pertains to the use of PEST in parameter estimation mode the most usual case of PEST usage Use of PEST in predictive analysis mode is described in Chapter 6 while use of PEST in regularisation mode is described in Chapter 7 Note also that whenever PEST is upgraded and additional variables are required in the PEST control file newer versions of PEST will always be capable of reading old versions of the PEST control file default values will simply be assigned to missing variables PET control data RSTFLE PESTMODE NPAR NOBS NPARGP NPRIOR NOBSGP NTPLFLE INSFLE PRECIS DPOINT NUMCOM JACFILE MESSFILE RLAMBDA1 RLAMFAC PHIRATSUF PHIREDLAM NUMLAM RELPARMAX FACPARMAX FACORIG PHIREDSWH NOPTMAX PHIREDSTP NPHISTP NPHINORED RELPARSTP NRELPAR ICOV ICOR IEIG parameter groups PARGPNME INCTYP DERINC DERINCLB FORCEN DERINCMUL DERMTHD one such line for each of the NPARGP parameter groups parameter data PARNME PARTRANS PARCHGLIM PARVAL1 PARLBND PARUBND PARGP SCALE OFFSET DERCOM one such line for each of t
118. using any of the above three options by clicking on the Run project button P on the WinPEST toolbar Immediately upon commencement of PEST execution a command line window appears outside of the WinPEST window WinPEST uses this window for running the model Any information that is normally written by the model to the screen is written to this window If WinPEST is shut down this window also disappears 3 3 2 The PEST Log Within the main WinPEST window a PEST Log window appears as soon as PEST execution is initiated As PEST runs information that is normally written to the screen by the command line version of PEST is written to this window If an error condition is encountered an appropriate error message is also written to this window and to the run record file If the PEST log is not the only WinPEST window open at any time it can be brought to the front of the screen by clicking on its tab While PEST is running PEST output should normally scroll upwards out of site as new text is added to the bottom of the PEST log However if for some reason the user wishes to disable autoscrolling click on the Autoscroll to the bottom of PEST log toolbar button Click on it again to re enable autoscrolling 3 3 3 Pausing and Stopping PEST PEST execution can be paused by clicking on the Pause button n or by selecting Run and then Pause from the WinPEST main menu Execution can then be re commenced by cli
119. weight Residuals for observation group group_2 Number of residuals with non zero weight 8 Mean value of non zero weighted residuals cal 119 Maximum weighted residual observation ar13 1 260 Minimum weighted residual observation arl0 3 424 Variance of weighted residuals 4 072 Standard error of weighted residuals 2 018 Note the above variance was obtained by dividing the sum of squared residuals by the number of items with non zero weight Residuals for observation group group_3 Number of residuals with non zero weight 5 Mean value of non zero weighted residuals 8 6256E 02 Maximum weighted residual observation ar15 0 3254 Minimum weighted residual observation ar19 4 0200E 03 Variance of weighted residuals 2 2300E 02 Standard error of weighted residuals 0 1493 Note the above variance was obtained by dividing the sum of squared residuals by the number of items with non zero weight Residuals for observation group group_4 Number of residuals with non zero weight 2 Mean value of non zero weighted residuals 2 585 Maximum weighted residual observation pi2 4 5451E 02 Minimum weighted residual observation pil 5 216 Variance of weighted residuals 13 61 Standard error of weighted residuals 3 689 Running PEST 5 9 Note the above variance was obtained by dividing the sum of squared residuals by the number of items with non zero weight Cova
120. where x is the model generated value corresponding to the sole observation belonging to observation group predict Obviously if you are not using PEST to undertake dual calibration you should avoid use of predict as an observation group name 6 1 5 Predictive Analysis Mode 39 66 PEST can run in three different modes parameter estimation mode regularisation mode and predictive analysis mode The first of these modes is used to find the objective function minimum min the second is most useful when many parameters require adjustment through the calibration process see the next chapter When run in predictive analysis mode PEST Predictive Analysis 6 8 finds the critical point depicted in Figures 6 4 and 6 5 and determines the model prediction associated with that point The theory underpinning PEST s calculations when run in predictive analysis mode is presented in Section 2 1 9 Model setup for predictive analysis is similar to that for dual calibration operations That is a composite model is constructed comprised of the model run under calibration conditions followed by the model run under predictive conditions There can be as many field observations corresponding to the former model component as desired these being the calibration observations However there should be only one output from the predictive model component which is used by PEST as an observation Furthermore so PEST can recognise
121. with respect to each of the n parameters To put it another way J is the derivative of the 7 th observation with respect to the j th parameter Equation 2 14 is a linearisation of equation 2 13 We now specify that we would like to derive a set of model parameters for which the model generated observations are as close as possible to our set of experimental observations in the least squares sense ie we wish to determine a parameter set for which the objective function defined by The PEST Algorithm 2 7 co J b bo Q c co J b bo 2 15 is a minimum where c in equation 2 15 now represents our experimental observation vector Comparing equation 2 15 with equation 2 8 it is apparent that the two are equivalent if c from equation 2 8a is replace by c co of equation 2 15 and b from equation 2 8a is replaced by b bo from equation 2 15 Thus we can use the theory that has been presented so far for linear parameter estimation to calculate the parameter upgrade vector b bo on the basis of the vector ce co which defines the discrepancy between the model calculated observations co and their experimental counterparts c Denoting u as the parameter upgrade vector equation 2 11 becomes u J QD SQ co 2 16 and equation 2 12 for the parameter covariance matrix becomes Cb 6 FQ 2 17 The linear equations represented by the matrix equation 2 16 are often referred to as the normal equations Th
122. 0 log rol 1 0 log ro2 0 0 1 0 regul tea 1 0 log ro3 1 0 log ro2 0 0 1 0 regul I seas 1 0 log rol5 1 0 log rol 0 0 1 0 regul Example 7 4 Regularisation observations contained in the prior information section of a PEST control file Note the following points e The observed value of each parameter difference is 0 0 e Each element of prior information pertains to the logarithm of the respective parameters rather than to the parameters themselves because the parameters are log transformed in the parameter estimation process If it is the user s desire that differences be formed between the parameters themselves rather than between the parameter logarithms then the differences could be calculated by the model and an observed value of 0 0 for each such observation supplied in the observation data section of the PEST control file e All of the items of prior information involved in the regularisation process have been assigned to the observation group regul e In the present instance all of the regularisation observations have been assigned the same weight These weights are multiplied internally by the regularisation weight factor before being used by PEST to calculate a new set of parameter values If it is the user s desire that PEST try harder to enforce uniformity in some areas rather than others then this could be requested by adopting a different weights assignment stra
123. 2 of the first number on the above model output file line The second whitespace instruction moves the cursor to the blank character preceding the first 4 of the second number on the above line processing of the third whitespace instruction results in PEST moving its cursor to the blank character just before the negative sign After the fourth whitespace instruction is implemented the cursor rests on the blank character preceding the last number the latter can then be read as a non fixed observation see below Tab The tab instruction places the PEST cursor at a user specified character position ie column number on the model output file line which PEST is currently processing The instruction syntax is tn where n is the column number The column number is obtained by counting character positions including blank characters from the left side of any line starting at 1 Like the whitespace instruction the tab instruction can be useful in navigating through a model output file line prior to locating and reading a non fixed observation For example consider the following line from a model output file The Model PEST Interface 3 19 TIME 1 A 1 34564E 04 TIME 2 A 1 45654E 04 TIME 3 A 1 54982E 04 The value of A at TIME 3 could be read using the instruction line 14 t60 a3 Here it is assumed that PEST was previously processing the fourth line prior to the above line in the model output file the marker de
124. 2 18 2 2 1 Parameter Transformation oireeseesi iersinii e iaeei 2 18 2 2 2 Fixed and Tied Parameters ssesesssesseesseseseseesereseresseesrressressressressressresssesssees 2 19 2 2 3 Upper and Lower Parameter Bounds seeseessseessssesssseeresressessrssrerresressesrrereses 2 19 2 24 Scale and Offset ecceri ii a EE nan 2 20 2 2 5 Parameter Ch nse TIMIS Sonriente ee aaeeea EE E E aE EEEE 2 21 2 2 6 Damping of Parameter Changes sessseeseessssessessssresresressessrsrrerreseessssresreses 2 23 2 2 7 Temporary Holding of Insensitive Parameters 0 0 escesseeseeseeeseeeeeeeeeeees 2 24 Table of Contents 2 2 8 Components of the Objective FUNCTION 20 0 ee eeecesceseeeeeeesneeeseeeeeeeeaeeeseeeaes 2 24 2 2 9 Termimation Criteria arobaini aa EA AAA 2 25 2 2 10 Operation in Predictive Analysis Mode eeeeseseseseeresressessresrerresressessrereees 2 26 2 2 11 Operation in Regularisation Mode eseseeseeeseseresressessssresresressessrssresress 2 26 2 3 The Calculation of Derivatives ccccccccccsscceesesscceeseseeccesseeccsssseeecessseseesaeesees 2 27 2 3 1 Forward and Central Differences cccccccccssccccessseccesssseceeesseeccsssseesesssseeeees 2 27 2 3 2 Parameter Increments for Derivatives Calculation ccccccccessseceeessteeeees 2 28 2 3 3 How to Obtain Derivatives You Can Trust ccccccccccsssscccessseccessseeeesssseees 2 30 2 3 4 Model Calculated Derivatives ccccccc
125. 23 69 gradient and update vectors out of bounds 0 89 of starting phi Running PEST 5 6 79 19 phi 0 88 of starting phi No more lambdas Lowest phi this iteration relative phi reduction between lambdas less than 0 0300 EQ dl9 Current parameter values Previous parameter values rol 0 500000 rol 0 500000 ro2 10 0000 ro2 10 0000 ro3 1 00000 ro3 1 00000 hl 0 157365 wl 0 472096 h2 44 2189 h2 34 3039 Maximum factor parameter change 3 000 hl Maximum relative parameter change 0 6667 h1 OPTIMISATION ITERATION NO 5 Model calls so far 22 Starting phi for this iteration 79 19 Contribution to phi from observation group group_1 6 920 Contribution to phi from observation group group_2 22 45 Contribution to phi from observation group group_3 14 88 Contribution to phi from observation group group_4 34 94 All frozen parameters freed Lambda 9 7656E 03 gt parameter ro2 frozen gradient and update vectors out of bounds phi 64 09 0 81 of starting phi Lambda 4 8828E 03 gt phi 64 09 0 81 of starting phi Lambda 1 9531E 02 gt phi 64 09 0 81 of starting phi No more lambdas relative phi reduction between lambdas less than 0 0300 Lowest phi this iteration 64 09 Current parameter values Previous parameter values rol 0 500000 rol 0 500000 ro2 10 0000 ro2 10 0000 ro3 1 00000 ro3 1 00000 hl 0 238277 hl 0 157365 h2 42 4176 h2 44 2189 Maximum facto
126. 28 2 32 5 26 Binary files 0 3 sai cciiiau doian aas 3 2 3 10 4 22 Calibration enres ninni iiaa 1 2 Central derivatives 1 8 2 26 2 28 2 29 4 10 4 15 Column numbers ccccccccccesssceeeeeessseeeeeees 3 18 3 20 Continuation character cc ccccccccsseeeesseeeeeee 3 25 4 26 Control datas ie aenta acarvin hacia iets 4 4 Control file eee 1 12 4 1 5 24 10 4 10 5 12 8 Control variables ccccccccccsssscseeessssseeeeeees 5 24 5 25 Correlation coefficient matrix wa 2 3 4 17 Covariance ccecceeesscceesesecececessceeeeeesssseeeeeees 2 15 5 39 Covariance Matix cccceeseesseeeeseeeeeee 2 3 4 17 5 36 Critical point evgar ree V NETE 6 3 CUO EE A ATOS ATEOA 3 14 3 23 Decimal pontis n E Ss 3 7 4 6 D pr esoffreedoMizsnrooniid ndia 2 3 DERCOM rosoisia 4 19 DERINC 20 cececcccceccccssccceseeeceseceesseeeeseeeesseeees 2 29 5 24 DERINCLB i333 fear wales 2 29 4 14 5 24 5 26 DERINCMUL Desi etn et 2 29 4 15 5 24 Derivatives essiri irsinin 2 27 4 13 8 8 DERMTHD 2 29 4 16 5 24 Distributed parameters cee 1 1 1 5 2 27 3 8 DPOINT cccceceeeeeeeee Dual calibration ccccccccccessceessceeeseecesseeeeseeeeeeeees Dummy Sroup 3 nanai sleiente aula Dummy observation Figenyali s mnene pea a a EigenyvectOt Snina niis Errorevelisene saii e a EA E e Excitations FACORIG 2 22 4 10 FACPARMAX 5 2 21
127. 3 0 01 3 1 1 1 parameter groups rol relative 0 01 0 0 switch 2 0 parabolic ro2 relative 0 01 0 0 switch 2 0 parabolic ro3 relative 0 01 0 0 switch 2 0 parabolic hl relative 0 01 0 0 switch 2 0 parabolic h2 relative 0 01 0 0 switch 2 0 parabolic parameter data rol none relative 1 00000 1 00000E 10 1 00000E 10 rol 1 0000 0 00000 1 ro2 none relative 40 0009 1 00000E 10 1 00000E 10 ro2 1 0000 0 00000 1 ro3 none relative 1 00000 1 00000E 10 1 00000E 10 ro3 1 0000 0 00000 1 BI none relative 1 00000 1 00000E 10 1 00000E 10 h1 1 0000 0 00000 1 h2 none relative 9 99978 1 00000E 10 1 00000E 10 h2 1 0000 0 00000 1 observation groups obsgroup observation data arl 152 1033 0 obsgroup ar2 1 51208 0 obsgroup ar3 2 07204 0 obsgroup ar4 2 94056 0 obsgroup ar5 4 15787 0 obsgroup ar6 5 77620 0 obsgroup ar7 7 78940 0 obsgroup ar8 9 99743 0 obsgroup ar9 1 8307 0 obsgroup ar10 2 3194 0 obsgroup AXIL 0 6003 0 obsgroup ar12 7 00419 0 obsgroup ar13 3 44391 0 obsgroup ar14 1 58278 0 obsgroup aris 10381 0 obsgroup ar16 1 03085 0 obsgroup arl7 01318 0 obsgroup ar18 00393 0 obsgroup arrg 00272 1 0 obsgroup model command line model model input output model tpl model inp model ins model out prior information Example 10 3 A PEST control file generated by PESTGEN Note that when viewing a PESTGEN generated PEST control file on your screen the OFFSET values in the para
128. 488000 0 640400 Example 12 3 A TWOLINE output file out dat 12 1 3 Preparing the Template File First a template file must be prepared This is easily accomplished by copying the file in dat listed in Example 12 2 to the file in tpl and modifying this latter file in order to turn it into a PEST template file Example 12 4 shows the resulting template file the value of each of the line parameters has been replaced by an appropriately named parameter space and the ptf header line has been added to the top of the file Because TWOLINE reads all parameters using free field format the width of each parameter space is not critical however where two parameters are found on the same line they must be separated by a space A parameter space width of 13 characters is employed in file in tpl in order to use the maximum precision available for representing single precision numbers As discussed in Section 3 2 5 while PEST does not insist that parameters be written with maximum precision to model input files it is a good idea nevertheless An Example 12 6 ptf sl yl XC T3 052 068 103 128 ek 12 195 230 275 315 3332 350 423 488 s2 See te io SO Or Os OO O Oi O O OO Example 12 4 The template file in tpl Now that in tpl has been prepared it should be checked using program TEMPCHEK run TEMPCHEK using the command tempchek in tpl Example 12 5 shows file in pmt written b
129. 5 and the discussion of the input variables RELPARMAX and FACPARMAX above PARCHGLIM must be provided with one of two possible values viz relative or factor For tied or fixed parameters this variable has no significance PARVALI PARVALI a real variable is a parameter s initial value For a fixed parameter this value remains invariant during the optimisation process For a tied parameter the ratio of PARVALI to the parent parameter s PARVALI sets the ratio between these two parameters to be maintained throughout the optimisation process For an adjustable parameter PARVALI is the parameter s starting value which together with the starting values of all other adjustable parameters is successively improved during the optimisation process To enhance optimisation efficiency you should choose an initial parameter value which is close to what you think will be the parameter s optimised value However you should note the following repercussions of choosing an initial parameter value of zero e A parameter cannot be subject to change limits see the discussion on RELPARMAX and FACPARMAX during the first optimisation iteration if its value at the start of that iteration is zero Furthermore FACORIG cannot be used to modify the action of RELPARMAX and FACPARMAX for a particular parameter throughout the optimisation process if that parameter s original value is zero e A relative increment for derivatives calculation cannot
130. 6 1 6 1 2 Some Solutions sesaria eo eee hea aE Eea ia TE SA E REEERE ERES 6 3 6 1 3 The Critical Point scsscocvee citiank oie vsusah odaoae a S e a e Ra aaa et 6 3 6 1 4 Dual Cali bration ciso eii a i aaa e aaa aS E i iei 6 6 6 1 5 Predictive Analysis Mode sesnsensenseseessesssssrenresesssessessesntesesssssntseesnessessesseese 6 7 6 2 Working with PEST in Predictive Analysis Mode ees eeseeseceeeceeeeeeeeeeeeeeateees 6 10 6 2 1 Structure of the PEST Control File eee eeccceneceneceeecececneeeeaeenaeeeseeeeaees 6 10 6 2 2 PEST Variables used for Predictive Analysis 6 12 O23 AMIE KAaMPle tiese etero A rae eo reee EEE EE a TaS aar E eSEE 6 16 Te Regularisat OM sanne annat a a aE A TEE e Ee EE A EE 7 1 T About Reg l ris ti onain ie e e A E E E E e Eaa 7 1 PAM General ennie aiia i e KAE AER a 7 1 7 1 2 Smoothing as a Regularisation Methodology eseessssseessrsereerrererrrssrrsrsreees 7 2 TELS TIRCOTY oraig aerie t r eE AE aE ona nenkcuedgesbaguva gen gs RE Aa 7 3 7 2Amplementation am PEST 3era cet i ate a hata tag eh cde cena aioe eeete 7 3 7 2 1 Regularisation MOde escescesccesceceeeceseeeseecscecsaeceaceesceesaeeseeseeesaeesaeeeseeeseees 7 3 7 2 2 The Observation Group Regul ooo cece eecescecseceeeceseeesceeseeesaeeeeeesaeesaeeeeeeeaees 7 4 7 3 Preparing for a PEST Run in Regularisation Mode 0 0 00 cee ceeeesceeeeceeeeeeeeceeeeeaeeenees 7 5 7 3 1 The PEST Control File Control Data Section eee
131. 6 User Intervention 5 6 1 An Often Encountered Cause of Aberrant PEST Behaviour Where many parameters are being estimated and some are far more insensitive than others it is not uncommon to encounter problems in the parameter estimation process PEST in response to the relative insensitivity of certain parameters may calculate an upgrade vector in which these insensitive parameters are adjusted by a large amount in comparison with other Running PEST 5 32 more sensitive parameters this large adjustment of insensitive parameters may be necessary if alterations to their values are to have any effect on the objective function However the magnitude of the change that can be incurred by any parameter during any particular optimisation iteration is limited by the values assigned to the PEST control variables RELPARMAX and FACPARMAX PEST reduces the magnitude but not the direction of the parameter upgrade vector such that no parameter undergoes a change that exceeds these limits Unfortunately if a particular insensitive parameter dominates the parameter upgrade vector restricting the magnitude of the upgrade vector such that the change to the value of the insensitive parameter is limited to RELPARMAX or FACPARMAX depending on its PARCHGLIM setting will result in much smaller changes to other more sensitive parameters Hence the objective function may be reduced very little if at all Under these circumstances increasing RELPARMAX and F
132. 6757 ar10 group_2 12 31940 8 895600 1 8875951 arll group_2 10 60030 8 402450 1 9690974 Example 5 4 Part of an observation sensitivity file 5 3 4 The Residuals File At the end of its execution PEST writes a residuals file listing in tabular form observation names the groups to which various observations belong measured and modelled observation values differences between these two ie residuals measured and modelled observation values multiplied by respective weights weighted residuals measurement standard deviations and natural weights This file can be readily imported into a spreadsheet for various forms of analysis and plotting Its name is case res where case is the current PEST case name A word of explanation is required concerning the last two data types presented in the residuals file As is explained in Section 2 1 2 of this manual after the parameter estimation process has been carried out and a value has been obtained for the reference variance o the standard deviation of each observation can be calculated as the inverse of its weight multiplied by the square root of the reference variance Care must be taken in interpreting this standard deviation for being dependent on the fit achieved between model outputs and corresponding field or laboratory measurements it is a valid measure of observation uncertainty only in so far as the model is a valid simulation of the processes that it is intended to rep
133. 9 2 30 4 17 4 25 5 12 5 27 Marker delimiter 0 cccccccsccccsseccesseeeeseeeeseeeeeeee 3 14 ETE SE 3 11 Marquardt lambda 2 10 4 7 5 10 5 24 5 27 Matix eee Measurement Measurements MESS FILE reri ner rE E RERA ES Modelu i Model input files 0 0 cece eee eeeeeeeeeeeeeeees 4 24 10 1 Model output files eee eeeeteeeee 3 9 4 24 10 3 Model zones Model calculated 0 cccceccsccceseeceeseceesseeeeseeees 2 32 8 1 Model generated errors ceeeeeeeeeeeeeeteeeeeseeeeeaeeees 5 23 Model generated observations 2 1 2 6 2 31 3 1 3 9 5 31 10 6 MULSCHPAC cccceccccceesscceseceeeseceesseeesseeeeseeeeenee 6 14 Non fixed observations Nonlinear models cccsccesseeeeeseeeeeee 2 6 2 32 4 10 NOMUMIQUENESS 2 0 0 eee eteeeeeseeeeteeeseeeeeeees 2 5 5 27 6 3 NOPTMA X onani 2 26 4 11 5 24 5 40 Index Normal matrix ccccccccssceeseeeeseees 2 7 2 10 4 5 5 27 NPA Roe csvcedecssned caved sontdesnletidecscdetiieecsisie tebe 4 5 11 5 NPARGP reer EA EE A 4 5 NPHINORED cccceccccceseceesseeesseeeesees 2 25 4 12 5 24 NPHIS TP iea y REEERE 2 25 4 12 5 11 NPREDMAXMIN cccccccccesceesseeeeseeeeseeeees 6 12 6 15 NPREDNORED es eere a EAEE 6 15 NPRIOR oehr an 4 5 4 24 NRELPAR pneri nocem iva 2 26 4 12 5 24 NSEARCH sketen a a EN 6 14 NSLAVE 9 8 9 9 NTPFLE 9 10 11 6 NIPPLE phita aia ea E
134. A D9 Model Input Output Tas o e a le nat ae Seas bo a E AA R E Eta 4 23 4 2 10 Prior Information cccccccccsesseeesecceseecesneecseseccsceeeseeceseecesaeecseeeeeseeeeeaeeesas 4 24 43 Observation COVAMMANCES Teena ereer e Eee eE EEEa E SEKE eae DE A EAEN 4 27 4 3 1 Using an Observation Covariance Matrix Instead of Weights eee 4 27 4 3 2 Supplying the Observation Covariance Matrix to PEST cece eeeeeeeeeeeee 4 27 43 3 PEST OUMU voeien fo o E E tonsa teat eee te chan eet aseetes 4 30 5 Runnin PES Teenie e eee oe EE ss ska EG Nia vane AEE a EA a eae a EE 5 1 Sel HOW tO RUN PES Tiot ernir ieia t sanee cue RE oee EE EE E EE T EE EEE 5 1 5 1 1 Checking PEST s Input Data eee ceecceeeceececeeeceseeeaeecseecsaecnaeeeeeesaeeeaeeees 5 1 5 1 2 Versions of PES Tirini en a e a E anh voles Rak ogtas dace E 5 1 5 2 The PEST Run Record onecrii niia eearri i a a E a i 5 2 IA A D01 0 E N AE E ET AAEN E AEE A 5 2 5 2 2 Echoing the Input Data Set oo eee ee cesceneecneeceseceseeeeeesaeesaeeeeeesaeesaeeeeeeeseees 5 9 5 2 3 The Parameter Estimation Record 0 0 0 ce cceeceeeccesecececeseeeaeecececeaeceaeeeseeesaeesaeeess 5 9 5 2 4 Optimised Parameter Values and Confidence Intervals eceeeseesneeeeeees 5 12 5 2 5 Observations and Prior Information ce eeeceeccesceeseceeceeeceeseeesaeeeeeeseeeeeees 5 13 5 2 6 Objective PUNCH ON irisse toea eeen Se eoeta Teea ri etp E seats Raae Erh 5 13 5 2 7 Correlation Coefficient nesci
135. ACPARMAX is not necessarily the solution to the problem for parameter change limits are necessary in order to avoid unstable behaviour in the face of problem nonlinearity In normal PEST usage the occurrence of this problem is easily recognised by the fact that either the maximum relative parameter change or the maximum factor parameter change for a particular optimisation iteration as printed to the screen and to the run record file is equal to RELPARMAX or FACPARMAX respectively and that the objective function is reduced very little PEST records the names of parameters that have undergone the largest factor and relative changes at the end of each optimisation iteration More often than not an inspection of the parameter sensitivity file see Section 5 3 2 will reveal that these same parameters possess a low sensitivity 5 6 2 Fixing the Problem The solution to the above problem is to hold parameters which are identified as being troublesome at their current values at least for a while With such recalcitrant parameters out of the road PEST can often achieve a significant improvement in the objective function Such temporarily held parameters can then be brought back into the parameter estimation process at a later date It may be found that quite a few parameters need to be held in this manner for once a particular troublesome parameter has been identified and held it may be found that the problem does not go away because another
136. E 2 OPTIMISATION ITERATION NO 2 SSS gt Parameter_name Group Current value Sensitivity Rel Sensitivity rol ro 3 79395 1 30721 0 756995 ro2 ro 15 0000 0 672146 0 790506 ro3 ro 4 57028 1 77164 1 16918 h1 hhh 2 85213 0 661729 0 301198 h2 hhh 4 00000 0 465682 0 280369 Example 5 3 Part of a parameter sensitivity file The relative composite sensitivity of a parameter is obtained by multiplying its composite sensitivity by the magnitude of the value of the parameter It is thus a measure of the composite changes in model outputs that are incurred by a fractional change in the value of the parameter It is important to note that composite sensitivities recorded in the parameter sensitivity file are sensitivities as PEST sees them Thus if a parameter is log transformed sensitivity is expressed with respect to the log of that parameter The relative composite sensitivity of a log transformed parameter is determined by multiplying the composite sensitivity of that parameter by the absolute log of the value of that parameter As is explained in Section 5 6 composite parameter sensitivities are useful in identifying those parameters which may be degrading the performance of the parameter estimation process through lack of sensitivity to model outcomes The use of relative sensitivities in addition to normal sensitivities assists in comparing the effects that different parameters have on the parameter estimation process when these param
137. E TERRE 4 6 NUMCOM eoe ESNE R ERER EEA 4 6 8 7 NUMLAM eerren ereire a ERENER 4 8 5 24 OBGNM Essin ear ea E EE A E 4 21 Objective function1l 7 2 2 2 6 2 10 2 25 4 8 4 20 5 9 5 11 5 25 6 5 11 11 Observationws hives deh Ri Eh 4 27 Observation groups cece ees cese ese eeeeeeeeecseneeeeee 4 20 Observation value file ccccceeeeeeee 10 4 10 6 12 7 Observations cceeccceceseeeesteeesseeeeteee 1 6 2 1 3 1 3 9 OBSNME viriito n iii 4 20 OBSVAL 4 20 OFFSET 2 20 4 19 5 16 5 29 10 2 OUTFEE s kA inde sick ia we 11 7 Outside POIMNtS parini iE Ri 2 28 Over parameterisation ssssssesesesesisisissessessererereses 5 39 PAR2PAR ccscesssccessceceseeceseceesseeesseeessaees 1 13 10 10 Parabolic method 2 28 2 32 5 26 Parallel PES T ga aki 5 1 9 1 13 1 PAtaMeter eer rie i E ORE N 10 14 Parameter bounds puesi nY 4 18 Parameter correlation 2 3 2 5 5 12 Parameter factor ccccccccscccessecesssceeseeeesseeeeseeeeseeees 4 25 Parameter groups ssssssssssesssrsesesesresees 2 27 2 29 4 13 Parameter hold file ccccccsccccssceessceesseeeeseeeesseeees 5 32 Parameter increments cccsceeeeee 4 14 5 26 5 28 Parameter sensitivity file 0 ccc eseeeseeeneeeee 5 16 Parameter SPpace cccccccccceessesseescssesseeseeseeeeeees 3 4 3 6 Parameter upgrade vector2 7 2 10 2 20 2 21 2 22 2 24 Parameter value file
138. EEE 20 AV RuN R cords e eA E Saad he E E Tasch end aa A E EE on 21 4 17 Prediction c scccssssasses scedascsesceassansenen oleeeconsaosbeacneacesesencbanta EKNE KEER EEIE SEAKS 21 5 Saving and Re Loadine PlotS s esir nennen ea e E n a aaa ae 22 SP Er lea rl AEE E E EEES 22 SL Printans PIOUS 2225 naa a as ta T E ath cies Sada it as NEN 22 6 Dual Calibration and Predictive Analysis ccsesccccessscecceeseeceeceeceessaeeecesaaeessesaeeeeeseas 23 6 1 Covariance Matrix and Related Plots 20 0 0 eeecesccsnseceeeceeesesneaeeeeceeseesneaeeeeeeeeeeeaaas 23 6 2 The Prediction Window cccccccsessscceeceesneneececeeceesennneceeecesesesaaaaeeeceeeeeesnnaeeeees 23 T User Intervention sireni a ca oh dia Sawer ed Sete chia seas ba See MSA Sa ean Seed Be 24 Ted EG E ral so icc caccsssisecsadasnceedecaava seven shone cessassoaeaaua sca seacadedeaaa canudncadedceasseeaene ooaecceaeaaabaaes 24 7 2 Holding Parameters in WinPEST 1 6cc ssscceecosssoacesescctesgsnbanbccasescvoossenenss T EEEE RE a es 24 73 RE Starting PEST vesccs cee ceed sc aesssseh ah enscdacaasbsdan eles nek cg vanades sahaiged senouseangsa se ieeied Seonteagerey 25 S Paralleli Wane EEE E tah cdi E E E eS tua ial ache ead A It aac 27 Introduction 2 1 Introduction 1 1 Installation The Visual PEST ASP installation program should run automatically when the Visual PEST ASP installation CD ROM is inserted into your CD ROM drive If the installation program
139. EP NY 497p Nash J C and Walker Smith M 1987 Nonlinear Parameter Estimation an Integrated System in Basic Marcel Dekker Inc Monticello NY 493p The Model PEST Interface 3 1 3 The Model PEST Interface 3 1 PEST Input Files PEST requires three types of input file These are e template files one for each model input file on which parameters are identified e instruction files one for each model output file on which model generated observations are identified and e an input control file supplying PEST with the names of all template and instruction files the names of the corresponding model input and output files the problem size control variables initial parameter values measurement values and weights etc This chapter describes the first two of these file types in detail the PEST control file is discussed in Chapter 4 Template files and instruction files can be written using a general purpose text editor following the specifications set out in this chapter Once built they can be checked for correctness and consistency using the utility programs TEMPCHEK INSCHEK and PESTCHEK these programs are described in Chapter 10 of this manual Note that in this and other chapters of this manual the word observations is used to denote those particular model outcomes for which there are corresponding laboratory or field data For clarity these numbers are often referred to as model generated observations to d
140. EST simply ignores the line If the pertinent line is removed from the parameter hold file or the parameter hold file itself is removed the parameter is then free to move in later optimisation iterations The format for the sixth line in Example 5 6 is hold group pargpnme lt x where pargpnme is the name of a parameter group and x is a positive number A line such as this directs PEST to hold any parameter in the named parameter group temporarily fixed if the sensitivity of that parameter is less than the supplied number ie x Held parameters can be freed again later by reducing x to zero if desired by deleting this line from the parameter hold file or by deleting the parameter hold file itself As is illustrated in the 7 line of Example 5 6 the n most insensitive parameters in a Running PEST 5 34 particular parameter group can be held at their current values using the command hold group pargpnme lowest n where n is a positive integer Such held parameters can be freed later in the parameter estimation process by reducing n to zero if desired by deleting this line from the parameter hold file or by deleting the parameter hold file itself The syntax of the 8 line of Example 5 6 is hold eigenvector n highest m This option has been included in PEST s parameter holding functionality to accommodate the fact that an observation dataset can be insensitive to certain groups of parameters varied in a specific rat
141. EST input dataset has been checked the parameter estimation process can begin Start PEST by selecting Run from the WinPEST main menu then Start Three starting options are available these being e Start new run e Restart current iteration r and e Restart using last Jacobian j As indicated the second and third options are equivalent to starting the command line versions of PEST with the r and j switches respectively These options are used to re start a PEST run which was prematurely stopped If the r option is selected PEST re commences execution at the beginning of the iteration in which its execution was terminated If the j option is selected PEST re commences execution at the point at which the Jacobian matrix was last calculated Selection of this option is an integral part of PEST user Preparing for a PEST Run 8 intervention functionality As is explained in Section 5 6 of the PEST manual a user can halt PEST execution hold recalcitrant parameters normally insensitive parameters at their current values and recalculate the parameter upgrade vector without the need for costly re computation of the Jacobian matrix User intervention in the WinPEST context is discussed below Note that a previously halted PEST run can only be re started if the variable RSTFLE on the third line of the PEST control file is set to restart PEST execution can also be initiated
142. FLE control variable on the PEST control file was set to restart on the previously halted PEST run Once PEST ceases execution whether it was stopped by the user or whether it judged of its own accord that the parameter estimation process had reached completion another window is automatically added to the set of open WinPEST windows This window displays the PEST run record file click on its tab to bring this window to the front of the screen Different parts of this record can be viewed using normal screen navigation keystrokes eg lt PgUp gt to scroll up a page lt Home gt to move to the top of the current screen lt Ctl Home gt to move to the top of the file etc or by using the scroll bar on the right side of this window Note that the run record file cannot be edited However it can be printed from within WinPEST WinPEST Graphical Windows 10 4 WinPEST Graphical Windows 4 1 Accessing the Windows As mentioned above as it runs WinPEST can display in graphical form the values of many quantities that are of interest from the point of view of monitoring the performance of the optimisation process These plots evolve during the course of the optimisation process as the values of the plotted variables change You can select the plots that you wish to display through the View item of the WinPEST main menu Once these plots have been selected for display any particular one of them can be brought to the front of the
143. IEIG are set to zero the corresponding matrix is not written to the matrix file If they are all set to zero then no matrices are written to this file If ICOV is set to 1 then as well as recording the covariance matrix to the matrix file PEST records current parameter values and standard deviations The standard deviation of a parameter is the square root of its variance parameter variances comprise the diagonal of the covariance matrix As with the elements of the covariance and associated matrices the standard deviation of a parameter actually pertains to the log of that parameter if the parameter is log transformed during the parameter estimation process The observant PEST user may notice slight differences between the matrices recorded to the final matrix file and those recorded to the run record file at the end of the PEST run If the lowest objective function achieved during the parameter estimation process was calculated by PEST during the final optimisation iteration then he she may expect that these two sets of matrices will be identical Nevertheless there are often differences between these two sets of matrices These differences result from the fact that the reference variance see equation 2 5 used in the calculation of matrices which are recorded in the matrix file is computed using the objective function calculated at the end of the previous optimisation iteration whereas for the covariance and related matrices recorded in
144. INCMUL and then recommence execution using the restart option Program PARREP one of the PEST utilities described in Chapter 10 provides a much safer means of restarting PEST execution with one or more control variables altered It allows a new PEST control file to be built from an existing PEST control file and a parameter value file the latter may contain values optimised by PEST on a previous run Thus a new PEST Running PEST 5 25 run can be restarted with or without altered control settings where an old one left off 5 4 3 Restarting PEST with the j Switch PEST can also be restarted with the j switch this is an integral part of the user interaction functionality provided by PEST It is discussed in Section 5 6 5 5 If PEST Won t Optimise 5 5 1 General PEST allows the user to follow closely the progress of an optimisation run both through its screen output and through the user s ability to inspect the run record file Through watching the value of the objective function referred to as phi on the PEST run record from optimisation iteration to optimisation iteration you can monitor PEST s ability and efficiency in lowering the objective function to the minimum which can be achieved within the user provided parameter domain There can be many reasons for a failure on the part of PEST to lower the objective function in most cases the problem can be easily overcome by adjusting one or a number of
145. It needs this room to move because of the fact that it bases its calculations on a linearity assumption that is only approximately satisfied Regularisation 7 7 Normally PHIMACCEPT should be about 5 to 10 greater than PHIMLIM However if PEST is performing well you may wish to make it closer to PHIMLIM than this In choosing the best parameter set at any stage of the optimisation process for recording in the parameter value file PEST looks at all parameter sets for which it has carried out model runs up to that point in the process If any of these runs have resulted in an objective function less than PHIMACCEPT it then searches from among these runs for the parameter set which gave rise to the lowest regularisation objective function If PHIMACCEPT is set too close to PHIMLIM PEST s selection of the best parameter set may be restricted somewhat for there may be some parameter sets for which the measurement objective function m is just above PHIMACCEPT but for which is quite low Alternatively if PHIMACCEPT is set too large then PEST might not try hard enough to reduce m to Dm preferring instead to work within the weaker constraint set by PHIMACCEPT When working in regularisation mode PEST prints out and m for every parameter upgrade attempt It will be apparent from this information whether PHIMACCEPT has been set correctly FRACPHIM PEST ignores the value supplied for FRACPHIM unless it is greater than zero A value
146. K is able to build correct input files for it Failure in this regard will normally result in a model generated error message Conversely if SENSAN or INSCHEK indicate a failure to read the model output file s a search should be made for a model generated error message If running SENSAN in verbose mode for cases where there are multiple template files the user may notice a message similar to the following scroll past on the screen Warning parameter rol from parameter value file t par not cited in template file ves tp2 This is of no concern for it is simply TEMPCHEK informing the user that it has been provided with a parameter value file ie t par written by SENSAN that contains the values of more parameters than are cited on any one template file 11 8 An Example Included in the pestex subdirectory of the directory into which PEST was installed are the files required to run the soil clod shrinkage example discussed in Chapter 12 of this manual Also included in this subdirectory are three files not discussed in Chapter 12 These are twofit sns a SENSAN control file out ins an instruction file identical to out ins discussed in Chapter 12 and parvar dat a parameter variation file An inspection of file twofit sns reveals that this SENSAN control file assumes the same number of parameters and observations as the PEST control file twofit pst As the parameter variation file parvar dat reveals parameter names are iden
147. O OT OPA OO E E Po BO OO Oo ro Example 4 6 Example of a covariance matrix file At the end of the inversion process the true covariance matrices pertaining to various observation groups can be calculated from user supplied covariance matrices through multiplication by the reference variance determined through the parameter estimation process ie o of equation 2 5 Recall from Section 2 1 2 that variances and covariances represented in covariance matrices supplied to PEST by the user will be related to true observation variances and covariances by a constant of proportionality that is unknown before completion of the inversion process For the sake of consistency with observations for which a covariance matrix is not supplied this constant of proportionality should be the same as that by which observation variances are related to the inverse square of observation weights In practice The PEST Control File 4 30 however the user should simply ensure that as always weights and covariance matrices are such that no observation group either dominates the parameter estimation process or is dominated by other observation groups 4 3 3 PEST Outputs When one or more observation covariance matrices are supplied to PEST as part of its input dataset PEST s output dataset is a little different from that which is recorded if no covariance matrices are supplied While PEST outputs are treated in detail in the next chapter these differences
148. PEST input variables The fact that PEST provides so many control variables by which it can be tuned to a particular model is one of the cornerstones of its model independence In other cases PEST s progress can be assisted by selectively holding either one or a few parameters at their current values the user may then re commence PEST execution at that spot at which the Jacobian matrix was last calculated in order to re compute the last parameter upgrade vector or simply continue execution with the selected parameters held fixed for a while See the next section for details If you are using a particular model for the first time with PEST you may wish to run a theoretical case first You should use the model to fabricate a sequence of observation values of the same type as that for which you have laboratory or field measurements and then use these fabricated observations as your field or laboratory data Then you should run PEST using as your initial parameter estimates the parameters that gave rise to the fabricated observation set PEST should terminate execution after the first model run with an objective function value of zero In some cases it will not be exactly zero because of roundoff errors nevertheless it should be extremely small In this way you can check that PEST is supplying correct parameter values to the model running the model correctly and reading observation values correctly Next you should vary the parameter initi
149. PEST Control File 4 16 parameters while being in theory continuously differentiable is often granular when examined under the microscope this granularity being a by product of the numerical solution scheme used by the model In such cases the use of parameter increments which are too small may lead to highly inaccurate derivatives calculation especially if the two or three sets of parameter observation pairs used in a particular derivative calculation are on the same side of a bump in the parameter observation relationship Parameter increments must be chosen large enough to cope with model output granularity of this type But increasing parameter increments beyond a certain amount diminishes the extent to which finite differences can approximate derivatives the definition of the derivative being the limit of the finite difference as the increment approaches zero However the deterioration in the derivative approximation as increments are increased is normally much greater for the forward difference method than for any of the central methods particularly the parabolic option Hence the use of one of the central methods with an enhanced derivative increment may allow you to calculate derivatives in an otherwise hostile modelling environment Whenever the central method is employed for derivatives calculation DERINC is multiplied by DERINCMUL no matter whether INCTYP is absolute relative or rel_to_max and whet
150. PEST template file The character following the space is the parameter delimiter In a template file a parameter space is identified as the set of characters between and including a pair of parameter delimiters When PEST writes a model input file based on a template file it replaces all characters between and including these parameter delimiters by a number representing the current value of the parameter that owns the space that parameter is identified by name within the parameter space between the parameter delimiters You must choose the parameter delimiter yourself however your choice is restricted in that the characters a z A Z and 0 9 are invalid The parameter delimiter character must appear nowhere within the template file except in its capacity as a parameter delimiter for whenever PEST encounters that character in a template file it assumes that it is defining a parameter space 3 2 4 Parameter Names All parameters are referenced by name Parameter references are required both in template files where the locations of parameters on model input files are identified and on the PEST control file where parameter initial values lower and upper bounds and other information are provided Parameter names can be from one to twelve characters in length any characters being legal except for the space character and the parameter delimiter character Parameter names are case insensitive Each parameter space is defined
151. PHIRATSUF which stands for phi ratio sufficient is a real variable for which a value of 0 3 is often appropriate If it is set too low model runs may be wasted in search of an objective function reduction which it is not possible to achieve given the linear approximation upon which the optimisation equations of Chapter 2 are based If it is set too high PEST may not be given the opportunity of refining lambda in order that its value continues to be optimal as the parameter estimation process progresses PHIREDLAM The PEST Control File 4 If a new old objective function ratio of PHIRATSUF or less is not achieved as the effectiveness of different Marquardt lambdas in lowering the objective function are tested PEST must use some other criterion in deciding when it should move on to the next optimisation iteration This criterion is partly provided by the real variable PHIREDLAM The first lambda that PEST employs in calculating the parameter upgrade vector during any one optimisation iteration is the lambda inherited from the previous iteration possibly reduced by a factor of RLAMFAC unless it is the first iteration in which case RLAMBDA1 is used Unless through the use of this lambda the objective function is reduced to less than PHIRATSUF of its value at the beginning of the iteration PEST then tries another lambda less by a factor of RLAMFAC than the first If the objective function is lower than for the first lambda and still
152. Predictive Amal ysis cceeceecccsncceseceseeesccesseesscessceeseeesseesseceseceseeseeeceneceaeecaeeesees 12 10 12 2 1 Obtaining the Model Prediction of Maximum Likelihood eee 12 10 12 2 2 The Composite Mod l nrar ui nren ie eaa a o R S aa h 12 11 12 2 3 The PEST Control File seten ine a e e E ERE e E e 12 13 12 2 4 Template and Instruction Files 20 0 0 ceescecesccesneceeeeeeesecceneeeeeeeceeneeeesneeees 12 14 1229 Runnin g PEST scones esa ee on siie case sosetvcsedacntisptncsteelocevele sade ERTES na 12 14 13 Frequently Asked Questions eccesccessccsseseesseeeseeeseceecceeeceneceaeecaeecsaecsaeesaceeseeenaees 13 1 jG Ae ci Wenn are ire e ee ete ea eee UE O rey Ae er Cee E 13 1 13 2 Parallel PES Vosssakssssssaviternenaasensnsvinay wes iatna sonaia a aes rade TS 13 1 T32 PEST and Windows NPs cssccretssestcatnceatoc cas eeri esn ei ene e a ieee 13 3 Introduction l 1 1 Introduction 1 1 Installation Installation instructions are provided on the printed sheet accompanying this manual follow these instructions to transfer PEST executable and support files to your machine s hard disk Your autoexec bat should be modified before you run PEST the PEST directory must be added to the PATH statement 1 2 The PEST Concept 1 2 1 A Model s Data Requirements There is a mathematical model for just about everything Computer programs have been written to describe the flow of water in channels the flow of electricity
153. ST PAR2PAR is run just before the actual model executable within a batch file comprising the composite model 1 5 4 Sensitivity Analysis SENSAN which stands for SENSitivity Analysis is a command line program which provides the capability to carry out multiple model runs without user intervention using different parameter values for each run Thus a computer can be kept busy all night undertaking successive model runs with key model outputs from each run being recorded in a format suitable for easy later analysis using a spreadsheet or other data processing package Any or all model output files for specific model runs can also be stored in their entirety under separate names if desired SENSAN uses the same model interface protocol as PEST does ie parameter values are supplied to a model through model input file templates and key model generated numbers are read from model output files using an instruction set In addition a special SENSAN control file must be built informing SENSAN of the names of all template instruction and model input output files the model command line and the parameter values that must be used for each model run SENSAN is accompanied by a checking program named SENSCHEK The role and operation Introduction 1 14 of SENSCHEK are very similar to those of PESTCHEK viz it checks all SENSAN input data to verify that it is consistent and correct SENSCHEK reads a SENSAN control file as well as
154. ST control file using PARREP as described above and set NOPTMAX to zero PARREP is run using the command parrep parfile pestfilel pestfile2 where parfile is the name of a parameter value file pestfilel is the name of an existing PEST control file and pestfile2 is the name for the new PEST control file When PARREP replaces parameter values in the existing PEST control file by those read from the parameter value file it does not check that each parameter value lies between its upper and lower bounds that log transformed parameters are positive etc Hence especially if using a manually created parameter value file it is as always a good idea to run PESTCHEK before running PEST to ensure that all is consistent and correct 10 6 JACWRIT JACWRIT is a utility program which allows the user to inspect the Jacobian matrix computed by PEST Recall from Chapter 2 that the Jacobian matrix contains the derivative of each model output for which there is a corresponding observation with respect to each parameter At the end of each optimisation iteration PEST records a binary file containing the Jacobian matrix corresponding to best parameters so far attained during the optimisation process The definition of best depends on the aim of the optimisation process When working in parameter estimation mode the best parameters are those for which the lowest objective function was obtained When working in predictive analysis mode th
155. ST converge to that global minimum Furthermore not only does a wise choice of initial parameter values reduce the PEST run time it may also make optimisation possible especially for highly nonlinear models or models for which there are local objective function minima at places removed in parameter space from the location of the global objective function minimum 5 5 9 Poor Choice of Initial Marquardt Lambda The PEST algorithm is such that PEST should find its way to a close to optimal Marquardt lambda at each stage of the parameter estimation process However if you supply an initial Marquardt lambda which is far from optimal the adjustment to optimal lambda may not occur After attempting a parameter upgrade with the initial lambda PEST searches for alternative lambdas using the input variable RLAMFAC to calculate them If the initial lambda was poor these alternative lambdas may be little better in terms of lowering the objective function than the first one Soon in accordance with the settings provided by PHIREDLAM and NUMLAM on the PEST control file PEST may move on to the next optimisation iteration having achieved little in lowering the objective function The story may then be repeated at the next optimisation iteration and perhaps the next as well Soon Running PEST 5 30 because the objective function has not been lowered or has been lowered very little over a number of iterations PEST will terminate execution in accordance wit
156. ST run management file and calib hld the parameter hold file are possible PEST input files Many of the data items in the PEST control file are used to tune PEST s operation to the case in hand such items include parameter change limits parameter transformation types termination criteria etc As is further discussed in Chapter 5 before using PEST on a real world problem you may wish to use it to estimate parameters for a case where your field data are in fact model generated in this way you know the answers that PEST should achieve Through a careful examination of the PEST run record file and perhaps a little experimentation with PEST input variables you should be able to determine what PEST settings are best for your particular problem You can also test whether the observation set that you provide affords the determination of a unique parameter set The PEST control file can be built in one of two ways It can be easily prepared using a text editor following the directions provided in this chapter Alternatively you can use the PEST utility PESTGEN to generate a PEST control file for your current case using default input variables this file can then be modified as you see fit using a text editor In either case the PEST control file can be checked for correctness and consistency using the utility program PESTCHEK Note also that some of the programs of the PEST Ground Water and Surface Water Utilities can be
157. STFLE PESTMODE PAR NOBS NPARGP NPRIOR NOBSGP TPLFLE NINSFLE PRECIS DPOINT NUMCOM JACFILE MESSFILE LAMBDA1 RLAMFAC PHIRATSUF PHIREDLAM NUMLAM ELPARMAX FACPARMAX FACORIG HIREDSWH OPTMAX PHIREDSTP NPHISTP NPHINORED RELPARSTP NRELPAR COV ICOR IEIG parameter groups PARGPNME INCTYP DERINC DERINCLB FORCEN DERINCMUL DERMTHD one such line for each of the NPARGP parameter groups parameter data PARNME PARTRANS PARCHGLIM PARVAL1 PARLBND PARUBND PARGP SCALE OFFSET DERCOM one such line for each of the NPAR parameters PARNME PARTIED one such line for each tied parameter observation groups OBGNME one such line for each observation group observation data OBSNME OBSVAL WEIGHT OBGNME one such line for each of the NOBS observations model command line write the command which PEST must use to run the model model input output TEMPFLE INFLE one such line for each model input file containing parameters INSFLE OUTFLE one such line for each model output file containing observations prior information PILBL PIFAC PARNME PIFAC log PARNME PIVAL WEIGHT OBGNME one such line for each of the NPRIOR articles of prior information predictive analysis NPREDMAXMIN PDO PD1 PD2 ABSPREDLAM RELPREDLAM INITSCHFAC MULSCHFAC NSEARCH ABSPREDSWH RELPREDSWH NPREDNORED ABSPREDSTP RELPREDSTP NPREDSTP R R P I Example 6 1 Construction details of the PEST contr
158. TERATION NO 6 etc Example 7 3 Extract from a PEST run record file PEST begins each optimisation iteration by recording the current value of the regularisation weight factor as calculated during the previous optimisation iteration and of the regularisation and measurement objective functions and m Note that user supplied regularisation weights are not multiplied by the current weight factor when calculating the regularisation objective function That is why the value of the regularisation objective function is comparable from optimisation iteration to optimisation iteration Of course no weight factor is used in calculation of the measurement objective function Next PEST fills the Jacobian matrix Then on the basis of the linearity assumption encapsulated in the Jacobian matrix PEST calculates the optimal value for the regularisation weight factor for the current iteration Once it has calculated the regularisation weight factor it can calculate an objective function ie phi for the current optimisation iteration using equation 2 33 PEST prints this phi as the starting objective function for this itn PEST then calculates one or a number of parameter upgrade vectors on the basis of one or a Regularisation 7 11 number of different Marquardt lambdas in an attempt to lower the objective function as much as possible As is apparent from equation 2 33 by lowering the objective function PEST will simultaneous
159. Visual PEST ASP Model Independent Parameter Estimation Watermark Numerical Computing waterloo maaa hydrogeologic eeeeee SOFTWARE CONSULTING TRAINING Preface License You have the non exclusive right to use the enclosed Licensed Programs You may not distribute copies of the Licensed Programs or documentation to others You may not assign sublicense or transfer this Licence without the written permission of Watermark Numerical Computing You may not rent or lease the Licensed Programs without the prior permission of Watermark Numerical Computing You may not incorporate all or part of the Licensed Programs into another product for the use of other than yourself without the written permission of Watermark Numerical Computing Term The Licence is effective until terminated You may terminate it by destroying the Licensed Programs and all copies thereof The Licence will also terminate if you fail to comply with any term or condition of the Licence Upon such termination you must destroy all copies of the Licensed Programs Disclaimer This manual and associated software are sold as is and without warranties as to performance or merchantability The seller s salespersons may have made statements about this software Any such statements do not constitute warranties This program is sold without any express or implied warranties whatsoever No warranty of fitness for a particular purpose is offer
160. When run in predictive analysis mode the current value of the model prediction is also important The situation is slightly different when PEST runs in regularisation mode Because the regularisation weight factor u in equation 2 33 is different from iteration to iteration the objective function calculated during one optimisation iteration is not directly comparable with that calculated by PEST during the previous optimisation iteration However the measurement and regularisation objective functions are comparable from optimisation iteration to optimisation iteration Example 7 3 shows part of a run record file produced by PEST when operating in regularisation mode Regularisation 7 10 OPTIMISATION ITERATION NO 5 Model calls so far 50 Current regularisation weight factor Current value of measurement objective function Current value of regularisation objective function Re calculated regularisation weight factor Starting objective function for this itn ie phi Lambda 3 90625E 02 gt Phi 2 0301 0 655 of starting phi Meas fn 1 6652 Regul fn 3 5520 Lambda 1 95313E 02 gt Phi 2 SLIT 0 815 of starting phi Meas fn 2 0957 Regul fn 4 1855 Lambda 7 81250E 02 gt Phi 2 2408 0 723 of starting phi Meas fn 1 9340 Regul fn 2 9866 No more lambdas phi rising Maximum factor change 1 650 PEST Maximum relative change 0 4643 ro8 OPTIMISATION I
161. While file polyp pst is very similar to poly pst there are some important differences Through this file PEST is asked to carry out predictive analysis minimising the value of the prediction while keeping the model calibrated to the extent that the objective function based on the match between model outputs and field data cited in the PEST control file remains at or below a value of 1 0 Starting parameter values are the same as those in poly pst As these are very different from optimal parameter values PEST effectively works in parameter estimation mode for a while concentrating on lowering the objective function until it sniffs the critical point ie the point at which the prediction is minimised while maintaining the objective function below the user supplied threshold which is 1 0 in the present case It then calculates the parameter values corresponding to this point Once again PEST receives derivatives from the model by reading the derivatives file poly_der out Check the PEST input dataset using PESTCHEK and then run PEST to obtain the minimum model prediction that satisfies calibration constraints This should be about 8 60 You can repeat these PEST runs with derivatives calculated by PEST using finite differences if you wish For each of the two PEST control files do the following 1 Alter the value of JACFILE sixth variable on the fifth line to 0 2 Alter the value of DERCOM f
162. able to read a number from a string which includes non numeric characters However by including the character as a secondary marker after the number representing the observation sws PEST will separate the character from the string before trying to read the number But note that if a post observation secondary marker of this type begins with a numerical character PEST cannot be guaranteed not to include this character with the observation number if there is no whitespace separating it from the observation 669 The fact that there is no whitespace between the character and the number we wish to read causes PEST no problems either After processing of the character as a secondary marker the PEST processing cursor falls on the character itself The search for the first non blank character initiated by the sws instruction terminates on the very next character after the viz the 2 character PEST then accepts this character as the left boundary of the string from which it must read the soil moisture content and searches forwards for the right boundary of the string in the usual manner The Model PEST Interface 3 25 After PEST has read a non fixed observation it places its cursor on the last character of the observation number It can then undertake further processing of the model output file line to read further non fixed fixed or semi fixed observations or process navigational instructions as direc
163. above PHIRATSUF of the starting objective function PEST reduces lambda yet again otherwise it increases lambda to a value greater by a factor of RLAMFAC than the first lambda for the iteration If in its attempts to find a more effective lambda by lowering and or raising lambda in this fashion the objective function begins to rise PEST accepts the lambda and the corresponding parameter set giving rise to the lowest objective function for that iteration and moves on to the next iteration Alternatively if the relative reduction in the objective function between the use of two consecutive lambdas is less than PHIREDLAM PEST takes this as an indication that it is probably more efficient to begin the next optimisation iteration than to continue testing the effect of new Marquardt lambdas Thus if j l amp lt PHIREDLAM 4 2 where amp is the objective function value calculated during optimisation iteration i using the J th trial lambda PEST moves on to iteration i 1 A suitable value for PHIREDLAM is often around 0 01 If it is set too large the criterion for moving on to the next optimisation iteration is too easily met and PEST is not given the opportunity of adjusting lambda to its optimal value for that particular stage of the parameter estimation process On the other hand if PHIREDLAM is set too low PEST will test too many Marquardt lambdas on each optimisation iteration when it would be better off starting on a new i
164. ach parameter difference to PEST as an extra observation or as an article of prior information the observed value for which is zero in each case Thus each non zero parameter difference makes a contribution to the objective function whose task it is for PEST to minimise Hence in minimising the objective function PEST seeks to minimise these parameter differences and in so doing increases parameter uniformity across the model domain or of sub areas within this domain see below Regularisation can be used in conjunction with most methods of defining a parameterisation scheme over a model domain It can be used in conjunction with zones of assumed parameter constancy for use of an appropriate regularisation scheme allows the modeller to use more zones than he she otherwise would Through enforcing either a single regional smoothing condition or a series of more local smoothing conditions over different model sub areas the model can be parameterised in such a way that it respects outside knowledge for example geological knowledge at the same time as it accommodates smaller scale property variations in order to provide a satisfactory fit between model outputs and field measurements Similarly if a pilot points parameterisation scheme is used in which parameter values are estimated for points lying within the model domain and these parameter values are then spatially interpolated to the cells or elements of the model grid or m
165. actor changes for all relative limited and factor limited parameters respectively Values for these variables are supplied at the beginning of the inversion process They can also be altered part of the way through a PEST run if desired see Section 5 6 Note that log transformed parameters must be factor limited The PEST Algorithm 2 22 Let f represent the user defined maximum allowed parameter factor change for factor limited parameters ie FACPARMAX f must be greater than unity Then if bo is the value of a particular factor limited parameter at the beginning of an optimisation iteration the value b of this same parameter at the beginning of the next optimisation iteration will lie between the limits bo f lt b lt fbo 2 44a if bo is positive and foo lt b lt bolf 2 44b if bo is negative Note that if a parameter is subject to factor limited changes it can never change sign Let r represent the user defined maximum allowed relative parameter change for all relative limited parameters ie RELPARMAX r can be any positive number Then if bo is the value of a particular relative limited parameter at the beginning of an optimisation iteration its value b at the beginning of the next optimisation iteration will be such that o bo bo lt r 2 45 In this case unless r is less than or equal to unity a parameter can indeed change sign However there may be a danger in using a relative limit for some types of parameters
166. ailable only when its window is displayed the menu item has the same name as the window Some options vary from window to window while other options are common to all graphical or text windows The following options are common to all graphical windows Copy to Clipboard This allows you to copy the graph displayed in the window to the Windows clipboard You can then paste the graph into another application such as a word processing program Overview 5 Save As Any graph can be saved as a BMP WMF or EMF file it can then be imported into other applications Print Any graph can be printed a suite of page layout and title block options is provided so that the printed page can be tailored to the user s specifications Properties A plethora of plotting options is available for each graph Whenever you click on a plotted data point bar or symbol a bubble appears in which the coordinates of that data point are listed Click anywhere in the bubble to remove it Where a legend appears on the side of a graph that legend can be moved to another part of the WinPEST graphical window by dragging it with the mouse Any part of the plotting area of a WinPEST graphical window can be viewed in greater detail by first clicking on the Zoom in button E on the button bar and then dragging a viewing rectangle across the part of the plot that is of particular interest The default plot view can be restored by clicking on the Zoom out
167. al sensitivities of log transformed parameters through division by 2 303 to account for the fact that logarithmic transformation takes place to base 10 There are two ways in which elements of the Jacobian matrix ie sensitivities can be plotted The first option the WinPEST default option is to group sensitivities according to parameters ie the sensitivities are plotted in clumps with each clump displaying the sensitivities of all observations with respect to one particular parameter Alternatively sensitivities can be grouped by observation In either case individual parameters and or observations can be selected deselected for display by clicking anywhere on the plot with the right mouse button and choosing the Select parameters or Select observations menu item as appropriate Where parameters and or observations are numerous you can select these quickly by sweeping down the list with the mouse while holding down the left mouse key Then click in any box while holding down the lt Ctl gt key and all boxes will be selected Note that if PEST is running in regularisation mode observations and or prior information comprising the observation group regul do not appear on this plot unless specifically selected through the Select groups item of the pop up menu pertaining to this window Similar to the functionality available in other graphical windows the actual sensitivity of a particular observation with re
168. al values and run PEST again It is at this stage while working with a theoretical dataset for which you know PEST should achieve a low objective function value that you can adjust PEST control variables in order to tune PEST to the model Note that it is unlikely that you will achieve an objective function value of zero again though depending on the number of observations and their magnitudes and weights the objective function should nevertheless be reduced to as close to zero as roundoff errors permit provided the model is not beset by severe nonlinearities and or the presence of local objective function minima In most cases PEST is able to solve a parameter estimation problem using substantially less than 20 optimisation iterations Running PEST 5 26 If PEST does not lower the objective function or lowers it slowly you should run through the following checklist of reasons for PEST s poor performance In most instances the problem can be rectified 5 5 2 Derivatives are not Calculated with Sufficient Precision Precise calculation of derivatives is critical to PEST s performance Improper derivatives calculation will normally be reflected in an inability on the part of PEST to achieve full convergence to the optimal parameter set Sometimes in such circumstances PEST will commence an optimisation run in spectacular fashion lowering the objective function dramatically in the first optimisation iteration or two But then it runs ou
169. alculating parameter increments using the rel_to_max option For the relative and rel_to_max options a DERINC value of 0 01 is often appropriate However no suggestion for an appropriate DERINC value can be provided for the absolute increment option the most appropriate increment will depend on parameter magnitudes DERINCLB If a parameter increment is calculated as relative or rel_to_max it is possible that it may become too low if the parameter becomes very small or in the case of the rel_to_max option if the magnitude of the largest parameter in the group becomes very small A parameter increment becomes too low if it does not allow reliable derivatives to be calculated with respect to that parameter because of roundoff errors incurred in the subtraction of nearly equal model generated observation values To circumvent this possibility an absolute lower bound can be placed on parameter The PEST Control File 4 15 increments this lower bound will be the same for all group members and is provided as the input variable DERINCLB Thus if a parameter value is currently 1000 0 and it belongs to a group for which INCTYP is relative DERINC is 0 01 and DERINCLB is 15 0 the parameter increment will be 15 0 instead of 10 0 calculated on the basis of DERINC alone If you do not wish to place a lower bound on parameter increments in this fashion you should provide DERINCLB with a value of
170. alculation of derivatives with respect to any parameter belonging to the group is calculated as a fraction of the current value of that parameter that fraction is provided as the real variable DERINC However if INCTYP is absolute the parameter increment for parameters belonging to the group is fixed being again provided as the variable DERINC Alternatively if INCTYP is rel_to_max the increment for any group member is calculated as a fraction of the group member with highest absolute value that fraction again being DERINC See Section 2 3 for a full discussion of the methods used by PEST to calculate parameter derivatives Thus if INCTYP is relative and DERINC is 0 01 a suitable value in many cases the increment for each group member for each optimisation iteration is calculated as 0 01 times the current value of that member However if INCTYP is absolute and DERINC is 0 01 the parameter increment is the same for all members of the group over all optimisation iterations being equal to 0 01 If INCTYP is rel_to_max and DERINC is again 0 01 the increment for all group members is the same for any one optimisation iteration being equal to 0 01 times the absolute value of the group member of highest current magnitude however the increment may vary from iteration to iteration If a group contains members which are fixed and or tied you should note that the values of these parameters are taken into account when c
171. alculations pertaining to these observations undertaken by PEST At the end of the parameter estimation process PEST records measurements their model generated counterparts residuals observation weights and a number of functions of these in a residuals file The format of this file is such that the data contained therein is suitable for importation into a spreadsheet for further mathematical analysis Where tabulated functions of those observations for which a covariance matrix is supplied involve observation weights these functions are not calculated and recorded by PEST for the weights supplied by the user for these observations are not used in the inversion process Instead the Cov Mat string is written in place of the redundant observation weight and na for not applicable is recorded in place of any functions which depend on these weights If at least one observation covariance matrix is supplied in its input dataset PEST records an The PEST Control File 4 3 additional residuals file called a rotated residuals file This has the same filename base as the ordinary residuals file ie the filename base of the PEST control file but is given an extension of rsr Whereas the normal residuals file tabulates measurements their model generated counterparts the residuals calculated therefrom and various functions of these quantities see Section 5 3 4 the rotated residuals file tabulates rota
172. alib gt errors chk If PESTCHEK finds one or a number of errors in your input dataset it is important that you re run PESTCHEK on the dataset after you have corrected the errors This is because PESTCHEK may not have read all of your input dataset on its first pass depending on the errors it finds it may not be worthwhile or possible for PESTCHEK to read an input dataset in its entirety once an error condition has been established Hence once you have rectified any problems that PESTCHEK may have identified in your input dataset you should submit it to PESTCHEK again being content that the data is fully correct and consistent only when PESTCHEK explicitly informs you that this is the case If you wish you can write a batch file which runs both PESTCHEK and PEST in sequence Because PESTCHEK terminates execution with a non zero errorlevel setting should it detect any errors you can program the batch process to bypass the running of PEST unless the input dataset is perfect In this way you can always be sure that PESTCHEK rather than PEST is the first to detect any input data errors A suitable batch file is shown in Example 10 2 echo off rem FILE RUNPEST BAT rem To run RUNPEST BAT type the command runpest case r j rem where case is the filename base of the PEST control file and rem r and j are optional restart switches pestchek 1 if errorlevel 1 goto end pest 1 32 end
173. alibration conditions copy inl dat in dat gt nul twoline gt nul copy out dat outl dat gt nul rem Next the model is run under predictive conditions copy in2 dat in dat gt nul twoline gt nul copy out dat out2 dat gt nul Example 12 12 A batch file encompassing a composite model The batch file is divided into three parts In the second part the model is run under calibration conditions First the calibration input file in dat is copied to the expected TWOLINE input file in dat TWOLINE is then run and its output file out dat is copied to another file OUT1 DAT for safekeeping The process is then repeated in the third part of file model bat for the predictive model run In this case in2 dat is the model input file and out2 dat is the model output file The first part of the batch file model bat illustrates a procedure that is recommended in the construction of all composite models In this section of the batch file all intermediate files used or produced during execution of the composite model are deleted Recall that PEST deletes all model output files that it knows about before it runs the model In this way it is ensured that if for some reason the model does not run then old model output files are not mistaken for new ones Thus if the model fails to run an error condition will be encountered and PEST will cease execution with an appropriate error message However in a composite model there are lik
174. all template and instruction files cited therein If any errors or inconsistencies are detected appropriate messages are written to the screen Note that SENSAN and SENSCHEK are both true WINDOWS executables 1 6 This Manual This introduction has provided an overview of the capabilities and components of PEST However to get the most out of PEST you should take the time to read this manual in its entirety Parameter estimation is a tricky business and will not work unless you know what you are doing If PEST does not appear to be able to calibrate your model or turn your model into a powerful data interpretation package the chances are that you are misusing it Thus even though it may be heavy going you should pay particular attention to Chapter 2 which provides details of the PEST algorithm As was explained above PEST may need to be tuned to your estimation or interpretation problem All that it may take to achieve this is the adjustment of a single optimisation control variable the log transformation of a single parameter or the setting of a single derivative increment Unless you are aware of the possibilities available to you for modifying PEST s operation to suit your particular problem you may never use it to its full potential Chapter 3 discusses the interface between PEST and your model describing how to make PEST template and instruction files Chapter 4 teaches you how to write a PEST control file and discusses the
175. ame of the file to which the model should write the derivatives matrix ie the name of the external derivatives file If JACFILE is set to 0 in the control data section of the PEST control file the derivatives command line section can be omitted 8 5 5 Model Command Line Section In previous versions of PEST the model command line section of the PEST control file contained only a single line this being comprised of the command that PEST must use to run the model Indeed if NUMCOM in the control data section of the PEST control file is set to 1 then the contents of this section are the same However if NUMCOM is set to n there must be n model command lines listed in this section of the PEST control file one under the other The DERCOM variable in the parameter data section of the PEST control file refers to these commands by number when indicating which of these commands is to be used when running the model to calculate derivatives using finite differences with respect to each parameter It is important to note that when the model is run in order to test a parameter upgrade and when the model is run for the first time in the optimisation process in order to obtain the objective function corresponding to the initial parameter set the first of the listed model commands is used to run the model 8 6 An Example A simple example is presented to demonstrate the use of PEST s external derivatives functional
176. ameter estimation becomes as straightforward as modelling itself 1 5 3 Parameter Preprocessing Sometimes it is useful to undertake complex mathematical operations on model parameters before actually providing them to the model This can help the parameter estimation process in a number of ways For example appropriate parameter transformation may render a model more linear with respect to one or more of its parameters in other circumstances the calculation of secondary parameters eg monthly variation of a particular model input type from a smaller number of primary parameters which describe the seasonal variation of the secondary parameter eg the mean amplitude and phase of that parameter s variation can bring stability to the parameter estimation process by allowing PEST to estimate a fewer number of parameters while incorporating the user s knowledge of the type of variation that the parameter undergoes directly into the parameter estimation process Parameter transformations of this type and many more can be undertaken using the parameter preprocessor PAR2PAR supplied with PEST PAR2PAR requires a text input file from which a template file is easily prepared describing the mathematical relationships which can be of arbitrary complexity that exist between parameters Like PEST it then writes its secondary parameters to a model input file using a template of that file During the calibration process undertaken by PE
177. ameter upgrade vector In optimisation iterations 3 and 4 parameter h1 limits the magnitude of the parameter upgrade vector through incurring the maximum allowed parameter factor change It is possible that convergence for this case would have been achieved much faster if FACPARMAX on the PEST control file were set higher than 3 0 Running PEST 5 11 At the beginning of the second optimisation iteration parameter ro2 is at its upper bound After calculating the Jacobian matrix and formulating and solving equation 2 23 PEST notices that parameter ro2 does not wish to move back into its domain so it temporarily freezes this parameter at its upper bound and calculates an upgrade vector solely on the basis of the remaining adjustable parameters The two step process by which PEST judges whether to freeze a parameter which is at its upper or lower limit is explained in Section 2 2 3 Note that at the beginning of optimisation iteration 3 parameter ro2 is released again in case with the upgrading of the other adjustable parameters during the previous optimisation iteration it wants to move back into the internal part of its domain In the third optimisation iteration only a single Marquardt lambda is tested the objective function having been lowered to below 0 4 times its starting value for that iteration through the use of this single lambda 0 4 is the user supplied value for the PEST control variable PHIRATSUF During the
178. an appropriate error message However a parameter value file may contain more parameters than are cited in the template file these extra parameters are ignored when generating the model input file This may occur if your model has a number of input files and you wish to optimise parameters occurring on more than one of them You must make a template file for each such model input file however you need to prepare only one parameter value file containing all the parameters for that particular problem PEST Utilities 10 3 10 2 INSCHEK Program INSCHEK assists in the construction of PEST instruction files Like TEMPCHEK it can be used in two modes In the first mode it simply checks that an instruction file has no syntax errors and obeys PEST protocol as set out in Section 3 3 In its second mode it is able to read a model output file using the directions contained in the instruction file it then writes a file listing all observations cited in the instruction file together with the values of these observations as read from the model output file In this way you can verify that not only is your instruction set syntactically correct but that it reads a model output file in the way it should INSCHEK is run using the command inschek insfile modfile where insfile is a PEST instruction file and modfile is a model output file to be read by INSCHEK optional The simplest way to run INSCHEK is to use the command inschek insfile When invo
179. an appropriate error message and abandons execution such errors may arise if for example INSCHEK finds a blank space where a number should be encounters the end of the model output file before locating all observations etc However if PEST Utilities 10 4 INSCHEK reads the file without trouble it lists all observations cited in the instruction set together with their values as read from modfile to file obf where file is the filename base of insfile Example 10 1 shows a typical observation value file arl 1 21038 ar2 1 51208 ar3 2 07204 ar4 2 94056 ars 4 15787 ar6 5 77620 ar7 7 78940 ar8 9 99743 ar9 11 8307 ar10 12 3194 arll 10 6003 ar12 7 00419 ar13 3 44391 ar14 1 58278 ar15 1 10381 ar16 1 03085 ar17 1 01318 ar18 1 00593 ar19 1 00272 Example 10 1 An observation value file 10 3 PESTCHEK PESTCHEK should be used when all preparations for a PEST run are complete ie when all template files instruction files and the PEST control file which brings it all together have been prepared PESTCHEK reads the PEST control file making sure that all necessary items of information are present on this file and that every item is consistent with every other item for example that logarithmically transformed parameters do not have negative lower bounds that RELPARMAxX is greater than unity if at least one parameter is free to change sign during the optimisation process etc As PEST does not carry out consistency checks
180. apter 10 of this manual PEST carries out some checking of its input dataset itself if there are any syntax errors in any of these input files or if some of the data elements are of the incorrect type for example real instead of integer integer instead of character PEST will cease execution with an appropriate error message However PEST does not carry out extensive consistency checks as the coding required to achieve this would take up too much memory this memory being reserved for array storage for PEST and possibly the model Hence unless you carry out input data checking yourself using the utility programs mentioned above PEST may commence execution on the basis of an erroneous data set Sometimes the error will be detected and PEST will terminate execution with an error message In other cases PEST may commence the optimisation process only to terminate execution at some later stage with a run time error message that may bear little relation to the inconsistency that gave rise to the problem in the first place 5 1 2 Versions of PEST As explained in Chapter 1 there are two versions of PEST Each can be run by typing the name of the pertinent executable file at the command prompt PEST The single window version of PEST contained in the pest exe executable program is the simpler version of PEST to use In this version of PEST the model and PEST share the same window Hence screen output from one will cause screen output fr
181. ar model run The problems are particularly frustrating because they occur so rarely Thus PEST may run happily for hours when suddenly one of the above conditions occurs Any one of them will cause immediate termination of execution In many cases where these problems have been encountered it has been found that the problem was rectified by re compiling the model source code using a modern compiler Old compilers used DOS memory extenders some of which had a memory leakage bug to gain access to all of a machine s memory very old compilers generated 16 bit executables and used no extender at all WINDOWS NT 2000 does not work well with either of these If the problem persists make sure that the latest service pack has been installed Also see if use of the command rather than the cmd command interpreter and vice versa makes any difference Index Index E e E cave anodic Sehr ccges sdk tess ca dies aus Shs 3 22 gt E dusranatss 1 4 5 22 10 1 10 3 10 5 ABSELE wescsee dvecesescostess vtesodeonesepsens is 11 6 11 10 ABSPREDEAM sanr a TAa 6 14 ABSPREDSTP ABSPREDSWH 6 15 Adjustable parameters 00 0 0 cece cece eeeeeeee 1 6 ASCH files ccccccccccsccceseeeesseeeeseeee 1 4 3 2 3 10 4 22 AUTOEXEC BAT s svescincisvacsveccsvessectsvsxaveiiavesvscsuereevets 1 1 Bat h ehanen enn ao aaro 10 19 Batch file rra a RENEE 3 2 4 21 10 5 Best fit method aosoysseorniorord ies 2
182. arameter change relative limited changes na Fraction of initial parameter values used in computing change limit for near zero parameters 1 00000E 03 Relative phi reduction below which to begin use of central derivatives 0 10000 Relative phi reduction indicating convergence 0 10000E 01 Number of phi values required within this range s 3 Maximum number of consecutive failures to lower phi 3 Maximum relative parameter change indicating convergence Number of consecutive iterations with minimal param change Maximum number of optimisation iterations 0 10000E 01 3 30 OPTIMISATION RECORD INITIAL CONDITIONS Sum of squared weighted residuals ie phi 523 8 Contribution to phi from observation group group_1 127 3 Contribution to phi from observation group group_2 117 0 Contribution to phi from observation group group_3 185 2 Contribution to phi from observation group group_4 94 28 Current parameter values group_1 Toropan NJ Op Ss group_4 rol 0 500000 ro2 5 00000 ro3 0 500000 h1 2 00000 h2 5 00000 OPTIMISATION ITERATION NO H d Model calls so far 1 Starting phi for this iteration 523 8 Contribution to phi from observation group Contribution to phi from observation group Contribution to phi from observation group Contribution to phi from observation group Lambda 5 000 gt phi 361 4 0 69 of starting phi Lambda 2 500 gt phi 357 3 0 68 of starting p
183. arameter upgrade vector to determine the point at which this vector crosses the o contour See Chapter 6 for further details 2 1 10 Regularisation The PEST Algorithm 2 13 The theory which underpins PEST s regularisation functionality bears some resemblance to that which underpins its predictive analysis functionality As is explained in Section 2 1 2 when calibrating a model it is PEST s task to minimise an objective function defined as m Xb Qu e Xb 2 30a The m subscript introduced to the left side of equation 2 30 which is otherwise equivalent to equation 2 8 denotes the fact that the objective function is comprised of the sum of weighted squared differences between model outputs and field measurements the m stands for measurement The vector is comprised of field measurements while the vector b is comprised of the parameters which must be estimated As has already been mentioned Qm is a diagonal matrix whose elements are the squares of measurement weights We wish to impose the requirement that a regularisation objective function be also minimised by the parameter set which is estimated by PEST r is defined by the equation d Zb Q d Zb 2 30b where Q is a diagonal matrix comprised of the squares of weights assigned to the various regularisation observations which collectively comprise the vector d The relationships by which the model generated count
184. arameters which describe the spatial distribution of some property over a two or three dimensional model domain for example a ground water or geophysical model The user is no longer required to subdivide the model domain into a small number of zones of piecewise parameter constancy Rather a large number of parameters can be used to describe the distribution of the spatial property and PEST s regularisation functionality can be used to estimate values for these parameters If supplementary information is provided in the Introduction 1 11 form of a preferentially smooth system state then only enough heterogeneity will be introduced to the system to guarantee a good fit between model outcomes and field data Furthermore PEST will determine the locations of areas of anomalous property values itself This is normally a vastly superior method by which to infer the distribution of a spatial parameter over a model domain than to estimate parameters associated with a pre defined zonation pattern PEST s regularisation functionality can also be useful when calibrating a number of models simultaneously for example rainfall runoff models in different watersheds PEST can be asked to preferentially estimate identical values for the same parameter types in the different model domains Differences in parameter values estimated through the regularised multi model calibration process will then be present because they must be present if all of the mo
185. ard differences Derivatives calculated in this way are only approximate If the increment is too large the approximation will be poor if the increment is too small roundoff errors will detract from derivatives accuracy Both of these effects will degrade optimisation performance To combat the problem of derivatives inaccuracy PEST allows derivatives to be calculated using the method of central differences Using this method two model runs are required to calculate a set of observation derivatives with respect to any parameter For the first run an increment is added to the current parameter value while for the second run the increment is subtracted Hence three observation parameter pairs are used in the calculation of any derivative the third pair being the current parameter value and corresponding observation value The derivative is calculated either by i fitting a parabola to all three points ii constructing a best fit straight line for the three points or iii by simply using finite differences on the outer two points its your choice It is normally best to commence an optimisation run using the more economical forward difference method allowing PEST to switch to central differences when the going gets tough PEST will make the switch automatically according to a criterion which you supply to it prior to the commencement of the run PEST s implementation of the Gauss Marquardt Levenberg method is extremely flexible man
186. are being run on the same machine in order to gain access to two different processors belonging to that machine there is no reason why each slave should not run the same model executable It is essential that for each slave engaged in the Parallel PEST optimisation process the model as run by that slave reads its input file s and writes its output file s to a different subdirectory or subdirectories from the model as run by any of the other slaves this will probably occur naturally when slaves reside on different machines If this is not done model output files generated during one parallel model run will be overwritten by those generated during another similarly model input files prepared by PEST for one particular model run will be overwritten by those that it prepares for another In many cases all input files for one particular model are housed in a single subdirectory also Parallel PEST 9 6 all model output files are written to this same subdirectory In this simple case preparation for model runs on different machines across a network consists in simply copying the entire model working directory from the master machine to an appropriate directory on each of the other machines As is explained below the structure of the run management file which Parallel PEST must read in order to ascertain the whereabouts of each of its slaves is particularly simple under these circumstances 9 2 4 Communications between Parallel PEST and
187. are now briefly outlined 4 3 3 1 Echoing of Covariance Matrices Before undertaking the parameter estimation process PEST records much of the information that it reads from the PEST control file to its run record file This information includes the contents of any covariance matrix files that are supplied to it in its input dataset When echoing observation weights to its run record file the weights supplied for observations belonging to an observation group for which a covariance matrix has been supplied are not recorded for these weights are not used in the inversion process Rather the character string Cov Mat is recorded in place of the pertinent weights to remind the user that an observation covariance matrix is used in their stead 4 3 3 2 Objective Function Calculation of the objective function and of the contribution to the objective function made by various observation groups takes account of the fact that a covariance matrix is supplied for at least one group of observations 4 3 3 3 Residuals Measurements together with their model generated counterparts calculated on the basis of best fit parameters are tabulated at the end of the run record file observation weights are also tabulated with this data For those observations which are associated with a covariance matrix the character string Cov Mat replaces the observation weight in this table this indicating once again that the latter are ignored in all c
188. at The external derivatives file must be an ASCII ie text file in which numbers are separated by spaces tabs or a comma it is read by PEST using FORTRAN free field input As mentioned above it must be headed by a line citing the number of columns and rows comprising the matrix The matrix itself must have NOBS rows and NPAR columns where NPAR is the number of parameters and NOBS is the number of observations cited in the PEST control file for the current case The ordering of parameters and observations in the external derivatives file must be the same as that in the PEST control file Note that a column must be included in the derivatives matrix for every parameter even for those parameters which are tied or fixed PEST ignores derivatives calculated with respect to fixed Model Calculated Derivatives 8 3 parameters Similarly there must be a row for every observation cited in the PEST control file even for observations which are assigned a weight of zero Where there are many parameters to be estimated each row representing the derivatives of a particular observation with respect to all parameters can be wrapped onto the next line or as many lines as you wish However derivatives for the next observation must begin on a new line 8 2 4 Derivatives Type As is documented elsewhere in this manual some parameters can be log transformed during the parameter estimation process PEST then estimates the log to base 10 of such paramet
189. ate both excitations and parameters A good deal of geophysical software falls into this category where sometimes very elegant mathematical models are developed in order to infer aspects of the earth s structure from measurements that are confined either to the earth s surface or to a handful of boreholes e Calibration If a natural or man made system is subject to certain excitations and numbers representing these same excitations are supplied to a model for that system it may be possible to adjust the model s parameters until the numbers which it generates correspond well with certain measurements made of the system which it simulates If so it may then be possible to conclude that the model will represent the system s behaviour adequately as the latter responds to other excitations as well excitations which we may not be prepared to give the system in practice A model is said to be calibrated when its parameters have been adjusted in this fashion Introduction 1 3 e Predictive Analysis Once a parameter set has been determined for which model behaviour matches system behaviour as well as possible it is then reasonable to ask whether another parameter set exists which also results in reasonable simulation by the model of the system under study If this is the case an even more pertinent question is whether predictions made by the model with the new parameter set are different Depending on the system under study and the type
190. ation criteria When PEST is used in parameter estimation mode the optimisation process is judged to be complete when the objective function can be reduced no further or if it is apparent that a continuation of the optimisation process will reduce it very little This is still the case if PEST when used in predictive analysis mode fails to lower the objective function below PD1 However if it has been successful in lowering it to this value which it should be if PDO and PD1 are chosen to be above min as determined from a previous parameter estimation run then termination criteria are based on improvements to the model prediction If NPREDMAXMIN is set to 1 and NPREDNORED optimisation iterations have elapsed since PEST has managed to raise the model prediction then it will terminate execution Alternatively if NPREDMAXMIN is set to 1 and NPREDNORED optimisation iterations have elapsed since PEST has managed to lower the model prediction then it will terminate execution A good setting for NPREDNORED is 4 ABSPREDSTP RELPREDSTP and NPREDSTP If NPREDMAXMIN is set to 1 and if the NPREDSTP highest predictions are within an absolute distance of ABSPREDSTP of each other or are within a relative distance of RELPREDSTP of each other PEST will terminate execution If NPREDMAXMIN is set to 1 and the NPREDSTP lowest predictions are within an absolute distance of ABSPREDSTP of each other or are within a relative distance of RELPREDSTP of each ot
191. ation group must be involved in the covariance matrix Example 4 6 illustrates a covariance matrix file 3 All diagonal elements of the covariance matrix must be positive While the matrix should theoretically be positive definite to qualify as a covariance matrix a symmetric matrix will be acceptable However the matrix must be such that it is possible to calculate eigenvectors and eigenvalues for that matrix without incurring numerical difficulties this will rarely be a problem 4 Elements of the covariance matrix as represented in the covariance matrix file must be space or comma delimited A line of this matrix can wrap around to the next line if it is too long However each row of the matrix must begin on a new line 5 Whether or not a covariance matrix is supplied for a particular observation group weights must still be supplied for members of that group in the observation data section of the PEST control file However these weights will be ignored by PEST including a weight of zero that may be assigned to a certain observation in order to take it out of the parameter estimation process 6 Observation groups used for prior information and those used for actual observations must be separate when one or more covariance matrices are supplied for use in the inversion process Thus a particular observation group cannot have members which are both observations and prior information equations DO QO H O O AO O OH O
192. ation into the objective function can change its structure in parameter space often making the global minimum easier to find depending on what weights are applied to the articles of prior information Once again this enhances optimisation stability and may reduce the number of iterations required to determine the optimal parameter set The PEST Algorithm 2 8 Initial parameter estimates Parameter 2 Contours of equal objective function value Parameter 1 Figure 2 1 Iterative improvement of initial parameter values toward the global objective function minimum 2 1 5 The Marquardt Parameter Equation 2 16 forms the basis of nonlinear weighted least squares parameter estimation It can be rewritten as u J QJ J Qr 2 18 where u is the parameter upgrade vector and r is the vector of residuals for the current parameter set Let the gradient of the objective function in parameter space be denoted by the vector g The 7 th element of g is thus defined as _ oP amp i Ob 2 19 ie by the partial derivative of the objective function with respect to the i th parameter The parameter upgrade vector cannot be at an angle of greater than 90 degrees to the negative of the gradient vector If the angle between u and g is greater than 90 degrees u would have a component along the positive direction of the gradient vector and movement along u would thus cause the objective function to rise which is the opposite of what we wan
193. ation is plotted under the same conditions that eigenvector information is plotted see the first paragraph of the preceding section for a description of when this plot is available The Eigenvalues window is divided into two compartments The upper compartment displays the magnitudes of the eigenvalues of the covariance matrix arranged in increasing order Components of the respective eigenvectors are displayed in the lower compartment Eigenvector components are linked to eigenvalues by colour Right mouse click on a particular eigenvalue in the upper compartment then click on Add Remove eigencomponents to have its eigenvector components displayed in the lower sub window a WinPEST C PEST2000 TEST1 VES11_PST File Run Options Eigenvalues View Yalidate oe i MERER TAI MENES Jacobian Correlation Covariance Eigenvectors Eigenvalues Uncertainties Run Record _4 Eigenvalues and Normalized Eigenvectors Eigenvalue 10s 105 10 10 107 4 me Vector Number 0 5 Component 0 0 0 5 1 rol ro2 ro3 hi h2 Parameters Not Running Figure 6 Eigenvalues are plotted in the top compartment while the components of the corresponding eigenvectors for one or more eigenvalues are plotted in the lower compartment WinPEST Graphical Windows 20 4 15 Uncertainties Information is available in the Uncertainties window under the same conditions as those for which in
194. ation observation is zero e All regularisation observations belong to the observation group regul e Each regularisation observation should pertain to a single difference between a particular parameter and its neighbour in either the row or column direction e For each parameter block differences should be taken in both the row and column directions resulting in two regularisation observations However where a geological boundary occurs between two neighbouring blocks the difference between parameter values on either side of that boundary should be omitted from the regularisation dataset In this way the regularisation scheme informs PEST that the preferred locations of hydraulic property contrasts are at recognised geological boundaries However heterogeneity elsewhere will be accommodated if required Other parameterisation schemes together with appropriate regularisation methodologies can be used For example a pilot points methodology is very attractive Using this technique PEST is asked to assign hydraulic conductivities to discrete points within the model domain The hydraulic conductivity at each cell or node of the numerical ground water model is then calculated from the hydraulic conductivities assigned to these pilot points using a spatial interpolation algorithm such as kriging If appropriate the interpolation algorithm can be tailored to the geology such that different subsets of pilot points are used as a basis for s
195. atrix through equation 2 7 The correlation coefficient matrix has the same number of rows and columns as the covariance matrix furthermore the manner in which these rows and columns are related to adjustable parameters or their logs is identical to that for the covariance matrix Like the covariance matrix the correlation coefficient matrix is symmetric The diagonal elements of the correlation coefficient matrix are always unity the off diagonal elements are always between 1 and 1 The closer that an off diagonal element is to 1 or 1 the more highly correlated are the parameters corresponding to the row and column numbers Running PEST 5 15 of that element Thus for the correlation coefficient matrix of Example 5 1 the logs of parameters ro2 and h2 show medium to high correlation as is indicated by the value of elements 1 3 and 3 1 of the correlation coefficient matrix viz 0 8756 This explains why individually these parameters are determined with a high degree of uncertainty in the parameter estimation process as evinced by their wide confidence intervals 5 2 11 The Normalised Eigenvector Matrix and the Eigenvalues The eigenvector matrix is composed of as many columns as there are adjustable parameters each column containing a normalised eigenvector Because the covariance matrix is positive definite these eigenvectors are real and orthogonal they represent the directions of the axes of the probability elli
196. be evaluated during the first iteration for a parameter whose initial value is zero If the parameter belongs to a group for which derivatives are in fact calculated as relative a non zero DERINCLB variable must be provided for that group e Ifa parameter has an initial value of zero the parameter can be neither a tied nor a parent parameter as the tied parent parameter ratio cannot be calculated PARLBND and PARUBND These two real variables represent a parameter s lower and upper bounds respectively For adjustable parameters the initial parameter value PARVAL1 must lie between these two bounds However for fixed and tied parameters the values you provide for PARLBND and PARUBND are ignored The upper and lower bounds for a tied parameter are determined by the upper and lower bounds of the parameter to which it is tied and by the ratio between the tied and parent parameters The PEST Control File 4 19 PARGP PARGP is the name of the group to which a parameter belongs As discussed already a parameter group name must be twelve characters or less in length and is case insensitive As derivatives are not calculated with respect to fixed and tied parameters PEST provides a dummy group name of none to which such tied and fixed parameters can be allocated Note that it is not obligatory to assign such parameters to this dummy group they can be assigned to another group if you wish However any group other than none whi
197. ber is produced by the second Predictive Analysis 6 7 component of the dual model PEST is thus asked to calibrate this dual model such that on the one hand the model produces numbers which match historical system behaviour while on the other hand it produces a certain best or worst case prediction If a parameter set can be found which allows the model to do this then the worst or best case prediction is compatible with calibration constraints and knowledge constraints if parameter bounds are imposed If it cannot then the hypothesised worst case or best case prediction is not possible The trick in implementing the dual calibration technique is the selection of a weight to assign to the single model prediction In general the weight applied to this prediction should be such that the contribution to the final objective function by the residual associated with the single prediction is of the same order as the contribution to the objective function by all of the residuals associated with the historical component of the dual model If this is the case then obviously the model will not be producing a prediction which is exactly equal to the user supplied best or worst case prediction Because of this the prediction may have to be supplied as a little worse than worst or better than best A strategy for choosing a suitable weight and predictive value will soon become apparent once the method is implemented on a particular case Dua
198. between the issuing of any of the above commands and a response from PEST There is however one particular situation that can result in a large elapsed time between the issuing of either of the above commands and the reception of a response from PEST If when PEST tries to read a model output file it encounters a problem it does not immediately terminate execution reporting the error to the screen Rather it waits for 30 seconds and then tries to read the file again If it is still unsuccessful it waits another 30 seconds and tries to read the file yet again This time if the error is still present it terminates execution reporting the error to the screen in the usual fashion By trying to read the model output file three times in this way before declaring that the model run was a failure for example because the parameter set that it was using was inappropriate in some way or an instruction file was in error it removes the possibility that network problems have been the cause of PEST s failure to read the model output file While it is engaged in this process however it does not check for the presence of file pest stp the file written by programs PPAUSE PSTOP and PSTOPST Hence if a user types any of these commands while PEST is thus engaged he she may have to wait some time for PEST to respond Meanwhile PEST records on the run management record that there is a problem either model or communications in reading results from slave
199. ble that the command used to start the model may be different for different slave machines for example if the model executable resides in a differently named subdirectory on each such machine and the full model pathname is used in issuing the system command hence PSLAVE prompts the user for the local model command as it commences execution Parallel PEST 9 5 Machine 2 writes model input files reads model output files pslave exe model input files model output files Machine 1 ppest exe template files instruction files Machine 3 writes model input files pslave exe model input files model output files reads model output files Figure 9 1 Relationship between PEST PSLA VE and the model Figure 9 1 illustrates in diagrammatic form the relationship between Parallel PEST PSLAVE and the model for the case where PEST resides on one machine and the model is run on each of two other machines Note that as is explained below this is an unusual case in that it is common practice for the master machine ie machine 1 in Figure 9 1 to also be a slave machine to avoid wastage of system resources 9 2 3 Running the Model on Different Machines Greatest efficiency can be achieved if an independent model executable program resides on each slave machine Thus when PSLAVE runs the model the executable program does not have to be loaded across the network Note however that if PEST and two incidences of PSLAVE
200. bs on the same line as a semi fixed observation because the column numbers provided with the semi fixed observation instruction determine the location of the observation on the line As always observations must be read from left to right on any one instruction line hence if more than one semi fixed observation instruction is provided on a single PEST instruction line the column numbers pertaining to these observations must increase from left to right For the case illustrated in Examples 3 6 and 3 7 all the fixed observations could have been read as semi fixed observations with the difference between the column numbers either remaining the same or being reduced to substantially smaller than that shown in Example 3 7 However it should be noted that it takes more effort for PEST to read a semi fixed observation than it does for it to read a fixed observation as PEST must establish for itself the extent of the number that it must read After PEST has read a semi fixed observation its cursor resides at the end of the number which it has just read Any further processing of the line must take place to the right of that position Non Fixed Observations Examples 3 11 and 3 13 demonstrate the use of non fixed observations A non fixed observation instruction does not include any column numbers because the number which PEST must read is found using secondary markers and or other navigational aids such as The Model PEST Interface 3 22 whitespace a
201. btained by using the noverbose option In this case the user should ensure that the model likewise produces no screen output by redirecting its output to the nul file using for example the command model gt nul to run the model If SENSAN is thus left to produce the only screen output the user can monitor progress and detect any SENSAN error messages if they are written to the screen NPAR This is the number of parameters It must agree with the number of parameters cited in the template file s used by SENSAN It must also agree with the number of parameters named in the parameter variation file provided to SENSAN NOBS NOBS is the number of observations ie the number of model outcomes used in the sensitivity analysis process It must agree with the number of observations cited in the instruction file s provided to SENSAN SENSAN 11 6 NTPFLE The number of template files to be used by SENSAN For each template file there must be a corresponding model input file see below Note that a given template file can be used to write more than one model input file however two templates cannot write the same model input file NINSFLE The number of instruction files used by SENSAN to read model outcomes For each instruction file there must be a matching model output file Note that the same instruction file cannot read more than one model output file observation values would be overwritten however two different in
202. by clicking on File then Open then Project and selecting the appropriate file through the file selection dialogue box Once this has been done the pertinent PEST control file does not appear in any WinPEST window However you can view it and edit it if you wish by clicking on File then Open then File from the WinPEST main menu Any text file opened in this way can be edited and then saved as in a normal text editor Note however that for a PEST control file to be actually useable by PEST in the optimisation process it must be read as a project As an alternative to using the WinPEST menu system a project can also be read by WinPEST through selection of the Open button on the WinPEST toolbar Alternatively click on the arrow next to this button to choose between opening a project or opening a file for editing in the WinPEST text editor window 3 2 Error Checking Before PEST is run its input dataset should be checked for correctness and consistency As is discussed in the PEST manual three error checking programs are available for this purpose Program TEMPCHEK checks template files program INSCHEK checks instruction files while program PESTCHEK checks the entire PEST input dataset All of these programs can be run from within WinPEST If Validate then Templates is selected from the WinPEST menu program TEMPCHEK is run to check all template files cited in the PEST control file previou
203. by the last Another possibility is that the model batch file may call the same model a number of times running it over different historical time periods so that measurements made through all these time periods can be simultaneously used in model calibration Furthermore as shown in the example above you can insert intelligence into the way component models are run through the use of the errorlevel variable 4 2 9 Model Input Output In this section of the PEST control file you must relate PEST template files to model input files and PEST instruction files to model output files You will already have informed PEST of the respective numbers of these files through the PEST control variables NTPLFLE and NINSFLE See Example 4 1 for the structure of the model input output section of the PEST control file For each model input file PEST template file pair there should be a line within the model input output section of the PEST control file containing two entries viz the character variables TEMPFLE and INFLE The first of these is the name of a PEST template file while the second is the name of the model input file to which the template file is matched Pathnames should be provided for both the template file and the model input file if they do not reside in the current directory Construction details for template files are provided in The PEST Control File 4 24 Chapter 3 of this manual It is possible for a single template file to be
204. cal across all machines that run the model In most cases all model input files will be placed into a single model working directory on each slave machine In most cases the model itself will need to be installed on each slave machine as well 9 3 2 Starting the Slaves Go to each of the slaves in turn open a command line window and transfer to the PSLAVE working directory on that machine Start PSLAVE execution by typing the command pslave PSLAVE immediately prompts the user for the command which it must use to run the model Type in the appropriate command Remember that as with the normal PEST the model may be a batch file running a series of executables alternatively the model may be a single executable Provide the model pathname if the model batch file or executable is not cited in the PATH environment variable on the local machine or does not reside in the PSLAVE working directory On each slave machine PSLAVE now waits for PEST to commence execution At this stage or at any other stage of PSLAVE execution the user can press the lt Ctl C gt keys to terminate Parallel PEST 9 13 its execution if this is desired 9 3 3 Starting PEST The next step is to start PEST Move to the machine on which PEST resides open a command line window transfer to the appropriate directory and type ppest case where case is the filename base of both the PEST control file and the Parallel PEST run management file PEST then com
205. cations between PEST and its slaves It will also result in a longer time between the issuing of a PPAUSE or PSTOP command and a response from PEST However it will ensure stable Parallel PEST performance across a busy network In general the busier is the network the higher should WAIT be set In most cases a value of 1 0 to 2 0 seconds will be adequate even for a relatively busy network however do not be afraid to set it as high as 10 0 seconds or even higher on an extremely busy network While this could result in elapsed times of as much as 1 minute between the end of one model run and the beginning of another if this is small in comparison with the model execution time then it will make little difference to overall Parallel PEST performance Also be aware that while networks may seem relatively quiet during the day they may become extremely busy at night when large backing up operations may take place 9 3 11 If PEST will not Respond As is mentioned above if the PPAUSE PSTOP or PSTOPST command is issued while the command line version of Parallel PEST is running PEST execution will be interrupted in the usual way However unlike the single window version of PEST Parallel PEST does not need to wait until the end of the current model run to respond to these commands rather there is only a short delay the length of this delay depending on the setting of the variable WAIT Hence if WAIT is set extremely high be prepared for a short wait
206. ccecccscceeeeseececceeees 7 5 Table of Contents 7 3 2 The PEST Control File Observation Groups essesseesceeseeeseeeeaeeeeeeeeees 7 5 7 3 3 Control File Regularisation Section cecesccesccssceeeeeeseeeeceeseeeeseeeseeeseees 7 5 7 3 4 The Control Variable FRACPHIM 0 0 ee eeeceeceeeeceeeceeeceneecaeeceeecaeenaeeeeeeesaees 7 8 7 4 Working with PEST in Regularisation Mode eee eeseesseeeseceeeceeeeeeeeceneceaeeeaeeenees 7 9 TAL Run Time Information cee eiie i a a a a 7 9 7 4 2 Composite Parameter Sensitivities cele eeeesseesseeseeceeeceeeeeeeeceeeeeeaeeeseeeues 7 11 7 4 3 Post Rum Information 0 ee eecescescessccescesseesceeseeesseesseceeceeeseeecesaeesaeesseeeaes 7 11 7 5 Other Considerations Related to Regularisation ecescescccsceceeeceeeeeeeeceneeeneeres 7 12 7 5 1 Using PEST in Two Different Modes ee eeeeesccssccesceesceeesceeseeesseesaeeeseeeaes 7 12 7 5 2 Initial Parameter Values ee ngee atia a e areis 7 13 7 6 Two Examples of Regularisation 20 0 0 ee eeecesscesseessceesceesseeeseeeseceeeeeeeceeeceneeeaeeras 7 13 7 6 1 A Layered Half Space eae aae taer Aaaa a e E E aeiiae REEE r SE EEEE EE 7 13 7 6 2 A Heterogeneous Aquifer eseseeseeeessesesesresresressessessrerresresesresreerresresesseesreses 7 15 8 Model Calculated Derivatives esseseeeeeeseseesesseseersessessesresresstssesstestestessessessresreereereseese 8 1 Bel Generalinei ire i iaa oraaa dis AEAEE a ESE
207. ccecsseccsessseeeceesseeeceseeeeeseeeeesesseeeeeeees 2 32 ZA Bibl OSraphy sosesc eee ea a a oa E a E ee spentestesoergesrarbeteens 2 32 2 4 1 Literature Cited in the Text da e E anaE EEEE i 2 32 2 4 2 Some Further Re din a eie eiae ee eeaeee rie Aona eE e E aeiiae KEE rE T EE TEE 2 33 3 The Mode PEST Interfaces a a ease edo ia T Eaa Esi 3 1 JE PEST Input Biles srrszonenn naii e ck ch bea aie A E E T RE RE O R E eaget 3 1 3 2 Template Files innen n anane iani a a Eie a E a ee ii ian 3 1 3 2 1 Model put Filesi nire pieeo eee ee eE REE anaE EEE EEEn Eei 3 1 322 AN Example e e ae e E e E A Ai aere a aE 3 2 3 2 3 The Parameter Delimiter cccccccccccessesccceeseseeceeeeseeeeeseseeeeeeesseeeesesseeeeeeaaes 3 3 3 24 Parameter NAMES e e ear a o eos 3 4 3 2 5 Setting the Parameter Space Width eee eeeceeeceneeceeeceeceaeceaeeeeeesaeeeaeeses 3 4 3 2 6 How PEST Fills a Parameter Space with a Numbe cece eeeceseeseeeeeeeeeeees 3 6 3 2 7 Multiple Occurrences of the Same Parameter 0 ee eeseesceceeeceneeeeeeeseeeeneeees 3 8 3 2 8 Preparing a Template File eee eeseseeceseceseecececececeaeeeaeecscecsaecsaeenseeeaeeaeeegs 3 9 SB ANStHUCtlONsPILES sss cece scr eebs ere e e Sess sac E es co NAA a oe oe Ube ARS Bose Se POT epee Ss 3 9 3 3 1 Precision in Model Output Files 2 0 00 ceccecceccccsseceesneceeseceeseeeeseeeeseeeeneeeeaaes 3 10 3 3 2 How PEST Reads a Model Output File cece eeceeeceneecececneeeeaeeeeeeeseeeeees 3 10 3 3 3 An
208. cersieesaunatseees 3 14 Prior information article ee eeeeeeseeeeeeeereeneeees 4 24 Prior information equation 0 0 0 cc eeeeeeeeeeseeeeteeees 4 25 Prior information label eeeseeeeeeeeeeeeeeeees 4 25 PSLAVE tensoro 9 3 9 12 13 2 PSTOP 1 11 5 23 9 13 PSTOPSTiscarevd wre AA 1 11 5 23 9 13 PONPA USE rara a a ONEA 1 11 5 23 Reference variance cccccssccesscceeseeeeeseeeeseeesseeeeseee 2 5 Regularisation cece 1 10 2 13 2 26 7 1 7 5 Relative limited parameters 2 22 4 9 5 10 5 29 REPL Ese i Siccvesoveiiedves ovek ict a 11 6 11 10 RELPARMAX cccccceseeees 2 21 4 9 5 10 5 28 5 33 REEPARS SP panyen 2 26 4 12 5 24 RELPREDLAM 6 14 RELPREDSTP 6 15 RELPREDSWH cccccccccssscesececeseceeseeesseeesseeeeenee 6 15 Residuals acessar Residuals file sstict 0 lt vecevetiieles abccstveetotactseacvertes Restarts witch erara e ces tod ist TRA RLAMBDAI1 REAMFAC horsie ertesi Rota ON rsisi orisii asen Roundoff errors RS TREE onc N E A AAEE a tives RUM Ass sivesvezevosssietels Run management file eee cece eeeseeeeeeeeeeeeeeee Run record file Run time errors 0 eeseceeeceesseceeseceeeseeeesseeeeseeees 3 25 5 1 SCALE v riiai 2 20 4 19 5 16 10 2 SCREENDISP a a a a E a as 11 5 Secondary marker 3 16 3 24 Section headers ose sisccssvestacevcsisscoswnciovenserkeosueeuntives 4 3 Semi fixed observations ccccccssceesseees
209. ces it should only be a little greater PDI The procedure by which PEST calculates the location of the critical point in parameter space is a complex one see Section 2 1 9 If PEST is asked to maximise minimise a certain model prediction while constrained to keep the objective function calculated on the basis of Predictive Analysis 6 13 calibration observations only as close as possible to PDO it will in the course of its iterative solution process wander in and out of allowed parameter space ie the area inside the o contour of Figures 6 1 6 2 6 4 and 6 5 Because the shape of the PDO contour can be so complex it is extremely hard for PEST to find a parameter set which lies exactly on this contour The value supplied for PD1 which must be slightly higher than PDO is a value which PEST will consider as being close enough when approaching this contour from the outside When approaching it from the inside any objective function value is good enough because values on the inside of the PDO contour are all less than PDO and hence all calibrate the model Thus PDO is the value of the objective function that PEST must aim for when maximising minimising the model prediction PD1 is the value that it will accept This should normally be about 5 higher than PDO However if you are not undertaking a line search for refinement of the parameter upgrade vector see below or if PEST appears to be having diffic
210. cess running smoothly and efficiently decreases with the number of parameters being estimated In highly parameterised problems some parameters are likely to be relatively insensitive in comparison with other parameters As a result of their insensitivity PEST may decide that large changes are required for their values if they are to make any contribution to reducing the objective function However as is explained in Section 2 2 5 limits are set on parameter changes These limits are enforced in such a way that the magnitude but not the direction of the parameter upgrade vector is reduced if necessary such that no parameter transgresses these limits The necessity for such limits has already been discussed If a parameter is particularly insensitive it may dominate the parameter upgrade vector ie the magnitude of the change calculated by PEST for this parameter may be far greater than that calculated for any other parameter When its change has been relative or factor limited in accordance with the user supplied settings for RELPARMAX or FACPARMAX and the magnitude of the parameter upgrade vector has thus been considerably reduced other parameters including far more sensitive ones may not change much at all with the result that at the end of the optimisation iteration the objective function may have been lowered very little The same process may then be repeated on the next iteration on account of the same or another insensitive parameter
211. ch is cited in the parameter data section of the PEST control file must be properly defined in the parameter groups section of this file SCALE and OF FSET Just before a parameter value is written to a model input file be it for initial determination of the objective function derivatives calculation or parameter upgrade it is multiplied by the real variable SCALE after which the real variable OFFSET is added The use of these two variables allows you to redefine the domain of a parameter Because they operate on the parameter value at the last moment before it is written to the model input file they take no part in the estimation process in fact they can conceal from PEST the true value of a parameter as seen by the model PEST optimising instead the parameter bp where bp bin o s 4 7 Here bp is the parameter optimised by PEST bm is the parameter seen by the model while s and o are the scale and offset for that parameter If you wish to leave a parameter unaffected by scale and offset enter the SCALE as 1 0 and the OFFSET as 0 0 DERCOM Unless using PEST s external derivatives functionality see Chapter 8 this variable should be set to 1 4 2 5 Parameter Data Second Part The second part of the parameter data section of the PEST control file consists of one line for each tied parameter if there are no tied parameters the second part of the parameter data section must be omitted
212. check that the entire PEST input dataset contained in these files is consistent and complete Once PEST has been run and an improved parameter set obtained you may wish to build a new PEST control file using the improved parameter estimates as initial estimates for another run This may occur if you wish to alter some facet of the model add prior information alter a PEST variable or two etc prior to continuing with the optimisation process Program PARREP allows you to replace initial parameter values as recorded on a PEST control file with those recorded in a parameter value file the latter having been written by PEST containing the best parameter values achieved on the previous PEST run Introduction 1 13 If you wish to generate a file containing the sensitivity of each model output for which there is a corresponding field or laboratory measurement with respect to each adjustable parameter use the JACWRIT utility program JACWRIT translates a binary file written by PEST containing this useful information into ASCII format for easy user inspection Note that because PEST input files are simple text files for which full construction details are provided in this manual they can be prepared by other software for example by a text editor or by a program that you may write yourself in order to automate PEST file generation for a specific application Thus if you wish you can integrate PEST into your modelling suite so that model par
213. check whether the system has properly interpreted and executed the command It simply reads the next set of parameters and undertakes the next model run after control has been returned back to it from the operating system after the latter has been provided with the command Care should be taken if SENSAN is executing in the noverbose mode for then the string gt nul is added to any command appearing in the parameter variation file This may cause problems if a command already uses the gt symbol to redirect command output to a file 11 3 SENSCHEK 11 3 1 About SENSCHEK Once all SENSAN input data has been prepared and before running SENSAN SENSCHEK should be run in order to verify that the entire SENSAN input dataset is correct and consistent SENSCHEK reads all SENSAN input files ie the SENSAN control file the parameter variation file as well as all template and instruction files It checks all of these files for correct syntax and for consistency between them Thus for example if the number of observations cited in all instruction files differs from the value supplied for NOBS in the SENSAN control file or if values are provided for parameters in the parameter variation file which are not cited in any template file SENSCHEK will detect the error and write an appropriate error message to the screen Though SENSAN itself carries out some error checking it has not been programmed to carry out extensive consistency chec
214. ched to the i th observation equation 2 3 defining the objective function is modified as follows c Xb Q c Xb 2 8a or to put it another way d wri 2 8b i where r the 7 th residual expresses the difference between the model outcome and the actual field or laboratory measurement for the i th observation If observation weights are correctly assigned it can be shown that equation 2 8a is equivalent to c Xb P c Xb 2 9 where P Q5 Coo 2 10 C c represents the covariance matrix of the m dimensional observation random vector of which our measurement vector is a particular realisation Because Q is a diagonal matrix so too is P its elements being the reciprocals of the corresponding elements of Q The assumption of independence of the observations is maintained through insisting that Q and hence P have diagonal elements only the elements of Q being the squares of the observation weights These weights can now be seen as being inversely proportional to the standard deviations of the field or laboratory measurements to which they pertain Note that the weights as defined by equation 2 8 are actually the square roots of the weights as defined by some other authors However they are defined as such herein because it has been found that users when assigning weights to observations find it easier to think in terms of standard deviations than variances especially when dealing with two or three diffe
215. cing model run 22 16 19 63 slave slave 2 finished execution reading results 22 16 35 95 slave slave 2 commencing model run 22 21 23 48 slave slave 2 finished execution reading results OPTIMISATION ITERATION NO A aee gt Calculating Jacobian matrix running model 5 times 22 21 45 07 slave slave 2 commencing model run 22 21 55 40 slave slave 3 commencing model run 22 22 25 05 slave slave 1 commencing model run 22 27 05 83 slave slave 2 finished execution reading results 22 27 28 20 slave slave 3 finished execution reading results 22 27 39 52 slave slave 2 commencing model run 22 27 52 63 slave slave 3 commencing model run 22 28 05 18 slave slave 1 finished execution reading results 22 33 06 55 slave slave 2 finished execution reading results 22 33 33 03 slave slave 3 finished execution reading results NDHNNDANHDAAnAUnHNA Testing parameter upgrades 22 34 11 46 slave slave 2 commencing model run 22 39 36 82 slave slave 2 finished execution reading results 22 39 48 09 slave slave 2 commencing model run 22 45 40 28 slave slave 2 finished execution reading results 22 45 51 38 slave slave 2 commencing model run 22 50 43 30 slave slave 2 finished execution reading results OPTIMISATION ITERATION NO e Soeae gt Calculating Jacobian matrix running model 10 times 22 50 54 46 slave slave 2 commencing m
216. citing problems as a result of these enhancements and that they will be able to harness the potential for more sophisticated and efficient use of models than ever before John Doherty October 1998 Preface to the Third Edition Production of the third edition of the PEST manual coincides with the release of Version 4 01 of PEST also known as PEST2000 The principal addition to PEST functionality encapsulated in PEST2000 is the provision of predictive analysis capabilities to complement PEST s existing parameter estimation capabilities With the increasing use of nonlinear parameter estimation techniques in model calibration there is a growing realisation among modellers of the extent of nonuniqueness associated with parameter values derived through the model calibration process This realisation is accompanied by a growing desire to examine the effect of parameter nonuniqueness on the uncertainty of predictions made by calibrated models The importance of quantifying predictive uncertainty cannot be understated It can be argued that model parameters are often something of an abstraction sometimes bearing only a passing resemblance to quantities that can be measured or even observed in real world systems However the same is not true of model predictions for these are the reason why the model was built in the first place If model predictions are being relied upon to serve as a basis for sound environmental management as they often are then
217. cking on the Run project button or by selecting Run and then Start from the WinPEST main menu PEST execution can be terminated altogether by clicking on either the Stop with statistics button m or the Stop without statistics button on the WinPEST toolbar alternatively select these same options through the Run and then the Pause or Stop menu items In the first case before it ceases execution PEST writes a plethora of statistical information to its run record file Also if any of the ICOV ICOR or IEIG variables was set to zero so that the covariance matrix correlation coefficient matrix or covariance matrix eigenvalues eigenvectors were not displayed and updated during the course of the PEST run they will now be displayed However if terminated using the Stop without statistics option PEST simply ceases execution dispensing with any statistical calculations and listings Nevertheless selection of the Stop without statistics option is preferable if you intend to re commence PEST execution at a later date with or without troublesome parameters held at their current values through PEST s user intervention functionality Preparing for a PEST Run 9 Whether PEST was stopped using the Stop with statistics or Stop without statistics option it can be re started using either of the Restart current iteration or Restart using last Jacobian options provided the RST
218. cobian matrix had just finished being calculated Thus depending on what PEST was doing when its execution was terminated it may re commence execution either within the same optimisation iteration as that in which it was interrupted or in the previous iteration In either case it moves straight into calculation of the parameter upgrade vector and the testing of different Marquardt lambdas It is through this restart mechanism that user assistance is possible with PEST Upon inspection of the run record file and the parameter sensitivity file a user may decide that PEST can do better in improving the objective function if it attempts the last parameter upgrade again with certain parameters or groups of parameters held fixed Thus the laborious calculation of the Jacobian matrix is not wasted for PEST is able to get a second chance at using this important information in calculating a better parameter set PEST can be stopped and restarted using the j switch as many times as is desired Thus in some over parameterised cases a user can progressively hold more and more parameters fixed until a significant improvement in the objective function is realised Then he she can let PEST move on to the next optimisation iteration 5 6 5 Maximum Parameter Change Running PEST 5 36 As has already been discussed at the end of each optimisation iteration PEST records on its run record file the maximum factor and relative changes undergone by
219. composite sensitivity of parameter i is defined as si S QD An 5 1 where J is the Jacobian matrix and Q is the cofactor matrix in most instances the later will be a diagonal matrix whose elements are comprised of the squared observation weights m in equation 5 1 is the number of observations with non zero weights Thus the composite sensitivity of the i th parameter is the normalised with respect to the number of observations magnitude of the column of the Jacobian matrix pertaining to that parameter with each element of that column multiplied by the weight pertaining to the respective observation Recall that each column of the Jacobian matrix lists the derivatives of all model generated observations with respect to a particular parameter Immediately after it calculates the Jacobian matrix PEST writes composite parameter sensitivities to a parameter sensitivity file called case sen where case is the current case name ie the filename base of the current PEST control file Example 5 3 shows an extract Running PEST 5 17 from a parameter sensitivity file PARAMETER SENSITIVITIES CASE VES4 OPTIMISATION ITERATION NO T gt Parameter_name Group Current value Sensitivity Rel Sensitivity rol ro 4 00000 1 25387 0 754905 ro2 ro 5 00000 0 327518 0 228925 ro3 ro 6 00000 2 09172 1 62768 h1 hhh 5 00000 0 176724 0 123525 h2 hhh 4 00000 4 718210E 02 2 840646
220. contains parameters which PEST must optimise a template can be built for it as if it were any other model input file A model may read many input files however a template is needed only for those input files which contain parameters requiring optimisation PEST does not need to know about any of the other model input files PEST can only write parameters to ASCII ie text input files If a model requires a binary input file you must write a program which translates data written to an ASCII file to binary form The translator program and then the model can be run in sequence by listing them in a batch file which PEST runs as the model The ASCII input file to the translator program will then become a model input file for which a template is required A model input file can be of any length However PEST insists that it be no more than 2000 characters in width The same applies to template files It is suggested that template files be provided with the extension tpl in order to distinguish them from other types of file 3 2 2 An Example A template file receives its name from the fact that it is simply a replica of a model input file except that the space occupied by each parameter in the latter file is replaced by a sequence of characters which identify the space as belonging to that parameter Consider the model input file shown in Example 3 1 this file supplies data to a program which computes the apparent resistivity on the
221. cording to how insensitive they are with respect to other members of the same parameter group This functionality is accessible through the Groups sub window of the Hold Status window This window lists all parameter groups If you wish that any parameter within a certain group be held if its composite sensitivity is below a certain threshold enter the value of this threshold in the first data column of this window beside the name of the pertinent parameter group If you wish that the N most insensitive parameters belonging to any particular group be held enter the value of N in the second column of the worksheet Parameters can also be held according to their ordering in selected eigenvectors of the covariance matrix click on the Eigenvectors tab of the Hold Status window to hold parameters according to this criterion Eigenvalues of the covariance matrix or at least approximations to these based on current parameter values are listed from lowest to highest eigenvalue Use the arrows on the right hand side of the right column to indicate how many parameters in the corresponding eigenvector are to be held As is explained in Section 5 6 of the PEST manual if it is requested that N parameters pertaining to a particular eigenvalue be held this will mostly be the largest eigenvalue then the parameters forming the N largest components of the corresponding eigenvector will be the parameters which are held as a result of this reque
222. d just before parameter upgrade vectors are calculated and tested PEST calculates a suitable regularisation weight factor ie from equation 2 33 by which all of the weights pertaining to the regularisation observations are multiplied prior to calculation of the parameter upgrade vector This regularisation weight factor is calculated in such a way as to try to ensure that the measurement objective function after parameter upgrade is equal to m defined in equation 2 31 ie the limiting measurement objective function below which the model is deemed to be calibrated However as it is calculated on the basis of a linearity assumption based on the Jacobian matrix that is not always a good approximation to reality measurement objective functions equal or nearly equal to m are not normally achieved until late in the parameter estimation process 7 2 2 The Observation Group Regul Regularisation observations are distinguished from measurement observations through being assigned to a special observation group named regul As was discussed in Section 4 2 6 prior information items as well as observations should be assigned to observation groups Thus regularisation observations can be supplied in either the observation data or prior information sections of the PEST control file or both All observations and or prior information equations which do not belong to the group regul are assumed to be part of the m
223. d to adapt your model to PEST By wrapping PEST around your model you can turn it into a non linear parameter estimator or sophisticated data interpretation package for the system which your model simulates The model can be simple or complex home made or bought and written in any programming language As far as I know PEST is unique Because of its versatility and its ability to meet the modeller where he or she is at rather than requiring the modeller to reformulate his her problem to suit the optimisation process I believe that PEST will place the nonlinear parameter estimation method into the hands of a wider range of people than has hitherto been possible and will allow its application to a wider range of problem types than ever before I sincerely hope that this will result in a significant enhancement in the use of computer modelling in understanding processes and interpreting data in many fields of study However you should be aware that nonlinear parameter estimation can be as much of an art as it is a science PEST or any other parameter estimator can only be used to complement your own efforts in understanding a system and inferring its parameters It cannot act as a substitute for Preface discernment it cannot extract more information from a dataset than the inherent information content of that dataset Furthermore PEST will work differently with different models There are many adjustments which you can make to PEST to
224. de vector is determined or if all parameters are simultaneously at their limits and the parameter upgrade vector points out of bounds PEST will take its deliberations no further unless it is currently calculating derivatives using forward differences and the option to use central differences is available to it in which case it will switch to the use of central differences for greater derivatives accuracy before moving on to the next iteration see Section 2 3 2 2 10 Operation in Predictive Analysis Mode Most aspects of PEST s operation when undertaking predictive analysis are identical to its operation when undertaking parameter estimation including the use of parameter bounds relative and factor change limits switching to the use of three point derivatives calculation prior information the linking and fixing of parameters the holding of parameters logarithmic transformation etc All termination criteria that are used in parameter estimation mode also apply to PEST s use in predictive analysis mode However as discussed in Section 6 2 2 a number of extra termination criteria applicable only to this mode of operation are available As is explained in Section 6 1 5 if the initial parameter estimates supplied to PEST at the commencement of a predictive analysis run are a long way from optimum ie the initial objective function is far above o of equation 2 28 PEST will work in parameter estimation mode until it is able to sni
225. dels are to match their corresponding field measurements well 1 5 How to Use PEST The PEST suite is comprised of two versions of PEST and six utility programs for building and checking PEST input files A sensitivity analyser and a parameter preprocessor are also supplied with PEST All of these programs are command line driven programs ie they can be run from a command line window by typing the name of the appropriate executable at the screen prompt Note however that they are all true WINDOWS executables A suite of utility programs is also available to enhance the use of PEST in certain modelling contexts See for example the PEST Ground Water and Surface Water Modelling Utilities 1 5 1 The Two Versions of PEST The two variants of PEST are the single window version of PEST and Parallel PEST In the single window version of PEST which is run through the pest command the model shares the same window as PEST with the result that screen output generated by the model is interspersed with that generated by PEST unless the former is re directed to a nul file see later Parallel PEST which is run through the ppest command is able to run multiple instances of a model in parallel either in different command line windows on the same machine or on different networked machines By undertaking simultaneous model runs enormous savings in overall optimisation time can be made particularly when calibrating larg
226. ditions Prior to running the composite model they must be written to both sets of input files together with other data specific to each component of the composite model 12 2 5 Running PEST Before running PEST run PESTCHEK to check that the entire input dataset is consistent and correct At the screen prompt type pestchek twofitl Then run PEST using the command pest twofitl There are two things to watch as PEST executes The first is the value of the objective function and the second is the value of the prediction Both of these are written to the screen on every occasion that PEST calculates a parameter upgrade vector these are easily seen when running PEST if screen output from the composite model is disabled as discussed earlier The objective function ie phi hovers around 5 0E 3 as it should though values on either side of this are recorded The value of the prediction slowly rises from iteration to iteration Note that information written to the screen during the course of PEST s execution is also recorded in the PEST run record file in this case twofitl rec When PEST ceases execution open file twofit rec and go to the bottom of the file Near the An Example 12 15 bottom of the file it is written that PEST achieved a maximum prediction value of 0 786 for a corresponding objective function value of 5 16E 3 This is a little above our target value of 5 0E 3 but is accepted due to the action of PD1 However due to th
227. ds the name of the parameter that underwent this change This information may be crucial in deciding which if any parameters should be temporarily held at their current values should trouble be encountered in the optimisation process For details of the options for user intervention see Section 5 6 of this manual The recording of the maximum factor and relative parameter changes at the end of each iteration allows you to judge whether you have set these vital variables ie FACPARMAX and RELPARMAX wisely In the present case only the maximum factor change is needed because no parameters are relative limited the maximum relative parameter change is recorded however because one of the termination criteria involves the use of relative parameter changes Note that had some of the parameters in Example 5 1 been relative limited this part of the run record would have been slightly different in that the maximum factor parameter change would have been provided only for factor limited parameters and the maximum relative parameter change would have been provided for relative limited parameters However a further line documenting the maximum relative parameter change for all parameters would have been added because of its pertinence to the aforementioned termination criterion The PEST run record of Example 5 1 shows that in iteration 2 one of the parameters viz h2 incurs the maximum allowed factor change thus limiting the magnitude of the par
228. e character and the echo off command Also model screen output can be redirected to the nul file See Example 11 4 SENSAN 11 7 echo off modell gt nul model2 gt nul Example 11 4 A batch file serving as a model all screen output has been disabled Care should be taken if SENSAN is executing in the noverbose mode for it then appends the string gt nul to the command recorded in the model command line section of its input file If the command already involves output redirection to a file using the gt symbol this may become confounded through use of the further gt symbol supplied by SENSAN to redirect command output to the nul file 11 2 9 Model Input Output TEMPFLE TEMPFLE is the name of a template file used to write a model input file INFLE The name of a model input file corresponding to the template file preceding it in the SENSAN control file INSFLE INSFLE is the name of a PEST instruction file OUTFLE The name of the model output file read by the instruction file whose name precedes it in the SENSAN control file 11 2 10 Issuing a System Command from within SENSAN SENSAN allows a user to issue a system command after each model run A system command is a direction to the operating system and is implemented by the system just as if the command were typed at the screen prompt The command can be an operating system command such as copy or
229. e for example to exploit that machine s dual processors then a separate PSLAVE working directory must be commissioned for each separate PSLAVE running on that machine Once again these directories may also be the working directories for each of the two distinct model runs Parallel PEST 9 7 File Written By Function pslave rdy PSLAVE Informs PEST that it has begun execution also informs it of the command line which it will use to run the model pest rdy PEST Informs PSLAVE that it has begun execution param rdy PEST Informs PSLAVE that it has just generated model input files s on the basis of a certain parameter set and that it must now run the model observ rdy PSLAVE Informs PEST that the model has finished execution and that it must now read the model output file s pslave fin PEST Informs PSLAVE that it must now cease execution D HH AT PEST Used to test whether PEST has access to all PSLAVE working directories Table 9 1 Files used by PEST and PSLAVE to communicate with each other Where Parallel PEST is used to run the model on different machines across a network it is likely that one such slave machine will be the machine that PEST is also running on This is because PEST is not in general a big consumer of system resources much of its role being to manage input and output to and from the model runs being initiated by its various slaves Hence the running of PEST leaves adequate sy
230. e the initial value of phi as read from the PEST run record file PEST writes this value after it has carried out just one model run e Inthe PEST control file the value of NOPTMAX should be set to zero Hence PEST runs the model only once before it terminates execution e There is only one template file and one instruction file e The template file is built from the PEST control file Parameters adjusted by SENSAN SENSAN 11 12 are initial parameter values as listed on the PEST control file e SENSAN s observation file is the run record file for the PEST case e As in normal SENSAN operation supply parameter values to be used by SENSAN through a parameter variation file e Use the single window version of PEST rather than Parallel PEST The instruction set by which the PEST control file is read is shown below the observation name is phi pif S ie phi S phi This instruction set simply instructs SENSAN to read the PEST run record file until it encounters the string ie phi followed by and then to read the observation named phi as a non fixed observation following that 11 7 SENSAN Error Checking and Run Time Problems As has already been discussed SENSAN does not carry out extensive error checking Comprehensive SENSAN input data error checking can be undertaken using SENSCHEK Hence if there are any problems encountered in SENSAN execution or if there are any suspicions regarding the
231. e Jacobian matrix 7 3 Re Starting PEST User Intervention 26 Whenever PEST is started or re started from within the WinPEST environment WinPEST first checks for the presence of a parameter hold file If such a file exists WinPEST asks the user to verify that he she wishes PEST to use the contents of this file in order to modify its behaviour If not WinPEST deletes the parameter hold file before PEST runs Note that when PEST is re started with the j switch for re calculation of the parameter upgrade vector with certain parameters held the objective function and parameter values will probably be different from those calculated on the previous parameter upgrade attempt This will be reflected in a forked appearance of the evolving Phi and Parameter plots Parallel WinPEST 27 8 Parallel WinPEST 8 1 General A special version of WinPEST named PWinPEST can be used for running Parallel PEST All aspects of PWinPEST functionality are identical to those of WinPEST it is started in a similar manner to WinPEST and all run time and post run displays are identical However before actually starting a PEST run by selecting Run then Start from the WinPEST main menu of by clicking on the Run project button the user must have prepared a run management file for the current case he she must also have commenced execution of all of the various slaves involved in the Parallel PEST run See Section
232. e amplitude and phase of the seasonal variation of the crop factor about this mean Thus twelve parameters are replaced with three This will lend stability to the parameter estimation process as it promulgates a more unique solution to it In implementing this strategy PEST will see three parameters while the model will still see PEST Utilities 10 12 the twelve parameters which it requires The task of transforming the three parameters seen by PEST to the twelve parameters seen by the model can be accomplished using PAR2PAR as a model preprocessor run by PEST just before the model on every occasion that the model is run Once again this can be accomplished by including both of the PAR2PAR and model executables in a batch file run by PEST as a composite model On the basis of the three parameters adjusted by PEST named for example mean amplitude and phase PAR2PAR will calculate the monthly crop factor parameters required by the model named for example crop crop2 crop12 using a series of relationships such as cropl mean amplitude sin 1 phase 2 0 3 142 12 0 crop2 mean amplitude sin 2 phase 2 0 3 142 12 0 etc In these equations phase is measured in months as is explained below the argument of the sin function must be supplied in radians where 27 radians is equal to a full cycle Seasonal parameter variation can be expressed in a number of different ways use of the sin
233. e and complex models Preparation for a Parallel PEST run requires the creation of an extra input file also a slave program PSLAVE must be run on each machine on which Parallel PEST runs the model PEST execution can be interrupted or stopped at any time To do this run one of the programs PPAUSE PUNPAUSE PSTOP or PSTOPST to achieve the desired effect 1 5 2 PEST Utilities PEST requires three types of input file These are Introduction 1 12 e template files one for each model input file which PEST must write prior to a model run e instruction files one for each model output file which PEST must read after a model run and e a PEST control file which brings it all together supplying PEST with the names of all template and instruction files together with the model input output files to which they pertain It also provides PEST with the model name parameter initial estimates field or laboratory measurements to which model outcomes must be matched prior parameter information and a number of PEST variables which control the implementation of the Gauss Marquardt Levenberg method You must prepare the template and instruction files yourself This can be easily done using a text editor full details are provided in Chapter 3 of this manual After you have prepared a template file you can use program TEMPCHEK to check that it has no syntax errors Furthermore if you supply TEMPCHEK with a set of parameter values it will wri
234. e downwards on the model output file though it can be instrumental in this see below rather it instructs PEST to move its cursor along the current model output file line until it finds the secondary marker string and to place its cursor on the last character of that string ready for subsequent processing of that line Example 3 10 shows an extract from a model output file while Example 3 11 shows the instructions necessary to read the potassium concentration from this output file A primary marker is used to place the PEST cursor on the line above that on which the calculated concentrations are recorded for the distance in which we are interested Then PEST is directed to advance one line and read the number following the K string in order to find an observation named kc the exclamation marks surrounding kc will be discussed shortly DISTANCE 20 0 CATION CONCENTRATIONS Na 3 49868E 2 Mg 5 987638E 2 K 9 987362E 3 Example 3 10 Extract from a model output file The Model PEST Interface 3 17 pif DISTANCE 20 0 IT aks kel Example 3 11 Extract from an instruction file A useful feature of the secondary marker is illustrated in Examples 3 12 and 3 13 of a model output file extract and a corresponding instruction file extract respectively If a particular secondary marker is preceded only by other markers including perhaps one or a number of secondary markers and certa
235. e increment is made too small accuracy of derivatives calculation will suffer because of the effect of roundoff errors as two possibly large numbers are subtracted to yield a much smaller number In most cases intuition and experience backed up by trial and error will be your best guide in reconciling these conflicting demands on The PEST Algorithm 2 29 increment size There are three PEST input variables viz INCTYP DERINC and DERINCLB by which you can set the manner in which increments are calculated for the members of a particular parameter group INCTYP determines the type of increment to use for which there are three options viz absolute relative and rel_to_max If the increment type for a parameter group is absolute the increment used for all parameters in the group is supplied as the input variable DERINC this increment is added and subtracted for central derivatives calculation directly to a particular group member when calculating derivatives with respect to that parameter However if the increment type is relative DERINC is multiplied by the current absolute value of a parameter in order to determine the increment for that parameter In this way the parameter increment is adjusted upwards and downwards as the parameter itself is adjusted upwards and downwards this may have the effect of maintaining significance in the difference between model outcomes calculated on the basis of the incrementall
236. e matrix J QJ is often referred to as the normal matrix Because equation 2 14 is only approximately correct so too is equation 2 16 in other words the vector b defined by adding the parameter upgrade vector u of equation 2 16 to the current parameter values bo is not guaranteed to be that for which the objective function is at its minimum Hence the new set of parameters contained in b must then be used as a starting point in determining a further parameter upgrade vector and so on until hopefully we arrive at the global minimum This process requires that an initial set of parameters bo be supplied to start off the optimisation process The process of iterative convergence towards the objective function minimum is represented diagrammatically for a two parameter problem in Figure 2 1 It is an unfortunate fact in working with nonlinear problems that a global minimum in the objective function may be difficult to find For some models the task is made no easier by the fact that the objective function may even possess local minima distinct from the global minimum Hence it is always a good idea to supply an initial parameter set bo that you consider to be a good approximation to the true parameter set A suitable choice for the initial parameter set can also reduce the number of iterations necessary to minimise the objective function for large models this can mean considerable savings in computer time Also the inclusion of prior inform
237. e one or a number of linear combinations of the parameters than the individual parameters themselves In such cases some parameter variances parameter variances constitute the diagonal elements of C b may be large even though the objective function is reasonably low If parameter correlation is extreme the matrix X X of equation 2 6 may become singular and parameter estimation becomes impossible There are two matrices both of which are derived from the parameter covariance matrix C b which better demonstrate parameter correlation than C b itself The first is the correlation coefficient matrix whose elements pj are calculated as Ojj p 2 7 4 01 O jj where oj represents the element at the ith row and j th column of C b The diagonal elements of the correlation coefficient matrix are always 1 off diagonal elements range between 1 and 1 The closer are these off diagonal elements to 1 or 1 the more highly are the respective parameters correlated The second useful matrix is comprised of columns containing the normalised eigenvectors of the covariance matrix C b If each eigenvector is dominated by one element individual parameter values may be well resolved by the estimation process However if predominance within each eigenvector is shared between a number of elements especially for those eigenvectors whose eigenvalues are largest the corresponding parameters are highly correlated See Section 5 2 11 for f
238. e parameters belonging to each parameter group The PEST Control File 4 11 Using the variable FORCEN you may wish to decree that for a particular parameter group derivatives will first be calculated using the forward difference method and later when PEST is faltering in its attempts to reduce the objective function calculated using one of the central methods Alternatively you may direct that no such switching take place the forward or central method being used at all times for the parameters belonging to a particular group In the former case you must provide PEST with a means of judging when to make the switch this is the role of the real variable PHIREDSWH If the relative reduction in the objective function between successive optimisation iterations is less than PHIREDSWH PEST will make the switch to three point derivatives calculation for those parameter groups for which the character variable FORCEN has the value switch thus if for the 7 th iteration i 1 Bi Bi 1 lt PHIREDSWH 4 5 where is the objective function calculated on the basis of the upgraded parameter set determined in the i th iteration PEST will use central derivatives in iteration i 1 and all succeeding iterations for all parameter groups for which FORCEN is switch A value of 0 1 is often suitable for PHIREDSWH If it is set too high PEST may make the switch to three point derivatives calculation before it needs to the result will be
239. e provides the means whereby values are assigned to a set of parameters These values can be provided either by the direct assignment of numbers or through mathematical expressions These expressions which may be of considerable complexity may cite parameters whose values were assigned in previous expressions The template and model input files section of the PAR2PAR input file provides the names of template files together with the names of the model input files to which they correspond Once it has determined values for all parameters appearing on the left sides of the expressions listed in the parameter data section of its input file PAR2PAR writes these parameter values to the nominated model input files using template files based on these model input files just like PEST does Note the following e Any parameter appearing in any of the template files listed in the template and model PEST Utilities 10 14 input files section of the PAR2PAR input file must be assigned a value in the parameter data section of the PAR2PAR input file e If there is more than one template model input file pair listed in the template and model input files section of the PAR2PAR input file any particular template file can be cited more than once if desired However each model input file can be cited only once for it would make no sense for a model input file generated on the basis of one template file to be overwritten by another
240. e rather subjective way in which an objective function value is selected at which the model is said to be calibrated this matters little While inspecting the run record file notice how observation ol4 is not listed with other observations in the section of this file which tabulates observed values corresponding model generated values and residuals This is because observation 014 is in fact the prediction PEST recognising it as such because it is the only observation assigned to the observation group predict Figure 12 4 shows a plot of the line segments calculated on the basis of the parameters derived by PEST during the above predictive analysis process The fit is not too bad though obviously not as good as that obtained on the basis of best fit parameters 1 0 Specific volume cu m Mg 0 0 0 2 0 4 0 6 Water content cu m Mg Figure 12 4 Soil clod shrinkage data with line segments superimposed In the present instance the worst case model prediction of 0 786 is not too different from the most likely model prediction of 0 756 This is comforting to know It is a frightening fact that in many instances of environmental modelling the worst case prediction can be hugely different from that calculated using parameters corresponding to the objective function minimum It is under these circumstances that predictive analysis becomes an absolute necessity Frequently Asked Questions 13 1 13
241. e surrounded by single quotes if it contains any blank spaces an apostrophe should be written twice This name is used for identification purposes only it need bear no resemblance to computer names or IP addresses as allocated by the network manager The second item on each of these lines ie SLAVDIR is the name of the PSLAVE working directory as seen by PEST This directory name should terminate with a backslash character if you do not terminate the name with a backslash PEST will add it automatically You can either provide the full path to the PSLAVE working directory or supply it in abbreviated form if this works Thus if having opened a command line window to run PEST you transfer to drive K in Example 9 2 this being a slave machine s disk and change the directory to the PSLAVE working directory and then change back to the local directory again eg by typing C then a path designation of K affixed to the filenames listed in Table 9 1 will Parallel PEST 9 10 ensure that these files are written correctly to the PSLAVE working directory no matter what is its full pathname Note however that the directory K without the dot is not the same directory as K with the dot if directories under the PSLAVE working directory are accessible from the machine on which PEST is run If there are any doubts provide the full PSLAVE path Note also from Example 9 2 that a slave on the local machine can work from the same direct
242. e to adjust then a template file must be prepared for each of the two input files containing adjustable parameters To construct a template file simply start with a model input file and replace each space occupied by a parameter by a set of characters that both identify the parameter and define its width on the input file Then whenever PEST runs the model it copies the template to the model input file replacing each parameter space with a parameter value as it does so PEST template files can be constructed from model input files using any text editor They can be checked for syntactical correctness and consistency using the utility programs PESTCHEK and TEMPCHEK An important point to note about template files is that a given parameter identified by a unique name of up to twelve characters in length can be referenced once or many times The fact that it can be referenced many times may be very useful when working with large numerical models For example a finite difference model may be used to calculate the electromagnetic fields within a half space the half space being subdivided into a number of zones of constant electrical conductivity The model may need to be supplied with a large two or even three dimensional array in which conductivity values are disposed in a manner analogous to their disposition in the half space Each half space zone is defined by that part of the array containing elements of a particular value this value providing t
243. e usual fashion on each slave machine before issuing either of these commands Note however that if PSLAVE is already running on each of these machines it does not have to be restarted This is because an already executing PSLAVE can detect the commencement of a new PEST run If PEST is restarted without the r or j switch it will commence the optimisation process from the very beginning Once again if PSLAVE is already running on each of the slave machines having been initially started for the sake of a previous Parallel PEST run it need not be restarted on any of these machines Such a re commencement of PEST execution from scratch will sometimes be warranted after PEST terminates execution with an error message or if the user terminates PEST execution with the Immediate stop option in neither case does PEST signal to its slaves to cease execution just in case the user wishes to restart PEST immediately after rectifying an error in the PEST control file or in a template or instruction file Note however that if changes are made to the run management file each of the slaves should be stopped and then re started If PEST is restarted with the r or j switch neither the PEST control file nor any template or instruction file should have been altered from that supplied to PEST on its original run It is important to note however that the same does not apply to the run management file Thus Parallel PEST
244. eafter When the program reads any of these numbers it is unconcerned as to what characters lie outside of the field on which its attention is currently focussed However the numbers to be read into variables D and E must be separated by whitespace or a comma in order that the program knows where one number ends and the next number begins Suppose all of variables A to E are model parameters and that PEST has been assigned the task of optimising them For convenience we provide the same names for these parameters as are used by the model code this of course will not normally be the case The template fragment corresponding to Example 3 4 may then be as set out in Example 3 5 Notice how the parameter space for each of parameters A B and C is 10 characters wide and that the parameter spaces abut each other in accordance with the expectations of the model as defined through the format specifier of Example 3 3 If the parameter space for any of these parameters were greater than 10 characters in width then PEST when it replaces each parameter space by the current parameter value would construct a model input file which would be incorrectly read by the model You could have designed parameter spaces less than 10 characters wide if you wished as long as you placed enough whitespace between each parameter space in order that the number which will replace each such space when PEST writes the model input file falls within the field expected by the model
245. easurement dataset The weights used by these measurement observations and or prior information equations are not varied during the parameter estimation process The user must also supply PEST with a value for the limiting measurement objective function Pm In some cases PEST will be run in regularisation mode only after it has been used on the same problem in parameter estimation mode If this is the case the user will know the lowest value for m that can be achieved without any regularisation constraints imposed and will set Pm a little higher than this Alternatively if no preceding parameter estimation run has been carried out m can be set at a level that is judged to be appropriate on the basis of assumed measurement standard deviations If PEST cannot lower the measurement objective function as low as m_ then it will simply lower it as far as it can and will normally calculate a low regularisation weight factor in order to achieve this While PEST s operation in regularisation mode is similar in many respects to its operation in parameter estimation mode there are some important differences In both modes of operation PEST attempts to lower an objective function however in regularisation mode the total objective function cannot be compared from iteration to iteration for the composition of the objective function changes with the regularisation weight factor u depicted in equation 2 33 However each of the separate components of the
246. ecision by the calibration process and hence that predictive nonuniqueness resulting from parameter nonuniqueness will be slight because of the absence of the latter Monte Carlo analysis is often used to examine uncertainty in model predictions Parameter sets can be generated at random for each such parameter set the model is run under calibration conditions If the resulting objective function is above Pmin the parameter set is rejected If it is below min 6 the model is then run under predictive conditions After many thousands of model runs have been undertaken a suite of predictions will have been built up all generated by parameter sets which satisfy calibration constraints In many cases some kind of probability distribution can then be attached to these predictions based on where corresponding calibration objective functions lie with respect to min and the Pmin contour This method of predictive analysis has many attractions however its main disadvantage is in the number of model runs required Where there are any more than a handful of adjustable parameters the dimensionality of the problem requires that millions of model runs be undertaken rendering the method intractable in many practical settings 6 1 3 The Critical Point The effect of parameter nonuniqueness on predictive nonuniqueness depends on the prediction being considered In many instances a model is calibrated under a very different stress regime from
247. ectory Table 12 1 shows the results of an experiment in which the specific volume of a soil clod the reciprocal of its bulk density is measured at a number of water contents as the clod is desiccated through oven heating The data are plotted in Figure 12 1 see also file soilvol dat We wish to fit two straight lines to this data In soil physics parlance the straight line segment of low slope fitted through the points of low water content is referred to as the residual shrinkage segment whereas the segment covering the remainder of the data with a slope near unity is referred to as the normal shrinkage segment Actually another segment of low slope is often present at high moisture contents this being the structural shrinkage segment this segment is not apparent in the data plotted in Figure 12 1 Specific volume cu m Mg 0 0 0 2 0 4 0 6 Water content cu m Mg Figure 12 1 Soil clod shrinkage data An Example 12 2 water content m Mg specific volume m Mg 0 052 0 501 0 068 0 521 0 103 0 520 0 128 0 531 0 172 0 534 0 195 0 548 0 230 0 601 0 275 0 626 0 315 0 684 0 332 0 696 0 350 0 706 0 423 0 783 0 488 0 832 Table 12 1 Soil clod shrinkage data 12 1 2 The Model Before we can use PEST there must be a model We will list the model program in a moment first we present the model algorithm Figure 12 2 shows two intersecting line segments Let the slope of the first s
248. ecution with a run time error message A run time error message will also occur if PEST encounters the end of a line while looking for the beginning of a non fixed observation Consider the output file fragment shown in Example 3 15 The species populations at different times cannot be read as either fixed or semi fixed observations because the numbers representing these populations cannot be guaranteed to fall within a certain range of column numbers on the model output file because iterative adjustment may be required in the calculation of any such population Hence we must find our way to the number using another method one such method is illustrated in Example 3 16 SPECIES POPULATION AFTER 1 YEAR 1 23498E5 SPECIES POPULATION AFTER 2 YEARS 1 58374E5 SPECIES POPULATION AFTER 3 YEARS ITERATIVE ADJUSTMENT REQUIRED 1 78434E5 SPECIES POPULATION AFTER 4 YEARS 2 34563E5 Example 3 15 Extract from a model output file The Model PEST Interface 3 23 pa fo SPECIES spl LI x lV sp2 11 sp3 11 sp4 Example 3 16 Extract from an instruction file A primary marker is used to move the PEST cursor to the first of the lines shown in Example 3 15 Then noting that the number representing the species population always follows a character the character is used as a secondary marker After it processes a secondary marker the PEST cursor always reside
249. ed regul See Chapter 7 for further details Note that if PESTMODE is supplied as estimation there is no need to include a predictive analysis section or a regularisation section in the PEST control file NPAR This is the total number of parameters used for the current PEST case including adjustable fixed and tied parameters NPAR must be supplied as an integer NOBS This integer variable represents the total number of observations used in the current case Note that when counting the number of observations to evaluate NOBS any dummy observations see Chapter 3 that may be referenced in one or a number of instruction files are ignored NPARGP This is the number of parameter groups parameter groups are discussed in detail below NPARGFP is an integer variable NPRIOR NPRIOR another integer variable is the number of articles of prior information that you wish to include in the parameter estimation process If there are no articles of prior information NPRIOR must be zero In general you should ensure that the number of adjustable parameters is less than or equal to the number of observations for which there are non zero weights plus the number of articles of prior information for which there are non zero weights If this is not the case the PEST normal matrix of equation 2 16 will not be positive definite in fact it will be singular and a unique solution to the parameter estimation problem will
250. ed The user is advised to test the program thoroughly before relying on it The user assumes the entire risk of using the program Any liability of seller or manufacturer will be limited exclusively to replacement of diskettes defective in materials or workmanship Waterloo Hydrogeologic Inc 460 Phillip Street Suite 101 Waterloo Ontario Canada N2L 5J2 Phone 519 746 1798 Fax 519 885 5262 E mail techsupport flowpath com Web www waterloohydrogeologic com Preface Visual PEST ASP is a trademark owned by Waterloo Hydrogeologic Inc WinPEST is a trademark owned by Waterloo Hydrogeologic Inc PEST is a trademark of Watermark Numerical Computing Visual PEST ASP User s Manual Copyright 2002 Waterloo Hydrogeologic Inc All Rights Reserved No part of this document may be photocopied reproduced or translated by any means without the prior written consent of Waterloo Hydrogeologic Inc Visual PEST ASP 2002 Waterloo Hydrogeologic Inc All Rights Reserved Preface Introduction 1 Table of Contents PTO CU CHONG sian e a decededagussiisasl cade aa EE a aeaa iiai 2 1 Installa onient eiere doa sas EER e eE spaneatscd ove tesa EEREN oke IP EPEE EEEE 2 T2 Other Software see citable r E E a a E a a Aa E Ea aE ASA 2 1 3 Upgrading PEST and its Utility Software cccsceecceeceeseeeneceeeceeeeeesneaeeeeeeeeeees 3 Di QV ELVIS Wo sine Ecce Sass aa aR Ze ea Shas gS A as ee a deed Teun haces Cot a iia 4 VUEI Pal
251. ed observation consists of two parts The first part consists of the observation name enclosed in square brackets while the second part consists of the first and last columns from which to read the observation Note that no space must separate these two parts of the observation instruction PEST always construes a space in an instruction file as marking the end of one instruction item and the beginning of another unless the space lies between marker delimiters pif PERIOD NO 3 SOLUTION VECTOR 11 obs1 1 9 obs2 10 18 o0bs3 19 27 obs4 28 36 obs5 37 45 amp obs6 46 54 Example 3 14 Extract from an instruction file Reading numbers as fixed observations is useful when the model writes its output in tabular form using fixed field width specifiers However you must be very careful when specifying the column numbers from which to read the number The space defined by these column The Model PEST Interface 3 20 numbers must be wide enough to accommodate the maximum length that the number will occupy in the course of the many model runs that will be required for PEST to optimise the model s parameter set if it is not wide enough PEST may read only a truncated part of the number or omit a negative sign preceding the number However the space must not be so wide that it includes part of another number in this case a run time error will occur and PEST will terminate execution with an appropriate error message Wh
252. ed without subdirectory name in the PEST control file and if each set of model input and output files lies within the PSLAVE working directory on each slave machine then a value of 0 can be supplied for IFLETYP and the model input and output filenames omitted from the run management file Example 9 3 shows such a file In this case Parallel PEST 9 9 PEST automatically affixes the working directory of each slave to the front of each model input and output file as named in the PEST control file The third item on the second line of the PEST run management file is the value for the variable WAIT As PEST and PSLAVE exchange information with each other and as PEST writes and reads model input and output files both PEST and PSLAVE pause at certain strategic places in the communications process for an amount of time specified as the value of the variable WAIT such pauses allow one machine to catch up in implementing the instructions supplied by another machine WAIT is the duration of each such pause expressed in seconds its value must be greater than zero Experience has demonstrated that if PEST and all of its slaves are running on the same machine a value of 0 2 seconds is normally adequate However for communications across a busy network values as high as 10 0 seconds or even higher may be appropriate By pausing at appropriate moments in the data exchange process sharing violation errors can be avoided In some cases such errors
253. ed with the width of the parameter domain However should this eventuality arise later in the course of the estimation process as may happen if certain parameter values grow large and increments calculated from them as relative or rel_to_max then exceed the parameter domain width divided by 3 2 PEST will automatically adjust the increment so that parameter limits are not transgressed as the increment is added and or subtracted from the current parameter value for the calculation of derivatives When choosing an increment for a parameter care must be taken to ensure that the parameter can be written to the model input file with sufficient precision to distinguish an incremented parameter value from one which has not been incremented Thus for example if a model input file template is such that a particular parameter value must be written to a space which is four characters wide and if the increment type for that parameter is absolute and the increment value is 0 0001 while the current parameter value is 01 it will not be possible to discriminate between the parameter with and without its increment added To rectify this situation you should either increase the parameter field width in the template file making sure that the model can still read the parameter or increase the increment to a value whereby the incremented parameter is distinguishable from the non incremented parameter in a field width of only four characters It sho
254. effects that different control settings have on PEST s performance Chapter 5 tells you how to run PEST it also discusses problems that may arise as PEST executes and how best to overcome them Chapter 6 discusses predictive analysis while Chapter 7 discusses regularisation Both or these aspects of parameterisation functionality are unique to PEST at the time of writing and help to make PEST so universally useful in calibrating models that simulate real world systems Chapter 8 discusses an advanced aspect of PEST s performance viz its ability to use derivatives calculated by the model instead of calculating them itself through the process of finite parameter differences In most cases of PEST usage the modeller need not be too familiar with the contents of this chapter for it is only in special circumstances that PEST is able to take advantage of the fact that the model is able to supply it with externally calculated derivatives Parallel PEST is described in Chapter 9 while Chapter 10 details the PEST utilities TEMPCHEK INSCHEK PESTCHEK PESTGEN PARREP JACWRIT and PAR2PAR Chapter 11 discusses the sensitivity analyser SENSAN together with its utility program SENSCHEK Chapter 12 presents an example of the use of PEST in solving a practical data interpretation problem Chapter 13 answers some frequently asked questions The PEST Algorithm 2 1 2 The PEST Algorithm This chapter discusses the mathematical foundations
255. egment be s and that of the second segment be s2 Let the intercept of the first segment on the y axis be y and the x coordinate of the point of intersection of the two line segments be xc The equation for the two line system is y s x y X lt Xe Y S2 X S1 S2 Xc y1 X gt Xe 12 1 where x is the water content and y represents the soil clod specific volume A simple FORTRAN program can be written based on this concept a listing is provided in Example 12 1 see also file twoline for in the pestex subdirectory Program TWOLINE begins by reading an input file named in dat which supplies it with values for s1 s2 y and xc as well as the water contents ie x values in equation 12 1 at which soil clod specific volumes are required TWOLINE writes a single output file named out dat listing both water contents and the specific volumes calculated for these water contents Example 12 2 shows a typical TWOLINE input file while Example 12 3 shows a corresponding TWOLINE An Example 12 3 output file An executable version of TWOLINE viz twoline exe is provided in the pestex subdirectory of the PEST directory Figure 12 2 Parameters of the two line model An Example 12 4 program twoline integer 4 i nx real 4 s1 s2 yl xc real 4 x 50 y 50 open unit 20 file in dat ol read the line parameters read 20 sl1 s2 read 20 yl read 20 xc read the abscissae at which there are measurement va
256. eight to those used in the previous parameter estimation run in which min was determined The single observation belonging to the predict group which obviously does not figure in the previous PEST calibration run can be assigned any weight at all this weight is ignored by PEST when run in predictive analysis mode When run in predictive analysis mode more model runs are normally required to achieve solution convergence than are required when PEST is run in parameter estimation mode because it is usually a more difficult matter to find the critical point than it is to find the objective function minimum Except for the value of one control variable the same control file can be used by PEST when run in both parameter estimation and predictive analysis modes provided the control file has a predictive analysis section see below Furthermore because of the functionality attached to the observation group predict it is a particularly simple matter to run PEST in predictive analysis mode on a particular problem and then switch to running PEST in parameter estimation mode on the same problem in order to implement a dual calibration exercise PEST screen output is slightly different when run in predictive analysis mode from its screen output when run in parameter estimation mode in that the value calculated for the prediction is written to the screen on every parameter upgrade The user should bear in mind when monitoring PEST per
257. el PEST Interface 3 11 selected observations can usually be linked to it in a simple manner For example if you are looking for the outcome of the above stress model s deliberations at an elapsed time of 100 milliseconds you may instruct PEST to read its output file looking for the marker STRESS CALCULATED AT FINITE ELEMENT NODES ELAPSED TIME 100 MSEC A particular outcome for which you have a corresponding experimental measurement may then be found for example between character positions 23 and 30 on the 4th line following the above marker or as the 5th item on the 3rd line after the marker etc Note that for simple models especially home made single purpose models where little development time has been invested in highly descriptive output files no markers may be necessary the default initial marker being the top of the file Markers can be of either primary or secondary type PEST uses a primary marker as it scans the model output file line by line looking for a reference point for subsequent observation identification or further scanning A secondary marker is used for a reference point as a single line is examined from left to right 3 3 3 An Example Instruction File Example 3 6 shows an output file written by the model whose input file appears in Example 3 1 Suppose that we wish to estimate the parameters appearing in the template file of Example 3 2 ie the resistivities
258. el output file unless it is a continuation character In the case of the primary marker PEST stops reading new lines when it finds the pertinent text string However for a line advance it does not need to examine model output file lines as it advances It simply moves forward n lines placing its processing cursor just before the beginning of this new line this point becoming the new reference point for further processing of the model output file Normally a line advance item is followed by other instructions However if the line advance item is the only item on an instruction line this does not break any syntax rules In Example 3 6 model calculated apparent resistivities are written on subsequent lines Hence before reading each observation PEST is instructed to move to the beginning of a new line using the 11 line advance item see Example 3 7 If a line advance item leads the first instruction line of a PEST instruction file the reference point for line advance is taken as a dummy line just above the first line of the model output file Thus if the first instruction line begins with 11 processing of the model output file begins on its first line similarly if the first instruction line begins with 18 processing of the model output file begins at its eighth line Secondary Marker A secondary marker is a marker which does not occupy the first position of a PEST instruction line Hence it does not direct PEST to mov
259. el output file from a slave machine 9 3 7 The Parallel PEST Run Management Record File Section 5 2 discusses in detail the PEST run record file which records the progress of the parameter estimation process Parallel PEST produces an identical run record file to that of the normal PEST It also writes a run management record file Like the run record file the run management record file possesses a filename base identical to that of the PEST control file However PEST provides this file with an extension of rmr for Run Management Record Example 9 4 shows part of a Parallel PEST run management record file PEST RUN MANAGEMENT RECORD FILE CASE VES2 SLAVE DETAILS Slave Name PSLAVE Working Directory slave 1 k model slave 2 1 model slave 3 m model Attempting to communicate with slaves slave Slave 2 has been detected slave Slave 3 has been detected slave Slave 1 has been detected SLAVE MODEL INPUT AND OUTPUT FILES Slave slave 1 gt Model input files on slave slave 1 k model ves inl k model ves in2 Model output files on slave Slave 1 k model ves ot1l k model ves ot2 k model ves ot3 Model command line for slave slave 1 ves Slave slave 2T earme gt Model input files on slave Slave 2 L model ves inl L model ves in2 Model output files on slave Slave 2 1 model ves ot1 1 model ves ot2 L model ves ot3 Model
260. ely to be intermediate files which are generated by one model to be read by another If any part of a composite model fails to execute the ensuing part of the composite model must be prevented from executing on the basis of intermediate files generated during previous model runs This can be ensured by deleting all such intermediate files The first line of the batch file model bat prevents the operating system from echoing batch file commands to the screen This relieves screen clutter when the model is run under the control of PEST in the same window as PEST To assist in this process screen output from all commands is directed to the nul file instead of the screen To find out more about batch files see the DOS help file on your Windows CD To satisfy yourself that the composite model runs correctly type model at the screen prompt Inspect the model output files out dat and out2 dat You may wish to delete the echo off line and remove gt nul from each command line before you run the model in order to see the appropriate model commands scroll past on the screen as they are executed An Example 12 13 12 2 3 The PEST Control File We would now like to obtain the maximum possible value for the soil specific volume at a water content of 0 4 compatible with the model being calibrated against our laboratory dataset Let us do this by assuming that the model can still be considered to be calibrated if the objective function u
261. ementary information provided for that parameter through the regularisation process A problem that arises when using such supplementary information as part of a traditional parameter estimation exercise is the determination of how much notice should be taken of this information in comparison to the notice taken of the observation dataset against which the model is being calibrated If the supplementary information is given too much weight in the parameter estimation process the observation dataset may be ignored On the other hand if it is given too little weight the stabilization potential of the supplementary dataset will not be realized When using PEST in regularisation mode PEST takes care of this problem for PEST calculates the relative weighting given to the two sets of information itself In this way the supplementary information is used only to the extent necessary to ensure stability of the parameter estimation process Or looked at another way if the information content of the calibration dataset is insufficient to provide unique estimation of certain parameters then PEST will automatically elevate the status of the supplementary information such that this provides the grounds for unique estimation of those parameters PEST s regularisation functionality is useful in many types of modelling particularly where many different parameters must be estimated for complex systems It is particularly useful in estimating values for p
262. ens tei eirias 5 13 5 2 8 Analysis Of Residuals sce seescesssessseessecseeceeeceececececeaeceaeecseecsaecaeeeaeeeseeeseees 5 13 5 2 9 The Parameter Covariance Matrix ceceecssecescceseeeseeeseeesceeseeesaeesaeeseeeeeees 5 14 5 2 10 The Correlation Coefficient Matrix ces eeeeeseeesseeescceseesneeeseeesaeesaeeeseeeues 5 14 5 2 11 The Normalised Eigenvector Matrix and the Eigenvalues e eee 5 15 5 3 Other PEST Output Files cccccccsssescseseccsensccsenceesssccunecebatecseedecsansdensecceedeesaees 5 15 5 3 1 The Parameter Value File mirer eenen eia ie eE E a E 5 15 5 3 2 The Parameter Sensitivity File cccccecccccsscceesseceenecceseeeeeeecesaeeeeneeeeeseeesas 5 16 5 3 3 Observation Sensitivity File eeesseeeeeeeseseereesesssrsrrsrerrersessessresrerressessessresres 5 19 5 3 4 The Residuals File 0 0 eeeeeccssccssecesceceseeeseecscecaccsscessceesaeesaeesseeeseeesaeeeseesses 5 20 D320 N VEA ET ED E EEEE EE E sees tietbvaaeon nets 5 21 NNO Other Filesi sre ee a a aai a cae a S T 5 21 9 3 7 PES T s Screen Output noi i e a e EE E i 5 22 5 3 8 R n tim EMrOrs vssscccsessassiccsssccuevsasavccccdadeeavnasnaceccesecesustaaseccceedeassetanacnaceduenasaaane 5 22 5 4 Stopping and Restarting PEST c esscsesesssecssenecoetonetsnecsccenenenenecsenenesenseeeees 5 23 5 4 1 Interrupting PEST Execution ceeessesseeescesseeceeceseeesaeesaceeeeesaeesaeeeaeeeees 5 23 5 4 2 Restarting PEST with the
263. ent values while it calculates the parameter upgrade vector As is discussed in Section 5 5 eigenvalues and eigenvectors are available at all stages of the parameter estimation process through the matrix file recorded by PEST during every optimisation iteration While the ability to hold parameters according to their component magnitudes in various eigenvectors can be of use in difficult cases care should be taken in using this functionality Where excessive parameter correlation is causing problems in the inversion process it is often sufficient for only 1 2 or a very few of the correlated parameters to be held rather than all of them when calculating an upgraded parameter set The parameters to which the held parameters were formerly correlated are then free to move as PEST searches for the objective function minimum using the reduced parameter set This may result in a greater objective function improvement than if all of the correlated parameters were held Unless all of the held parameters have values close to optimal with optimal in this case covering a broad range of values by virtue of their correlated state there will be little benefit in holding all of them for at least some of them will need to be assigned different values if the objective function is Running PEST 5 35 to be minimised 5 6 4 Re calculating the Parameter Upgrade Vector In normal PEST operation the user will probably not be aware that his her interventio
264. equation may also yield beneficial results 5 5 6 Discontinuous Problems The equations derived in Chapter 2 upon which the Gauss Marquardt Levenberg algorithm is based are predicated on the assumption that observations are continuously differentiable functions of parameters If this assumption is violated for a particular model PEST will have extreme difficulty in estimating parameters for that model However it may have some success if the dependence is continuous if not continuously differentiable 5 5 7 Parameter Change Limits Set Too Large or Too Small As discussed above with respect to highly nonlinear problems a suitable choice for relative Running PEST 5 29 and factor parameter change limits ie RELPARMAX and FACPARMAX may allow optimisation to be carried out under hostile circumstances However if these change limits are set too low minimisation of the objective function may be hampered as the upgrade vector is continually shortened in order to conform to the demands of these limits An inspection of the run record file should reveal immediately whether parameter upgrades are being limited by these variables If the maximum relative and or factor parameter changes per optimisation iteration are consistently equal to the respective user supplied limits then it is possible that these limits could be increased however if your model is highly nonlinear or messy it may be better to keep RELPARMAX and FACPARMAX low as the
265. equired only if IFLETYP is nonzero the following group of lines is to be repeated once for each slave INFLE 1 INFLE 2 to NTPFLE lines where NTPFLE is the number of template files O O UTFLE 1 UTFLE 2 to NINSFLE lines where NINSFLE is the number of instruction files Example 9 1 Structure of the Parallel PEST run management file prf S 25 0 my machine steve s machine k jerome s machine 1 model 600 600 720 model inl model in2 model ol model o2 model o3 model inl model in2 model ol model o2 model o3 model model inl model model in2 model model ol model model o2 model model o3 PREP EP AA Example 9 2 A typical Parallel PEST run management file The first line of a run management file should contain only the character string prf identifying the file as a PEST run management file The next line must contain four items the first of which is the number of slaves NSLAVE involved in the current Parallel PEST run The second item on this line is the value of the variable IFLETYP which must be either 1 or O If it is 1 then all model input and output files on the various slave machines must be individually named as is demonstrated in Example 9 2 However if the names of all corresponding input and output files are the same across all slaves being identical to the names provid
266. er is ignored this being interpreted as a comment line If any lines are in error they are also ignored for PEST does not pause in its execution or clutter up either its screen display or its run record file with error messages pertaining to the parameter hold file However it does report any alterations that it makes to its behaviour on the basis of directives obtained from the parameter hold file to its run record file A user is permitted to alter the values of three PEST control variables using the parameter hold file These are RELPARMAX FACPARMAX and LAMBDA The syntax is shown in Example 5 6 ie the name of the variable must be followed by its new value It is important to note that if a parameter hold file is left lying around any lines altering the value of lambda should be removed or commented out or PEST will be prevented from making its normal adjustment to lambda from iteration to iteration This may severely hamper the optimisation process Note that once RELPARMAX and FACPARMAX have been altered using a parameter hold file they stay altered even if the file is removed or the lines pertaining to RELPARMAX and FACPARMAX are subsequently deleted or commented out To hold a parameter at its current value while the parameter upgrade vector is being calculated use a line such as the fourth appearing in Example 5 6 ie the string hold parameter followed by the parameter s name If the parameter name is incorrect P
267. er by selecting View then Hold parameters from the PEST main menu or by clicking on the Hold button R from the WinPEST toolbar Parameters Groups Eigenvectors Controls Composite Sensitivity Adjustable Held by User ca Held by Threshold En m Held by Sensitivity F Held by Eigenvector iz E Ez Se o E oO s 0 rol ro2 rog roS ro ro10 Parameter Parameter 3 45 Composite sensitivity 0 278 Figure 6 The WinPEST Hold Status window The Hold Status window has four sub windows all are accessible by clicking on their respective tabs The Parameters sub window contains a bar chart of parameter composite sensitivities these are equivalent to the natural sensitivities available through the WinPEST Sensitivity window As is explained in Section 5 6 of the PEST manual insensitive parameters are often the most troublesome parameters To hold a parameter click on the User Intervention 25 pertinent bar with the right mouse button and select Select on off from the popup menu or simply click on the bar with the left mouse button The parameter can be unheld by repeating this procedure Once a parameter is held by individual selection in this manner it is coloured blue as is indicated by the legend that accompanies this window As is explained in the PEST manual parameters can also be held according to a sensitivity threshold or ac
268. er derivatives calculation with respect to that parameter Also as has already been discussed even though PEST adjusts its internal representation of a parameter value to the precision with which the model can read it so that PEST and the model are using the same number in general more precision is better Two user supplied control variables PRECIS and DPOINT affect the manner in which PEST writes a parameter value to a parameter space Both of these variables are provided to PEST through the PEST control file see Section 4 2 2 for details PRECIS is a character variable which must be supplied as either single or double It determines whether single or double precision protocol is to be observed in writing parameter values unless a parameter space is greater than 13 characters in width it has no bearing on the precision with which a parameter value is written to a model input file as this is determined by the width of the parameter space If PRECIS is set to single exponents are represented by the letter e also if a parameter space is greater than 13 characters in width only the last 13 spaces are used in writing the number representing the parameter value any remaining characters within the parameter space being left blank For the double alternative up to 23 characters can be used to represent a number and the letter d is used to represent exponents also extremely large The Model PEST Interface 3 7 and
269. ere a model writes its results in the form of an array of numbers it is not an uncommon occurrence for these numbers to abut each other Consider for example the following FORTRAN code fragment A 1236 567 B 8495 0 C 900 0 WRITE 10 20 A B C 20 FORMAT 3 F8 3 The result will be 1236 5678495 000 900 000 In this case there is no choice but to read these numbers as fixed observations Both of the alternative ways to read an observation require that the observation be surrounded by either whitespace or a string that is invariant from model run to model run and can thus be used as a marker Hence to read the above three numbers as observations A B and C the following instruction line may be used 11 A 1 8 B 9 16 C 17 24 If an instruction line contains only fixed observations there is no need for it to contain any whitespace or tabs nor will there be any need for a secondary marker unless the secondary marker is being used in conjunction with a primary marker in determining which model output file line the PEST cursor should settle on see above This is because these items are normally used for navigating through a model output file line prior to reading a non fixed observation see below such navigation is not required to locate a fixed observation as its location on a model output file line is defined without ambiguity by the column numbers included within the fixed observation instruction Semi Fixed Observations
270. erences Even greater complexity can arise For example a model may be able to calculate derivatives with respect to a certain parameter for some observations but not for others In this case derivatives with respect to the pertinent parameter are actually obtained twice by PEST First PEST undertakes model runs in the usual manner to calculate derivatives for that parameter using finite differences Then after all necessary finite difference model runs have been undertaken for the purposes of finite difference derivatives calculation for those parameters which need it PEST completes the Jacobian calculation process by running the derivatives model to calculate external derivatives As is discussed above for those observations where derivatives can be calculated by the model such derivatives are usually more accurate than those calculated by PEST using finite differences and hence should be used in preference to those calculated by PEST However in reading the derivatives file it must be ensured that previously calculated finite difference derivatives are not overwritten by those elements of the external derivatives matrix that could not be calculated by the model To prevent this from happening the respective elements of the external derivatives file should be assigned a value of 1 11e33 by the model Wherever PEST encounters such a value it does not use it Rather it Model Calculated Derivatives 8 4 uses the derivative value that already
271. erparts to these observations are calculated from the parameter values constituting the vector b are encapsulated in the matrix Z Note that if these relationships are not linear a linear approximation to them can be calculated by taking derivatives with respect to parameters in the same way that the Jacobian matrix is calculated as a linear approximation to the model function As an example each row of the matrix Z may be comprised of 0 s except for two elements which are and 1 the 1 and 1 elements occur at those of its columns which pertain to two parameters whose difference is taken The observed value of this difference ie the element of the corresponding row of the vector d might be zero indicating that PEST should minimise this parameter difference insofar as this is possible while still allowing the model to fit the field data If there are as many rows in the matrix Z as there are parameters in the model domain and if each such row represents a single parameter difference and if the observed difference is zero in each case then imposition of the regularisation criterion is the imposition of a maximum homogeneity condition There may in fact be more differences represented in the matrix Z than there are parameters in the model domain This may occur if for example differences are formed in both the horizontal and vertical directions or row and column directions of a particular model domain to ensure max
272. ers rather than the parameters themselves For such parameters respective elements of the Jacobian matrix used by PEST contain derivatives with respect to the logs of these parameters rather than to the parameters themselves PEST calculates derivatives with respect to parameter logs internally from derivatives with respect to native parameters When writing the external derivatives file the model need not concern itself with whether a parameter is log transformed by PEST or not The model must simply supply derivatives with respect to untransformed parameters and let PEST take care of the calculations required to convert these derivatives to derivatives with respect to parameter logs Similarly if the SCALE and OFFSET values for a particular parameter differ from 1 and 0 respectively the model need not concern itself with this PEST modifies the derivatives cited in the external derivatives file to take account of this 8 2 5 Use of Derivatives Information A complex model often consists of many different parameter types It may be possible to compute derivatives with respect to some of these parameters inside the model yet it may be necessary to compute derivatives with respect to others using PEST s traditional method of finite differences As is discussed below the user is able to indicate to PEST the parameters for which derivatives information is supplied externally and those for which derivatives must be computed by PEST using finite diff
273. ertainties with weights multiplied by the optimised regularisation weight factor If you would like to calculate parameter uncertainties with this observation group removed from consideration follow the directions provided in Section 7 5 1 of the PEST manual WinPEST Graphical Windows 21 4 16 Run Record Once PEST has finished execution WinPEST displays the PEST run record file in a window of its own The file can be viewed but not edited in this window 4 17 Prediction See Section 6 below Saving and Re Loading Plots 22 5 Saving and Re Loading Plots 5 1 General Through the Options item of the WinPEST main menu you can instruct WinPEST to save its plots at the end of a run the default condition is for this option to be selected Plots are saved to a file named case plots plt where case is the filename base for the current case ie the filename base of the PEST control file If you shut down WinPEST after completion of the parameter estimation process and open it later with the same project ie case all of the plots generated during the previous WinPEST run can be re loaded by selecting File from the main menu then Load graphs then All plots Plots can also be saved at intermediate stages of a PEST run by selecting File then Save graphs then All plots These are once again saved under the name case plots plt where case is the current case name Individual plots can also be saved for
274. ervation Measured Calculated Residual Weight Group value value arl 1 21038 1 64016 0 429780 1 000 group_1 ar2 1 51208 2 25542 0 743340 1 000 group_1 ar3 2 07204 3 03643 0 964390 000 group_1 ar4 2 94056 3 97943 1 03887 000 group_1 ar5 4 15787 5 04850 0 890630 1 000 group_1 ar6 5 77620 6 16891 0 392710 1 000 group_1 ar7 7 78940 7 23394 0 555460 000 group_2 arg 9 99743 8 12489 1 87254 000 group_2 ard 11 83 07 8 72561 Sy LOSTI 1 000 group_2 ar10 2 3194 8 89590 3 42350 1 000 group_2 arll 0 6003 8 40251 219779 000 group_2 ar12 7 00419 6 96319 4 100000E 02 1 000 group_2 ar13 3 44391 4 70412 1 26021 000 group_2 ar14 1758279 2 56707 0 984280 1 000 group_2 ar15 10380 1 42910 0 325300 000 group_3 arl 03086 1 10197 7 111000E 02 000 group_3 arl7 1 01318 1 03488 2 170000E 02 1 000 group_3 ar18 1 00593 1 01498 9 050000E 03 1 000 group_3 ar19 00272 1 00674 4 020000E 03 1 000 group_3 Running PEST 5 8 Prior information gt Prior Provided Calculated Residual Weight Group information value value pil 2 00000 0 261177 1 73882 3 000 group_4 pi2 2 60260 2 62532 2 271874E 02 2 000 group_4 See file TEMP3 RES for more details of residuals in graph ready format See file TEMP3 SEO for composite observation sensitivities Objective Function gt Sum of squared weighted residuals ie phi 63 59 Contribution to phi from observation group group_1 3 686 Contribution to phi from observation group gro
275. ervations which reflects their interdependence rather than to ignore this interdependence through the assignment of separate individual weights when undertaking the inversion process 4 3 2 Supplying the Observation Covariance Matrix to PEST The design of PEST is such that if PEST is supplied with a covariance matrix that matrix must pertain to a specific observation group Because prior information items can also be The PEST Control File 4 28 assigned to one or more observation groups this allows a covariance matrix to be supplied for a group of prior information items just as it can for a group of observations More than one covariance matrix can be supplied to PEST for use in the parameter estimation process In fact a covariance matrix can be supplied for every observation group However more often than not it will be supplied for only one or two such groups with weights being used for the remainder of the groups Example 4 5 shows a simple PEST control file in which two covariance matrices are supplied one for the observation group obsgp1 and the other for the observation group obsgp2 the latter being used for prior information pcf control data restart estimation LOL B 3 1 single point 1 0 0 2 0 3 0 01 10 3 0 001 0 O O 801 SS 005 1 1 parameter groups ro relative 0 001 0 0001 switch 2 parabolic parameter data rol log factor 4 00 0 1 10000 ro 101 ro2 log factor 5 00 0 1 10000 ro 101
276. ery important to note that if PEST is used in predictive analysis mode and at least some derivatives are supplied externally then the sole member of the observation group predict must be the last observation cited in the PEST control file Because the ordering of observations in the PEST control and external derivatives files must be the same then derivatives for this observation must also comprise the last row of the derivatives matrix contained in the external derivatives file 8 2 9 Parallel PEST PEST cannot receive derivatives through an external file if it is being run as Parallel PEST 8 3 Sending a Message to the Model PEST has the ability to send a small message to the model prior to running it This is useful if some aspect of the model s deployment depends on whether it is being run to test parameter upgrades to calculate derivatives by forward or central differences or to fill an external derivatives file The message sent by PEST resides in a file which is always named pest mmf The contents of a typical message file are shown in Example 8 2 Model Calculated Derivatives 8 5 derivative_increment 2 4 20 hcondl 5 005787 1 hcond2 9 850230 0 storl 5 660591 2 stor2 8 257257 10000 Example 8 2 A PEST to model message file pest mmf The first line of a message file contains a character string which tells the model why PEST is running it The various strings used by PEST are as follows forwa
277. es Before it runs the model PEST deletes all model output files that it knows about ie the model output files cited in the PEST control file Hence if the model fails to run PEST will not read old model output files produced on previous model runs mistaking them for new ones Thus if PEST generates an error message saying that it cannot find a particular model output file this is a sure sign that for some reason the model failed to run In most cases the matter is then easily rectified by taking some simple measure such as altering the contents of the model command line section of the PEST control file Where a model is comprised of multiple executable programs listed in a batch file similar considerations apply to intermediate model files ie to files generated by one or more of the executable programs comprising the composite model and read by one or more succeeding executable programs cited in the model batch file If for some reason an executable program which generates such an intermediate file fails to run then later executable programs of the composite model may read old intermediate files mistaking them for new ones If this happens model outputs will not reflect current parameter values in fact because they are independent of current parameter values PEST will probably declare that at least some model outputs are insensitive with respect to some parameter values This problem can be avoided if commands are included i
278. es at least one other parameter through the optimisation process e Derivatives reflect the transformation status of a parameter Thus if a parameter is log transformed the derivative with respect to the log of that parameter is presented 10 7 PAR2PAR 10 7 1 General On many occasions of model calibration there is a need to manipulate parameters before providing them to a model There can be a number of reasons for this two of them are now outlined 10 7 1 1 Parameter Ordering Suppose that a particular model has three parameters named infilt infilt2 and infilt3 For purposes of illustration let it be assumed that these parameters govern infiltration of water into different parts of a catchment in this case into subareas 1 2 and 3 respectively Soil property data may suggest that infiltration increases with subarea index that is that infilt lt infilt2 lt infilt3 Thus during the parameter estimation process it would be desirable for the lower bound of infilt2 to be the current value for infilt and for the lower bound of infilt3 to be the current value of infilt2 Unfortunately it would be very difficult to incorporate parameter dependent bounds into the PEST inversion algorithm However an alternative path can be taken which accomplishes the same thing This alternative path consists of estimating infilt together with two other parameters named infiltrat2 and infiltrat3 infiltrat stands for infiltration ratio
279. es Fortunately on most occasions this problem can be circumvented by a wise choice of those input variables which determine how PEST evaluates derivatives for a particular model The PEST input variables affecting derivatives calculation pertain to parameter groups In the PEST control file each parameter must be assigned to such a parameter group The assignment of derivative variables to groups rather than to individual parameters introduces savings in memory and complexity Furthermore in many instances parameters naturally fall into one or more categories for example if the domain of a two or three dimensional spatial model is subdivided into zones of constant parameter value and the parameters pertaining to all of these zones are being estimated parameters of the same type for each such zone would normally belong to the same group However if you wish to treat each parameter differently as far as derivatives calculation is concerned this can be achieved by assigning each parameter to a group of its own The simplest way to calculate derivatives is through the method of forward differences To calculate derivatives in this manner PEST varies each parameter in turn by adding an increment to its current value unless the current parameter value is at its upper bound in which case PEST subtracts the increment runs the model reads the altered model generated observations and then approximates the derivative of each observation wi
280. es any errors it finds to the screen These errors can be redirected to a file using the gt symbol on the TEMPCHEK command line Thus to run program TEMPCHEK directing it to write any errors found in the template file model tpl to PEST Utilities 10 2 the file errors chk use the following command tempchek model tpl gt errors chk If no errors are encountered in the template file TEMPCHEK informs you of this through an appropriate screen message This message also informs you of the number of parameters that TEMPCHEK identified in the template file TEMPCHEK lists these parameters in a file named file pmt where file is the filename base of tempfile ie the filename minus its extension If tempfile has no extension TEMPCHEK simply adds the extension pmt to tempfile By supplying a parameter value as well as a scale and offset for each parameter file tmp can be transformed into a PEST parameter value file which TEMPCHEK can then use to generate a model input file see below Note that if a parameter is cited more than once in a template file the parameter is nevertheless written only once to file pmt also it is counted only once as TEMPCHEK sums the total number of parameters cited in the template file If you wish TEMPCHEK to generate a model input file you must supply it with the name of the template file upon which the model input file is based the name of the model input file which it must generate and the values of all
281. ese files containing parameter values appropriate to that model run After the model has finished execution Parallel PEST reads one or more files generated by the model in order to obtain values calculated by the model for a set of outcomes for which there are corresponding field or laboratory measurements Operation of Parallel PEST assumes that PEST can write model input files and read model output files even though these files may reside on a different machine to that on which PEST itself is running Access to files on other machines is achieved through the use of modern Parallel PEST 9 4 network software Input and output files for a particular model may reside on the machine which actually runs the model or on a network server to which both PEST and the model s machine have access The only provisos on where these files reside is that both PEST and the model must be able to read and write to these files and that these files are named using normal filename protocol This is easily accomplished through the use of the shared folder concept available across local area networks Parallel PEST writes input file s for the models running on the various networked machines using one or more templates residing on the machine on which PEST is running Similarly Parallel PEST reads the output file s produced by the various models using the instructions contained in one or a number of instruction files residing on the machine on which PEST runs
282. esh using kriging or some other spatial interpolation method the degree of spatial parameter variation between these pilot points or of various subgroups of them can be limited through the imposition of appropriate uniformity criteria As stated above regularisation can take place across the entire model domain or across sub areas within the domain In the latter case if there are boundaries within the model domain where property discontinuities are known to exist then no parameter differences should be taken across these boundaries for inclusion in the regularisation observations used to Regularisation 7 3 enforce the smoothing condition Regularisation criteria other than uniform smoothing can also be used For example individual parameter differences might be weighted according to inter parameter distance Or a minimum curvature condition could be imposed each relationship comprising this regularisation scheme will cite at least 3 parameters stating that the difference between the inner parameter value and the average of at least two outer parameter values is zero Or a regularisation scheme might be comprised of a series of differences between many or all of the model s parameters and some preferred value for each of these parameters Even more sophisticated techniques might be developed for example a regularisation condition imposed on the calibration of a transient model might be that the same parameter distrib
283. ess is based are not altogether consistent Such extremely large or small values may depending on the model result in floating point errors or numerical convergence difficulties Again carefully chosen parameter bounds will circumvent this problem If a parameter upgrade vector u is determined which would cause one or more parameters to move beyond their bounds PEST adjusts u such that this does not occur placing such parameters at their upper or lower bounds On later iterations special treatment is then provided for parameters which are at their allowed limits If the components of both the upgrade vector and the negative of the gradient vector pertaining to a parameter at its upper or lower limit are such as to take the parameter out of bounds then the parameter is temporarily frozen and the parameter estimation problem reformulated with that parameter fixed at its limit hence the new upgrade vector will not result in any adjustment to that parameter If after reformulation of the problem in this manner there are parameters at their limits for which the parameter upgrade vector still points outward the negative of the gradient vector pointing inward then these parameters too are temporarily frozen This process continues until a parameter upgrade vector is calculated which either moves parameters from their bounds back into the allowed parameter domain or leaves them fixed The strength of this strategy is that it allows PEST to search
284. ess is numerically very stable avoiding the deleterious effects on this process of parameter insensitivity or excessive correlation that often accompanies an attempt to estimate too many parameters The decision to place PEST in the public domain was not taken lightly However two factors made the decision almost impossible to avoid One of these was the advent of competing public domain software which while not having anything like the functionality of PEST is nevertheless highly visible and has US government auspice The other consideration was of a more philanthropic nature My instincts tell me that the biggest issue in environmental modelling over the next decade will be that of predictive uncertainty analysis PEST has a substantial contribution to make in this regard It is my hope that by making PEST freely available to all modellers at zero cost it will make an even more important contribution to environmental management based on computer simulation of real world systems than it already has to date Other new features found in PEST ASP that were not available in previous versions of PEST include the following e All names pertaining to parameters parameter groups observations observation groups and prior information items can now be up to twelve characters in length e Prior information items must now be assigned to observation groups e Uncertainties in observations and prior information equations used in the inversion process can
285. eter groups section of the PEST control file the user informs PEST whether derivatives of model outcomes with respect to the members of each parameter group are to be calculated using two points three points or two points at first and then three points later When run in parameter estimation mode PEST makes the switch between two point and three point derivatives calculation if it fails to lower the objective function by a relative amount equal to PHIREDSWH between successive optimisation iterations The value for PHIREDS WH is supplied in the control data section of the PEST control file When used in predictive analysis mode the role of PHIREDSWH is unchanged if the current objective function is above PD1 However if it is below PD1 PEST s decision to switch from two point derivatives calculation to three point derivatives calculation is based on improvements to the model prediction If between two successive optimisation iterations the model prediction is raised lowered by no more than a relative amount of RELPREDSWH or by an absolute amount of ABSPREPSWH PEST makes the switch to three point derivatives calculation A setting of 0 05 is often appropriate for RELPREDSWH The setting for ABSPREDSWH is context dependent Supply a value of 0 0 for either of these variables if you wish that it has no effect on the optimisation process On most occasions ABSPREDSWH should be set to 0 0 NPREDNORED The last four variables are termin
286. eter value thus it measures the sensitivity of all observations with respect to a relative change in the value of that parameter It is easily shown that for a log transformed parameter the composite scaled WinPEST Graphical Windows 12 sensitivity can be calculated from the composite natural sensitivity by dividing the latter by 2 303 4 5 Parameters Selection of the Parameters tab allows you to see how each parameter changes in the course of the parameter estimation process By clicking on the right mouse button while viewing this window you can select deselect different parameters for representation in this plot Any parameter can be plotted in this manner no matter what its transformation status and no matter whether it is fixed or tied to another parameter during the inversion process The default condition is for all parameters to be represented 4 6 Lambda By clicking on the Lambda tab you can view a plot of the Marquardt lambda plotted against parameter upgrade attempt There is at least one parameter upgrade attempt per optimisation iteration often more In most cases the value of the Marquardt lambda will gradually fall as the optimisation process progresses 4 7 Calc vs Obs Choose this window to view a plot of field or laboratory measurements against their model generated counterparts optionally both calculated and observed values can be multiplied by respective observation weights before plotting this i
287. eters are of different type and possibly of very different magnitudes Information is appended to the parameter sensitivity file during each optimisation iteration immediately following calculation of the Jacobian matrix In the event of a restart the parameter sensitivity file is not overwritten rather PEST preserves the contents of the file appending information pertaining to subsequent iterations to the end of the file In this manner the user is able to track variations in the sensitivity of each parameter through the parameter estimation process When inspecting the parameter sensitivity file keep the following points in mind e If PEST is working in predictive analysis mode it assumes that the weight assigned to the observation constituting the sole member of the observation group predict is zero Thus there is no contribution to the composite sensitivity of any parameter from the sole member of this observation group However the situation is slightly different for information written to the parameter sensitivity file at the end of the simulation Running PEST 5 18 see below e If PEST is working in regularisation mode the weights assigned to members of the observation group regul vary from optimisation iteration to optimisation iteration Composite parameter sensitivities for any optimisation iteration are calculated using the optimal weight factor calculated on an iteration by iteration basis by PEST for members
288. eters must be made A common assumption is that the model domain is zoned According to this assumption the system is subdivided into a number of areas or volumes in each of which a certain physical property is constant Hence while the input arrays will still contain hundreds maybe thousands of elements each element will be one of only n different numbers where n is the number of zones into which the model domain has been subdivided The Model PEST Interface 3 9 It is a simple matter to construct a PEST template file for a model such as this Firstly prepare for a model run in the usual way Using the model preprocessor assign n different values for a particular property to each of the different model zones writing the model input arrays to the model input files in the usual manner Then using the search and replace facility of a text editor edit the model input file such that each occurrence within a particular array of the number representing the property of a certain zone is altered to a parameter space identifier such as rol remember to make the parameter space as wide as the model will allow in order to ensure maximum precision If this is done in turn for each of the n different numbers occurring in the array using a different parameter name in place of each different number the array of numbers will have been replaced by an array of parameter spaces When PEST writes the model input file it will as usual replace each
289. eviated pathnames the abbreviations being sufficient for PEST to write and read the respective files from the directory in which it is run Data for the various slaves must be supplied in the same order as that in which the slaves were introduced earlier in the file For each slave the names of NTPFLE model input files must be followed by the names of NINSFLE model output files where NTPFLE is the number of template files and NINSFLE is the number of instruction files pertaining to the present PEST case Note that the PEST template and instruction file corresponding to each of these model input and output files is identified in the PEST control file read by Parallel PEST Example 9 3 shows a Parallel PEST run management file equivalent to that of Example 9 2 but with the value of NFLETYP set to 0 Use of an abbreviated run management file such as that shown in Example 9 3 is only possible where all model input and output files on each slave reside in the one subdirectory and this subdirectory is also the PSLAVE working directory on that machine prf 1301 5 my machine steve s machine k jerome s machine 1 model 600 600 720 Example 9 3 A Parallel PEST run management file equivalent to that of Example 9 2 but with NFLETYP set to zero Parallel PEST 9 11 9 2 6 More on Partial Parallelisation of the Marquardt Lambda Testing Process The algorithm used by PEST to undertake parallel model
290. evious versions of PEST only one command line could be used to run the model However for versions of PEST from 5 0 onwards different commands can be used to run the model depending on the purpose of the model run Note that when counting the number of available model command lines when assigning a value to NUMCOM the command that is used to run the model in order to fill the external derivatives file should not be included in the count This command is listed in a separate section of the PEST control file to the model command line section as will be discussed shortly If there is only one command listed in the model command line section of the PEST control file which will most often be the case then NUMCOM should be supplied with a value of 1 JACFILE Provide this integer variable with a value of 1 if a special model run is to be undertaken during each optimisation iteration for the purpose of external derivatives calculation Otherwise provide JACFILE with a value of 0 Note that if JACFILE is provided with a value of 1 and an attempt is made to run Parallel PEST PEST will cease execution with an appropriate error message MESSFILE Provide this integer variable with a value of 1 if PEST is required to write a PEST to model message file prior to each model run Otherwise provide it with a value of 0 8 5 3 Parameter Data Section As is described in Section 4 2 4 each line of the parameter data section of
291. exists in its internal derivatives matrix this having been calculated by finite differences In summary model calculated derivatives are read by PEST after derivatives are calculated by finite differences for those parameters for which the user has requested finite difference derivatives calculation for at least one observation Externally supplied derivatives override those already calculated by finite differences except where a value of 1 11e33 is supplied for the derivative value 8 2 6 Tied Parameters If a parameter is tied to a parent parameter and derivatives of the former parameter for a particular observation are supplied externally then derivatives of the tied parameter for that same observation must also be supplied externally If this does not occur PEST will cease execution with an appropriate error message 8 2 7 Name of the Derivatives File The user must inform PEST of the name of the external derivatives file through the PEST control file for the current case see Section 8 5 The external derivatives file can have any legal name except for the following names which are already used by PEST If the filename root of the current project is case the names to be avoided are case hld case jac case jco case jst case par case pst case res case rmf case rmr case rst case rsr Case sen case mtt and case seo Other names which must be avoided are pest hit pest mmf and pest stp 8 2 8 Predictive Analysis Mode It is v
292. ey are those for which the prediction was maximised minimised compatible with the objective function being below the user supplied calibration threshold When working in regularisation mode the best parameters are those for which the regularisation objective function is the least provided that the measurement objective function is below the user supplied measurement objective function limit The name of the binary file in which the Jacobian matrix is stored is case jco where case is the filename base of the current PEST control file jco stands for Jacobian optimised PEST Utilities 10 10 The Jacobian file is stored in binary rather than text format to save space To translate it to text format you must run JACWRIT by typing the command jacwrit jacfilel jacfile2 where jacfilel is the name of the binary Jacobian file written by PEST and jacfile2 is the name of the text file to which JACWRIT should write the Jacobian matrix in a form which is fit for human consumption Note the following e Parameter and observation names are listed in the text file written by JACWRIT so that it becomes an easy matter to link a sensitivity ie a derivative to a particular parameter observation pair e Only adjustable parameters are represented in the file written by JACWRIT fixed and tied parameters are not represented e The sensitivity of a parameter to which another parameter is tied reflects the fact that this parameter carri
293. f that parameter rather than to the parameter itself Furthermore the parameter name must be placed in brackets and preceded by log note that there is no space between log and the following opening bracket Thus in the above example if parameter parl is log transformed the prior information article should be rewritten as pil 1 0 log parl 362671 1 0 pr_info Note that logs are taken to base 10 Though not illustrated you will also need to review the weight which you attach to this prior information article by comparing the extent to which you would permit the log of par1 to deviate from 0 362671 with the extent to which model generated observations are permitted to deviate from their corresponding measurements The left side of a prior information equation can be comprised of the sum and or difference of a number of factor parameter pairs of the type already illustrated these pairs must be separated from each other by a or sign with a space to either side of the sign For example pi2 1 0 par2 3 43435 par4 2 389834 par3 1 09e3 3 00 group_pr Prior information equations which include log transformed parameters must express a relationship between the logs of those parameters For example if you would like the ratio between the estimated values of parameters parl and par2 to be about 40 0 the prior information article may be written as The PEST Control File 4 26 pi3 1
294. f the Marquardt lambda The maximum number of model runs that PEST will devote to this line search for any value of lambda is equal to the user supplied value of NSEARCH set NSEARCH to 1 if you wish that no line search be undertaken Otherwise a good value is between 6 and 8 When undertaking the line search the initial model run is undertaken at that point along the parameter upgrade vector which is a factor of INITSCHFAC along the line of the distance that PEST would have chosen using the theory of Section 2 1 9 alone Unless there is a good reason to do otherwise a value of 1 0 is appropriate here Then PEST moves along the parameter upgrade vector increasing or decreasing the distance along this vector by a factor of MULSCHFAC as appropriate A value of 1 5 to 2 0 is suitable for this variable in most cases Then once the min contour has been subtended by two different model runs PEST uses a bisection algorithm to find the intersection point with greater precision It may seem at first sight that implementation of a line search algorithm may prove very costly in terms of model runs However experience to date is such as to suggest that inclusion of the line search option may result in a dramatic reduction in overall model runs as fewer optimisation iterations are required to find the critical point even though each iteration may individually require more model runs Predictive Analysis 6 15 ABSPREDSWH and RELPREDSWH In the param
295. f the highest to lowest eigenvalue constitutes another significant item of information that is forthcoming as a by product of the parameter estimation process The square root of this ratio is related to the condition number of the matrix that PEST must invert when solving for the parameter upgrade vector see equation 2 23 If the condition number of a matrix is too high then inversion of this matrix becomes numerically difficult or even impossible In the present instance this is an outcome of the fact that solution of the inverse problem approaches nonuniqueness as elongation of the probability ellipsoid increases In general if the ratio of the highest to lowest eigenvalue is greater than about 10 there is a strong possibility that PEST is having difficulty in calculating the parameter upgrade vector because of parameter insensitivity and or correlation Its performance may be seriously degraded as a result 5 3 Other PEST Output Files 5 3 1 The Parameter Value File Running PEST 5 16 At the end of each optimisation iteration PEST writes the best parameter set achieved so far ie the set for which the objective function is lowest if PEST is running in parameter estimation mode to a file named case par where case is the filename base of the PEST control file this type of file is referred to as a PEST parameter value file At the end of a PEST run the parameter value file contains the optimal parameter set Example 5 2 illustra
296. f untransformed numbers The uncertainty thus obtained provides only an indication of the uncertainty pertaining to the untransformed parameter estimate Apart from the fact that it is difficult to relate standard deviations in the log domain to those in the untransformed domain the linearity assumption upon which calculation of parameter uncertainty is based may not extend as far into parameter space as the uncertainties estimated for log transformed parameters Uncertainties in the whisker plot are represented by a bar through the parameter value spanning the parameter uncertainty interval For untransformed parameters the uncertainty interval extends from one standard deviation below the parameter value to one standard deviation above it hence it is symmetrical about the estimated value of the parameter in the untransformed domain For log transformed parameters the uncertainty interval is symmetric in the log domain extending one standard deviation as estimated by PEST either side of the optimised log of the parameter value The user is reminded that all parameter statistics must be considered as approximate only Their calculation is based on a linearity assumption that is violated sometimes to a large degree by most environmental models Note also that if PEST is used in regularisation mode then observations and or prior information items belonging to the observation group regul are used in the calculation of parameter unc
297. ferent groups are represented in this plot using different symbols As the optimisation process progresses the plotting region adjusts to the range of values represented in this plot If you wish you can set the plotting region to your own specifications by selecting the Properties option from the pop up menu that appears after clicking with the right mouse button anywhere on the plot These specifications can be maintained during subsequent optimisation iterations by selecting the Fixed plot region option from the same popup menu If PEST is running in regularisation mode observations and or prior information comprising the observation group regul do not appear on this plot unless specifically selected through the Select groups item of the pop up menu pertaining to this window 4 8 Residuals Click on the Residuals tab to view a plot of residuals vs observed values Both the residuals and the observed values can be multiplied by respective observation weights if desired click using the right mouse button and select Weighted residuals and or Weighted obs values from the pop up menu Ideally when the optimisation process is complete weighted residuals should be normally distributed about zero Also neighbouring residuals should be uncorrelated requiring that WinPEST Graphical Windows 14 there be few runs of successively positive or negative residual values appearing on a plot of this natu
298. ff the po contour The transition to predictive analysis mode as it approaches this contour is a gradual one unseen by the user 2 2 11 Operation in Regularisation Mode Within each optimisation iteration PEST s task when working in regularisation mode is identical to its task when working in parameter estimation mode ie it must minimise an objective function using a linearised version of the model encapsulated in a Jacobian matrix However just before calculating the parameter upgrade vector PEST calculates the appropriate regularisation weight factor to use for that iteration This is the factor by which all of the weights pertaining to regularisation information are multiplied in accordance with equation 2 33 prior to formulating the overall objective function whose task it is for PEST to minimise on that iteration As parameters shift and the Jacobian matrix changes an outcome of the nonlinear nature of most models the regularisation weight factor also changes Hence it needs to be re calculated during every optimisation iteration Use of PEST in regularisation mode is fully described in Chapter 7 of this manual As is discussed in that Chapter the user is required to supply a few extra control variables to govern PEST s operation in this mode One of these is the target measurement objective function ie Pm of equation 2 31 Other variables govern the procedure by which y of equation 2 33 is calculated and allow sligh
299. fifth optimisation iteration three lambdas are tested The second results in a raising of the objective function over the first though this is not apparent in the run record because phi the objective function is not written with sufficient precision to show it so PEST tests a lambda which is higher than the first For the case illustrated in Example 5 1 when lambda is raised or lowered it is adjusted using a factor of 2 0 this being the user supplied value for the PEST control variable RLAMFAC For optimisation iteration 6 the first lambda tested is the same as the most successful one for the previous iteration viz 1 9531E 02 However for each of the previous iterations where the objective function was improved through lowering lambda during the iteration prior to that the starting lambda is lower by a factor of 2 0 ie REAMFAC than the most successful lambda of the previous iteration At the end of optimisation iteration 6 PEST calculates that the relative reduction in the objective function from that achieved in iteration 5 is less that 0 1 ie it is less than the user supplied value for the PEST control variable PHIREDSWH Hence as the input variable FORCEN for at least one parameter group both groups in the present example is set to switch PEST records the fact that it will be using central differences to calculate derivatives with respect to the members of those groups from now on Note that in Example 5 1 the use of ce
300. formance through watching its screen output that successful PEST execution is no longer measured in terms of how much it can reduce the objective function It is now measured by how high or low depending on the user s request the prediction can be made while keeping the objective function as close as possible to o After completion of a predictive analysis run the highest or lowest model prediction for which the objective function is equal to or less than o is recorded on the PEST run record file Corresponding parameter values are also recorded on this file as well as in file case par where case is the filename base of the PEST control file Predictive Analysis 6 10 As has already been mentioned predictive analysis should only be undertaken after PEST has been used to undertake parameter estimation with the model run under calibration conditions alone The predictive analysis and parameter estimation runs are closely related All parameters parameter transformations parameter linkages observations observation weights prior information equations and prior information weights must be the same for the two runs in order to ensure consistency in objective function values 6 2 Working with PEST in Predictive Analysis Mode 6 2 1 Structure of the PEST Control File The PEST control file used for running PEST in predictive analysis mode is shown in Example 6 1 An example of this file is provided in Example 6 2 pct control data R
301. formation is available in the Covariance Matrix window Two options are available for the display of parameter uncertainties in the Uncertainties window In the first selectable as Standard deviation in the popup options menu the standard deviations of all adjustable parameters are displayed in bar chart form In the second option selectable as Whisker plot estimated parameter values are displayed together with whiskers representing the magnitude of the standard deviation on either side of the parameter value A few words of explanation are required for each of these plots The standard deviations represented in bar chart form if the Standard deviation option is selected pertain to untransformed parameters irrespective of whether a parameter is log transformed or not through the inversion process If a parameter is not log transformed then its standard deviation is calculated as the square root of the pertinent diagonal element of the parameter covariance matrix However if a parameter is log transformed the size of the bar representing its standard deviation is calculated as the length of the interval between its optimised value and its upper uncertainty bound The latter is calculated by adding one standard deviation of the log transformed parameter square root of the parameter variance as calculated by PEST to the estimated log parameter value and then transforming back from the log domain to the domain o
302. formation must be assigned to at least two different observation groups one of which must be named regul All observations and or prior information items belonging to the group regul comprise the regularisation observations all observations and or prior information equations belonging to all other groups comprise the measurement observations Weights must be assigned to individual observations and prior information equations within each group in the normal manner or an observation covariance matrix can be assigned see Section 4 3 However as was explained above weights assigned to the regularisation observations and or prior information equations are multiplied internally by a regularisation weight factor prior to formulation of the total objective function during each optimisation iteration 7 3 3 Control File Regularisation Section Example 7 1 shows the format of the regularisation section of the PEST control file This section should be placed at the end of the file regularisation PHIMLIM PHIMACCEPT FRACPHIM WFINIT WFMIN WFMAX WFFAC WFTOL Example 7 1 Format of the regularisation section of the PEST control file An example of the regularisation section of a PEST control file is provided in Example 7 2 Regularisation 7 6 regularisation 135 0 140 0 0 0 1 0e 2 1 0e 6 1 0e6 1 3 1 0e 2 Example 7 2 An example of the regularisation
303. from representations of parameter values on model input files if the decimal point is redundant thus making room for the use of one extra significant figure If DPOINT is supplied as point PEST will ensure that the decimal point is always present See Section 3 2 6 NUMCOM JACFILE and MESSFILE These variables are used to control the manner in which PEST can obtain derivatives directly from the model if these are available see Chapter 8 For normal operation these should be set at 1 O and 0 respectively RLAMBDAI The PEST Control File 4 7 This real variable is the initial Marquardt lambda As discussed in Section 2 1 7 PEST attempts parameter improvement using a number of different Marquardt lambdas during any one optimisation iteration however in the course of the overall parameter estimation process the Marquardt lambda generally gets smaller An initial value of 1 0 to 10 0 is appropriate for most models though if PEST complains that the normal matrix is not positive definite you will need to provide a higher initial Marquardt lambda As explained in Section 2 1 7 for high values of the Marquardt lambda the parameter estimation process approximates the steepest descent method of optimisation While the latter method is inefficient and slow if used for the entirety of the optimisation process it often helps in getting the process started especially if initial parameter estimates are poor The Marquardt lambda used by PEST
304. fter it has run the model Hence while a user must supply SENSAN with the instruction files required to read one or more model output files it is actually INSCHEK which reads these SENSAN 11 35 files SENSAN then reads the observation value files written by INSCHEK in order to ascertain current model outcome values Because SENSAN must run INSCHEK at least once every time it runs the model it is essential that the executable file inschek exe reside either in the current directory or within a directory cited in the PATH environment variable Before running SENSAN you should check the integrity of all instruction files which you supply to it by running INSCHEK yourself outside of SENSAN 11 2 4 The Parameter Variation File SENSAN s task is to run a model as many times as a user requires providing the model with a user specified set of parameter values on each occasion As discussed above the parameters which are to be varied from model run to model run are identified on one or a number of template files The values which these parameters must assume on successive model runs are provided to SENSAN in a parameter variation file an example of which is presented below depl dep2 resl res2 res3 EgO 10 0 5 0 2 0 10 0 2 0 10 0 5 0 2 0 10 0 E0 11 0 53 0 20 10 0 1 0 10 0 6 0 2 0 10 0 1 0 10 0 5 0 3 0 10 0 LsO 10 0 5 0 2 0 11 0 Example 11 1 A parameter variation file The file shown in Example 11 1 provides 6 sets of val
305. function is just one of them Another method would be to use seasonal ratios in this case only one parameter may require estimation this being the factor by which all such ratios are multiplied to achieve model calibration 10 7 2 Using PAR2PAR 10 7 2 1 Running PAR2ZPAR PAR2PAR is run using the command par2par infile where infile is a PAR2PAR input file which must be prepared by the user 10 7 2 2 The PAR2PAR Input File The structure of the PAR2PAR input file is shown in Example 10 4 An example of such a file is provided in Example 10 5 PEST Utilities 10 13 parameter data PARNME expression PARNME expression template and model input files TEMPFLE INFLE TEMPFLE INFLE control data PRECIS DPOINT Example 10 4 Structure of the PAR2PAR input file parameter data infiltl 0 3456 infiltrat2 1 0453 infiltrat3 1 5432 infilt2 infiltl infiltrat2 infilt3 infilt2 infiltrat3 template and model input files modell tpl modell in model2 tpl model2 in control data single point Example 10 5 An example of a PAR2PAR input file A PAR2PAR input file must contain at least a parameter data section and a template and model input files section The control data section is optional if it is omitted the default values of single and point are supplied for the variables PRECIS and DPOINT The parameter data section of the PAR2PAR input fil
306. g out a list of observation values read from that file In this way you can be sure that your instruction set works before it is actually used by PEST See Chapter 10 of this manual for more details The PEST Control File 4 4 The PEST Control File 4 1 The Role of the PEST Control File Once all the template and instruction files have been prepared for a particular case a PEST control file must be prepared which brings it all together Unlike template and instruction files for which there is no naming convention there are some conventions associated with the name of the PEST control file In particular the file must have an extension of pst Its filename base is referred to as the PEST case name PEST uses this same filename base for the files which it generates in the course of its run Thus for example if you name the PEST control file calib pst PEST will generate files calib rec the run record file calib par best parameter values achieved calib rst restart information stored at the beginning of each optimisation iteration calibjac the Jacobian matrix for a possible restart calib jst the same file from the previous optimisation iteration calib jco the Jacobian matrix pertaining to best parameters for access by the JACWRIT utility calib sen parameter sensitivities calib seo observation sensitivities and calib res tabulated observation residuals User supplied files calib rmf Parallel PE
307. g the machine which PEST itself is running on If model run times are large and the number of parameters is greater than four or five overall PEST run times can be reduced by a factor almost equal to the number of machines over which Parallel PEST is able to distribute model runs As well as allowing a user to distribute model runs across a network Parallel PEST can also manage simultaneous model runs on a single machine This can realise significant increases in PEST efficiency when carrying out parameter optimisation or predictive analysis on a multi processor computer by keeping all processors simultaneously busy carrying out model runs The optimisation including regularisation and predictive analysis algorithms used by Parallel PEST are no different from those used by the normal PEST Preparation of template files instruction files and the PEST control file is identical in Parallel PEST to that of the normal PEST However use of Parallel PEST requires that one extra file be prepared prior to undertaking an optimisation run viz a run management file This file informs Parallel PEST of the machines to which it has access of the names of the model input and output files residing on those machines and of the name of a subdirectory it can use on each of these machines to communicate with a slave which carries out model runs on request see below 9 1 2 Parallelisation of the Jacobian Matrix Calculation Process When PEST calcula
308. ger code that informs the model of the parameter s status in the inversion process A value of 0 denotes that the parameter is adjustable and is not logarithmically transformed A value of 1 indicates that the parameter is adjustable and is logarithmically transformed A value of n indicates that the parameter is tied to parameter number n while a value of 10000 indicates that the parameter is fixed The PEST to model message file is always written to the current working directory it is written just before each model run is undertaken However in the case of Parallel PEST the message file is written to each slave working directory just before the pertinent model run is Model Calculated Derivatives 8 6 initiated by the slave 8 4 Multiple Command Lines As has already been discussed when PEST runs a model for the purpose of external derivatives calculation it can use a different command to that which it uses for ordinary model runs The same command can be used for both of these types of model run if desired In this case it may be necessary for the model to acquaint itself with PEST s expectations It can do this by reading the PEST to model message file With PEST it is possible to use different commands to run the model when calculating finite difference derivatives with respect to different parameters Recall that when PEST calculates the derivatives of all model outputs with respect to a particular parameter it runs the model
309. h as electrical conductivity and solute concentration should never be provided with negative values Also if a parameter occurs as a divisor anywhere in the model s code it can never be zero For many models it has been found that if the logarithms of certain parameters are optimised rather than the parameters themselves the rate of convergence to optimal parameter values can be considerably hastened PEST allows such logarithmic transformation of selected parameters Often there is some information available from outside of the parameter estimation process about what value a parameter should take Alternatively you may know that the sum or difference of two or more parameters or their product or quotient in the case of logarithmically transformed parameters should assume a certain value PEST allows you to incorporate such prior information into the estimation process by increasing the value of the objective function ie the sum of squared deviations between model and observations see the next section in proportion to the extent to which these articles of prior information are transgressed Finally parameters adjusted by PEST can be scaled and offset with respect to the parameters actually used by the model Thus you may wish to subtract 273 15 from an absolute temperature before writing that temperature to a model input file which requires Celcius degrees or you may wish to negate a model parameter which never becomes positive so that
310. h one of its termination criteria In most cases the choice of an initial Marquardt lambda of between 1 0 and 10 0 works well Nevertheless if PEST spends the first few optimisation iterations adjusting this to a vastly higher value or a vastly lower value but remember that lambda is reduced in the normal course of the optimisation process anyway before making great gains in objective function reduction then you will probably need to reconsider your choice of initial lambda in subsequent uses of PEST in conjunction with the same model However if the parameter estimation process simply does not get off the ground you should start again with an entirely different lambda try a much greater one first especially if PEST has displayed messages to the effect that the normal matrix is not positive definite To help PEST search farther afield for a suitable Marquardt lambda perhaps you should set the input variable RLAMFAC high for a while However it is bad practice to keep it high through the entirety of an optimisation run hence if PEST finds a lambda which seems to work you should terminate PEST execution supply that lambda as the initial lambda reset RLAMFAC to a reasonable value eg 2 0 and start the optimisation process again Experience has shown that if the initial parameter set is poor PEST may need a high Marquardt lambda to get the parameter estimation process started Also the Marquardt lambda may need to be greater for hig
311. hat the directory containing each executable is cited in the PATH environment variable on the machines from which the model is run Can PEST and PPEST be used interchangeably Yes all versions of PEST use identical template and instruction files and an identical PEST control file Parallel PEST terminated execution with an error message to the effect that it could not find the model output file What could be wrong Furthermore it waited an unduly long time after commencement of execution to inform me of this If Parallel PEST encounters any problems whatsoever in reading a model output file for example premature termination of the file data errors poor instructions or file not present it tries to read the file three times at 30 second intervals in order to be sure that the problems are not network related If after making three attempts to read the file it is still unsuccessful it reports the error to the screen and terminates execution If a model output file cannot be found then either the model did not write it the model did not run or an erroneous name was provided for the model output file in the Parallel PEST run management file In the first case a poor parameter set may have been provided to the model resulting in a model run time error being generated before the model wrote any data to its output file Or perhaps model input files were misnamed in the Parallel PEST run management file Alternatively if the model did not ru
312. he NPAR parameters PARNME PARTIED one such line for each tied parameter observation groups OBGNME one such line for each observation group observation data OBSNME OBSVAL WEIGHT OBGNME one such line for each of the NOBS observations model command line write the command which PEST must use to run the model model input output TEMPFLE INFLE one such line for each model input file containing parameters INSFLE OUTFLE one such line for each model output file containing observations prior information PILBL PIFAC PARNME PIFAC log PARNME PIVAL WEIGHT OBGNME one such line for each of the NPRIOR articles of prior information T Example 4 1 Construction details of the PEST control file The PEST Control File 4 3 pcf control data restart estimation DLS De 26 4 1 1 single point 1 0 0 5 0 2 0 0 4 0 03 10 3 0 3 0 1 0e 3 ok 30 S01 33 20s 3 dl R parameter groups ro relative 001 00001 switch 2 0 parabolic h relative 001 00001 switch 2 0 parabolic parameter data rol fixed factor 0 5 1 10 none 1 0 0 0 1 ro2 log factor 5 0 1 10 ro 1 0 0 0 1 ro3 tied factor 0 5 sT 10 ro 1 20 0 2 0 J h1 none factor 2 0 05 100 h 1 0 0 0 1 h2 log factor 5 0 05 100 h 10
313. he conductivity pertaining to that zone It may be these zone conductivity values that we wish to optimise Fortunately creating a template for the model input file holding the array is a simple matter for each occurrence of a particular zone defining number in the original input file can be replaced by the parameter identifier specific to that zone Hence every time PEST rewrites the array all array elements belonging to a certain zone will have the same value specific to that zone On a particular PEST run a parameter can remain fixed if desired Thus while the parameter may be identified on a template file PEST will not adjust its value from that which you supply at the beginning of the parameter estimation process Another feature is that one or a number of parameters can be tied to a parent parameter In this case though all such tied Introduction 1 6 parameters are identified on template files only the parent parameter is actually optimised the tied parameters are simply varied with this parameter maintaining a constant ratio to it PEST requires that upper and lower bounds be supplied for adjustable parameters ie parameters which are neither fixed not tied this information is vital to PEST for it informs PEST of the range of permissible values that a parameter can take Many models produce nonsensical results or may incur a run time error if certain inputs transgress permissible domains For example parameters suc
314. he thickness parameter is log transformed again it would be surprising if the log transformation of elevation improved optimisation performance due to the arbitrary datum with respect to which an elevation must be expressed PEST can thus optimise thickness converting this thickness to elevation every time it writes a model input file by adding the reference elevation stored as the parameter offset The scale variable is equally useful A model parameter may be such that it can only take on negative values such a parameter cannot be log transformed However if a new parameter is defined as the negative of the model required parameter PEST can optimise this new parameter log transforming it if necessary to enhance optimisation efficiency Just before it writes the parameter to a model input file PEST multiplies it by its SCALE variable 1 in this case so that the model receives the parameter it expects If you do not wish a parameter to be scaled and offset enter its scale as 1 and its offset as Zero It should be stressed that PEST is oblivious to a parameter s scale and offset until the moment it writes its value to a model input file It is at this point and only this point that it first multiplies by the scale and then adds the offset the scale and offset take no other part in the parameter estimation process Note also that fixed and tied parameters must each be supplied with a scale and offset just like their adjustable log transfo
315. her PEST will terminate execution A good setting for RELPREDSTP is 0 005 The setting for ABSPREDSTP is context dependent set ABSPREDSTP to 0 0 if you wish it to have no effect which is normally the case A good setting for NPREDSTP is 4 Predictive Analysis 6 16 Note that the maximum allowed number of optimisation iterations is set by the value of the NOPTMA lt X variable provided in the control data section of the PEST control file Consider setting this higher than you would for PEST s usage in parameter estimation mode 6 3 An Example See Section 12 2 for an example of PEST s use in Predictive Analysis Mode Regularisation 7 1 7 Regularisation 7 1 About Regularisation 7 1 1 General PEST and other nonlinear parameter estimation software sometimes encounters difficulties in minimising the calibration objective function where too many parameters must be simultaneously estimated Such situations often arise when calibrating models that represent two and three dimensional spatial processes or when using such models to infer the properties of a two or three dimensional model domain as part of a data interpretation process In many cases the user may not wish to observe the conventional wisdom of parameter parsimony for fear of losing important system details However the use of too many parameters leads to numerical instability and nonuniqueness of parameter estimates In an attempt to reduce the number of parame
316. her FORCEN is always_3 or switch If you do not wish the increment to be increased you must provide DERINCMUL with a value of 1 0 Alternatively if for some reason you wish the increment to be reduced if three point derivatives calculation is employed you should provide DERINCMUL with a value of less than 1 0 Experience shows that a value between 1 0 and 2 0 is usually satisfactory DERMTHD There are three variants of the central ie three point method of derivatives calculation each method is described in Section 2 3 If FORCEN for a particular parameter group is always_3 or switch you must inform PEST which three point method to use This is accomplished through the character variable DERMTHD which must be supplied as parabolic best_fit or outside_pts If FORCEN is always_2 you must still provide one of these three legal values for DERMTHD however for such a parameter group the value of DERMTHD has no bearing on derivatives calculation for the member parameters 4 2 4 Parameter Data First Part For every parameter cited in a PEST template file up to ten pieces of information must be provided in the PEST control file Conversely every parameter for which there is information in the PEST control file must be cited at least once in a PEST template file The parameter data section of the PEST control file is divided into two parts in the first part a line must appear for each pa
317. her slave if the latter is standing idle Similarly if a slave has just become free and Parallel PEST calculates that a model run which is currently being undertaken on a certain slave can be completed on the newly freed slave in a significantly faster time it reassigns the run to the new slave As it allocates model runs to different slaves it writes a record of what it does to its run management record file see below Parallel PEST execution continues until either the optimisation process is complete or the user interrupts it by typing the PPAUSE PSTOP or PSTOPST command while situated in the PPEST working directory within another command line window see Section 5 4 1 for further details In the former case PEST execution can be resumed if the PUNPAUSE command is typed Meanwhile the run record file can be examined by opening it with any text editor 9 3 4 Re Starting Parallel PEST If the RSTFLE variable on the PEST control file is set to restart a terminated Parallel PEST run may be restarted at any time from the beginning of the optimisation iteration during which it was interrupted This can be achieved through entering the command Parallel PEST 9 14 ppest case r where case pst is the PEST control file for the current case Alternatively execution can be re commenced at that point at which calculation of the Jacobian matrix had most recently been completed by typing the command ppest case j PSLAVE must be started in th
318. hi No more lambdas Lowest phi this 12743 11730 1852 94 28 relative phi reduction between lambdas less than 0 0300 iteration BITE Current parameter values 0 500000 rol Previous parameter values 0 500000 rol Running PEST 5 5 ro2 10 0000 ro2 5 00000 ro3 1 00000 ro3 0 500000 hl 1 94781 h1 2 00000 h2 10 4413 h2 5 00000 Maximum factor parameter change 2 088 h2 Maximum relative parameter change 1 088 h2 OPTIMISATION ITERATION NO 2 Model calls so far 6 Starting phi for this iteration 357 3 Contribution to phi from observation group group_1 77 92 Contribution to phi from observation group group_2 103 8 Contribution to phi from observation group group_3 121 3 Contribution to phi from observation group group_4 54 28 Lambda 1 250 gt parameter ro2 frozen gradient and update vectors out of bounds phi 252 0 0 71 of starting phi Lambda 0 6250 gt phi 243 6 0 68 of starting phi Lambda 0 3125 wnead gt phi 235 9 0 66 of starting phi Lambda 0 1563 gt phi 230 1 0 64 of starting phi No more lambdas relative phi reduction between lambdas less than 0 0300 Lowest phi this iteration 230 1 Current parameter values Previous parameter values rol 0 500000 rol 0 500000 ro2 10 0000 ro2 10 0000 ro3 1 00000 ro3 1 00000 h1 1 41629 h1 1 94781 h2 31 3239 h2 10 4413 Maximum factor parameter change 3 000 h2 Maximum rela
319. hly nonlinear problems than for well behaved problems Note that the Marquardt lambda is one of the input variables that can be adjusted in mid run through user intervention see Section 5 6 5 5 10 Observations are Insensitive to Initial Parameter Values For some types of models observations can be insensitive to initial parameter values if the latter are not chosen wisely For example if you wish to optimise the resistivities and thicknesses of a three layered half space on the basis of electric current and voltage measurements made on the surface of that half space it would be a mistake to provide all the layer resistivities with the same initial value If you did the model would be insensitive to the thicknesses of either of the upper layers the lowest layer extends to infinity because the half space is uniform no matter what these thicknesses are Hence PEST will set about calculating derivatives of observations with respect to these thicknesses and discover that the derivatives of all observations with respect to all thicknesses are zero It will then issue a message such as parameter h1 has no effect on model output This problem can be easily circumvented by choosing initial layer resistivities which are different from each other In other cases the solution may not be as obvious For some models a certain parameter may have very little effect on model outcomes over part of its domain yet it may have a much greater effect
320. i i ee sieeg tee een canade a io deena en cee 9 12 9 3 1 Preparing for a Parallel PEST Run oo cee eeeceeeseceenecceseeeeseeeeeeeeeseeeeeeeeses 9 12 9 3 2 Startine the Slaves n seinna ta erecta atte caine 9 12 O73 23 Startins PES Taoa en syne E E A A towed E e EEE aS 9 13 9 3 4 Re Starting Parallel PEST ssssseenseeeseeesssesseesssessssesseeesseesseesseessresssesssesssesssees 9 13 93 5 Parallel PES PEO S erre ee eeaeee Akae PE Aoa AEE E E aeiiae REEE rE TEET EKE EEEE 9 14 9 JO Losing Slaye Sienen int a a alin a a Aaa T h 9 14 9 3 7 The Parallel PEST Run Management Record File 0 0 ees eeeeseeeeeeeeeeeeeeees 9 15 9 3 8 Running PSLAVE on the Same Machine as Parallel PEST eee 9 17 9 3 9 Running Parallel PEST on a Multi Processor Machine 9 17 9 3 10 The Importance of the WAIT Variable ceeccceesecesecceeteeeeseeeeeneeeeseeees 9 17 9 3 11 If PEST will not Respond 0 eee eeeceseceseecececeseceseeeceesaeesaeesneeesaeesaeeeseesues 9 18 9 3 12 The Model minne nnna i E sath sen sedsseeassvaviestevedeesenes 9 19 DALAM Example AEEA E EE E NA E E EE 9 19 9 5 Frequently Asked Questions ccccscccsssccessseeesseceseeeeeaeceeseeceaeeeseeceaeesesaeeeeaeeesaas 9 19 T0 PEST Utilitie Surasi ge ea ere 10 1 10 1 TEMPCHEK ciii oae nea ccs e ae E E EE E E a 10 1 TO 2AINS CHEK a aE ees Aeee e A ee conve AE AE Ea EREE E E AEREE ERRER AS 10 3 10 3 PESTCHEK na a a a cena te donee TE ie wa Sean ie 10 4 TOA PESTGEN s oee erea E Aa EE E a E E Ea
321. icial or downhole measurements Though not referenced specifically the following discussion is just as applicable to PEST s usage in that role as it is to PEST s usage in model calibration PEST calibrates a model by reducing the discrepancies between model outputs and field observations to a minimum in the weighted least squares sense The differences between field measurements and model outputs are encapsulated in an objective function defined as the weighted sum of squared deviations between field observations and corresponding model outputs As PEST executes it progressively reduces this objective function until it can reduce it no more In many cases the landscape of the objective function in parameter space is not comprised of a discrete bowl shaped depression with the objective function minimum lying neatly at the bottom of that depression Rather especially if parameters number more than just a few the objective function minimum often lies at the bottom of a long narrow valley of almost equal depth along its length Any parameters for which the objective function lies within the valley can be considered to calibrate or almost calibrate the model Figure 6 1 illustrates this situation for a simple linear two parameter model for a linear model contours of the objective function in parameter space are always elliptical The situation in a more complex nonlinear case is schematised in Figure 6 2 In both of these figu
322. if PEST is run in predictive analysis mode then you must ensure that the PEST control file contains a predictive analysis section You must also ensure that there are at least two observation groups one of which is named predict and that the predict group has just one observation In most cases this will correspond to an output of the predictive component of a composite model The observation can have any name value and weight the latter two are ignored by PEST when run in predictive analysis mode Naturally you must supply an instruction file to read this extra observation from the pertinent model output file NPREDMAXMIN When PEST is used in predictive analysis mode its task is to maximise or minimise the single model prediction while maintaining the objective function at or below min 6 Cie o If NPREDMAXMIN is set to 1 PEST will maximise the prediction if NPREDMAXMIN is set to 1 PEST will minimise the prediction PDO PDO is a value for the objective function which under calibration conditions is considered sufficient to just calibrate the model It is equal to min ie Po see Section 6 1 1 A PEST predictive analysis run should be preceded by a parameter estimation run in which min is determined The user then decides on a suitable value for and hence o before supplying the latter as PDO for a PEST predictive analysis run Naturally PDO should be greater than min however in most circumstan
323. ights for the concentration measurements should be a factor of 10 greater than those for the head measurements so that both sets of measurements are equally effective in determining model parameters Some parameter estimation packages offer a log least squares option whereby the objective function is calculated as the sum of squared deviations between the logarithms of the measurements and the logarithms of their respective model generated counterparts Note that provided the linearity assumption upon which the estimation process is based is reasonably well met it can be shown that the same effect can be achieved by providing a set of weights in which each weight is inversely proportional to the measurement to which it pertains provided all measurements are of the same sign OBGNME OBGNME is the name of the observation group to which the observation is assigned When recording the objective function value on the run record file PEST lists the contribution made to the objective function by each observation group It is good practice to assign observations of different type to different observation groups In this way the user is in a position to adjust observation weights in order that one measurement type does not dominate over another in the inversion process by virtue of a vastly greater contribution to the objective function The observation group name supplied here must be one of the group names listed in the observation groups
324. ile PEST lists a number of pieces of information pertaining to the residuals taken as a whole and to the residuals pertaining to each observation group Much can be learned about residuals and about the efficacy of the parameter estimation process that gave rise to these residuals by different types of graphical inspection Graphs can be readily obtained using commercial plotting software based on the information contained in the residuals file and perhaps the rotated residuals file written by PEST at the end of its run These files can also be used as a basis for more sophisticated residuals analysis using commercial statistical software Ideally a plot of weighted rotated residuals against weighted or unweighted rotated observation values or weighted or unweighted simulated values should reveal no dependence of one upon the other unless weights were purposefully chosen to accentuate certain observation types Furthermore a plot of weighted rotated residuals against the normal variate should ideally reveal that residuals are normally distributed Perhaps the most important types of residuals analyses are those that can only be undertaken in a case specific manner For example if the model being calibrated is a steady state ground water model and the measurements are of water levels in various bores spread throughout the model domain then it is important that residuals be plotted at the locations of their respective boreholes and s
325. ile refer to the actual parameter You should never ask PEST to logarithmically transform a parameter with a negative or zero initial value or a parameter that may become negative or zero in the course of the estimation process Hence a log transformed parameter must be supplied with a positive lower bound see below PEST is informed if a parameter is log transformed through the parameter variable PARTRANS in the PEST control file see Section 4 2 4 Note that more complex parameter transformations can be undertaken using the parameter preprocessor PAR2PAR see Section 10 7 2 2 2 Fixed and Tied Parameters A parameter can be identified in a template file see Chapter 3 yet take no part in the parameter estimation process In this case it must be declared as fixed so that its value does not vary from that assigned to it as its initial estimate in the PEST control file PEST allows one or more parameters to be tied ie linked to a parent parameter PEST does not estimate a value for a tied parameter rather it adjusts the parameter during the estimation process such that it maintains the same ratio with its parent parameter as that provided through the initial estimates of the respective parameters Thus tied parameters piggyback on their parent parameters Note that a parameter cannot be tied to a parameter which is either fixed or tied to another parameter itself PEST is informed whether a parameter is fixed or tied thr
326. imental measurements satisfy the m equations expressed by equation 2 1 when the n optimal parameter values are substituted for the elements of b We define this optimal parameter set as that for which the sum of squared deviations between model generated observations and experimental observations is reduced to a minimum the smaller is this number referred to as the objective function the greater is the consistency between model and observations and the greater is our confidence that the parameter set determined on the basis of these observations is the correct one Expressing this mathematically we wish to minimise where is defined by the equation c Xb c Xb 2 3 and now contains the set of laboratory or field measurements the t superscript indicates the matrix transpose operation It can be shown that the vector b which minimises of equation 2 3 is given by b X X Xe 2 4 Provided that the number of observations m equals or exceeds the number of parameters n the matrix equation 2 4 provides a unique solution to the parameter estimation problem Furthermore as the matrix X X is positive definite under these conditions the solution is relatively easy to obtain numerically The vector b expressed by equation 2 4 differs from b of equation 2 1 the equation which defines the system in that the former is actually an estimate of the latter because ec now contains measured data In fact b of equa
327. imum parameter uniformity in both of these directions Similar matrices can be formed in order to impose a minimum curvature constraint or to implement any other type of regularisation criteria that the user may wish to impose see Chapter 7 for more details When working in regularisation mode PEST s task is to minimise while ensuring that m is suitably low Such a suitably low value will normally be slightly above the minimum value for m that could have been achieved if regularisation conditions had not been imposed It is the user s responsibility to select this value it will be denoted as Om ie the limiting measurement objective function It is this ability which PEST gives the user to indicate in advance of the parameter estimation process the extent of model to measurement misfit that is tolerable in order to attempt to satisfy imposed regularisation constraints on parameter values that makes the regularisation methodology as implemented by PEST so The PEST Algorithm 2 14 powerful Thus the regularisation process must minimise while enforcing the condition that Pm lt Om or in practice that Pm Pm 2 31 because a decrease in will nearly always require an increase in m where parameter values are close to optimum The constrained minimisation problem described above can be formulated as an unconstrained minimisation problem through the use of a Lagrange multiplier y Thus with regular
328. in length These twelve characters can be any ASCII characters except for or the marker delimiter character As discussed above a parameter name can occur more than once within a parameter template file PEST simply replaces each parameter space in which the name appears with the current value of the pertinent parameter However the same does not apply to an observation name Every observation is unique and must have a unique observation name In Example 3 6 observations are named arl ar2 etc These same observation names must also be cited in the PEST control file where measurement values and weights are provided see the next chapter for further details There is one observation name however to which these rules do not apply viz the dummy observation name dum This name can occur many times if necessary in an instruction file it signifies to PEST that although the observation is to be located as if it were a normal observation the number corresponding to the dummy observation on the model output file is not actually matched with any laboratory or field measurement Hence an observation named dum must not appear in the PEST control file where measurement values are provided and observation weights are assigned As is illustrated below the dummy observation is simply a mechanism for model output file navigation 3 3 6 The Instruction Set Each of the available PEST instructions is now described in
329. in parameters are highly correlated with each other each member of the correlated set could take on an infinite number of values provided other members of the set take on complementary values Imposition of a properly constructed regularisation condition will result in the selection of just one set of values for these correlated parameters ie the set of values that is most in harmony with the natural state of the system as defined by the regularisation conditions 7 1 3 Theory The theoretical underpinnings of the regularisation methodology provided in PEST is presented in Section 2 1 10 of this manual 7 2 Implementation in PEST 7 2 1 Regularisation Mode Regularisation 7 4 To introduce regularisation into the parameter estimation process in the manner described in Section 2 1 10 PEST must be run in regularisation mode This mode of operation is not too different from PEST s traditional parameter estimation mode in that an objective function is minimised the objective function being defined as the sum of squared weighted differences between observations and corresponding model outputs However when used in regularisation mode observations must be subdivided into measurement observations and regularisation observations so that PEST can calculate the contribution made to the total objective function by each of these Furthermore during each optimisation iteration just after the Jacobian matrix has been filled an
330. ing the operation of the Gauss Marquardt Levenberg method in determining the optimum upgrade vector can also be adjusted prior to repeating the calculation Thus you can interact with PEST assisting Introduction 1 9 it in its determination of optimum parameter values in difficult situations if you so desire At the end of the parameter estimation process the end being determined either by PEST or by you PEST writes a large amount of useful data to its run record file PEST records the optimised value of each adjustable parameter together with that parameter s 95 confidence interval It tabulates the set of field measurements their optimised model calculated counterparts the difference between each pair and certain functions of these differences These are also recorded on a special file ready for immediate importation into a spreadsheet for further processing Then it calculates and prints displays three matrices viz the parameter covariance matrix the parameter correlation coefficient matrix and the matrix of normalised eigenvectors of the covariance matrix 1 4 4 Predictive Analysis When used to calibrate a model the traditional use of PEST PEST is asked to minimise an objective function comprised of the sum of weighted squared deviations between certain model outcomes and their corresponding field measured counterparts When undertaking this task PEST is run in parameter estimation mode It is a sad fact of model usage
331. ing the progress of the parameter estimation process to its run record file PEST also prints some of its run time information to the screen through this means the user is informed of the status of the estimation process at any time If you are using the single window version of PEST and the model of which PEST has control writes its own output to the screen this will interfere with PEST s presentation of run record information to the screen Perhaps this will not worry you because it allows you to check that the model is running correctly under PEST s control in any case you can interrupt PEST execution to inspect the run record file at any time see the following section However if you find it annoying you may be able to redirect the model screen output to a file using the gt symbol in the model command line this will leave the screen free to display PEST s run time information only Thus if program VES cited in the PEST control file of Example 4 2 produces a verbose screen output which is of no real use in the parameter estimation process the model command line cited in the model command line section of the PEST control file could be replaced by ves gt temp dat or ves gt nul In the latter case screen output is simply lost for there is no nul file 5 3 8 Run time Errors As was discussed above PEST performs limited checking of its input dataset In the event of an error or inconsistency in its inpu
332. inite difference based derivatives Chapter 8 of this manual describes the mechanism by which PEST can receive derivatives calculated internally by a model In summary this kind of model PEST interaction requires that the model generate a file in which these derivatives are recorded Because the calculation of derivatives by the model may place an extra computational burden on the model s shoulders it is sometimes necessary that the model be run in a slightly different manner when calculating derivatives from that in which it is run when undertaking normal simulation This can be accommodated through the use of multiple model command lines and through PEST to model messaging See Chapter 8 for a discussion of both of these As is also described in Chapter 8 there will be some situations especially those involving calibration and predictive analysis for complex models in which it is possible for a model to calculate some of its derivatives but not others In cases such as this PEST can accept those derivatives from the model which the model is capable of calculating while computing the remaining derivatives itself by the traditional method of finite differences 2 4 Bibliography 2 4 1 Literature Cited in the Text Cooley R L 1983 Some new procedures for numerical solution of variably saturated flow problems Water Resources Research v19 no 5 p1271 1285 Cooley R L and Naff R L 1990 Regression modeling of ground water fl
333. inly a primary marker and the text string corresponding to that secondary marker is not found on a model output file line on which the previous markers strings have been located PEST will assume that it has not yet found the correct model output line and resume its search for a line which holds the text from all three markers Thus the instruction TIME STEP 10 will cause PEST to pause on its downward journey through the model output file at the first line illustrated in Example 3 12 However when it does not find the string STRAIN on the same line it recommences its perusal of the model output file looking for the string TIME STEP 10 again Eventually it finds a line containing both the primary and secondary markers and having done so commences execution of the next instruction line TIME STEP 10 13 ITERATIONS REQUIRED STRESS gt X 1 05 STRESS 4 35678E 03 X 1 10 STRESS 4 39532E 03 TIME STEP 10 BACK SUBSTITUTION STRAIN gt X 1 05 STRAIN 2 56785E 03 X 1 10 STRAIN 2 34564E 03 Example 3 12 Extract from a model output file It is important to note that if any instruction items other than markers precede an unmatched secondary marker PEST will assume that the mismatch is an error condition and abort execution with an appropriate error message The Model PEST Interface 3 18 o pif 5 STIME STEP
334. insensitive parameter then dominates the parameter upgrade vector When that parameter is held yet another troublesome parameter may be identified and so on All such parameters can be temporarily held at their current values if desired A user may hold such parameters one by one as they are identified in the manner described above or he she may prefer instead to take the pre emptive measure of temporarily holding at their current values all parameters identified in the parameter sensitivity file as being particularly insensitive and hence potentially troublesome 5 6 3 The Parameter Hold File After it calculates the Jacobian matrix and immediately before calculating the parameter upgrade vector PEST looks for a file named case hld where case is the filename base of the PEST control file in its current directory If it does not find it PEST proceeds with its execution in the normal manner However if it finds such a file it opens it and reads its Running PEST 5 33 contents in order to ascertain the user s wishes for the current optimisation iteration A parameter hold file is shown in Example 5 6 relparmax 10 0 facparmax 10 0 lambda 200 0 hold parameter thickl hold parameter thick2 hold group conduct lt 15 0 hold group thicknss lowest 3 hold eigenvector 1 highest 2 Example 5 6 Part of a parameter hold file Entries in a parameter hold file can be in any order Any line beginning with the charact
335. inty Hence parameter uncertainty is in itself not a big issue Parameter uncertainty is important only in so far as it effects predictive uncertainty and this depends on the prediction Unless of course the role of the inversion process is actually to infer parameters for their own sakes in this case the parameters themselves are the predictions The discussion that follows is directly applicable to this situation when parameters are considered in that light Figure 6 3 illustrates the dependence of a particular model prediction on parameter values for the two parameter system represented in Figure 6 2 As can be seen from Figure 6 3 the prediction of interest increases with both p and p2 the values of the two parameters If it is our desire to find the maximum prediction of this type that is compatible with the fact that model parameters must be constrained such that the model correctly simulates system performance under calibration conditions then this prediction is seen to correspond to the critical point depicted in Figure 6 4 This is the point of maximum model prediction on the min contour The model prediction made with parameter values corresponding to this critical point is the maximum model prediction that is compatible with calibration imposed constraints on parameter values In many instances of model usage this particular prediction is of greater importance than a model prediction made with best fit parameter values
336. io even though observations might be individually sensitive to each parameter comprising the group As is discussed in Chapter 2 this is a manifestation of the phenomenon of parameter correlation The damage done to the parameter estimation process by such an insensitive parameter combination can be every bit as bad as that done by insensitive parameters on their own Hence in some circumstances the parameter estimation process may benefit from the temporary removal of some or all members of such a damaging parameter combination In interpreting the above parameter holding command eigenvectors are counted in order of the magnitude of their corresponding eigenvalues starting from the highest eigenvalue and working down Thus eigenvector number 1 is the eigenvector associated with the highest eigenvalue of the covariance matrix This will be the eigenvector most commonly cited in parameter hold files supplied by the user because the magnitude of an eigenvalue is a measure of the insensitivity of the objective function to parameters varied in ratios specified by the elements of the corresponding eigenvector Once the desired eigenvector has been identified ie eigenvector number n in the above command PEST then selects the parameters which comprise the m largest components of that eigenvector These are the parameters which are collectively those to which the observation dataset is least sensitive PEST then holds these parameters at their curr
337. ion 7 4 3 Post Run Information At the end of the parameter estimation process PEST records information to its run record file to its residuals file to its parameter value file and to its observation sensitivity file Information recorded to the run record file is similar to that recorded to this file when PEST operates in parameter estimation mode Included in this information are parameter uncertainties and the parameter covariance and associated matrices Regularisation weights are multiplied by the optimised regularisation weight factor prior to computation of these matrices Caution should thus be exercised in interpreting the information contained in these matrices Regularisation information serves the very useful purpose of imposing a reality check on an overparameterised calibration problem and of adding numerical stability to the solution of that problem However it does not constitute part of the measurement dataset and hence should not really be used in the calculation of the uncertainty associated with each parameter that is estimated on the basis of that dataset Like the run record file produced by PEST when run in parameter estimation mode the run record file produced as an outcome of a regularisation run contains a listing of residuals and respective weights together with a brief statistical summary of the residuals pertaining to each observation group It should be noted that wherever weights are cited or are used in an
338. ion mode differs from a PEST control file suitable for use by PEST in parameter estimation mode in three ways viz 1 the PESTMODE variable must be set to regularisation rather than estimation 2 an observation group named regul must be present and 3 a regularisation section must be present at the end of the PEST control file Once a PEST control file has been built for PEST usage in regularisation mode it is a simple matter to use PEST in parameter estimation mode on the basis of the same input dataset All that needs to be done is to change PESTMODE from regularisation to estimation PEST will then run happily in parameter estimation mode ignoring the redundant regularisation section at the end of the PEST control file It will treat the observation group regul just like any other group and members of this group just like any other observations However if you wish to dispense with this regularisation information altogether assign all members of this group a weight of zero As was mentioned above parameter uncertainties and other statistics recorded at the end of the run record file after PEST was run in regularisation mode are calculated on the basis of user supplied weights for members of the observation group regul multiplied by the optimised regularisation weight factor However if you have just undertaken a regularisation run and you would like to calculate parameter uncertainties
339. ion 2 2 4 and the discussion on the SCALE and OFFSET variables below If a parameter is fixed taking no part in the optimisation process PARTRANS must be supplied as fixed If a parameter is linked to another parameter this is signified by a PARTRANS value of tied In the latter case the parameter takes only a limited role in the estimation process However the parameter to which the tied parameter is linked this parent parameter must be neither fixed nor tied itself takes an active part in the parameter estimation process the tied parameter simply piggy backs on the parent parameter the value of the tied parameter maintaining at all times the same ratio to the parent parameter as the ratio of their initial values Note that the parent parameter for each tied parameter must be provided in the second part of the parameter data section of the PEST control file If a parameter is neither fixed nor tied and is not log transformed the parameter transformation variable PARTRANS must be supplied as none The PEST Control File 4 18 Note that if a particular parameter estimation problem will benefit from a more complex parameter transformation type than logarithmic this can be accomplished using the parameter preprocessor PAR2PAR see Section 10 7 for further details PARCHGLIM This character variable is used to designate whether an adjustable parameter is relative limited or factor limited see Section 2 2
340. ion and predictive conditions PEST answers the question by attempting to minimise a new objective function which incorporates not just the differences between model generated and observed quantities under calibration conditions but the difference between the key prediction and the user specified value of this prediction when the model is run under predictive conditions Thus PEST is run in its traditional parameter estimation mode with a slightly altered objective function When run in this manner PEST s run time outputs are adjusted such that information on the key model prediction is recorded together with information on all other aspects of the parameter estimation process 1 4 5 Regularisation In its broadest sense regularisation is a term used to describe the process whereby a large number of parameters can be simultaneously estimated without incurring the numerical instability that normally accompanies parameter nonuniqueness Numerical stability is normally achieved through the provision of supplementary information to the parameter estimation process Such supplementary information often takes the form of preferred values for parameters or for relationships between parameters Thus if for a particular parameter the information content of the observation dataset is such that a unique value cannot be estimated for that parameter on the basis of that dataset alone uniqueness can nevertheless be achieved by using the suppl
341. ion groups as well as of each individual observation group The composite parameter sensitivity of each observation group is evaluated by calculating the magnitude of the respective column of the weighted Jacobian matrix using Equation 5 1 with the summation confined to members of that particular observation group The magnitude is then divided by the number of members of that observation group which have non zero weights When PEST is run in predictive analysis mode the observation group predict deserves special attention As was mentioned above it is not included in the computation of overall parameter sensitivities when PEST is run in this mode However because it is a separate observation group PEST lists the composite sensitivity to each parameter of the member of this group together with composite sensitivities of other observation groups at the end of the parameter sensitivity file The observation weight used in this calculation is the weight assigned to the observation comprising the sole member of the observation group predict in the PEST control file When working in predictive analysis mode this weight is ignored by PEST in actual predictive analysis calculations However it is not ignored in calculating the sensitivity of the sole member of this group to each adjustable parameter for the purpose of recording composite sensitivities pertaining to each observation group at the end of the parameter sensitivity file The user sh
342. is message demands an answer which if not provided can temporarily remove a particular slave from the Parallel PEST optimisation process If the message access denied appears on the PEST or PSLAVE screen this is a sure indication that WAIT needs to be set larger This occurs when PEST or PSLAVE attempts to delete one of the signal files of Table 9 1 after they have acted on the signal If they do this before the program which wrote the signal file has closed it as can happen if WAIT is set too small then the above message appears This is not serious however as no user response is required and both PEST and PSLAVE ensure that a particular signal file has in fact been deleted before they proceed with their execution The more serious case of a model trying to read or write a file that is still opened by PEST results in the operating system generated message Sharing violation reading drive C Parallel PEST 9 18 Abort Retry Fail to which a response is demanded If this occurs and you are there to respond simply press the r key If you are not there to respond the model cannot run furthermore the affected slave can take no further part in the optimisation process until the r key is pressed Experience will dictate an appropriate setting for WAIT However it is important to note that a user should err on the side of caution rather than setting WAIT too low A high setting for WAIT will certainly slow down communi
343. isation constraints imposed the parameterisation problem consists in determining the elements of the vector b ie the values of the adjustable parameters which minimise the total objective function defined by the equation P O ym 2 32 while simultaneously finding the value for y which causes equation 2 31 to be satisfied AS was mentioned above this problem is somewhat similar to the predictive analysis problem discussed in the previous section in that one function is minimised in this case the regularisation objective function while the objective function based on field or laboratory measurements is held at some upper limit selected by the user below which the model is deemed to be calibrated An inspection of equation 2 32 reveals that the regularisation problem can be viewed from a slightly different angle It can be formulated as a traditional nonlinear parameter estimation problem which attempts to estimate the parameter set b that provides the best fit between a set of observations and corresponding model outputs with the observations consisting of both measurement observations and regularisation observations The Lagrange multiplier y can then be considered as a factor by which to multiply the weights pertaining to the measurement observations in order to ensure that equation 2 31 is satisfied when the total objective function is minimised Alternatively if equation 2 32 is divided by y the parameter esti
344. isation weight factor that is u of equation 2 33 of the PEST manual This varies from optimisation iteration to optimisation iteration as PEST attempts to lower the regularisation objective function as far as it can while still maintaining the measurement objective function at or below a value at which the user has deemed the model to be calibrated 4 4 Parameter Sensitivity This window contains a plot of composite parameter sensitivities The composite sensitivity of any parameter describes the effect that an alteration to that parameter has on all model generated observations with observation weights applied employed in the optimisation WinPEST Graphical Windows Ll process see Section 5 3 2 of the PEST manual for further details This is a very useful variable for assessing the degree to which a parameter is capable of estimation on the basis of the observation dataset available for model calibration It can also be very useful in deciding which parameters to temporarily hold fixed in order to improve PEST performance through user intervention see Section 7 below Ry WinPEST C PEST2000 TESTISSOUNDING PST File Run Options Sensitivity View Yalidate eo w one 0v 8 amp 8 SS ARAR PEST Log Phi Sensitivity Parameters Lambda Calc vs Obs Residuals Jacobian gt Composite Sensitivity vs Iteration Number 2A T Sensitivity 133 33 Iteration Number Iteration Number 1 7 Sensiti
345. istinguish them from their laboratory or field acquired counterparts which are referred to as measurements or laboratory or field observations 3 2 Template Files 3 2 1 Model Input Files Whenever PEST runs a model as it must do many times in the course of the optimisation process it must first write parameter values to the model input files which hold them Whether the model is being run to calculate the objective function arising from user supplied initial parameter values to test a parameter upgrade or to calculate the derivatives of observations with respect to a particular parameter PEST provides a set of parameter values which it wants the model to use for that particular run The only way that the model can access these values is to read them from its input file s Some models read some or all of their data from the terminal the user being required to supply these data items in response to model prompts This can also be done through a file If you write to a file the responses which you would normally supply to a model through the terminal you can redirect these responses to the model using the lt symbol on the model command line Thus if your model is run using the command model and you type your responses in advance to the file file inp then you and PEST can run the model without having to supply terminal input using the command model lt file inp The Model PEST Interface 3 2 If file inp
346. ith the right mouse button anywhere within the window and selecting Close from the resulting pop up menu As is documented in the PEST manual TEMPCHEK allows a user to write a model input file on the basis of a template file and a parameter value file This aspect of TEMPCHEK functionality can be activated by selecting Options from the main WinPEST menu then Checking options and then Template file checking options A dialogue box appears in which you may select the template file which is to be checked and the parameter value file wherein parameter values are listed The model output file to be written by TEMPCHEK can also be denoted Then next time the user invokes TEMPCHEK by activation of WinPEST template file checking functionality the nominated model input file will be written on the basis of the selected template and parameter value files Similarly if you select Options then Checking options then Instruction file checking options from the WinPEST main menu you can instruct INSCHEK to generate an observation value file by reading selected numbers from a specified model output file next time it is run The name of the observation value file does not need to be supplied a name of file obf is automatically provided for this file file being the filename base of the selected instruction file 3 3 Running PEST 3 3 1 Starting the Parameter Estimation Process Once a project has been read and the entire P
347. its Slaves Parallel PEST must communicate with each of its slaves many times during the course of the optimisation process It must inform each slave that it has begun execution it must command various slaves to run the model and it must receive signals from its slaves informing it that different model runs have reached completion It must also inform all slaves to shut down under some circumstances of run termination and be informed by each slave when it commences execution that the slave itself is up and running Such signalling is achieved through the use of short shared signal files These files whether originating from PEST or PSLAVE are written to the directory from which PSLAVE is run on each slave machine PSLAVE provides these signal files with no path prefix thus ensuring that they are written to the directory from which its execution was initiated PEST however must be informed of the names of these various PSLAVE working directories most of which will reside on other machines through its run management file Note that there is no reason why a PSLAVE working directory should not also be a model working directory on any slave machine as long as filename conflicts are avoided In fact such an arrangement being simpler mitigates against making mistakes Table 9 1 shows the names of the signal files used by PEST and PSLAVE to communicate with each other If Parallel PEST carries out two simultaneous model runs on the one machin
348. ity Files pertaining to this example can be found in the edpestex subdirectory of the PEST directory after installation See Chapter 12 for a more fully discussed example of the use of PEST and its utilities In that example derivatives are calculated using finite differences File polymod f contains the source code for a simple program which computes the ordinates of a third degree polynomial at a number of different abscissae That is it computes the Model Calculated Derivatives 8 10 function year b text d 8 1 for different values of x It reads these values of x from a file named poly_x in and writes its computed values of y to a file named poly_val out Parameter values ie the values of the polynomial coefficients a b c and d are read from a file named poly_par in As well as computing values of y corresponding to different values of x POLYMOD also computes a prediction in this case a function of the parameter values given by the equation p a 2b 3c 4d 8 2 The prediction is written to the end of file poly_val out following the computed polynomial values POLYMOD also computes a Jacobian matrix this is a matrix of the derivative of y with respect to each parameter ie a b c and d at each value of x This is stored internally in the JACOB matrix and written to a derivatives file in the format expected by PEST The name of this file is poly_der out A template file named poly_par tpl has been pre
349. ivatives Command Line Section cece eeeceececeeeceeeeesceceeceaeeeeeeseeesaeeees 8 8 8 5 5 Model Command Line Section 0 ceccceessseesscceeneecesneeeeeeeceneeeeseeeeeeeesaeeees 8 9 S O AMEXaMple c4ccciteetese tet E E EE aul baad E T 8 9 9 Parallel PEST gt niet annseo naras nete E AEE EE EEES Aiie a a 9 1 9 1 Intr d ct onan ienest eene RS T EEE EA E eaa NEEE Ea E EEEE 9 1 O GOMeral E E SS 9 1 9 1 2 Parallelisation of the Jacobian Matrix Calculation Process ccccsscesseeeeees 9 1 9 1 3 Parallelisation of the Marquardt Lambda Testing Process s eeeesenceeeeeeereeee 9 2 ITAA Wali eene oreren a eea esee E eree E EEEa OERE 9 3 9 1 5 Installing Parallel PEST 00 ec cescesccessecencecseeceaeceaeeeceesaeeeaeeeeeesaeesaeeeaeeeseees 9 3 9 2 How Parallel PEST Works 0 cece ccccccccsssecseccccccccessscescesccseeeessscescesseuseaesesceseesees 9 3 Table of Contents 9 2 1 Model Input and Output Files eee eecesecseeceneceseeeseeeseeeeaceeseeesaeesaeeeseeeeeees 9 3 9 2 2 The PEST Slave Program ninesini eia a iE 9 4 9 2 3 Running the Model on Different Machines ce seeccessceseceeeceeeeeeeseeeeeeseees 9 5 9 2 4 Communications between Parallel PEST and its Slaves cceeececsssceeeteeeteees 9 6 9 2 5 The Parallel PEST Run Management File cee eecceseeeeneeeseeeeeeeeeeeeeeeeseees 9 7 9 2 6 More on Partial Parallelisation of the Marquardt Lambda Testing Process 9 11 9 3 Using Parallel PEST orie
350. ive analysis tasks for a wide range of model types Thus PEST can turn just about any existing computer model be it a home made model based on an analytical solution to a simple physical problem a semi empirical description of some natural process or a sophisticated numerical solver for a complex boundary value problem into a powerful nonlinear parameter estimation package for the system which that model simulates 1 3 What Pest Does Models produce numbers If there are field or laboratory measurements corresponding to some of these numbers PEST can adjust model parameter and or excitation data in order that the discrepancies between the pertinent model generated numbers and the corresponding measurements are reduced to a minimum It does this by taking control of the model and running it as many times as is necessary in order to determine this optimal set of parameters and or excitations You as the model user must inform PEST of where the adjustable parameters and excitations are to be found on the model input files Once PEST is provided with this information it can rewrite these model input files using whatever parameters and excitations are appropriate at any stage of the optimisation process You must also teach PEST how to identify those numbers on the model output files that correspond to observations that you have made of the real world Thus each time it runs the model PEST is able to read those model outcomes which must be matched to
351. jective function by a relative amount equal to PHIREDLAM on successive parameter upgrade attempts using successive Marquardt lambdas PEST will move on to the next optimisation iteration When PEST is run in predictive analysis mode as the objective function approaches PDO the relative change in the objective function between Marquardt lambdas may be small however the relative reduction in o ie the objective function minus PDO may be sufficient to warrant testing the efficacy of another Marquardt lambda The objective function value at which PEST stops testing for a relative objective function reduction and begins testing for a relative reduction in o is PD2 Generally this should be set at 1 5 to 2 times PDO In either case the decision as to whether to try another lambda or move on to the next optimisation iteration is made through comparison with PHIREDLAM Predictive Analysis 6 14 ABSPREDLAM and RELPREDLAM During each iteration after it has filled the Jacobian matrix PEST tests the ability of a number of different values of the Marquardt lambda to achieve its objective Its exact objective depends on the current value of the objective function If the objective function is above PD1 PEST s highest priority is to lower it if the objective function is less than PD1 PEST s highest priority is to raise or lower depending on the value of NPREDMAXMIN the model prediction In either case PEST is constantly faced with
352. ked in this way INSCHEK simply reads the instruction file insfile checking that every instruction is valid and that the instruction set is consistent If it finds any errors it writes appropriate error messages to the screen You can redirect this screen output to a file if you wish by using the gt symbol on the command line Thus to run INSCHEK such that it records any errors found in the instruction file model ins to the file errors chk use the command inschek model ins gt errors chk If no errors are found in the instruction file insfile INSCHEK informs you of how many observations it identified in the instruction set and lists these observations to file obf where file is the filename base ie the filename without its extension of insfile if insfile has no extension the extension obf is simply appended to the filename For an instruction set to be useable by PEST it must do more than simply obey PEST protocol it must also read a model output file correctly You can check this by invoking INSCHEK with the command inschek insfile modfile When run in this way INSCHEK first checks insfile for syntax errors if any are found it writes appropriate error messages to the screen and does not proceed to the next step Alternatively if the instruction set contained in insfile is error free INSCHEK reads the model output file modfile using the instruction set If any errors are encountered in this process INSCHEK generates
353. ks in the way that SENSCHEK does In fact SENSAN may run happily if provided with certain erroneous datasets however SENSAN s results under such conditions will be misleading Thus it is most important that SENSCHEK be run prior to SENSAN once all SENSAN input files have been prepared 11 3 2 Running SENSCHEK SENSCHEK is run using the command SENSAN 11 9 senschek infile where infile is the name of the SENSAN control file If the latter possesses an extension of sns then this extension can be omitted from the filename in the same manner that the pst extension can be omitted from the name of the PEST control file when running PEST and PESTCHEK SENSCHEK writes its error messages to the screen It is important to note that if SENSCHEK detects certain errors early in the SENSAN control file it may not proceed with its checking of the remainder of this file nor of the template and instruction files cited in the SENSAN control file nor of the parameter variation file Thus it is important to ensure that once a SENCHEK identified error has been rectified SENSCHEK is run again Only when SENSCHEK explicitly informs the user that no errors have been detected in the entire SENSAN input dataset is it safe to run SENSAN 11 4 Running SENSAN 11 4 1 SENSAN Command Line SENSAN is run using the command sensan infile where infile is the name of a SENSAN control file If the latter possesses an extension of sns it is not
354. l need to write a simple program which reads this binary data and rewrites it in ASCII form PEST can then read the ASCII file for the observations it needs Note that as described in Section 4 2 8 when PEST runs a model this model can actually consist of a number of programs run in succession Unfortunately observations cannot be read from model output files using the template concept This is because many models cannot be relied upon to produce an output file of identical structure on each model run For example a model which calculates the stress regime in an aircraft wing may employ an iterative numerical solution scheme for which different numbers of iterations are required to achieve numerical convergence depending on the boundary conditions and material properties supplied for a particular run If the model records on its output file the convergence history of the solution process and the results of its stress calculations are recorded on the lines following this the latter may be displaced downwards depending on the number of iterations required to calculate them So instead of using an output file template you must provide PEST with a list of instructions on how to find observations on an output file Basically PEST finds observations on a model output file in the same way that a person does A person runs his her eye down the file looking for something which he she recognises a marker if this marker is properly The Mod
355. l will receive the highest priority in the allocation of system resources 9 3 9 Running Parallel PEST on a Multi Processor Machine Parallel PEST can be used to harness the full potential of a multi processor machine Such a machine can either be used on its own or as part of a network of other machines some of which may possess multiple processors and some of which may not On a single dual processor machine operation of Parallel PEST is identical to that described above for machines across a network Three command line windows must be opened two of which are used to run PSLAVE and one of which is for the use of PEST A different set of model input files residing in a different subdirectory must be prepared for the use of each instance of PSLAVE Each PSLAVE must also have its own separate working directory which for convenience may just as well be the subdirectory holding its set of model input files Note that each PSLAVE can invoke the same model executable 9 3 10 The Importance of the WAIT Variable The role of the WAIT variable was briefly discussed in Section 9 2 5 As was outlined in that section an appropriate value for this variable gives machines across the network time to respond to the information sent to them by other machines If WAIT is set too small the potential exists for conflicts to occur resulting in a message on the PEST or PSLAVE screen sent by the operating system In some cases as mentioned in Section 9 2 5 th
356. l associated with that prediction Most aspects of PEST usage in predictive analysis mode are identical to PEST s usage in parameter estimation mode In particular bounds can be placed on adjustable parameters one parameter can be tied to another parameters can be logarithmically transformed for greater problem linearity troublesome parameters can be temporarily held while the parameter upgrade vector is re calculated etc However the end point of the iterative solution process is no longer a minimised objective function it is a maximised or minimised prediction with the objective function being as close as possible to that defining the acceptable limit for model calibration Once a key model prediction has been identified it is also possible to ask another important question of PEST The question is is it possible to find a parameter set for which the key model prediction is a certain value while still maintaining the calibration objective function at Introduction 1 10 or below the acceptable calibration limit This is similar to the question that is answered by running PEST in predictive analysis mode However it differs slightly from that question in that a maximum or minimum prediction is no longer being sought rather the acceptability of a certain prediction in terms of the model s ability to satisfy calibration constraints is being tested Once again a dual model is required in which the model is run under both calibrat
357. l calibration can be used with any nonlinear parameter estimator However it is particularly easy to implement with PEST One reason for this is the model independent nature of PEST which allows it to be used as easily with a composite model encapsulated in a batch file as with a model comprised of a single executable Another reason lies in the robust nature of the PEST inversion algorithm Yet another reason lies in the fact that PEST prints out the contribution made to the objective function by different observation groups Thus if historical observations used with the calibration component of the composite model are assigned to one observation group and the single model prediction made under future conditions is assigned to another observation group the user can see at a glance what the calibration component of the objective function is and what the predictive component of the objective function is Together these add up to the total objective function whose task it is for PEST to minimise during the dual calibration process If one of the names of the observation groups supplied to PEST is predict and if there is only one observation belonging to that group this observation can have any name then PEST prints out the model calculated number corresponding to that observation every time it calculates a new parameter update vector Thus whenever it displays a new objective function it also prints out the line prediction x
358. lated on the basis of the optimised parameter set This is because it was fabricated to demonstrate a number of aspects of the parameter estimation process that are discussed in the following pages Note also that PEST was run in parameter estimation mode in order to produce the run record demonstrated in Example 5 1 As will be discussed in Sections 6 and 7 the run record produced as an outcome of a PEST run in predictive analysis or regularisation modes is slightly different Running PEST 5 3 Example 5 1 A PEST run record file Example 4 2 shows the corresponding PEST control file PEST RUN RECORD CASE manual Case dimensions Number of parameters Number of adjustable parameters Number of parameter groups Number of observations t A Number of prior estimates NONW Model command line ves Model interface files Templates ves tpl for model input files ves inp Parameter values written using single precision protocol Decimal point always included Instruction files ves ins for reading model output files ves out Derivatives calculation Param Increment Increment Increment Forward or Multiplier Method group type low bound central central central ro relative 1 0000E 03 1 0000E 05 switch 2 000 parabolic h relative 1 0000E 03 1 0000E 05 switch 2 000 parabolic Parameter definitions Name Trans Change Initial Lower Upper Group formation limit value bound bound rol fixed na 0 500000 na
359. later re use they are saved under the name case plottype pit where case is the current casename and plottype is a description of the type of plot being currently saved As is discussed in Section 2 2 individual plots can be saved as pictures by selecting Save as in the popup menu accessible by clicking on the right mouse button within each graphical window A plot can also be saved as a picture by selecting File from the main WinPEST menu then Save graphs then As picture 5 2 Printing Plots Graphs can be printed either individually or in groups by selecting Print from the WinPEST File menu Dual Calibration and Predictive Analysis 23 6 Dual Calibration and Predictive Analysis 6 1 Covariance Matrix and Related Plots Even though PEST s aim when running in predictive analysis mode is not to lower the objective function to a minimum but rather to maximise minimise a key model prediction while maintaining the objective function at or below a specified threshold covariance and related plots are nevertheless available in respective WinPEST windows The user should be aware of the fact that these plots will only be approximate in nature as strictly they should be calculated on the basis of optimised parameter values ie parameter values which correspond to the minimum objective function Nevertheless these plots are made available when PEST is run in predictive analysis mode as they can provide valuable
360. lculated by its preconditioned conjugate gradient matrix solution package As discussed in Section 2 3 3 variables such as this should be set small so that heads can be calculated with high precision the accurate calculation of head derivatives depends on this However if HCLOSE is set too low the conjugate gradient method may never converge to a point where the maximum head correction between successive conjugate gradient iterations is less than HCLOSE roundoff errors causing slight oscillatory behaviour In this case MODFLOW will terminate execution with an error message Unfortunately it may be very difficult to predict when this will occur behaviour of the solution method may be perfect for one parameter set and unsatisfactory for another Hence as PEST continually adjusts parameters for derivatives calculation and parameter upgrades there is a good chance that on at least one occasion there will be a solution failure When this happens PEST will not find the observations it expects on the model output file and will terminate execution with an appropriate error message One solution to this problem may be to set HCLOSE high enough such that convergence failure will never occur However this may result in mediocre PEST performance because of inaccurate derivatives calculation A better solution would be to recode MODFLOW slightly such that it reads HCLOSE from a tiny file called hclose dat and such that if it terminates execution because of so
361. le that a number lacking a decimal point may be misinterpreted see Section 3 2 6 5 5 12 Incorrect Instructions If the objective function cannot be lowered it is possible that PEST is reading the model output file incorrectly If a non numeric string is present on the model output file at a place where PEST has been lead to expect through its instruction set a number PEST will terminate execution with an appropriate error message However if a number is present where PEST expects one but this number is not the one that you intended PEST to read when you built the instruction set then it is unlikely that PEST s performance in lowering the objective function will be good You can check that PEST is reading the correct numbers by terminating PEST execution using the Stop with statistics option PEST will then list on its run record file the model generated observations corresponding to its best parameter set achieved so far As PEST also lists the residual corresponding to each model generated observation an incorrectly read model outcome may be apparent as that for which there is an unusually high residual Note that program INSCHEK can also be used to check that an instruction file does indeed read the correct numbers from a model output file See Chapter 10 5 5 13 Upgrade Vector Dominated by Insensitive Parameters This is a common cause of poor PEST performance It is discussed in greater detail in the following section 5
362. let its proposed relative change be p For factor limited parameters which are not log transformed define q for parameter j as qi Bu fb bj if u and b have the same sign and 2 46 qi Buj bj bj f ifuj and b have the opposite sign where b is the current value for the j th parameter and f is the maximum allowed factor change for all factor limited parameters Let the parameter for which the absolute value of q is greatest be parameter and let q for this parameter be q Finally let the log transformed parameter for which the absolute value of Bu is greatest be parameter k and let the element of Bu pertaining to this parameter be Bux Let io lo ko poi qor and Bouok define these same quantities for the previous iteration except that for the previous iteration they are defined in terms of actual parameter changes rather than proposed ones Now define s7 s2 and s3 such that 1 pi lpoi if i io 5 0 otherwise 2 47a s2 qi qoi if l lo s2 0 otherwise and 2 47b 53 Bux Bouok if k ko s3 0 otherwise 2 47c Let s be the minimum of s s2 and s3 and define p as p 34 s 3 s ifs gt 1 2 48a 0 1 2 s otherwise 2 48b Then oscillatory behaviour of the parameter estimation process can be mitigated by defining a new parameter upgrade vector v by v ppu 2 49 The PEST Algorithm 2 24 2 2 7 Temporary Holding of Insensitive Parameters The possibility of a parameter estimation pro
363. limiter character is assumed to be Implementation of the t60 instruction places the cursor on the following the TIME 3 string for the colon is in the sixtieth character position of the above line PEST is then directed to find the next character From there it can read the last number on the above line as a non fixed observation see below Fixed Observations An observation reference can never be the first item on an instruction line either a primary marker or line advance item must come first in order to place PEST s cursor on the line on which one or more observations may lie If there is more than one observation on a particular line of the model output file these observations must be read from left to right backward movement along any line being disallowed Observations can be identified in one of three ways The first way is to tell PEST that a particular observation can be found between and including columns n and n2 on the model output file line on which its cursor is currently resting This is by far the most efficient way to read an observation value because PEST does not need to do any searching it simply reads a number from the space identified Observations read in this way are referred to as fixed observations Example 3 14 shows how the numbers listed in the third solution vector of Example 3 8 can be read as fixed observations The instruction item informing PEST how to read a fix
364. ll lowering of as the estimation process progresses Testing the effects of a few different s in this manner requires that PEST undertake a few extra model runs per optimisation iteration however this process makes PEST very robust If the optimisation procedure appears to be bogged the adjustments made to in this fashion often result in the determination of a parameter upgrade vector that gets the process moving again 2 1 8 Optimum Length of the Parameter Upgrade Vector Inclusion of the Marquardt parameter in equation 2 23 has the desired effect of rotating the parameter upgrade vector u towards the negative of the gradient vector However while the The PEST Algorithm 2 11 direction of u may now be favourable its magnitude may not be optimum Under the linearity assumption used in deriving all equations presented so far it can be shown that the optimal parameter adjustment vector is given by Bu where u is determined using equation 2 23 and B is calculated as m 2 Vici coi wi Y i Pra Lewis 2 24 where once again the vector c represents the experimental observations co represents their current model calculated counterparts w is the weight pertaining to observation i and y is given by y X Ocoi Uj i Dos 2 25a ie y Ju 2 25b where J represents the Jacobian matrix once again If bo holds the current parameter set the new upgraded set is calculated using the equation b
365. log par3 1 342 1 00 obgpl RHI D SHI HM HSM HMM SI SI SM I KS The PEST Control File 4 27 However the following prior information article is illegal because of the break between log and par2 pil 2 0 Log parl E 2 5 log amp par2 3 5 log par3 1 342 1 00 obgpl 4 3 Observation Covariances 4 3 1 Using an Observation Covariance Matrix Instead of Weights As was discussed in Section 2 1 2 the use of observation weights in calculating the objective function is based on the premise that observations are independent ie that the uncertainty pertaining to any one observation bears no relationship to the uncertainty pertaining to any other observation However if residuals are likely to show consistency over space and or time for certain observation types then it may not be appropriate to assume statistical independence of these observation types In such cases it may be preferable to describe the uncertainties associated with these observations using an observation covariance matrix or a matrix that is proportional to this matrix rather than using a set of individual observation weights See Section 2 1 11 for more details Use of an observation covariance matrix can be particularly useful when prior information is employed in the inversion process especially if this prior information comprises the regularisation observations used by PEST when running in regularisation mode In many ca
366. low However if desired a line search procedure along the Predictive Analysis 6 9 direction of the parameter upgrade vector can be used to improve calculation of the maximum or minimum model prediction for any value of the Marquardt lambda This in fact is the recommended procedure When run in predictive analysis mode PEST can be stopped and restarted at any time it can be re started using the r or j switch just as in parameter estimation mode provided that the PEST RSTFLE variable was set to restart on the previous run Relative and factor change limits are just as important when PEST is used in predictive analysis mode as when it is used in parameter estimation mode Prior information can also be used naturally if it is used in a predictive analysis run following a parameter estimation run the prior information equations and weights should be the same for both runs Observation and observation group functionality in predictive analysis mode is identical to that in parameter estimation mode However as was mentioned above when PEST is used in predictive analysis mode there must be at least two observation groups one of which is named predict This group must contain only one observation of any name this being the observation corresponding to the single model output for which a maximum or minimum is sought within the constraints of o All other observations for this PEST run must be identical in value and w
367. lues read 20 nx do 100 i 1 nx read 20 x i 100 continue close unit 20 evaluate y for each x do 200 i 1 nx if x i le xc then y i s1 x i y1 else y 1 s2 x 1i sl s2 xctyl end if 200 continue c write the y values to the output file open unit 20 file out dat do 300 i 1 nx write 20 x 1i y i 300 continue close unit 20 end Example 12 1 A listing of program TWOLINE We would like TWOLINE to calculate specific volumes at water contents corresponding to our experimental dataset as set out in Table 12 1 The input file of Example 12 2 ensures that this will indeed occur Hence TWOLINE is now our system model We would like PEST to adjust the parameters of this model such that the discrepancies between laboratory and model generated specific volumes are as small as possible The parameters in this case are the four line parameters viz 51 52 y1 and xc Now that our model is complete our next task is to prepare the TWOLINE PEST interface An Example 12 5 0 8 w amp Ww Ws 052 068 103 128 172 295 230 219 s315 2332 350 423 488 O OO OG OOG OGO a O O 0 0 HOOO Example 12 2 A TWOLINE input file in dat 0 520000E 01 0 415600 0 680000E 01 0 420400 0 103000 0 430900 0 128000 0 438400 0 172000 0 451600 0 195000 0 458500 0 230000 0 469000 0 275000 0 482500 0 315000 0 502000 0 332000 0 515600 0 350000 0 530000 0 423000 0 588400 0
368. lution convergence failure it does so with a non zero errorlevel The PEST Control File 4 23 setting of say 100 See a DOS manual or help file for a description of this setting most compilers allow you to set the errorlevel variable on program run completion through an appropriate exit function call Then write two small programs one named HMUL which reads hclose dat multiplies HCLOSE by 2 and then rewrites hclose dat with the increased HCLOSE value the second program named SETORIG should write the original low value of HCLOSE to hclose dat A suitable model batch file may then be as shown in Example 4 3 echo off rem Set hclose to a suitably low value SETORIG rem Now run the model model MODF LOW rem Did MODFLOW converge if errorlevel 100 goto adjust goto end rem Multiply HCLOSE by 2 adjust HMUL rem Now run model goto model end Example 4 3 A batch file called by PEST as the model Note that there are alternative simpler solutions to the MODFLOW convergence problem discussed here the purpose of this example is to demonstrate the type of batch processing that may be useful as a PEST model run The variations on the content of a model batch file are endless You can call one model followed by another then by another The third model may or may not require the outputs of the other two PEST may read observations from the files generated by all the models or just from the file s generated
369. ly lower one or both of and m Marquardt lambdas are selected using a similar procedure to that used when PEST is working in parameter estimation mode For each parameter upgrade vector that it tests PEST lists the measurement and regularisation objective functions as well as the total objective function calculated on the basis of a model run undertaken with the trial parameters Note that the sum of the measurement and regularisation objective functions will not equal the total objective function unless the regularisation weight factor is unity see equation 2 33 7 4 2 Composite Parameter Sensitivities As described in Section 5 3 2 of this manual in the course of its execution PEST records the composite sensitivities of all adjustable parameters to a parameter sensitivity file The composite sensitivity of any parameter can be considered as the magnitude of the vector comprising the weighted column of the Jacobian matrix corresponding to that parameter divided by the number of observations Where some of the observations taking part in the parameter estimation process are regularisation observations their weights will change from optimisation iteration to optimisation iteration in accordance with the current value of the regularisation weight factor The changing weight factor alters the values of the composite parameter sensitivities Hence these sensitivities are not directly comparable from optimisation iteration to optimisation iterat
370. main into a large number of parameter blocks or rectangles arranged in rows and columns in a grid like structure and to then estimate the hydraulic conductivity within each such block Due to the large number of parameters requiring estimation a suitable regularisation scheme is essential One such scheme is to request that differences between block hydraulic conductivities or their logarithms in both the row and column directions assuming that the blocks are arranged in a grid pattern be reduced to a minimum The regularisation information can be supplied as prior information or the parameter differences comprising this information can be calculated by the model In the latter case they should be provided to PEST as a series of observations Hence whenever the model is run it provides not just the model generated counterparts to the heads measured at boreholes but also a series of hydraulic conductivity differences based on the current set of parameters used by the model If the pertinent regularisation observations are named d1 d2 etc part of the observation data section of the PEST control file may appear as in Example 7 5 Regularisation 7 16 o1 0 0 1 0 regul o2 0 0 1 0 regul 03 0 0 1 0 regul etc Example 7 5 Regularisation observations contained within the observation data section of a PEST control file The following points should be noted e The observed value of each regularis
371. mation problem can also be seen as the problem of minimising defined as P up pdr On 2 33 where yu is the reciprocal of y With the objective function defined in this way the reciprocal of the Lagrange multiplier can be seen to be equivalent to a regularisation weight factor ie the factor by which all regularisation observations are multiplied in formulation of the overall objective function now defined as amp Once again the value of u must be such that equation 2 31 is satisfied When undertaking problem regularisation PEST minimises the objective function defined by equation 2 33 During each optimisation iteration it calculates the regularisation weight factor y that results in equation 2 31 being satisfied It does this using an iterative procedure based on linearised model and regularisation conditions Once u has been determined all The PEST Algorithm 2 15 regularisation weights ie weights assigned by the user to the regularisation observations are multiplied by this factor a parameter upgrade vector is then calculated in the normal way The problem is then linearised again through calculation of a new Jacobian matrix and the process is repeated 2 1 11 Use of an Observation Covariance Matrix In the theory presented in Section 2 1 2 the squares of user supplied observation weights comprise the elements of the diagonal matrix Q often referred to as the cofactor matrix The inverse of Q is p
372. matrix to a binary file named case jac on every occasion of this matrix being filled This allows re commencement of execution with the j switch which as is explained in Section 5 6 may be useful if it is desired that PEST re calculate the parameter upgrade vector with certain recalcitrant parameters temporarily held fixed If the RSTFLE variable is set to norestart PEST will not intermittently dump its array or Jacobian data hence a later re commencement of execution after premature termination is impossible PESTMODE This character variable must be supplied as either estimation prediction or regularisation If set to estimation PEST will run in parameter estimation mode its traditional mode of operation if set to prediction PEST will run in predictive analysis mode if set to regularisation PEST will run in regularisation mode If PEST is run in predictive analysis mode you must ensure that the PEST control file contains a predictive analysis section You must also ensure that there are at least two observation groups one of which is named predict and that the predict group has just one observation See Chapter 6 for further details The PEST Control File 4 5 If PEST is run in regularisation mode you must ensure that the PEST control file contains a regularisation section You must also ensure that there are at least two observation groups one of which is nam
373. me values Under these circumstances SENSAN reports to the screen that it cannot find an INSCHEK generated observation value file records the parameter values to its output file and moves on to the next parameter set Another reason why SENSAN may report that it cannot open a temporary observation file ie an INSCHEK generated file is that it was unable to run INSCHEK and or TEMPCHEK because their directories were not cited in the PATH environment variable Alternatively it SENSAN 11 13 may not have been able to run the model for the same reason If a parameter appears to be totally insensitive on SENSAN output files make sure that it has been provided with the same name in the parameter variation file as that provided for this same parameter in any template file in which it appears If parameter names are not identical between these two file types some parameter values as supplied to SENSAN in the parameter variation file cannot be written to model input file s Note however that such an error will be detected and recorded by SENSCHEK When undertaking a SENSAN run for the first time it is a good idea to set SCREENDISP to verbose so that TEMPCHEK and INSCHEK can report what they are doing to the screen After any errors have been corrected SCREENDISP can then be set to noverbose for routine SENSAN usage Similarly model output should not be directed to the nul file until it is verified that SENSAN through TEMPCHE
374. mely powerful aid to surface water model calibration Using TSPROC calibration can be undertaken based on one or more observed time series as well as important attributes of these time series such as volumes accumulated over various times certain flow statistics and flow exceedence fractions Functionality is included for automatic construction of PEST input files even for complex parameterisation problems 1 3 Upgrading PEST and its Utility Software The command line version of PEST all of its support software and all of the groundwater and surface water utility software discussed above is now public domain If at any stage you would like to obtain an upgrade go to either of the following site and follow the links http www sspa com http www waterloohydrogeologic com Overview 4 2 Overview 2 1 General WinPEST is a version of PEST that contains all of the functionality of the command line version of PEST However unlike the command line version of PEST WinPEST allows the user to run PEST in a graphical Windows environment where many of the variables and outcomes of the inversion process can be viewed in graphical format For those quantities which change value during the course of the optimisation process for example parameter values the objective function composite parameter sensitivities etc an evolving run time display of the values of these variables can be viewed in the WinPEST interface For those variables which a
375. mences execution First it reads the PEST control file and then the run management file Then it attempts to write to and read from the PSLAVE working directory on each slave machine in order to verify that it has access to each such directory It also informs each PSLAVE that it has commenced execution and waits for a response from each of them Once it has received all necessary responses it commences the parameter optimisation process Operation of Parallel PEST is very similar to that of PEST However whenever a model run must be carried out Parallel PEST selects a slave to carry out this run If model runs are to be conducted one at a time as do for example the initial model run and the sequence of model runs in which parameter upgrades are tested if the user has decided not to parallelise the lambda search procedure by setting the PARLAM variable to zero Parallel PEST selects the fastest available slave to carry out each run Initially it knows which slave is fastest from the estimated model run times supplied in the run management file However once it has completed a few model runs itself Parallel PEST is able to re assess relative slave machine execution speeds and will use these upgraded machine speed estimates in allocating model runs to slaves The Parallel PEST run manager is intelligent to the extent that if a model run is significantly late in completion Parallel PEST fearing the worst allocates that same model run to anot
376. meter If JACFILE in the control data section of the PEST control file is set to 1 then the model will be called specifically to calculate external derivatives after PEST has calculated derivatives using finite differences for all parameters with a non zero DERCOM value As is stressed above to ensure that a finite difference calculated derivative is not overwritten when PEST reads the derivatives matrix from the external derivatives file the model should fill all elements of the derivatives matrix for which it has not actually calculated an external derivative with a value of 1 1le33 It is important to note that if JACFILE is provided with a value of 1 then the external derivatives command will be issued and PEST will read derivatives from the external derivatives file whether or not any parameter has been assigned a DERCOM value of 0 Thus if JACFILE is set to 1 PEST can only assume that for at least one parameter with a non zero DERCOM value finite difference calculated derivatives for some observations are to be supplemented by external derivatives for others It is again emphasised that when writing code to fill the external derivatives matrix the user should take particular care to provide all elements of this matrix with a value of 1 11e33 unless an external derivative is actually calculated for that element Thus derivatives with respect to parameters for which external derivatives are not required will not be overwritten by spu
377. meter data section of the file may not be visible as they are written beyond the 80th column of the file to bring them into view move your editor s cursor over them Example 10 3 shows the default values used by PESTGEN in generating a PEST control file The following features in particular should be noted e PESTGEN assumes that PEST will be run in parameter estimation mode Neither a predictive analysis nor a regularisation section is included in the PEST control PEST Utilities 10 8 file e PESTGEN generates a separate parameter group for each parameter the name of the group is the same as that of the parameter For each of these groups derivatives are calculated using a relative increment of 0 01 with no absolute lower limit provided for this increment At the beginning of the optimisation process derivatives will be calculated using the forward method switching to the three point parabolic method on the iteration following that for which the objective function fails to undergo a relative reduction of at least 0 1 ie PHIREDSWH The derivative increment to be used in implementing the parabolic method is twice the increment used in implementing the forward method of derivatives calculation e No prior information is supplied e No parameters are tied or fixed no parameters are log transformed and changes to all parameters are relative limited with a RELPARMAX value of 3 0 The upper bound for each
378. meter values b can also be The PEST Algorithm 2 17 calculated using the equation b Y TY Y Td 2 38 where d is given by equation 2 35 and Y is calculated from the model matrix X using the equation Y RX 2 39 The matrix T in equation 2 38 is related to the inverse of the matrix D the covariance matrix of d by the same constant of proportionality that occurs in the relationship between the matrix Q of equation 2 37 and the inverse of the matrix C the covariance matrix of c Equation 2 38 thus demonstrates that the vector b can be calculated from the rotated measurement vector d using exactly the same mathematics as that used to compute b from the non rotated measurement vector c provided that the model matrix X is also rotated and that the cofactor matrix of d is used rather than the cofactor matrix of c What is important however is that for the reasons outlined above T is a diagonal matrix whose elements are proportional to the inverse of the eigenvalues of C and can thus be expressed as a set of weights Equation 2 38 with appropriate modifications for use in the context of a nonlinear model is the one used by PEST to calculate optimised parameter values however this is transparent to the user The relationship between the parameter covariance matrix C b and the observation cofactor matrix Q is expressed by equation 2 12 which is repeated below C b o X QX 2 40 Recall that o is the reference variance E
379. meters in the vector b Common sense tells us that we should be able to use the elements of c to infer the elements of b Unfortunately we cannot do this by recasting equation 2 1 as another matrix equation with b on the right hand side as X is not a square matrix and hence not directly invertible But you may ask Have we not made a rod for our own back by measuring the system response at more times than there are parameter values ie elements of b If b was of the same order as c X would indeed be a square matrix and may well be invertible If so it is true that an equation could be formulated which solves for the elements of b in terms of those of c However what if we then made just one more measurement of the system at a time not already represented in the n x n matrix X We would now have n 1 values of c which n of these would we use in solving for b And what would we do if we obtained as we probably would slightly different estimates for the components of b depending on which n of the n 1 values of we used in solving for b The problem becomes even more acute if the information redundancy is greater than one Actually as intuition should readily inform us redundancy of information is a bonus rather than a problem for it allows us to determine not just the elements of b but some other numbers which describe how well we can trust the elements of b This trustworthiness is based on the consistency with which the m exper
380. method of inferring half space properties is to assume that the earth under the measurement site is comprised of a small number of layers each of uniform resistivity PEST can then be asked to estimate the resistivity and thickness of each of these layers from the surficial voltage measurements An alternative method is to assume that there are a large number of geoelectric layers and that the boundaries between these layers are situated at logarithmically increasing depths below the surface PEST can then be asked to estimate the resistivities of these layers However because of the large number of layers and hence parameters to be estimated the level of parameter correlation and hence of nonuniqueness Regularisation 7 14 will be high Thus a regularisation constraint must be enforced to stabilise the problem This can take the form of a series of differences between the resistivities of successive model layers the observed value for each such difference being zero Assume that the names provided to PEST for the resistivity of each model layer are rol ro2 etc Assume also that these parameters are log transformed during the parameter estimation process If regularisation observations are added to the prior information section of the PEST control file this section will look something like Example 7 4 It is assumed that the subsurface has been subdivided into 15 different layers prior information pil 1
381. model input file generated on the basis of the same or another template file If either of these rules are violated PAR2PAR will inform you of this through an appropriate error message All template files cited in the template and model input files section of the PAR2PAR input file should be checked for correctness using TEMPCHEK While PAR2PAR will detect and report any errors that it finds in these files it will only report the first error that it encounters then it will cease execution TEMPCHEK on the other hand attempts to examine the entirety of a template file reporting all errors to the screen 10 7 2 3 Parameter Relationships The relationships by which parameter values are calculated from numbers or from values previously assigned to other parameters may be mathematical expressions of complex form They can include any or all of the and operators as well as brackets Note that the operator raises the number in front of the symbol to a power equal to the number trailing the symbol this operation can also be designated using the symbol as in the FORTRAN programming language Mathematical operations of equal rank are evaluated in the order followed by and followed by and as is the usual convention This order can be overridden by the use of brackets The following mathematical functions are supported in
382. n the model command line as provided to PSLAVE may have been inappropriate To find out more about the cause of the problem ascertain the name of the slave where the trouble occurred this should be apparent from the PEST error message issued on termination of its execution then shut down that slave and run the model from the slave working directory using the same command as that used by PSLAVE and watch what happens But before doing this check that the model output files are not in fact present after all If they are then they must have been misnamed in the Parallel PEST run management file If after following the steps outlined above the reason for the problem is still not apparent increase the value of the WAIT variable and try running Parallel PEST again 13 3 PEST and Windows NT A few problems have been encountered when using PEST and Parallel PEST with WINDOWS NT and WINDOWS 2000 The problems are not caused by PEST However this is not much consolation if PEST does not perform correctly Problems encountered include the following e A command line window used by PEST or PSLAVE closes in the middle of a model run Frequently Asked Questions 13 4 e PEST reports that it cannot find a model output file when the model output file is actually present e In a composite model written as a batch file and containing a number of executables run in succession one of the executable programs is not actually executed on one particul
383. n PEST s ability to estimate large numbers of parameters can be achieved by running PEST in regularisation mode see Chapter 7 for full details 5 5 4 Inappropriate Parameter Transformation PEST allows adjustable parameters to be either log transformed or untransformed A suitable choice for or against log transformation for each parameter can make the difference between a successful PEST run and an unsuccessful one Running PEST 5 28 Trial and error is often the only means by which to judge whether certain parameters should be log transformed or not There is no general rule governing which parameters are best log transformed however experience has shown that parameters whose values can vary over one or a number of orders of magnitude often benefit from log transformation Log transformation of these parameters will often linearise the relationship between them and the observations making the optimisation process more amenable to the linearity assumption upon which the equations of Chapter 2 are based Use of suitable SCALE and OFFSET variables can be used to change the domain of a parameter such that logarithmic transformation and with it the possible benefits of increased linearity becomes a possibility The use of parameter scaling and offsetting is discussed in Sections 2 2 4 and 4 2 4 More complex parameter transformations than logarithmic which may in some circumstances decrease the nonlinearity of a particular parameter estima
384. n aid to identification of recalcitrant parameters PEST now records the parameters that underwent maximum factor and relative changes during any parameter upgrade event these often being the parameters that create problems It is important to note that even though PEST has changed somewhat and includes a number of new and powerful features file protocols used with previous versions of PEST are identical to those used by the latest version of PEST with one exception this is the addition of observation group data to the PEST control file However the new version of PEST is able to recognise a PEST control file written for an older PEST version and will read it without complaint assigning a dummy group name to all observations PEST has stood the test of time When it was initially released it offered entirely new possibilities for model calibration and data interpretation Slowly but surely the PEST user base is expanding as more and more scientists and engineers are realising the benefits that can be gained through the use of these possibilities The latest version of PEST which includes Parallel PEST and the options for user intervention briefly outlined above allows the use of PEST to be extended to the calibration of large and complex models to which the application of nonlinear parameter estimation techniques would have hitherto been considered impossible It is hoped that Preface new and existing PEST users can apply PEST to new and ex
385. n be zero if you wish meaning that the observation takes no part in the calculation of the objective function but it must not be negative If observations are all of the same type weights can be used to discriminate between field or laboratory measurements which you can trust and those with whom a greater margin of uncertainty is associated the trustworthy measurements should be given a greater weight Weights should in general be inversely proportional to measurement standard deviations If observations are of different types weights are vital in setting the relative importance of The PEST Control File 4 2 each measurement type in the overall parameter estimation process For example a ground water model simulating pollution plume growth and decay within an aquifer may produce outputs of ground water head and pollutant concentration Field measurements of both of these quantities may be available over a certain time period If both sets of measurements are to be used in the model calibration process they must be properly weighted with respect to each other Head measurements may be expressed in meters and pollutant concentrations may be expressed in meg l Heads may be of the order of tens of meters with model to measurement discrepancies of up to 0 1 m being tolerable however pollutant concentrations may be of the order of 10 meq l with model to measurement discrepancies of 0 1x10 meq 1 being tolerable In such a case the we
386. n by automating most of the tasks involved in this process Introduction 3 Included in the Groundwater Data Utilities are a number of programs which can be used to parameterise a MODFLOW model using pilot points and to introduce geostatistically based regularisation constraints into the parameterisation process When used in conjunction with the regularisation functionality of PEST ASP this software is more powerful by far than any other groundwater parameterisation software available today Note that two versions of some of these utilities are provided use the alternative version if you have trouble reading MODFLOW unformatted files This problem often arises because of disparities in unformatted file protocol used by different FORTRAN compilers Also included is the MODFLOW ASP suite This includes a MODFLOW2000 to PEST translator that translates a MODFLOW 2000 input dataset such as that produced by your favourite graphical user interface to a format readable by PEST ASP It also includes a version of MODFLOW 2000 that exchanges sensitivity information directly with PEST for extremely efficient MODFLOW 2000 calibration using PEST If you are wondering Why should I use PEST with MODFLOW 2000 instead of using parameter estimation functionality already supplied with MODFLOW 2000 please read the MODFLOW ASP documentation and find out The Surface Water Utilities include a comprehensive time series processor named TSPROC which is an extre
387. n curve should be plotted on the basis of all weighted residuals or just those which have been selected for display in the frequency distribution histogram choose Full distribution or Selected distribution from the pop up menu The y axis variable is also selectable choose Population Frequency or Relative frequency from the pop up menu In a Frequency plot the height of each bar of the WinPEST Graphical Windows 17 histogram represents the number of occurrences of a residual falling within the bounds represented by the bar In the Relative frequency plot each frequency is divided by the total number of samples represented in the plot so that the sum of relative frequencies adds up to one In the Population plot the area under the curve is unity If PEST is running in regularisation mode observations and or prior information comprising the observation group regul do not appear in the histogram plot unless specifically selected through the Select groups item of the pop up menu pertaining to this window 4 11 Covariance This plot will be available at all stages of the parameter estimation process if 1 the ICOV variable of the PEST control file has been set to 1 and 2 the normal matrix which must be inverted in calculation of the covariance matrix is not singular as a result of parameter non uniqueness If ICOV is set to 0 and the normal matrix is not singular then a plo
388. n file 3 3 4 The Marker Delimiter The Model PEST Interface 3 13 The first line of a PEST instruction file must begin with the three letters pif which stand for PEST instruction file Then after a single space must follow a single character the marker delimiter The role of the marker delimiter in an instruction file is not unlike that of the parameter delimiter in a template file Its role is to define the extent of a marker a marker delimiter must be placed just before the first character of a text string comprising a marker and immediately after the last character of the marker string In treating the text between a pair of marker delimiters as a marker PEST does not try to interpret this text as a list of instructions You can choose the marker delimiter character yourself however your choice is limited A marker delimiter must not be one of the characters A Z a z 0 9 6 or amp the choice of any of these characters may result in confusion as they may occur elsewhere in an instruction file in a role other than that of marker delimiter Note that the character you choose as the marker delimiter should not occur within the text of any markers as this too will cause confusion 3 3 5 Observation Names In the same way that each parameter must have a unique name so too must each observation be provided with a unique name Like a parameter name an observation name must be twelve characters or less
389. n is required until after at least one optimisation iteration has elapsed Even if it has met with little success in lowering the objective function PEST moves on to the next optimisation iteration commencing once again its time consuming calculation of the Jacobian matrix possibly having switched to the use of three point derivatives However the functionality exists within PEST to halt its execution at any time and restart it at that place at which it last commenced calculation of the parameter upgrade vector ie at the place at which it last completed its calculation of the Jacobian matrix Provided the PEST control variable RSTFLE is set to 1 PEST stores the Jacobian matrix in a binary file each time it is calculated the Jacobian matrix is easily retrieved if PEST is asked to re calculate the parameter upgrade vector Re commencement of PEST execution for upgrade vector re calculation is effected by running PEST using the command pest case j or if using Parallel PEST ppest case j 66599 where case is the current PEST case name ie the filename base of the PEST control file j stands for Jacobian Whether PEST was terminated while testing the efficacy of different Marquardt lambdas in lowering the objective function or whether it was terminated after the iteration counter had ticked over and PEST was engaged in calculation of a new Jacobian matrix PEST will re commence execution at the place at which the last Ja
390. n is assigned must also be cited in the observation groups section of the PEST control file As was discussed above the name of an observation group must be twelve characters or less in length If desired a particular item of prior information can belong to the same group as an observation cited in the observation data section of the PEST control file However it is difficult to see why this would be done because under normal circumstances the user will want to know the relative contributions made to the objective function by observations and prior information separately When writing articles of prior information you should note that no two prior information equations should say the same thing Thus the following pair of prior information lines is illegal 1 342 1 00 obgpl pil 2 0 bogi parl F 2 57 X 0 2 684 1 00 obgp2 2 log par3 pi2 4 0 log parl Sr log par3 Rao 0 If you wish to break a single prior information article into more than one line use the continuation character amp This must be placed at the beginning of each continuation line separated from the item which follows it by a space The line break must be placed between individual items of a prior information article not within an item Thus the following lines convey the same information as does the first of the above pair of prior information lines pil amp 2 0 log parl 23 5 log par2 345
391. n process during the final optimisation iteration but not enough to warrant the undertaking of another optimisation iteration then sensitivities will not correspond exactly to optimised parameter values as PEST does not compute another Jacobian matrix before ceasing execution under these conditions Nevertheless sensitivities computed by PEST on the basis of the near optimal parameter values which it uses at the beginning of the last iteration will be a very close approximation to sensitivities calculated for PEST s final parameter estimates However if you would like to ensure that sensitivities correspond exactly to optimised parameter values you can do the following 1 Use program PARREP see Chapter 10 to build a new PEST control file based on optimised parameter values from the present run 2 Set NOPTMAX in that file to 1 thus requesting that PEST compute sensitivities and then cease execution 3 Perhaps set the FORCEN variable for each parameter group to always_3 thus ensuring that PEST calculates derivatives with maximum precision 4 If working in regularisation mode set the initial weight factor WFINIT to the optimal weight factor determined on the present optimisation run Running PEST 5 19 Then run PEST When writing completion parameter sensitivities to the end of the parameter sensitivity file PEST lists the composite sensitivity and relative composite sensitivity to each parameter of all observat
392. n the model batch file to delete all intermediate files before any of the executable programs comprising the model are run If PAR2PAR is one such executable program then all model input files cited in the template and model input files section of the PAR2PAR input file should be deleted prior to running PAR2PAR Example 10 11 shows an example of a model batch file in which this precaution is taken rem Model input files written by PAR2PAR are deleted del modell in del model2 in rem PAR2PAR is run par2par par2par in rem The model is run model Example 10 11 A model batch file which includes PAR2PAR as one of the model executable programs In the batch file depicted in Example 10 11 file par2par in is the PAR2PAR input file If it is desired that screen output from all programs comprising the composite model including the model batch file itself be suppressed so that the model s screen output does not interfere with PEST Utilities 10 20 that of PEST the batch file shown in Example 10 11 could be altered to that shown in Example 10 12 echo off rem Model input files written by PAR2PAR are deleted del modell in gt nul del model2 in gt nul rem PAR2PAR is run par2par par2par in gt nul rem The model is run model gt nul Example 10 12 The batch file of Example 10 11 with all screen output suppressed SENSAN 11 1 11 SENSAN 11 1 Introduction In many modelling applications an analy
393. n the optimised parameter set to the run record file PEST records the measured observation values together with their model generated counterparts calculated on the basis of the optimised parameter set The differences between the two ie the residuals are also listed together with the user supplied set of observation weights Following the observations the user supplied and model optimised prior information values are listed a prior information value is the number on the right side of the prior information equation As for the observations residuals and user supplied weights are also tabulated Tabulated residuals and weighted residuals can also be found in file case res see Section 5 3 4 Composite observation sensitivities can be found in file case seo see Section 5 3 3 5 2 6 Objective Function Next the objective function is listed together with the contribution made to the objective function by the different observation groups 5 2 7 Correlation Coefficient The correlation coefficient pertaining to the current parameter estimation problem calculated using equation 2 43 is next listed 5 2 8 Analysis of Residuals The next section of the run record file lists a number of statistics pertaining to observation residuals first to all residuals and then separately to each observation group including any observation groups to which prior information was assigned Ideally after the parameter estimation process is complete weigh
394. ncy is multiplied by the squared weight pertaining to that article of prior information prior to inclusion in the objective function 2 1 4 Nonlinear Parameter Estimation Most models are nonlinear ie the relationships between parameters and observations are not of the type expressed by equations 2 1 and 2 2 Nonlinear models must be linearised in order that the theory presented so far can be used in the estimation of their parameters Let the relationships between parameters and model generated observations for a particular model be represented by the function M which maps n dimensional parameter space into m dimensional observation space For reasons which will become apparent in a moment we require that this function be continuously differentiable with respect to all model parameters for which estimates are sought Suppose that for the set of parameters comprising the vector bo the corresponding set of model calculated observations generated using M is Co ie co M bo 2 13 Now to generate a set of observations c corresponding to a parameter vector b that differs only slightly from bo Taylor s theorem tells us that the following relationship is approximately correct the approximation improving with proximity of b to bo c co J b bo 2 14 where J is the Jacobian matrix of M ie the matrix comprised of m rows one for each observation the n elements of each row being the derivatives of one particular observation
395. nd tabs which precede the non fixed observation on the instruction line If you do not know exactly where on a particular model output file line a model will write the number corresponding to a particular observation but you do know the structure of that line then you can use this knowledge to navigate your way to the number In the PEST instruction file a non fixed observation is represented simply by the name of the observation surrounded by exclamation marks as usual no spaces should separate the exclamation marks from the observation name as PEST interprets spaces in an instruction file as denoting the end of one instruction item and the beginning of another When PEST encounters a non fixed observation instruction it first searches forward from its current cursor position until it finds a non blank character PEST assumes this character is the beginning of the number representing the non fixed observation Then PEST searches forward again until it finds either a blank character the end of the line or the first character of a secondary marker which follows the non fixed observation instruction in the instruction file PEST assumes that the number representing the non fixed observation finishes at the previous character position In this way it identifies a string of characters which it tries to read as a number if it is unsuccessful in reading a number because of the presence of non numeric characters or some other problem PEST terminates ex
396. nder calibration conditions is as high as 5 0E 3 this is thus the value assigned to the variable PDO The PEST control file used in the parameter estimation process was named twofit pst This file should be copied to file twofitl pst and the following alterations made actually this has already been done for you 1 2 Replace the word estimation with the word prediction on the third line of this file When undertaking a predictive analysis run there is an extra observation this being the model prediction Hence the number of observations ie NOBS must be increased from 13 to 14 on the 4 line of file twofitl pst There will now be two observation groups so alter the 55 entry on line 4 of file twofitl pst ie NOBSGP to 2 There are now two model input files and two model output files so alter the first two entries on the 5 line of file twofitl pst ie NTPLFLE and NINSFLE to 2 and 2 respectively In the observation groups section of the PEST control file add the observation group predict In the observation data section of the PEST control file add an extra observation named 014 Assign this to the observation group predict Provide whatever observation value and weight that you like as these are ignored by PEST when run in predictive analysis mode It is probably best to make both of these 0 0 just in case you wish to run PEST later using the same file
397. neceneeececceceseceseeesceeaeesaeesseeesaeesaeesseesues 1 13 1 5 4 Sensitivity Amal ysis ceson neniani i i a a ii 1 13 1 6 This Manual ieoa eaei ie E EE E E AEREE ENARE PEE RETE 1 14 2 The PEST Algorithms eraa a e a a aaie ar a aa i Eaa 2 1 2 1 The Mathematics of PEST ce eeccesccssccescesscesscessceesaeessecsseceseeeeeeeeeseneceaeecaeeesaes 2 1 2 1 1 Parameter Estimation for Linear Models 00 cee scesccsscceeseeesceesceeeeeesaeeeseeeseees 2 1 2 1 2 Observation Weights ee eoero eaaet aea E anea ao E EE e EEE E EE Eia eaS 2 3 2 1 3 The Use of Prior Information in the Parameter Estimation Process 2 5 2 1 4 Nonlinear Parameter Estimation sseesseeseeesessesresresressessresressessrsrrerressessreereseese 2 6 2 1 5 The Marquardt Parameter sseseeeeeeeeeeessesesresressessessresrerressesstssrenresressessrssresees 2 8 PNE O E T E E E A A E E A 2 9 2 1 7 Determining the Marquardt Lambda sessessseeseeseseessseeresrsssessessrerresrsssrssrsreses 2 10 2 1 8 Optimum Length of the Parameter Upgrade Vector seeeseeenereereeeeeserreeeeee 2 10 21 9 Predictive Analysis cscc cc sieceiisegereesenstecseedeueeeedens lecess sede oa Ea En SE ah 2 11 2 LAO SSE ETTET OM E E EE EE 2 12 2 1 11 Use of an Observation Covariance Matrix ccccessseeeseceeteeceeeeeeeseeeeeaeeeees 2 15 AI IPA EJTETTEK K Bb AE E NEE E E E E 2 17 2 2 PEST s Implementation of the Method 0 eee eeseeseeeseeeseeeeceeeseeceeeeceeeceaeeeaeeees
398. necessary to include this extension in the SENSAN command line for SENSAN automatically appends sns to a filename supplied without extension It is important to ensure before SENSAN is run that the executable files tempchek exe and inschek exe are either in the current directory or are in a directory cited in the PATH environment variable As is mentioned above SENSAN runs both of these programs in the course of its execution the first to generate model input files and the second to read model output files 11 4 2 Interrupting SENSAN Execution To interrupt SENSAN type lt Ctl C gt 11 5 Files Written by SENSAN 11 5 1 SENSAN Output Files SENSAN produces three output files each of which is easily imported into a spreadsheet for subsequent analysis In each of these files the first NPAR columns contain the parameter values supplied to SENSAN in the parameter variation file The subsequent NOBS columns pertain to the NOBS model outcomes ie observations cited in the instruction file s supplied to SENSAN The first row of each of these output files contains parameter and observation names The last NOBS entries on each line of the first SENSAN output file ABSFLE simply list the SENSAN 11 10 NOBS model outcomes read from the model output file s after the model was run using the parameter values supplied as the first NPAR entries of the same line The second SENSAN output file RELFLE lists the relative differences bet
399. ner It is suggested that instruction files be provided with the extension ins in order to distinguish them from other types of file 3 3 1 Precision in Model Output Files As was discussed in the previous chapter if there are any model input variables which allow you to vary the precision with which its output data are written they should be adjusted for maximum output precision Unlike parameter values for which precision is important but not essential precision in the representation of model generated observations is crucial The Gauss Marquardt Levenberg method of nonlinear parameter estimation upon which the PEST algorithm is based requires that the derivative of each observation with respect to each parameter be evaluated once for every optimisation iteration PEST calculates these derivatives using the finite difference technique or one of its three point variants In all cases the derivative value depends on the difference between two or three observations calculated on the basis of incrementally varied parameter values Unless the observations are represented with maximum precision this is a recipe for numerical disaster 3 3 2 How PEST Reads a Model Output File PEST must be instructed on how to read a model output file and identify model generated observations For the method to work model output files containing observations must be text files PEST cannot read a binary file If your model produces only binary files you wil
400. nly one of these constraints ie a particular parameter must be either relative limited or factor limited in its adjustments Parameters are denoted as either relative limited or factor limited through the character variable PARCHGLIM supplied for each parameter see below The relative change in parameter b between optimisation iterations i 1 and i is defined as bia bi bi 4 3 If parameter b is relative limited the absolute value of this relative change must be less than RELPARMAX If a parameter upgrade vector is calculated such that the relative adjustment for one or more relative limited parameters is greater than RELPARMAX the magnitude of the upgrade vector is reduced such that this no longer occurs The factor change for parameter b between optimisation iterations i 1 and i is defined as bi bi if bia gt bi or bi bi t if b gt bi 4 4 If parameter b is factor limited this factor change which either equals or exceeds unity according to equation 4 4 must be less than FACPARMAX If a parameter upgrade vector is calculated such that the factor adjustment for one or more factor limited parameters is greater than FACPARMAX the magnitude of the upgrade vector is reduced such that this no longer occurs Whether a parameter should be relative limited or factor limited depends on the parameter However you should note that a parameter can be reduced from its current value right down to zero for a relative change of
401. not achievable Nevertheless PEST will probably still determine a parameter vector which minimises the objective function using the Marquardt parameter to make the normal matrix positive definite see equation 2 23 However a parameter vector determined in this way is not unique furthermore PEST will be unable to determine the parameter covariance matrix and to calculate parameter uncertainty levels because the Marquardt lambda is not added to the normal matrix prior to the determination of these quantities NOBSGP NOBSGBP another integer variable is the number of observation groups used in the current case Each observation and each prior information equation must be assigned to an observation group they can all be assigned to the same group if desired When PEST evaluates the objective function it also evaluates the contribution made to the objective The PEST Control File 4 6 function by the observations and prior information equations belonging to each group NTPLFLE This is an integer variable informing PEST of the number of model input files which contain parameters PEST must write each of these files prior to a model run As there must be one template file for each such model input file NTPLFLE is also equal to the number of template files which PEST must use in writing the current parameter set A model may have many input files however PEST is concerned only with those which it needs to rewrite prior to each model run ie
402. ntral derivatives does not result in a significant further lowering of the objective function nor in a dramatic change in parameter values the objective function having been reduced nearly as far as possible through the use of forward derivatives only However in other cases especially those involving a greater number of adjustable parameters than in the above example the introduction of central derivatives can often get a stalled optimisation process moving again The optimisation process of Example 5 1 is terminated at the end of optimisation iteration 7 after the lowest 3 ie NPHISTP objective function values are within a relative distance of 0 01 ie PHIREDSTP of each other Note that where PEST lists the current objective function value at the start of the optimisation process and at the start of each optimisation iteration it also lists the contribution made to the objective function by each observation group including the observation group group_4 comprised solely of prior information This is valuable information for if a user notices that one particular group is either dominating the objective function or is not seen as a result of dominance by another contributor he she may wish to adjust observation or prior information Running PEST 5 12 weights and start the optimisation process again 5 2 4 Optimised Parameter Values and Confidence Intervals After completing the parameter estimation process PEST prints the
403. o affect on model output Try changing initial parameter value increasing derivative increment holding parameter fixed or using it in prior information KKEKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KK KKK KKK KKK KKK KKK KKK KKK KKKKKKKKKKKKKKKKK Example 5 5 A PEST run time error message PEST run time errors are written both to the screen and to the PEST run record file 5 4 Stopping and Restarting PEST 5 4 1 Interrupting PEST Execution At the end of every model run PEST checks for the presence of a file named pest stp in the directory from which it was invoked If this file is present PEST reads the first item in the file If this item is 1 PEST ceases execution immediately If it is 2 PEST ceases execution after it prints out parameter statistics If it is 3 PEST pauses execution to resume PEST execution after a pause rewrite file pest stp with a 0 as the first entry File pest stp can be written by the user using any text editor while positioned in another window to that in which PEST is running However this file can also be written using programs PPAUSE PUNPAUSE PSTOP and PSTOPST supplied with PEST simply by typing the name of the appropriate program as a command while situated in the PEST working directory in another command line window As the names suggest PPAUSE writes a 3 to pest stp in order to tell PEST to pause execution PUNPAUSE writes a 0 to pest stp to tell it to resume exec
404. o agree exactly with the values of the PEST variables NPAR and NOBS in the PEST control file The derivative matrix is listed next in the file Some further aspects of this file are now discussed in detail These should be noted carefully for if the external derivatives file does not meet PEST s specifications it will not be read by PEST or what is worse may be read incorrectly by PEST 8 2 2 File Management As is further discussed below it is not necessary that the model supplies an external derivatives file if you do not want it to However if you notify PEST that the model will supply such a file then you must also inform PEST of the name of the external derivatives file and of the command which PEST must use to run the model in such a way that it writes this file Just before issuing this command it is issued once every optimisation iteration PEST first checks to see whether a derivatives file already exists If such a file does exist PEST deletes it Thus if the model fails to run PEST does not read the old file mistaking it for the new one instead it writes an error message to the screen informing the user that the derivatives file cannot be found Alternatively if PEST issues an error message to the effect that it encountered a premature end to the external derivatives file this indicates that either the model did not run to completion or that there is an error in the code added to the model to write this file 8 2 3 File Form
405. o every model run This can be accomplished by including the commands to run both programs in a batch file called by PEST as the model hence PEST can now write input files for the preprocessor rather than for the model itself Similarly the model output file may be voluminous in fact often models of this kind write their data to binary files rather than ASCII files relying on the user s postprocessing software to make sense of the abundance of model generated information You may have a postprocessing program which interpolates the model generated stress array to the locations of your stress sensors In this case PEST should read the postprocessor output file rather than the model output file Hence to use PEST in the parameterisation of the above stress field model a suitable model command line may be stress where stress bat is a batch file containing the following sequence of commands prestres stres3d postres Here PRESTRES and POSTRES are the model pre and postprocessors respectively STRES3D is the stress model itself You can get even more complicated than this if you wish For example a problem that can arises in working with large numerical models is that they do not always converge to a solution according to the model convergence criteria which you the user must supply The popular United States Geological Survey ground water model MODFLOW requires a variable HCLOSE which determines the precision with which heads are ca
406. observations while they could have been read as fixed observations we may have been unsure of just how wide a number can ever get in the second column of file out dat for example if a number becomes negative very large or very small pif 11 01 19 26 02 19 26 03 19 26 04 19 26 05 19 26 06 19 26 07 19 26 08 19 26 09 19 26 010 19 26 011 19 26 012 19 26 013 19 26 n n n n n L n n l H H H H H H H H H H H H fee E a e as pe eis pe es Example 12 7 The instruction file out ins Program INSCHEK should now be used to check that file out ins contains a legal instruction set Run INSCHEK using the command inschek out ins If no errors are encountered you should then run INSCHEK again this time directing it to read a TWOLINE output file using the instruction set use the command inschek out ins out dat INSCHEK will produce a file named out obf listing the values it reads from file out dat for the observations cited in file out ins see Example 12 8 An Example 12 8 ol 0 415600 o2 0 420400 o3 0 430900 o4 0 438400 o5 0 451600 o6 0 458500 o7 0 469000 o8 0 482500 o9 0 502000 0o10 0 515600 oll 0 530000 012 0 588400 ols 0 640400 Example 12 8 File out obf 12 1 5 Preparing the PEST Control File The PEST TWOLINE interface is now complete as PEST can now generate a TWOLINE input file and read a TWOLINE output file The next step is to generate a PEST control file
407. odel its parameterisation and its Running PEST 5 37 appropriateness for a specific application from an inspection of these quantities High levels of parameter uncertainty can result from a poor fit between model outcomes and field observations from a high level of parameter correlation from insensitivity on the part of certain parameters or from all of these Provided PEST has not faltered in the nonlinear parameter estimation process see Section 5 5 the first condition indicates that either field data is poor or that the model is inappropriate Or it may indicate that certain model parameters that were held fixed during the optimisation process may have been held at inappropriate values Fortunately the reason for the poorness of fit is often easy to identify though not necessarily easy to rectify especially if it requires alterations to model design Parameter uncertainty resulting from high levels of parameter correlation may or may not be a defect depending on the problem to which the model will be applied Highly correlated parameters are recognised through an inspection of the correlation coefficient matrix and through an inspection of the eigenvalues and eigenvectors of the parameter covariance matrix Whether indeterminacy of these parameters affects a particular model prediction depends on the prediction The ultimate test of this is to use PEST s predictive analyser see Chapter 6 in order to test the range of predictive
408. odel run 22 51 14 68 slave slave 3 commencing model run Example 9 4 Part of a Parallel PEST run management record file As is explained elsewhere in this manual if PEST or Parallel PEST execution is re commenced through use of the r or j command line switches the newly re activated PEST appends information to the run record file created on the previous PEST or Parallel PEST run The same is not true for the run management record file however for it is overwritten by a newly re activated Parallel PEST This is because as was mentioned above there is no necessity for Parallel PEST to employ the same slaves when it recommences Parallel PEST 9 17 execution as those which it employed in its previous life 9 3 8 Running PSLAVE on the Same Machine as Parallel PEST On many occasions of Parallel PEST execution at least one of the slaves will be run on the same machine as that on which PEST is run In most cases of PEST usage PEST s tasks are small compared with those of the model hence there is adequate capacity on one single processor machine to run both a model and PEST without serious diminution of the performance of either However it is wise to ensure that the command line window running PSLAVE and hence the model is the active window unless of course the user is running other interactive applications at the same time In this way the window undertaking most of the work ie the window running the mode
409. of between zero and 1 0 but normally less than about 0 3 can be supplied for this variable if you are unsure what value to use for PHIMLIM See Section 7 3 4 below for a full discussion of this variable WFINIT This is the initial regularisation weight factor During every optimisation iteration PEST calculates a suitable regularisation weight factor to use during that optimisation iteration using an iterative numerical solution procedure its initial value when implementing this procedure for the first optimisation iteration is WFINIT If there are many adjustable parameters calculation of the regularisation weight factor for the first optimisation iteration can be very time consuming if WFINIT is far from optimal Hence if you have any idea of what the weight factor should be for example from a previous PEST run then you should provide WFINIT with this value WFMIN WFMAX These are the minimum and maximum permissible values that the regularisation weight factor is allowed to take If a regularisation scheme is poor and does not lend too much stability to an already unstable parameter estimation process selection of appropriate values for WFMIN and WFMAX may be quite important for these can prevent PEST from calculating outrageous values for the regularisation weight factor in an attempt to compensate for inadequacies of the regularisation scheme A regularisation scheme should be such that even if there are no field measurements i
410. of model being used to study this system the ramifications of such differences may be extremely important The purpose of PEST which is an acronym for Parameter ESTimation is to assist in data interpretation model calibration and predictive analysis Where model parameters and or excitations need to be adjusted until model generated numbers fit a set of observations as closely as possible then provided certain continuity conditions are met see the next section PEST should be able to do the job PEST will adjust model parameters and or excitations until the fit between model outputs and laboratory or field observations is optimised in the weighted least squares sense Where parameter values inferred through this process are nonunique PEST will analyse the repercussions of this nonuniqueness on predictions made by the model The universal applicability of PEST lies in its ability to perform these tasks for any model that reads its input data from one or a number of ASCII ie text input files and writes the outcomes of its calculations to one or more ASCII output files Thus a model does not have to be recast as a subroutine and recompiled before it can be used within a parameter estimation process PEST adapts to the model the model does not need to adapt to PEST Thus PEST as a nonlinear parameter estimator can exist independently of any particular model yet can be used to estimate parameters and or excitations and carry out various predict
411. of parameter values Thus it is essential that the directory in which the executable file tempchek exe resides is either the current directory or is a directory cited in the PATH environment variable You can provide SENSAN with the name of a single template file in order that it can generate a single model input file Alternatively you may provide SENSAN with the names of many template files in order to generate multiple input files prior to running the model In either case before it runs the model SENSAN writes a parameter value file using the current set of parameter values as provided in the parameter variation file see below Then SENSAN runs TEMPCHEK for each model input file which must be produced It then runs the model Before running SENSAN you should always check the integrity of all template files which you supply to it by running TEMPCHEK yourself outside of SENSAN 11 2 3 Instruction Files Instruction files are discussed in Section 3 3 of this manual Model generated numbers can be read from one or many model output files as long as at least one instruction file is provided for each model output file The integrity of an instruction file can be checked using the PEST utility INSCHEK described in Section 10 2 INSCHEK is also capable of actually reading values from a model output file on the basis of a user supplied instruction file In order to avoid duplication of this functionality SENSAN runs INSCHEK to read model output files a
412. of the group regul e If an observation covariance matrix is supplied instead of observation weights for any observation group this is automatically taken into account when computing composite parameter sensitivities 5 3 2 2 Sensitivity Information Recorded on Termination of PEST Execution At the end of the parameter estimation process or if PEST is halted prematurely using the stop with statistics option PEST provides a complete listing of composite parameter sensitivities based on the best sensitivity matrix ie Jacobian matrix computed during the optimisation process Best is defined in terms of the aim of the optimisation process this may be to minimise the objective function parameter estimation mode to maximise minimise a prediction subject to objective function constraints predictive analysis mode or to minimise the regularisation component of the objective function subject to constraints imposed on the measurement component of the objective function regularisation mode The point within the parameter estimation process where the best Jacobian matrix was computed will vary from run to run It may have been computed during the last optimisation iteration or it may have been computed some iterations ago subsequent attempts to improve the outcome of the optimisation process since that iteration having met with no success Note also that if there was a marginal improvement in the outcome of the optimisatio
413. of the maximum relative parameter change RELPARSTP is a real variable for which a value of 0 01 is often suitable NRELPAR is an integer variable a value of 2 or 3 is normally satisfactory ICOV ICOR and IEIG The PEST Control File 4 13 As is explained in Section 5 3 5 at the end of each optimisation iteration PEST writes a matrix file containing the covariance and correlation coefficient matrices as well as the eigenvectors and eigenvalues of the covariance matrix based on current parameter values The settings of the ICOV ICOR and IEIG variables determine which if any of these data are recorded on the matrix file A setting of 1 for each of these variables will result in the corresponding data being recorded on the matrix file On the other hand a setting of 0 will result in the corresponding data not being recorded If all of these variables are set to zero none of this data will be recorded Where a large number of parameters are being estimated as might happen for example when PEST is being used in regularisation mode setting all of these variables to 0 may result in some savings in computation time for there is then no need for PEST to calculate the covariance matrix until the end of the parameter estimation process when these matrices and eigenvalues eigenvectors are recorded on the run record file irrespective of the settings of ICOV ICOR and IEIG 4 2 3 Parameter Groups Each adjustable parameter ie each parameter
414. of the three half space layers and the thicknesses of the upper two by comparing apparent resistivities generated by the model with a set of apparent resistivities provided by field measurement Then we need to provide instructions to PEST on how to read each of the apparent resistivities appearing in Example 3 6 An appropriate instruction file is shown in Example 3 7 The Model PEST Interface 3 12 SCHLUMB ERGE RE H Apparent 1 00 1 47 205 3 16 4 64 6 81 10 0 w PR Oo A 68 100 147 215 316 464 681 1000 resistivities calculated using the linear filter method electrode spacing I OA NNE TSs Lgs lef 16 12 aw oo ao ECTRIC SOUNDING apparent resistivity 21072 aoe oles 07536 95097 19023 87513 08115 8029 8229 5158 7689 4943 8532 79979 30746 40524 15234 06595 02980 Example 3 6 A model output file 11 n n n n n n n n n n n H H H H H H H H H H H H H H H H H pnp a fa f pN BBP aao ee pa o p G fasi F pi ipai pif electrode PAE Zils Zils 21 2i 21s 2i Zils 213 1 27 L225 Le27 TPs27 1 27 LEZ 1 27 1 27 T27 T 27 arl ar2 ar3 ar4 ard ar6 ar7 ar8 ar9 arl arl ard arl arl aril arl arl arl arl Te gE e o DEE E o gt S C a E E Eo SES E O N NDN N NA AI ORS TONI 27 27 21 27 27 27 27 24 27 Example 3 7 A PEST instructio
415. ol file for use in predictive analysis mode Predictive Analysis 6 11 pcf control data restart prediction 5 04 2 03 3 3 single point 1 0 0 5 2 0 3 0 01 10 2 3 0 001 0 1 30 0 01 5 5 0 01 5 MT parameter groups ro relative 0 001 0 0001 switch 2 parabolic hhh relative 0 001 0 0001 switch 2 parabolic parameter data rol log factor 4 000000 le 10 10000 ro 1 0 1 ro2 log factor 5 000000 le 10 10000 ro 1 0 1 ro3 log factor 6 000000 1le 10 10000 ro 101 hl log factor 5 000000 1le 10 100 hhh 10 1 h2 log factor 4 000000 le 10 100 hhh 1 0 1 observation groups obsgpl obsgp2 predict observation data arl 1 21038 1 obsgpl ar2 1 51208 1 obsgp ar3 2 07204 1 obsgpl extra 5 0 0 0 predict ar4 2 94056 1 obsgpl ar5 4 15787 1 obsgpl 1 1 ar6 5 7762 1 obsgp ar7 7 7894 1 obsgp ar8 9 99743 1 obsgpl ar9 11 8307 1 obsgp2 model command line model bat model input output vesl tpl a_model inl ves2 tpl a_model in2 extra tpl extra dat vesl ins a_model otl ves2 ins a_model ot2 extra ins extral dat prior information predictive analysis 1 0 1 05 2 0 0 00 0 7005 Ls 0 2 40 8 0 00 0 05 4 0 0 0 005 4 wornuen Example 6 2 Example of a PEST control file for use in predictive analysis mode Differences between the PEST control file used by PEST for running in predictive analysis mode and that used for work in parameter estimation mode are few The PESTMODE variable on the
416. ol file is set to 1 or displayed only after this process is complete if IEIG is set to 0 The former option can be put to use if the information forthcoming from a study of eigenvectors and eigenvalues is used in the user intervention process see below However if there are many parameters re calculation of eigenvectors and eigenvalues during every optimisation iteration can slow the parameter estimation process down The matrix displayed in the Eigenvectors window is comprised of columns representing the components of the normalised eigenvectors of the covariance matrix Vectors are arranged from left to right in order of increasing eigenvalue Dominant components of normalised eigenvectors with large eigenvalues are likely to indicate highly correlated parameters For example Figure 5 indicates that parameters h1 and rol represented in that plot are highly correlated a WinPEST C PEST2000 TESTINYES11 PST File Run Options Eigenvectors View Validate eR ue TEEN Parameters Lambda Calc vs Obs Residuals Jacobian Correlation Covariance Eigen_4 Normalized Eigenvectors of Covariance Matrix Positive amp Negative Parameter Vector Number Vector Number n a Parameter n a Not Running Figure 5 Graphical representation of the elements of the normalised covariance matrix eigenvectors WinPEST Graphical Windows 19 4 14 Eigenvalues Eigenvalue inform
417. om the other to scroll away out of sight The single window version of PEST is run using the command pest case r j where case is the filename base of the PEST control file PEST automatically adds the extension pst and r or j is an optional restart switch PPEST PPEST is Parallel PEST the operation of which is fully described in Chapter 9 Parallel Running PEST 5 2 PEST is contained in the ppest exe executable When a user runs Parallel PEST he she must also run one or a number of slaves which in turn run the model These slaves can reside on the same machine as Parallel PEST or on other machine s with which the PEST machine is networked Because model runs can be undertaken simultaneously on different machines during calculation of the Jacobian matrix the savings in overall optimisation time through the use of Parallel PEST can be considerable It should be noted however that due to the overheads involved in communicating with one or a number of slaves parameter estimation for a small model with a short run time may actually be larger when using Parallel PEST than when using the single window version of PEST The considerable efficiencies involved in parallelisation of the parameter estimation process are only fully realised where model run times are of the order of 30 seconds or greater Parallel PEST is a little more complex to run than the single window version of PEST because an extra PEST in
418. on the hydraulic conductivity values that we assign to different parts of the model domain in order to achieve this match may be correct this is fortunate as it is often difficult or expensive to measure rock hydraulic conductivities directly Similarly if the resistivities and thicknesses that we assign to a layered half space reproduce voltage current ratios measured at various electrode spacings on the surface of the half space then perhaps these resistivities and thicknesses represent a facet of reality that it may not have been possible to obtain by direct observation 1 2 2 The Role of PEST PEST is all about using existing models to infer aspects of reality that may not be amenable to direct measurement In general its uses fall into three broad categories These are e Interpretation In this case an experiment is set up specifically to infer some property of a system often by disturbing or exiting it in some way A model is used to relate the excitations and system properties to quantities that can actually be measured An interpretation method may then be based on the premise that if the excitation is known it may be possible to estimate the system properties from the measurement set Alternatively if system properties are known it may be possible to use the model to infer something about the excitation by adjusting model input excitation variables until model outcomes match measurements In some cases it may even be possible to estim
419. on after listing both the updated parameter values as well as those from which the updated parameter set was calculated viz those at the commencement of the optimisation iteration Note that the only occasion on which the previous parameter values recorded at the end of an optimisation iteration do not correspond with those determined during the previous optimisation iteration is when the switch to three point derivatives calculation has just been made and the previous iteration failed to lower the objective function on such an occasion PEST adopts as its starting parameters for the new optimisation iteration the parameter set resulting in the lowest objective function value achieved so far At the end of each optimisation iteration PEST records either two or three depending on the input settings very important pieces of information in the case of Example 5 1 it is two These are the maximum factor parameter change and the maximum relative parameter change As was discussed in previous chapters each adjustable parameter must be designated as either factor limited or relative limited in Example 5 1 all adjustable parameters are factor limited with a factor limit of 3 0 A suitable setting for the factor and relative change limits Ge FACPARMAX and RELPARMAX may be crucial in achieving optimisation stability Note that along with the value of the maximum factor or parameter change encountered during the optimisation iteration PEST also recor
420. on exactly where it left off rather it will recommence the parameter estimation process at the beginning of the optimisation iteration in which it was previously interrupted In general it is unwise to interfere with any of PEST s input files ie the PEST control file as well as its template and instruction files between interrupting and restarting PEST While PEST reads all of the data previously contained in its storage arrays from the binary file case rst in the event of a restart it still needs to obtain the problem dimensions and many of its settings from the PEST control file the parameter templates from the respective template files and its instructions from the respective instruction files If any information in any of these files is inconsistent with the information stored in file case rst PEST s behaviour will be unpredictable However if you are very very careful you can alter a number of control variables with impunity The variables which you may alter are RLAMFAC PHIRATSUF PHIREDLAM and NUMLAM which affect the way in which PEST selects Marquardt lambdas and NOPTMAX PHIREDSTP NPHINORED RELPARSTP and NRELPAR which are termination criteria You can also alter the derivative variables DERINC DERINCLB DERINCMUL and DERMTHD for any parameter group Thus if for example PEST terminates execution with a run time error message such as shown in Example 5 5 you can edit the PEST control file altering DERINC DERINCLB and or DER
421. on mode one of these group names must be regul 4 2 7 Observation Data For every observation cited in a PEST instruction file there must be one line of data in the observation data section of the PEST control file Conversely every observation for which data is supplied in the PEST control file must be represented in an instruction file Each line within the observation data section of the PEST control file must contain four items Each of these four items is discussed below refer to Example 4 1 for the arrangement of these items OBSNME This is a character variable containing the observation name As discussed in Section 3 3 5 an observation name must be twelve characters or less in length Observation names are case insensitive but must be unique to each observation OBSVAL OBSVAL a real variable is the field or laboratory measurement corresponding to a model generated observation It is PEST s role to minimise the difference between this number and the corresponding model calculated number the difference being referred to as the residual over all observations by adjusting parameter values until the sum of squared weighted residuals ie the objective function is at a minimum WEIGHT This is the weight attached to each residual in the calculation of the objective function The manner in which weights are used in the parameter estimation process is discussed in Section 2 1 2 An observation weight ca
422. once maybe twice with the value of the parameter incrementally varied Thus a different command can now be used to run the model for each such incrementally varied parameter Use of a variable command line strategy may allow a reduction in overall PEST run time to be achieved in some circumstances For example if a composite model is comprised of a sequence of executable programs encapsulated in a batch file it may not be necessary to run the earlier programs of the sequence when parameters pertaining to the later programs are being incrementally varied for the purpose of derivatives calculation for outputs of the earlier programs will not vary between subsequent model runs Hence the model run by PEST when incrementally varying these later parameters may replace the earlier submodel commands with commands by which pertinent output files for these earlier models stored under separate names are copied to the model output files expected by PEST recall that these are deleted by PEST prior to each model run Caution should be exercised in doing this however for it must be ensured that the model output files that are copied in this way pertain to un incremented parameter values for the current optimisation iteration Thus it may be necessary to undertake a full model run for the first of those parameters which affect only the later submodels and as part of this run copy model output files from the earlier models to the files which are to be used
423. ontent of 0 4 is of particular interest to us this can now be easily calculated with our calibrated model An appropriate TWOLINE input file in2 dat is provided this is easily prepared from the new in dat file created using TEMPCHEK by alteration with a text editor in conformity with the expectations of program TWOLINE As the first two lines of this file contain parameter values to be used by TWOLINE it was necessary to run TEMPCHEK first to ensure that the parameter values contained in this file were the optimal parameter values As TWOLINE expects a file named in dat in2 dat must be copied to in dat before TWOLINE is run But before doing this copy the existing in dat to inI dat for safekeeping After running program TWOLINE by typing twoline at the screen prompt open file out dat to obtain the model predicted specific volume It should be 0 756 This is thus our best estimate of the soil clod specific volume at a water content of 0 4 12 2 2 The Composite Model Before undertaking predictive analysis we must construct a composite model comprised of the model run under calibration conditions followed by the model run under prediction conditions This model must be encompassed in a batch file an appropriate file named model bat is supplied Example 12 12 shows a printout of model bat An Example 12 12 echo off rem Intermediate files are deleted del in dat del out dat rem First the model is run under c
424. ontinuously differentiable PEST must be provided with a set of input files containing the data which it needs in order to effectively take control of a particular model Specifications for these files will be described later in the manual their preparation is a relatively simple task Amongst the data which must be supplied to PEST is the name of the model of which it must take control In the simplest case this may be the name of a single executable file ie a program that simulates a system for which parameterisation or excitation estimation is required In more complex cases the name may pertain to a batch program which runs a number of executable programs in succession Thus output data from one model can feed another or a model postprocessor may extract pertinent model outputs from a lengthy binary file and place them into a smaller tidy file in ASCII format for easy PEST access Models which receive their data directly from the user through keyboard entry and write their results directly to the screen can also be used with PEST Keyboard inputs can be typed ahead of time into a file and the model directed to look to this file for its input data using the lt symbol on the model command line likewise model screen output can be redirected to a file using the gt symbol PEST can then be instructed to alter parameters and or excitations on the input file and read numbers matching observations from the output file Thus it can
425. opriate Two possible settings for NOPTMAX have special significance If NOPTMAX is set to 0 PEST will not calculate the Jacobian matrix Instead it will terminate execution after just one model run This setting can thus be used when you wish to calculate the objective function corresponding to a particular parameter set and or to inspect observation residuals corresponding to that parameter set The PEST Control File 4 12 If NOPTMAX is set to 1 PEST will terminate execution immediately after it has calculated the Jacobian matrix for the first time The parameter covariance correlation coefficient and eigenvector matrices will be written to the run record file and parameter sensitivities will be written to the sensitivity file these are based on the initial parameter set supplied in the PEST control file PHIREDSTP and NPHISTP PHIREDSTP is a real variable whereas NPHISTP is an integer variable If in the course of the parameter estimation process there have been NPHISTP optimisation iterations for which Bi min Di lt PHIREDSTP 4 6 being the objective function value at the end of the 7 th optimisation iteration and min being the lowest objective function achieved to date PEST will consider that the optimisation process is at an end For many cases 0 01 and 4 are suitable values for PHIREDSTP and NPHISTP respectively However you must be careful not to set NPHISTP too low if the optimal values for some parameters
426. or crop growth processes in agricultural areas crop factor may be one such parameter Crop factor is also a parameter that together with other parameters often requires adjustment through the calibration process in order that the model can replicate measured crop water usage observed crop growth or some other system response for which historical records are available Many models require that the crop factor be provided on a monthly basis However while monthly crop factors may indeed require estimation through the calibration process it would generally be unwise to attempt to estimate each monthly crop factor independently of every other monthly crop factor through the calibration process for this would ignore an inherent relationship between these parameters this being the fact that variation of crop factor with season may show a regular perhaps sinusoidal pattern To ignore this pattern in parameterising the model would be to ignore an important facet of system behaviour Furthermore in many model calibration contexts it would be unlikely that 12 different monthly crop factors could be independently estimated with any degree of uniqueness because of the high degree of correlation that is likely to exist between these individual parameters especially where the data available for model calibration is limited For a case such as this a suitable parameterisation strategy may be to estimate the mean monthly crop factor together with th
427. or each parameter last variable on each line of the parameter data section to 1 Check your work with PESTCHEK and then run PEST Parallel PEST 9 9 Parallel PEST 9 1 Introduction 9 1 1 General In the course of optimising parameters for a model or of undertaking predictive analysis PEST runs the model many times Some model runs are made in order to test a new set of parameters Others are made with certain parameters temporarily incremented as part of the process of calculating the Jacobian matrix ie the matrix of derivatives of observations with respect to parameters unless derivatives are supplied to PEST directly by the model in accordance with PEST s external derivatives functionality In calculating the Jacobian matrix PEST needs to run the model at least as many times as there are adjustable parameters and up to twice this number if derivatives for some of the adjustable parameters are calculated using central differences In most cases by far the bulk of PEST s run time is consumed in running the model It follows that any time savings that are made in carrying out these model runs will result in dramatic enhancements to overall PEST performance Parallel PEST can achieve a high degree of performance enhancement by carrying out model runs in parallel If installed on a machine that is part of a local area network Parallel PEST can carry out model runs on the different machines which make up the network includin
428. or further details Thus the predictive analysis problem can be formulated as follows Find b such as to maximise minimise Zb 2 28a subject to c Xb Q c Xb bo 2 28b where Po Pmin 2 28c It can be shown that the solution to this problem is b x Qx x Qe 5 2 29a where is defined by the equation 2 t t t rity l 1 _ amp c Qce e Qx x QX X Qe 2 29b 2a Z x Qx Z Where predictive analysis is carried out for a nonlinear model the same equations are used However in this case X is replaced by the model Jacobian matrix J and a parameter upgrade vector is calculated instead of a solution vector The solution process is then an iterative one in which the true solution is approached by repeated calculation of an upgrade vector based on repeated linearisation of the problem through determination of the Jacobian matrix For further details see Cooley and Vecchia 1987 and Vecchia and Cooley 1987 Use of the Marquardt lambda in solving the nonlinear parameter estimation problem is discussed above Its use in solving the nonlinear predictive analysis problem is very similar As in the parameter estimation problem PEST continually adjusts the Marquardt lambda through the solution process such that its value is optimal at all stages of this process As a further numerical measure to solve the predictive analysis problem for a nonlinear model PEST undertakes a line search in the direction of the p
429. ory as PEST This may be desirable if all model input files are in this same directory which is often the case and this is also the current PEST working directory A designation of is sufficient to identify this directory The next line of the run management file must contain as many entries as there are slaves Each entry is the expected run time of the model on the respective slave the ordering of slaves on this line being the same as that in which slave data was supplied earlier in the run management file Run times should be supplied in seconds if you are unsure of the exact run times simply provide the correct run time ratios There is no need for stopwatch precision here as PEST continually updates model run time estimates in the course of the parameter estimation process However it is better to overestimate rather than underestimate these run times so that PEST will not re instigate the initial model run which is not part of the Jacobian calculation on an alternative machine if the initial model run takes much longer than you have indicated and PEST comes to the conclusion that a mishap may have occurred on the machine to which that run was initially assigned If the value supplied for IFLETYP is 0 then the run management file is complete However if it is supplied as 1 the names of all model input files and all model output files on all slave machines must next be supplied individually Either full pathnames can be supplied or abbr
430. ough the parameter variable PARTRANS in the PEST control file see Section 4 2 4 2 2 3 Upper and Lower Parameter Bounds As well as supplying an initial estimate for each parameter you are also required to supply parameter upper and lower bounds These bounds define the maximum and minimum values which a parameter is allowed to assume during the optimisation process They are provided through the parameter variables PARLBND and PARUBND in the PEST control file It is important that upper and lower parameter bounds be chosen wisely For many models parameters can lie only within certain well defined domains determined by the theory on which the model is based In such cases model generated floating point errors may result if The PEST Algorithm 2 20 PEST is not prevented from adjusting a parameter to a value outside its allowed domain For example if at some stage during a model run the logarithm or square root of a particular parameter is taken then that parameter must be prevented from ever becoming negative or zero if the model takes the log of the parameter If the reciprocal is taken of a parameter the parameter must never be zero In some cases where a large number of parameters are being estimated based on a large number of measurements PEST may try to force a fit between model and measurements by adjusting some parameters to extremely large or extremely small values especially if the measured values upon which the estimation proc
431. ould be noted that when PEST is run in predictive analysis mode the sole observation belonging to observation group predict does not appear in any observation related or residual related plot This is because it is not really an observation at all furthermore there are no field or laboratory measurements associated with it However when PEST is run in dual calibration mode a member of the observation group predict is treated just like any other observation for in this case there is indeed a field or laboratory measurement associated with it User Intervention 24 7 User Intervention 7 1 General One of PEST s most powerful assets is the ability that it gives the user to hold recalcitrant parameters fixed for a while as it re calculates the parameter upgrade vector without having to re calculate the Jacobian matrix As is explained in Section 5 6 of the PEST manual user intervention is implemented by first terminating PEST execution preferably without statistics denoting certain parameters as held through the parameter hold file and re starting PEST execution using the j switch 7 2 Holding Parameters in WinPEST User intervention functionality is also available through WinPEST However WinPEST allows you to created and edit the parameter hold file in an intuitive fashion from within WinPEST itself Editing of the parameter hold file is done through the Hold Status window which can be accessed eith
432. ould consider this when assigning a weight to the sole member of the observation group predict when preparing the PEST control file for a predictive analysis run When using PEST in regularisation mode weights assigned to the observation group regul are multiplied by the optimal regularisation weight factor determined as part of the parameter estimation process before recording composite sensitivities with respect to the members of this group of each adjustable parameter 5 3 3 Observation Sensitivity File The composite observation sensitivity of observation oj is defined as s QUIS In 5 2 That is the composite sensitivity of observation j is the magnitude of the j th row of the Jacobian multiplied by the weight associated with that observation this magnitude is then divided by the number of adjustable parameters It is thus a measure of the sensitivity of that observation to all parameters involved in the parameter estimation process At the end of its run PEST lists all observation values and corresponding model calculated values as well as composite sensitivities for all observations to the observation sensitivity file This file is named case seo Though composite observation sensitivities can be of some use they do not in general convey as much useful information as composite parameter sensitivities In fact in some instances the information that they provide can even be a little deceptive Thus while a high
433. outcomes of this process to the third section of the run record file First it lists the optimised parameter values It does this in three stages the adjustable parameters then the tied parameters and finally any fixed parameters PEST calculates 95 confidence limits for the adjustable parameters However you should note carefully the following points about confidence limits e Confidence limits can only be obtained if the covariance matrix has been calculated If for any reason it has not been calculated eg because J QJ of equation 2 17 could not be inverted confidence limits will not be provided e As noted in the PEST run record itself parameter confidence limits are calculated on the basis of the same linearity assumption which was used to derive the equations for parameter improvement implemented in each PEST optimisation iteration If the confidence limits are large they will in all probability extend further into parameter space than the linearity assumption itself This will apply especially to logarithmically transformed parameters for which the confidence intervals cited in the PEST run record are actually the confidence intervals of the logarithms of the parameters as evaluated by PEST from the covariance matrix If confidence intervals are exaggerated in the logarithmic domain due to a breakdown in the linearity assumption they will be very much more exaggerated in the domain of non logarithmically transformed numbers This i
434. ow U S Geological Survey Techniques in Water Resources Investigations book 3 chap B4 232p Cooley R L and Vecchia A V 1987 Calculation of nonlinear confidence and prediction intervals for ground water flow models Water Resources Bulletin Vol 23 No 4 pp581 599 Hill M C 1992 A Computer Program MODFLOWP for Estimating Parameters of a The PEST Algorithm 2 33 Transient Three Dimensional Ground Water Flow Model using Nonlinear Regression U S Geological Survey Open File Report 91 484 Hill M C 1998 Methods and Guidelines for Effective Model Calibration U S Geological Survey Water Resources Investigations Report 98 4005 Marquardt D W 1963 An algorithm for least squares estimation of nonlinear parameters Journal of the Society of Industrial and Applied Mathematics v11 no 2 p431 441 Levenberg K 1944 A method for the solution of certain non linear problems in least squares Q Appl Math v 2 p164 168 Vecchia A V and Cooley R L 1987 Simultaneous confidence and prediction intervals for nonlinear regression models with application to a ground water flow model Water Resources Research vol 23 no 7 pp1237 1250 2 4 2 Some Further Reading Bard Jonathon 1974 Nonlinear parameter estimation Academic Press NY 341p Koch K 1988 Parameter Estimation and Hypothesis Testing in Linear Models Springer Verlag Berlin 377p Mikhail E M 1976 Observations and Least Squares I
435. parameter input variable PARGP in the parameter data section of the PEST control file the properties for that group must be defined in the parameter groups section of the PEST control file Note that an adjustable parameter cannot be assigned to the dummy group none As Example 4 1 shows one line of data must be supplied for each parameter group Seven entries are required in each such line the requirements for these entries are now discussed in detail PARGPNME This is the parameter group name all PEST names viz parameter observation prior information and group names are case insensitive and must be a maximum of twelve The PEST Control File 4 14 characters in length If derivative data is provided for a group named by PARGPNME it is not essential that any parameters belong to that group However if in the parameter data section of the PEST control file a parameter is declared as belonging to a group that is not featured in the parameter groups section of the PEST control file an error condition will arise Note that derivative variables cannot be defined for the group none as this is a dummy group name reserved for fixed and tied parameters for which no derivatives information is required INCTYP and DERINC INCTYP is a character variable which can assume the values relative absolute or rel_to_max If it is relative the increment used for forward difference c
436. pared to complement the model input file poly_par in This file provides spaces for the four parameters a b c and d An instruction file named poly_val ins reads polynomial values and the prediction value from the model output file poly_val out Two PEST control files have been prepared In one of these poly pst PEST is asked to run in parameter estimation mode while in the other polyp pst it is asked to run in predictive analysis mode In the former case the model prediction plays no part in the parameter estimation process as it is assigned a weight of zero PEST is thus asked to estimate values for the parameters a b c and d by matching computed polynomial values at different abscissae to the field data contained in the PEST control file Because this field data was in fact model generated PEST is able to achieve a very low objective function On the fifth line of file poly pst the values of the PEST control variables NUMCOM JACFILE and MESSFILE are set to 1 1 and 0 respectively Thus PEST is asked to look to a derivatives file to obtain its Jacobian matrix no PEST to model message file is requested As is evident in the derivatives command line section of poly pst the expected name of the derivatives file is poly_der out Note also from the contents of the derivatives command line and model command line sections of the PEST control file that the command used by PEST to run the model for the
437. paring for a Parallel PEST Run Before running Parallel PEST a PEST run should be set up in the usual manner This will involve preparation of a set of template and instruction files as well as a PEST control file When preparing the PEST control file it is important for template and instruction files to be properly identified However the names of the corresponding model input and output files are not used if the value of IFLETYP in the run management file is set to 1 for these are then supplied to PEST in the run management file itself Similarly the model command line as provided in the PEST control file is ignored by Parallel PEST as the model command line is supplied directly to each slave by the user on commencement of the latter s execution see below Once the entire PEST input dataset has been prepared PESTCHEK should be used to ensure that it is all consistent and correct Next the Parallel PEST run management file should be prepared in the manner discussed in Section 9 2 5 Before Parallel PEST is started care should be taken to ensure that the model runs correctly on each slave machine A set of model input files should be transferred from the master machine to each slave machine or TEMPCHEK can be used to construct such files on the basis of a set of template files and a parameter value file Where any model input files are not generated by Parallel PEST because they contain no adjustable parameters these files should be identi
438. patial interpolation within different mapped geological units For each subset of pilot points a series of differences can be formed between the parameter values assigned to these points in order to create a regularisation scheme not unlike that described above A Delauney triangulation of the model domain based on pilot point locations can be used to define the set of neighbours to each such point a parameter difference will then be taken for each neighbouring pair of points The weight used for each regularisation observation could be independent of the distance between the points or could be defined as a function of this distance Where a geological boundary passes between two points then either the difference would not be taken or the weight assigned to the pertinent regularisation observation would be zero A suite of utility software that implements the use of pilot points for spatial parameter definition in conjunction with the United States Geological Survey ground water flow model MODFLOW is available through the Ground Water Data Utilities supplied with PEST Model Calculated Derivatives 8 1 8 Model Calculated Derivatives 8 1 General Accuracy in the calculation of model outputs with respect to adjustable parameters is essential for good PEST performance especially when working with highly parameterised systems As well as accuracy efficiency of derivatives calculation is also important for run times can be very long when PEST
439. prior information equation the constant of proportionality being the same as used for the observations comprising the other elements of the vector c of equation 2 1 In practice the user simply assigns the weights in accordance with the extent to which he she The PEST Algorithm 2 6 wishes each article of prior information to influence the parameter estimation process It is sometimes helpful to view the inclusion of prior parameter information in the estimation process as the introduction of one or more penalty functions The aim of the estimation process is to lower the objective function defined by equation 2 8 to its minimum possible value this is done by adjusting parameter values until a set is found for which the objective function can be lowered no further If there is no prior information the objective function is defined solely in terms of the discrepancies between model outcomes and laboratory or field measurements However with the inclusion of prior information minimising the discrepancy between model calculations and experimental measurements is no longer the sole aim of the parameter estimation process To the extent that any article of prior information is not satisfied there is introduced into the objective function a penalty equal to the squared discrepancy between what the right hand side of the prior information equation should be and what it actually is according to the current set of parameter values This discrepa
440. process This information is extremely helpful in the assignment of weights to different measurement types PEST no longer ceases execution with an error message if a parameter has no effect on observations rather it simply holds the offending parameter at its initial value PEST can be asked to run a model only once and then terminate execution In this way PEST can be used simply for objective function calculation Alternatively it can be asked to run the model only as many times as is necessary in order to calculate the parameter covariance matrix and related statistics based on initial parameter estimates Two new programs have been added to the PEST suite These are SENSAN a model independent sensitivity analyser and PARREP a utility that facilitates the commencement of a new PEST run using parameter values generated on a previous PEST run However by far the most important changes to PEST are the improved capabilities that it offers for user intervention in the parameter estimation process Every time that it calculates the Jacobian matrix PEST now stores it on file for possible later use It records on another file the overall sensitivity of each parameter this being defined as the magnitude of the vector comprising the column of the Jacobian matrix pertaining to that parameter divided by the number of observations Thus at any stage of the optimisation process sensitive and insensitive parameters can be distinguished It is the insen
441. protocol for only observations of this type can be dummy observations An alternative to the use of whitespace in locating the observation obs in the above example could involve using the dummy observation more than once Hence the instruction line below would also enable the number representing obs1 to be located and read 110 dum dum dum obs1 However had the numbers in the above example been separated by commas instead of whitespace the commas should have been used as secondary markers in order to find obs1 A number not surrounded by whitespace can still be read as a non fixed observation with the proper choice of secondary markers Consider the model output file line shown below SOIL WATER CONTENT NO CORRECTION 21 345634 It may not be possible to read the soil water content as a fixed observation because the NO CORRECTION string may or may not be present after any particular model run Reading it as a non fixed observation appears troublesome as the number is neither preceded nor followed by whitespace However a suitable instruction line is 15 sws Notice how a secondary marker viz is referenced even though it occurs after the observation we wish to read If this marker were not present a run time error would occur when PEST tries to read the soil water content because it would define the observation string to include the character and naturally would be un
442. provided in the control data section of the PEST control file is not zero PEST expects NPRIOR articles of prior information Prior information is written to this section of the PEST control file in a manner not unlike the way in which you would write it down on paper yourself however certain strict protocols must be observed Refer to Example 4 2 for an instance of a PEST control file containing prior information Each item on a prior information line must be separated from its neighbouring items by at least one space Each new article of prior information must begin on a new line No prior information line is permitted to exceed 300 characters in length however a continuation character amp followed by a space at the start of a line allows you to write a lengthy prior information article to several successive lines Prior information lines must adhere to the syntax set out in Example 4 1 The protocol is repeated here for ease of reference PILBL PIFAC PARNME PIFAC log PARNME PIVAL WEIGHT OBGNME one such line for each of the NPRIOR articles of prior information Example 4 4 The syntax of a prior information line Each prior information article must begin with a prior information label the character The PEST Control File 4 25 variable PILBL in Example 4 4 Like all other names used by PEST this label must be no more than twelve characters in length is case insensitive and must be uniq
443. psoid in the n dimensional space occupied by the n adjustable parameters In the eigenvector matrix the eigenvectors are arranged from left to right in increasing order of their respective eigenvalues the eigenvalues are listed beneath the eigenvector matrix The square root of each eigenvalue is the length of the corresponding semiaxis of the probability ellipsoid in n dimensional adjustable parameter space If the ratio of a particular eigenvalue to the lowest eigenvalue pertaining to the parameter estimation problem is particularly large then the respective eigenvector defines a direction of relative insensitivity in parameter space The eigenvector pertaining to the highest eigenvalue is worthy of attention in most parameter estimation problems for this defines the direction of maximum insensitivity and hence of greatest elongation of the probability ellipsoid in adjustable parameter space If this eigenvector is dominated by a single element then the parameter associated with that element may be quite insensitive the magnitude of its insensitivity being defined by the square root of the magnitude of the corresponding eigenvalue However if this eigenvector contains a number of significant components rather than just one then this is an indication of insensitivity associated with a group of parameters ie parameter correlation The correlated parameters are those whose eigenvector components are significantly non zero The ratio o
444. ption upon which their calculation is based is valid The diagonal elements of the covariance matrix are the variances of the adjustable parameters for Example 5 1 the variances pertain from top left to bottom right to the parameters log ro2 h1 and log h2 in that order The variance of a parameter is the square of its standard deviation With log h2 having a variance of 0 866 and hence a standard deviation of 0 931 and bearing in mind that the number 1 in the log domain represents a factor of 10 in untransformed parameter space it is not hard to see why the 95 confidence interval cited for parameter h2 is so wide The off diagonal elements of the covariance matrix represent the covariances between parameter pairs thus for example the element in the second row and third column of the above covariance matrix represents the covariance of h1 with log h2 If there are more than eight adjustable parameters the rows of the covariance matrix are written in wrap form ie after eight numbers have been written PEST will start a new line to write the ninth number Similarly if there are more than sixteen adjustable parameters the seventeenth number will begin on a new line Note however that every new row of the covariance matrix begins on a new line of the run record file 5 2 10 The Correlation Coefficient Matrix The correlation coefficient matrix is calculated from the covariance m
445. purpose of derivatives calculation is the same as the command that it uses to run the model in order to simply obtain model outputs The final entry on each line of the parameter data section of file poly pst is the value of the variable DERCOM In the present instance DERCOM is zero for all parameters this indicates that for each parameter cited in the control file PEST will obtain derivatives of all model outputs with respect to that parameter from the derivatives file poly_der out ie no supplementary model run is required to calculate some derivatives with respect to this Model Calculated Derivatives 8 11 parameter using finite differences Note that an OFFSET value of 1 0 is provided for parameter d this circumvents problems that can sometimes arise in the parameter estimation process when a parameter approaches zero see the discussion of RELPARMA X and FACPARMAX in Section 4 2 2 Check the input dataset contained in file poly pst and the template and instruction file cited therein by typing the command pestchek poly at the screen prompt PESTCHEK should report no errors or inconsistencies just a warning that the command used to run the model for the purpose of derivatives calculation is the same as that used to run the model for the purpose of obtaining normal model outputs Then run PEST using the command pest poly PEST should quickly reduce the objective function to a very low value Now inspect file polyp pst
446. pushing the number in which we are interested to the right then we have no alternative but to read the number as a non fixed observation However if the previous numbers vary from model run to model run we cannot use a secondary marker either nor can a tab be used Fortunately whitespace comes to the rescue with the following instruction line taking PEST to the fourth number 110 ww w obsl1 Here it is assumed that prior to reading this instruction the PEST cursor was located on the 10th preceding line of the model output file As long as we can be sure that no whitespace will ever precede the first number there will always be three incidences of whitespace preceding the number in which we are interested However if it happens that whitespace may precede the first number on some occasions while on other occasions it may not then we can The Model PEST Interface 3 24 read the first number as a dummy observation as shown below 110 dum w w w _ obs1 As was explained previously the number on the model output file corresponding to an observation named dum is not actually used nor can the name dum appear in the observation data section of the PEST control file see the next chapter The use of this name is reserved for instances like the present case where a number must be read in order to facilitate navigation along a particular line of the model output file The number is read according to the non fixed observation
447. put file called the run management file must be prepared Also as well as starting PEST the user must also start each of the slaves However it is more than worth the extra trouble where model run times are large and adjustable parameters are many While Parallel PEST was built for the purpose of running a model simultaneously on a number of different machines across a network it can also be used to run a single instance of the model on a single machine Doing this has the advantage that the model and PEST operate in different windows hence the screen output of one does not interfere with the screen output of the other Parallel PEST is run using the command ppest case r j where case is the filename base of the PEST control file PPEST automatically adds the extension pst and r and j are optional restart switches For more information on running Parallel PEST see Chapter 9 and the Frequently Asked Questions in Chapter 13 5 2 The PEST Run Record 5 2 1 An Example As PEST executes it writes a detailed record of the parameter estimation process to file case rec where case is the filename base of the PEST control file to which it is directed through the PEST command line Example 5 1 shows such a run record file the PEST control file corresponding to Example 5 1 is that shown in Example 4 2 Note that this example does not demonstrate a very good fit between measurements and model outcomes calcu
448. put file to the observation value file instruct obf where instruct is the filename base of the instruction file provided to INSCHEK Hence for any instruction file provided to SENSAN use of a file with the same filename base but with an extension of obf will result in that file being overwritten 11 6 Sensitivity of the Objective Function SENSAN allows a user to undertake many model runs without user intervention The sensitivity of certain model outputs to certain parameters can be tested However SENSAN does not compute an objective function because it does not read an observation dataset and hence cannot compare model outputs with corresponding observations to calculate residuals However once all PEST input files have been prepared for a particular case SENSAN can be used in conjunction with PEST to study the dependence of the objective function on certain parameters Where there are only two parameters this can be used to contour the objective function in parameter value space A SENSAN control file implementing this is shown in Example 11 6 scf control data verbose 6 1 1 1 single point sensan files parvar dat outl txt out2 txt out3 txt model command line pest ves4 model input output pst tpl ves4 pst rec ins ves4 rec Example 11 6 A SENSAN control file with PEST as the model Note the following points e There can be as many parameters as you like but only one observation This should b
449. put files is lost when parameter values are written to those files using program PAR2PAR because PEST has no way of adjusting its internal representation of parameters based on PAR2PAR outputs Thus unless the formatting requirements of the model input file are such that it allows model input parameters to be supplied with full numerical precision which is normally about 7 significant figures slight errors will be incurred in the derivatives calculation process Note that where a number is small or large enough for exponential notation to be required for its representation up to 13 characters may be required for the PEST Utilities 10 19 representation of that number using 7 significant figures Imprecision in derivatives calculation can have a profound effect on the outcome of an inversion process Thus if the model permits it you should make absolutely sure that the template files used by PAR2PAR to write model input files use parameter space widths which are as large as the model will tolerate up to a maximum of the 13 characters if using single precision arithmetic or 23 characters if using double precision arithmetic If model input file formatting requirements are too restrictive to allow a parameter value to be written without some loss of significance then you should at least be aware of the fact that use of PAR2PAR under these circumstances has the potential to reduce the efficacy of PEST s performance 10 7 3 3 Intermediate Fil
450. quation 2 40 is applicable whether weights or a covariance matrix are used to specify observation uncertainties ie whether Q is a diagonal matrix or not It is easily shown that the parameter covariance matrix can also be calculated from the rotated observation cofactor matrix T using the formula C b 5 Y TY 2 41 Recall that T is diagonal because rotation of the vector to yield the vector d results in an uncorrelated set of stochastic variables It can also be shown that calculation of the objective function using equation 2 8a is equivalent to calculating it using the equation d Yb T d Yb 2 42 Once again calculations made using equation 2 42 are based on the use of a rotated observation dataset ie the vector d complemented by a rotated model matrix vector ie the matrix Y and a rotated diagonal cofactor matrix T 2 1 12 Goodness of Fit When it is being used in parameter estimation mode PEST aims to lower the objective function as far as it can be lowered When used in predictive analysis mode PEST aims to The PEST Algorithm 2 18 maximise or minimise a specified prediction while maintaining the model in a calibrated state ie while ensuring that the objective function rises no higher than a specified level When working in regularisation mode PEST aims to maximise adherence to a certain regularisation condition by minimising a regularisation objective function while ensuring that the measuremen
451. r estimation problem is recorded for inspection The PAR2PAR utility is used to undertake mathematical operations of arbitrary complexity between existing parameters in order to generate new parameters It is normally used in a model preprocessing capacity being run by PEST as part of a composite model encapsulated in a batch file Its many uses include complex parameter transformation perhaps to improve model linearity and the generation of a large number of secondary parameters as used by the model from a smaller number of primary parameters as estimated by PEST 10 1 TEMPCHEK Program TEMPCHEK checks that PEST template files obey PEST protocol If provided with a set of parameter values TEMPCHEK can also be used to generate a model input file from a template file It builds the model input file in the same way that PEST does you can then run your model checking that it is able to read such a PEST generated input file without any difficulties TEMPCHEK is run using the command tempchek tempfil modfile parfile where tempfile is the name of a template file modfile is the name of a model input file to be generated by TEMPCHEK optional and parfile is the name of a PEST parameter value file also optional The simplest way to run TEMPCHEK is to use the command tempchek tempfil When invoked in this way TEMPCHEK simply reads the template file tempfile checking it for breaches of PEST protocol It writ
452. r allowed domains in which case you will have to adjust their upper and or lower bounds accordingly on the PEST control file Another model related error which can lead to PEST run time errors of this kind will occur if the subdirectory which contains the model executable file is not cited in either the PATH environment variable or in the model command line section of the PEST control file In this case after PEST attempts to make the first model run you will receive the message Running model Bad command or file name prior to a PEST run time error message informing you that a model output file cannot be opened Note however that the model path is not required if the model executable resides in the current directory It is normally an easy matter to distinguish PEST errors from model errors as PEST informs you through its screen output when it is running the model A model generated error if it occurs will always follow such a message Furthermore a PEST run time error message is clearly labelled as such as shown in Example 5 5 If you are using Parallel PEST the model window will be different from the PEST window In this case it will be much easier to distinguish an error originating from the model from an error originating from PEST KKEKKK KKK KKK KK KKK KKK KKK KKK KKK KK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKK KKEKKEKKKKKKK Error condition prevents continued PEST execution Varying parameter parl has n
453. r on the second line of the parameter variation file immediately under the parameter names As will be discussed below SENSAN produces two output files in which variations from base value outputs are recorded base value outputs being defined as model outputs calculated on the basis of base parameter values SENSAN 11 4 Items on each line of a parameter value file can be space comma or tab delimited 11 2 5 SENSAN Control File It is recommended though it is not essential that the SENSAN control file be provided with a filename extension of sns Use of this default extension avoids the need to type in the entire SENSAN control filename when running either SENSAN or SENSCHEK Example 11 2 shows a SENSAN control file Example 11 3 shows the structure of the SENSAN control file As is apparent the SENSAN control file resembles to some extent the PEST control file Like the PEST control file the SENSAN control file must begin with a three character code viz scf identifying it as a SENSAN control file Like the PEST control file the SENSAN control file is divided into sections by lines beginning with the character And like the PEST control file the SENSAN control file provides information to SENSAN through the values taken by certain input variables many of which are also used by PEST Where such variables are indeed used by PEST they are provided with identical names to the corresponding PEST variables
454. r parameter change 1 514 h1 Maximum relative parameter change 0 5142 h1 OPTIMISATION ITERATION NO 6 Model calls so far 28 Starting phi for this iteration 64 09 Contribution to phi from observation group group_1 6 740 Contribution to phi from observation group group_2 18 98 Contribution to phi from observation group group_3 10 53 Contribution to phi from observation group group_4 27 84 All frozen parameters freed Lambda 1 9531E 02 gt parameter ro2 frozen gradient and update vectors out of bounds phi 63 61 0 99 of starting phi Lambda 9 7656E 03 gt phi 63 61 0 99 of starting phi No more lambdas relative phi reduction between lambdas less than 0 0300 Lowest phi this iteration 63 61 Relative phi reduction between optimisation iterations less than 0 1000 Switch to central derivatives calculation Current parameter values Previous parameter values rol 0 500000 rol 0 500000 ro2 10 0000 ro2 10 0000 ro3 1 00000 ro3 1 00000 KI 0 265320 h1 0 238277 h2 42 2249 h2 42 4176 Maximum factor parameter change 1 113 h1 Maximum relative parameter change 0 1135 J h1 OPTIMISATION ITERATION NO 7 Model calls so far 4 33 Starting phi for this iteration 63 61 Contribution to phi from observation group group_1 3 679 Running PEST 5 7 Contribution to phi from observation group group_2 32 58 Contribution to phi from observation group group_3 0 111 Contribution
455. r performance of this software Preface Preface to First Edition This document describes the use of PEST a model independent parameter optimiser Nonlinear parameter estimation is not new Many books and papers have been devoted to the subject subroutines are available in many of the well known mathematical subroutine libraries many modelling packages from all fields of science include parameter estimation as a processing option most statistical and data analysis packages allow curve fitting to a user supplied data set However in order to take advantage of the nonlinear parameter estimation facilities offered by this software you must either undertake a modelling task specific to a particular package you must alter the source code of your model so that it meets the interface requirements of a certain optimisation subroutine or you must re cast your modelling problem into a language specific to the package you are using While PEST has some similarities to existing nonlinear parameter estimation software it uses a powerful yet robust estimation technique that has been extensively tested on a wide range of problem types it has been designed according to a very different philosophy What is new about PEST is that it allows you to undertake parameter estimation and or data interpretation using a particular model without the necessity of having to make any changes to that model at all Thus PEST adapts to an existing model you dont nee
456. rameter In the second part a little extra data is supplied for tied parameters viz the name of the parameter to which each such tied parameter is linked If there are no tied parameters the second part of the parameter data section of the PEST control file is omitted Each item of parameter data is now discussed in detail refer to Example 4 1 for the arrangement on the PEST control file of the PEST input variables discussed below The PEST Control File 4 17 PARNME This is the parameter name Each parameter name must be unique and of twelve characters or less in length the name is case insensitive PARTRANS PARTRANS is a character variable which must assume one of four values viz none log fixed or tied If you wish that a parameter be log transformed throughout the estimation process the value log must be provided As discussed in Section 2 2 1 logarithmic transformation of some parameters may have a profound affect on the success of the parameter estimation process If a parameter is log transformed PEST optimises the log of the parameter rather than the parameter itself Hence the column of the Jacobian matrix pertaining to that parameter actually contains derivatives with respect to the log of the parameter likewise data specific to that parameter in the covariance correlation coefficient and eigenvector matrices computed by PEST pertains to the log of the parameter However when you supply
457. rd_model_run This string informs the model that it is being run either to test a parameter upgrade or as the first model run of the PEST optimisation process derivative_increment The model is being run as part of the finite difference derivatives calculation process undertaken by PEST external_derivatives The model is being run in order to write an external derivatives file If the character string on the first line of the PEST to model message file is derivative_increment then the integer on the second line of this file has significance A value of n for this integer indicates that the model run is being undertaken with the value of the n parameter incremented for the purpose of derivatives calculation by forward differences or as the first of two runs by which derivatives will be calculated using central differences A value of n indicates that the n parameter is decremented in the second of two runs undertaken for the purpose of derivatives calculation by central differences The third line of the message file lists the number of parameters PEST variable NPAR and number of observations PEST variable NOBS involved in the parameter estimation process Following this are NPAR lines of data with three entries on each line The first entry on each line is a parameter name recall that this name can contain up tol2 characters Then follows the value of the parameter used on the current model run Following that is an inte
458. rders packets of model runs to be completed parameters used for such runs are all based on decreasing Marquardt lambdas or on increasing Marquardt lambdas depending on the results of the previous package of parallel runs The lambda search procedure is such that parallelisation inevitably results in some model runs being wasted Hence although PEST might inform the user through its screen output that n parallel model runs are being carried out it may not display the results ie the objective function and perhaps the model prediction of all of these model runs It simply processes the results of that packet of runs in accordance with its lambda search algorithm If the demands of that algorithm are such that more Marquardt lambdas must then be tested another packet of runs is carried out Parallel PEST 9 12 Nevertheless there will be some occasions when the path taken by the parameter estimation process is slightly different when the lambda search procedure is parallelised from that which is taken when the lambda search is conducted on the basis of serial model runs If an unexpected and significant advance in the parameter estimation process is achieved in a run that would not have been undertaken on the basis of the usual Marquardt lambda testing procedure based on serial model runs PEST will not ignore this the results of that run will be assimilated into the parameter estimation process 9 3 Using Parallel PEST 9 3 1 Pre
459. re Optionally markers can be added to this plot select the appropriate pop up menu item this allowing near zero residuals to be more easily identified Furthermore by selecting the Absolute residuals option the absolute values of the residuals rather than the residuals themselves can be plotted Naturally symmetry about zero is lost in this case Where observations and or prior information items belong to different groups residuals pertaining to these different groups are plotted using different colours If PEST is running in regularisation mode observations and or prior information comprising the observation group regul do not appear on this plot unless specifically selected through the Select groups item of the pop up menu pertaining to this window 4 9 Jacobian Selection of the Jacobian tab allows a graphical representation of the Jacobian matrix to be displayed Recall that the Jacobian matrix otherwise known as the sensitivity matrix contains the derivative of each observation including items of prior information with respect to each parameter Where a parameter is log transformed the derivative is taken with respect to the log to base 10 of that parameter WinPEST C PEST2000 TEST1 VES11_PST File Run Options Jacobian View Validate S AB uw Av E F R RAR Phi Sensitivity Parameters Lambda Calc vs Obs Residuals Jacobian Correlation 4 gt Jacobian na
460. re of interest upon completion of the inversion process for example parameter uncertainties and covariances graphical displays can be inspected when the optimisation process is at an end alternatively provided pertinent settings are correct in the PEST control file these quantities can also be inspected as they evolve iteration by iteration Optimisation results calculated by WinPEST are also written in text form to the normal PEST output files such as the run record file parameter value file residuals file parameter and observation sensitivity files etc Use of WinPEST does not remove the necessity to understand the workings of PEST Prior to using WinPEST you will need to familiarize yourself with the PEST manual in order to understand how to prepare the PEST input files 2 2 WinPEST Windows Once WinPEST has been started see the next section different windows within the main WinPEST window are selectable by clicking on appropriately labelled tabs at the tops of these windows While some of WinPEST s windows are used for displaying text most are used for plotting graphs these latter windows will be referred to as graphical windows in the discussion that follows No matter what window you are viewing a list of available options pertinent to that window is available by clicking with the right mouse button anywhere within that window Alternatively click on the main menu item pertinent to that window each such item becomes av
461. recision as the model is capable of reading them with so that they can be provided to the model with the same precision with which PEST calculates them Generally models read numbers from the terminal or from an input file in either of two ways The Model PEST Interface 3 5 viz from specified fields or as a sequence of numbers each of which may be of any length in FORTRAN the latter method is often referred to as free field input If the model uses the former method then somewhere within the model program the format ie field specification for data entry is defined for every number which must be read in this fashion The FORTRAN code of Example 3 3 directs a program to read five real numbers The first three are read using a format specifier whereas the last two are read in free field fashion READ 20 100 A B C 100 FORMAT 3F10 0 READ 20 D E Example 3 3 Formatted and free field input The relevant part of the input file may be as illustrated in Example 3 4 6 32 1 42E 05123 456789 34 567 1 2E17 Example 3 4 Numbers read using the code of Example 3 3 Notice how no whitespace or comma is needed between numbers which are read using a field specifier The format statement labelled 100 in Example 3 3 directs that variable A be read from the first 10 positions on the line that variable B be read from the next 10 positions and that variable C be read from the 10 positions ther
462. rent observation The PEST Algorithm 2 5 types of vastly different magnitude The quantity o is known as the reference variance if all observation weights are unity it represents the variance of each experimental measurement If the weights are not all unity the measurement covariance matrix is determined from equation 2 10 with o given by equation 2 5 and given by equation 2 8 With the inclusion of observation weights equation 2 4 by which the system parameter vector is estimated becomes b X QX X Qc 2 11 while equation 2 6 for the parameter covariance matrix becomes C b o X QX 2 12 2 1 3 The Use of Prior Information in the Parameter Estimation Process It often happens that we have some information concerning the parameters that we wish to optimise and that we obtained this information independently of the current experiment This information may be in the form of other unrelated estimates of some or all of the parameters or of relationships between parameters expressed in the form of equation 2 2 It is often useful to include this information in the parameter estimation process both for the philosophical reason that it is a shame to withhold it and because this information may lend stability to the process The latter may be the case where parameters as determined solely from the current experiment are highly correlated This can lead to nonunique parameter estimates because certain pairs or groups of parameter
463. represent any of these numbers to the same degree of precision with which they are cited in the PEST control file The fact that PEST adjusts its internal representations of parameter values such that they are expressed with the same degree of precision as that with which they are written to the model input files has already been discussed see Section 3 2 For consistency PEST s internal representation of parameter bounds is adjusted in the same way 5 2 3 The Parameter Estimation Record After echoing its input data PEST calculates the objective function arising out of the initial parameter set it records this initial objective function value on the run record file together with the initial parameter values themselves Then it starts the estimation process in earnest beginning with the first optimisation iteration After calculating the Jacobian matrix PEST attempts objective function improvement using one or more Marquardt lambdas As it does this it records the corresponding objective function value both in absolute terms and as a Running PEST 5 10 fraction of the objective function value at the commencement of the optimisation iteration During the first iteration of Example 5 1 PEST tests two Marquardt lambdas because the second lambda results in an objective function fall of less than 0 03 ie PHIREDLAM relative to the first one tested PEST does not test any further lambdas Instead it progresses to the next optimisation iterati
464. res p and p2 are the values of the two parameters Predictive Analysis 6 2 min Figure 6 1 Objective function contours in parameter space linear model P 2 Figure 6 2 Objective function contours in parameter space nonlinear model Both of Figures 6 1 and 6 2 depict situations where there is a high degree of parameter correlation that is one parameter can be varied in harmony with another with virtually no effect on the objective function Thus the solution to the inverse problem ie the model calibration problem is nonunique If the minimum of the objective function is denoted as min and if all parameters for which the objective function is less than min 6 where is relatively small can also be considered to calibrate the model then the range of parameter values which can be considered to calibrate the model can be quite large indeed In Figures 6 1 and 6 2 the set of allowable parameter values from the model calibration point of view occupies the area that lies within the min contour in parameter space In most instances of model calibration only a single set of parameters lying within the min 6 contour of Figures 6 1 and 6 2 is calculated Model predictions are then made with this single parameter set An obvious question is this what would have been the model s Predictive Analysis 6 3 predictions if another set of parameters lying within the min 6 contour were used for predictive purpose
465. resent Natural weights as represented on the observation residuals file are the inverse of measurement standard deviations as determined above If these weights are used in the parameter estimation process the reference variance will be 1 0 Where a covariance matrix is supplied for one or more observation groups instead of weights the residuals file is slightly modified As well as this an extra file called a rotated residuals file is generated by PEST See Section 4 3 3 for details Running PEST 5 21 5 3 5 The Matrix File During each optimisation iteration just after it has calculated the Jacobian matrix if any of the ICOV ICOR or IEIG variables supplied in the PEST control file are set to 1 PEST calculates the covariance and correlation coefficient matrices as well as the eigenvalues and normalised eigenvectors of the covariance matrix for the current set of parameter values Depending on the settings of the ICOV ICOR and IEIG variables these matrices will then be written to a special file named a matrix file This file is named case mtt where case is the current case name ie the filename base of the PEST control file Each time this file is written the previous file of the same name is overwritten Hence the matrices contained in the matrix file pertain to the current optimisation iteration only or at the end of the parameter estimation process to the last optimisation iteration If any of ICOV ICOR or
466. riance Matrix gt ro2 h1 h2 ro2 0 3136 4 8700E 03 0 4563 h1 4 8700E 03 0 3618 1 3340E 02 h2 0 4563 1 3340E 02 0 8660 Correlation Coefficient Matrix gt ro2 hl h2 ro2 1 000 1 4457E 02 0 8756 h1 1 4457E 02 1 000 2 3832E 02 h2 0 8756 2 3832E 02 1 000 Normalized eigenvectors of covariance matrix gt Vector I Vector_2 Vector_3 ro2 0 8704 3 6691E 02 0 4909 hi 3 5287E 02 0 9993 1 2121E 02 h2 0 4910 6 7718E 03 0 8711 Eigenvalues gt 5 6045E 02 0 3621 1 123 The various sections of the PEST run record file are now discussed in detail 5 2 2 Echoing the Input Data Set PEST commences execution by reading all its input data As soon as this is read it echoes most of this data to the run record file Hence the first section of this file is simply a restatement of most of the information contained in the PEST control file Note that the letters na stand for not applicable in Example 5 1 na is used a number of times to indicate that a particular PEST input variable has no effect on the optimisation process Thus for example the type of change limit for parameter rol is not applicable because this parameter is fixed It is possible that the numbers cited for a parameter s initial value and for its upper and lower bounds will be altered slightly from that supplied in the PEST control file This will only occur if the space occupied by this parameter in a model input file is insufficient to
467. rious values 8 5 4 Derivatives Command Line Section If a non zero value is supplied for JACFILE in the control data section of the PEST control file the PEST control file must contain a derivatives command line section This must be situated just above the model command line section Contents of the derivatives command line section of the PEST control file are illustrated in Example 8 3 while an example is provided in Example 8 4 derivatives command line command to run the model EXTDERFLE Example 8 3 Structure of the derivatives command line section of a PEST control file Model Calculated Derivatives 8 9 derivatives command line model_d bat derivs dat Example 8 4 An example of the derivatives command line section of a PEST control file Like all other sections of the PEST control file the beginning of the derivatives command line section must be denoted using a special header line the first character of which is an asterisk Following the header line the next line of this section consists of the command used to run the model when it is required to calculate external derivatives If appropriate this can be the same command as that used to run the model for the purpose of testing parameter upgrades or for the calculation of derivatives using finite differences The final line of the derivatives command line section consists of the n
468. rk Numerical Computing for versions of SENSAN TEMPCHEK and INSCHEK in which these restrictions are lifted SENSAN 11 2 A comprehensive SENSAN input data checker named SENSCHEK is provided with SENSAN Its role is similar to that of PESTCHEK and should be run after all SENSAN input data has been prepared prior to running SENSAN itself 11 2 SENSAN File Requirements 11 2 1 General SENSAN requires four types of input file The first two are the SENSAN control file and the parameter variation file The former file provides SENSAN with the structural details of a particular sensitivity analysis The latter provides SENSAN with the parameter values to be used in the succession of model runs which it must undertake The other two file types are PEST template and instruction files These latter two kinds of file are dealt with briefly first 11 2 2 Template Files Section 3 2 of this manual provides a detailed discussion of how PEST writes parameter values to model input files After a user has prepared a template file prior to running PEST he she can check its integrity using the PEST utility TEMPCHEK As explained in Chapter 10 TEMPCHEK also provides the functionality to generate a model input file on the basis of a template file and a corresponding user supplied list of parameter values Rather than reproduce this functionality within SENSAN SENSAN simply runs TEMPCHEK whenever it wishes to prepare a model input file on the basis of a set
469. rmed and untransformed counterparts 2 2 5 Parameter Change Limits As has already been discussed no parameter can be adjusted by PEST above its upper bound or below its lower bound However there is a further limit on parameter changes determined by the amount by which a parameter is permitted to change in any one optimisation iteration If the model under PEST s control exhibits reasonably linear behaviour the updated parameter set determined by equations 2 23 2 24 and 2 26 will result in a lowering of the objective function However if the model is highly nonlinear the parameter upgrade vector Bu may overshoot the objective function minimum and the new value of may actually be worse than the old one This is because equations 2 23 and 2 24 are based on a linearity assumption which may not extend as far into parameter space from the current parameter estimates as the magnitude of the upgrade vector which they predict To obviate the possibility of overshoot it is good practice to place a reasonable limit on the maximum change that any adjustable parameter is allowed to undergo in any one optimisation iteration Such limits may be of two types viz relative and factor You must inform PEST through the parameter variable PARCHGLIM on the PEST control file which type of change limit applies to each adjustable parameter Two other PEST input variables RELPARMAX and FACPARMAX provide the maximum allowed relative and f
470. rmination are set through PEST input variables PHIREDSTP NPHISTP and NPHINORED If the lowest NPHISTP s achieved in all iterations carried out to date are within a distance of PHIREDSTP of each other relative to the lowest achieved so far or if NPHINORED iterations have elapsed since the lowest was achieved then PEST execution will cease The second indicator of either convergence to the objective function minimum or of the unlikelihood of achieving it is the behaviour of the adjustable parameters If successive iterations are effecting little change in parameter values there is probably little to gain in continuing with PEST execution Input variables RELPARSTP and NRELPAR set the exact criterion if the largest relative parameter change over the last NRELPAR iterations has been RELPARSTP or less PEST will not proceed to the next iteration The input variable NOPTMAX sets an upper limit on the number of optimisation iterations which PEST carries out PEST will terminate execution after NOPTMAX iterations no The PEST Algorithm 2 26 matter what the current status of the objective function or of the parameter values Other termination criteria are set internally As has already been mentioned PEST will terminate the optimisation process if it calculates a parameter set for which the objective function is zero Also if the gradient of the objective function with respect to all parameters is zero or if a zero valued parameter upgra
471. rmore if there is sufficient space up to 23 characters can be used to record the value of the parameter instead of the usual maximum of PEST Utilities 10 16 13 Setting the DPOINT variable to nopoint instructs PEST to write a parameter s value to a model input file without the decimal point if this can be accomplished through numerical formatting thereby gaining one extra significant figure of precision more will be said about precision shortly As is stated above the control data section of the PAR2PAR input file is optional if it is omitted default values of single and point are supplied for PRECIS and DPOINT respectively 10 7 3 Using PAR2PAR with PEST 10 7 3 1 The Composite Model As was discussed above when used with PEST PAR2PAR will normally be run as part of a composite model encapsulated in a batch file Thus whenever PEST runs the model it first runs PAR2PAR and any other model preprocessors cited in the batch file followed by the model followed by any model postprocessors cited in the batch file As for any other model executable program which uses parameters which require estimation by PEST a template file must be built based on a PAR2PAR input file Just before it runs the model PEST will then write current parameter values to the PAR2PAR input file using the corresponding template file An example of such a template file based on the PAR2PAR input file shown in Example 10 5 is pro
472. ro3 log factor 6 00 0 1 10000 ro 101 observation groups obsgp EEU CEN UE W obsgpl covl dat obsgp2 cov2 dat observation data arl 1 21 1 0 obsgp ar2 1 51 1 0 obsgp ar3 20 7 0 obsgp ar4 2 94 0 obsgp ard 4 15 0 obsgp ar6 5 77 1 0 obsgp ar7 7 78 1 0 obsgpl ar8 9 99 1 0 obsgpl arg 11 8 0 obsgpl arlO 12 3 0 obsgpl model command line model bat model input output ves tpl model inl ves ins model out prior information pil 1 0 log rol 1 32 1 0 pi2 1 0 log ro2 0 45 1 0 obsgp2 pi3 1 0 log ro3 0 89 1 0 Example 4 5 A PEST control file citing two covariance matrices The following rules must be obeyed when using one or more observation covariance matrices in a PEST run 1 The name of a text file containing an observation covariance matrix or rather a The PEST Control File 4 29 matrix related to an observation covariance matrix by an unknown constant of proportionality can be provided in the PEST control file following the name of the observation group to which the matrix pertains in the observation groups section of the PEST control file in Example 4 5 the names of these observation covariance matrix files are cov dat and cov2 dat 2 A covariance matrix file must contain a square symmetric matrix of dimension n where n is the number of observations belonging to the observation group to which the covariance matrix pertains Thus every observation belonging to the pertinent observ
473. roportional to the covariance matrix of the observations the constant of proportionality being the reference variance Because Q is a diagonal matrix so too is the observation covariance matrix The use of observation weights in calculating the objective function is based on the premise that observations are independent ie that the uncertainty pertaining to any one observation bears no relationship to the uncertainty pertaining to any other observation In practice observation uncertainty in a calibration context is determined by the level of misfit between these observations and corresponding model outputs ie by the model to measurement residuals calculated at the end of the inversion process If these residuals are expected to be uncorrelated then observation uncertainties can be expressed in terms of individual observation weights However if residuals are likely to show consistency over space and or time for certain observation types then it may not be appropriate to assume statistical independence for these observation types In such cases it may be preferable to describe the uncertainties associated with these observations using an observation covariance matrix or a matrix that is proportional to this matrix rather than using a set of individual observation weights The theory underpinning the use of observation weights presented in Section 2 1 2 is also applicable to the use of an observation covariance matrix in
474. roup can then be individually monitored BA WinPEST D COURSETESTSUNSAT2RA PST File Run Options Histograms View Validate Help S B s Ov a PEST Log Phi Sensitivity Parameters Lambda Calc vs Obs Residuals Jacobian all Calibration Residuals Histogram weighted Mean Value 0 003708841 g Frequency Normal Distribution Zero gt oO o D gt ie 20 So 0 36 0 16 0 04 0 24 Normalized Residuals Calc Obsi Max Min weight Normalized Residuals Calc Obs Max Min weight n a Frequency nia Pest suspended Figure 4 Ideally residuals should be normally distributed about a mean of zero after completion of the parameter estimation process The Histograms window allows the user to assess the degree to which residuals either taken as a whole or belonging to individual groups are normally distributed about a mean of zero This is achieved by superimposing a normal distribution curve over a frequency distribution histogram for the residuals The normal distribution curve possesses the same standard deviation as the residuals but has a mean of zero Observation groups for representation in the residuals histogram can be selected by first clicking with the right mouse button anywhere within the window and then selecting the Select groups submenu item from the resulting pop up menu The user can also decide whether the superimposed normal distributio
475. rt PEST from a previously optimised set of parameter values If this is done the measurement objective function m will approach PHIMLIM ie Pm from the inside rather than from the outside Sometimes this appears to be a more stable procedure Where run times are long it may also prove to be a faster procedure as PEST can normally minimise the regularisation objective function subject to the PHIMLIM constraint using a fewer number of model runs when starting from an optimal or near optimal set of parameter values than when starting from parameter values that are far from optimal 7 6 Two Examples of Regularisation 7 6 1 A Layered Half Space Electrical soundings are often undertaken by geophysicists in order to infer the variation of electrical resistivity with depth in the ground A sounding is carried out by passing electrical current through the ground between two current electrodes placed at varying distances apart from each other and measuring the voltage gradient at the surface induced by this current flow Interpretation of data gathered in this way is based on the premise that the earth can be simulated as a layered half space A model can be used to simulate the electrical response of this half space to the currents that flow through it for different current electrode separations PEST can then be used in conjunction with this model to infer the electrical properties of the half space from the sounding data A traditional
476. rty January 2002 Bugs In the unlikely event that you discover a bug in PEST please report it to me John Doherty at the following email address ekaa Table of Contents Table of Contents Te INtrOGUCtION oenn e ae E SEE A E E E NEA debs bla sod sodadeuetwaboas Saepe ES 1 1 1 1 Tristall ation siina noisa aetas eE EE KESE EE TIAA E R EEEa 1 1 T2 The PEST Concepten et e vies sete aaa e A E a a ates a 1 1 1 2 1 A Model s Data Requirements seseeseseessesessessesssesresresressessrssresresressessrsseeeresees 1 1 1 2 2 The Role of PEST rient i a e aa a Mee a os 1 2 K3 What Pest Does merone oeh ea eea EREE E A TEE Ee ERAR meet 1 3 1 4 An Overview of PEST anioi ae aa a ET AE EE E E i 1 4 1 4 1 Parameter Definition and Recognition sssseseseeesssessssesressresresresresressrssreerese 1 5 1 4 2 Observation Definition and Recognition sesseseeseesessessessresresresressessessreereses 1 6 1 4 3 The Parameter Estimation Algorithm seseeseseesereeresressessresresresressessrssreereseese 1 7 1 4 4 Predictive Analysis ooieoe on i eaer a ea Se sobs TA ep TE Eia inae 1 9 1 4 5 Regularisation snenie n nnii n a i a e e iea a aiei 1 10 1 5 How to Use PES Tonini iae dete eee hen Sag ate ah va a E aaoi iR ieat 1 11 1 5 1 The Two Versions Of PEST ooo cee eeceeceseccscecseeceseceeeeseeesaeesaceeeeesseesaeeeseesees 1 11 1 5 2 PEST Utilities ieda a ether ace bata cise eee 1 11 1 5 3 Parameter Preprocessing sceecccescce
477. s Notwithstanding the above limitations the presentation of 95 confidence limits provides a useful means of comparing the certainty with which different parameter values are estimated Running PEST 5 13 by PEST In Example 5 1 it is obvious that parameters ro2 and h2 particularly h2 are estimated with a large margin of uncertainty This is because these two parameters are well correlated this means that they can be varied in harmony and provided one is varied in a manner that properly complements the variation of the other there will be little effect on the objective function Hence while the objective function may be individually sensitive to each one of these parameters it appears to be relatively insensitive to both of them if they are varied in concert This illustrates the great superiority of using covariance and eigenvector analysis over the often used sensitivity analysis method of determining parameter reliability Confidence limits are not provided for tied parameters The parent parameters of all tied parameters are estimated with the tied parameters riding on their back hence the confidence intervals for the respective parent parameters reflect their linkages to the tied parameters Note that at the end of a PEST optimisation run a listing of the optimised parameter values can also be found in the PEST parameter value file case par 5 2 5 Observations and Prior Information After it has writte
478. s The contribution that this makes to network traffic may result in a user having to employ a higher value for WAIT than would otherwise be the case further slowing down Parallel PEST s operations However with disk caching these problems may be mitigated somewhat When I came to work one morning after Parallel PEST had been running all night one of my slaves had ceased carrying out model runs displaying instead the following prompt Sharing violation reading drive C Abort Retry Fail Why did this happen This message was sent by the operating system not by PSLAVE The model was probably trying to read an input file that had not yet been closed by PEST or PEST was trying to read a model output file that had not yet been closed by the model Theoretically such things should not happen However on busy machines connected to busy networks it may take some time for one process on a particular machine to get the message that a file has been closed by a process running on another machine To prevent this error from happening increase the value of WAIT the length in seconds of the pause made by PSLAVE and PEST at certain strategic stages of run management and data exchange Can PSLAVE run a batch file containing multiple executables as the model just like PEST Frequently Asked Questions 13 3 can Yes However make sure that either the full directory of each executable is included within its name as cited in the batch file or t
479. s for example solute concentration and solvent flow rates in a chemical process model the weights assigned to the two observation types should reflect the relative magnitudes of the numbers used to express the two quantities in this way the set of larger numbers will not dominate the parameter estimation process just because the numbers are large A particular observation can be provided with a weight of zero if you do not wish it to affect the optimisation process at all Like parameters you must provide each observation with a name of up to twelve characters in length PEST uses this name to provide you with information about that observation 1 4 3 The Parameter Estimation Algorithm The Gauss Marquardt Levenberg algorithm used by PEST is described in detail in the next chapter For linear models ie models for which observations are calculated from parameters through a matrix equation with constant parameter coefficients optimisation can be achieved in one step However for nonlinear problems most models fall into this category parameter estimation is an iterative process At the beginning of each iteration the relationship between model parameters and model generated observations is linearised by formulating it as a Taylor expansion about the currently best parameter set hence the derivatives of all observations with respect to all parameters must be calculated This linearised problem is then solved for a better parameter set and the ne
480. s if varied in concert in a certain linear combination may effect very little change in the objective function In some cases this nonuniqueness can even lead to numerical instability and failure of the estimation process However if something is known about at least one of the members of such a troublesome parameter group this information if included in the estimation process may remove the nonuniqueness and provide stability Parameter estimates will also be nonunique if there are less observations then there are parameters equation 2 11 is not solvable under these conditions as the matrix X QX is singular Note that PEST will nevertheless calculate parameter estimates for reasons discussed later in this chapter However the inclusion of prior information being mathematically equivalent to taking extra measurements may alter the numerical predominance of parameters over observations and thus provide the system with the ability to supply a unique set of parameter estimates Prior information is included in the estimation algorithm by simply adding rows containing this information to the matrix equation 2 1 This information must be of a suitable type to be included in equation 2 1 both simple equality and linear relationships of the type described by equation 2 2 are acceptable A weight must be included with each article of prior information this weight being inversely proportional to the standard deviation of the right hand side of the
481. s encountering their relative or factor limits thus restricting alterations made to other parameters to ineffectual levels See Section 5 6 for further details 2 2 8 Components of the Objective Function As has already been discussed the objective function is calculated as the squared sum of weighted residuals including prior information If is often of interest to know what contribution certain observations or groups of observations make to the objective function This is possible through the use of observation groups Each observation and each item of The PEST Algorithm 2 25 prior information must be assigned to a group the number and names of such groups are specified by the user The ability to calculate the contribution made by individual observations or groups of observations to the objective function is useful in situations where the user wishes that different types of information contribute an approximately equal amount to the value of the objective function This ensures that no observation groupings are either drowned by other information or dominate the inversion process 2 2 9 Termination Criteria PEST updates parameters using equations derived on the basis of a linearity assumption which is not met if the model is nonlinear Nevertheless by iteratively updating the parameters in accordance with these equations as many times as is necessary an optimal parameter set will mostly be obtained in the end When
482. s instead particularly if this alternative set is relatively distant from the original parameter set as measured in parameter space 6 1 2 Some Solutions A number of different methods are available for answering this question However traditional sensitivity analysis cannot be used The term sensitivity analysis is often used to describe the technique whereby parameter values are individually varied from calibration values in order to determine the effects of these changes on model predictive outcomes This is an unacceptable method of predictive analysis in most instances because unless parameters are varied in certain discrete ratios that vary with parameter value for nonlinear models the model is immediately uncalibrated as soon as any parameter is varied from its calibrated value To fully explore the repercussions of parameter nonuniqueness on predictive nonuniqueness parameters must be varied in such a way that the objective function hardly changes As is apparent from Figures 6 1 and 6 2 parameter values can often vary enormously while the objective function changes very little If parameters are simply incremented and or decremented one by one and the effect of this variation tested on both model predictions and on the objective function calculated under calibration conditions it is likely that the latter will rise very quickly giving the modeller the false impression that parameter values are estimated with a high degree of pr
483. s on the last character of that marker in this case on the character itself Hence after reading the character PEST is able to process the sp1 instruction by isolating the string 1 23498ES in the manner described above After it reads the model calculated value for observation sp1 PEST moves to the next instruction line In accordance with the 11 instruction PEST reads into its memory the next line of the model output file It then searches for a character and reads the number following this character as observation sp2 This procedure is then repeated for observations sp3 and sp4 Successful identification of a non fixed observation depends on the instructions preceding it The secondary marker tab and whitespace instructions will be most useful in this regard though fixed and semi fixed observations may also precede a non fixed observation remember that in all these cases PEST places its cursor over the last character of the string or number it identifies on the model output file corresponding to an instruction item before proceeding to the next instruction Consider the model output file line shown below as a further illustration of the use of non fixed observations 4 33 20 3 23 392093 3 394382 If we are interested in the fourth of these numbers but we are unsure as to whether the numbers preceding it might not be written with greater precision in some model runs hence
484. s readily apparent in Example 5 1 e No account is taken of parameter upper and lower bounds in the calculation of 95 confidence intervals Thus an upper or lower confidence limit can lie well outside a parameter s allowed domain In Example 5 1 the upper confidence limits for both ro2 and h2 lie well above the allowed upper bounds for these parameters as provided by the parameter input variable PARUBND for each of these parameters similarly the lower confidence limit for parameter h1 lies below its lower bound PARLBND of 0 05 PEST does not truncate the confidence intervals at the parameter domain boundaries so as not to provide an unduly optimistic impression of parameter certainty e The parameter confidence intervals are highly dependent on the assumptions underpinning the model If the model has too few parameters to accurately simulate a particular system the optimised objective function will be large and then so too through equations 2 5 and 2 17 will be the parameter covariances and with them the parameter confidence intervals However if a model has too many parameters the objective function may well be small but some parameters may be highly correlated with each other due to an inability on the part of a possibly limited measurement set to uniquely determine each parameter of such a complex model this will give rise to large covariance values and hence large confidence intervals for the correlated parameter
485. s the default condition Ideally at the end of the optimisation process the relationship between calculated and measured values should be that of a straight line with a slope of 1 and an intercept of zero For reference purposes this line appears on the plot as well If the plotted points comprising this graph appear to fall on a line with a slope different from unity or with an intercept different from zero this indicates an inability on the part of the model to simulate significant aspects of system behaviour On the other hand the degree of scatter of plotted points around the 1 1 line indicates either an inability on the part of the model to simulate fine details of system behaviour or the presence of a high amount of measurement noise WinPEST Graphical Windows 13 ae WinPEST C PEST2000 TESTISSOUNDING PST File Run Options Calc vsObs View Validate S i B gt in Ov E F RAAR PEST Log Phi Sensitivity Parameters Lambda Calc vs Obs Residuals Jacobian 4 gt Calculated vs Observed weighted Calc Obs on g E obsgpi obsgp2 iS gt D ka o o wm Ss 2 O 0 3 5 9 10 9 Observed weight Calculated weight 7 5 Dbserved weight 10 4 Pest suspended Figure 2 Observations can be plotted against their model generated counterparts by selecting the Calc vs Obs tab Where there is more than one observation group cited in the PEST control file dif
486. s to carry out model runs However if this slave is then re started PEST will not recognise it Even if you start a slave just after PEST has started before it has actually undertaken any model runs PEST will still not recognise it as it must be started before PEST commences execution Our office network is such that we have a single server to which all of our individual machines have access as drive O Our machines are not able to communicate individually with each other in a peer to peer sense What is the best way to set up Parallel PEST under these circumstances Install the model on each machine Create as many subdirectories on drive O as there are slaves calling these subdirectories for example subl sub2 etc On each slave machine make one such directory the current working directory making sure to use a different subdirectory in each case Make sure the model can run from that subdirectory once PEST generated input files are placed there ie do a dummy model run on each slave machine Frequently Asked Questions 13 2 using normal model input files placed in each of the sub sub2 etc subdirectories Then start up PSLAVE on each slave machine one of which will also probably be the machine running PEST The PEST template and instruction files can reside in any directory on the master machine However for each slave Parallel PEST should be informed that model input files and model output files reside in the subl sub2 e
487. sary to represent their value the decimal point does not need to be included if it is redundant If exponentiation is required this can be accomplished with either the d or e symbol note however that all real numbers are stored internally by PEST as double precision numbers The data which must reside in the PEST control file is now discussed in detail section by section Refer to Example 4 1 for the location within the PEST control file of each input variable discussed below 4 2 2 Control Data The data provided in the control data section of the PEST control file are used to set internal array dimensions tune the optimisation algorithm to the problem at hand and set some data output options RSTFLE This character variable must be assigned one of two possible values viz restart or norestart Note that for this and other character variables in the PEST control file PEST is case insensitive If it takes the value restart PEST will dump the contents of many of its data arrays to a binary file named case rst where case is the current case name at the beginning of each optimisation iteration this allows PEST to be restarted later if execution is prematurely terminated If subsequent PEST execution is initiated using the r command line switch see Section 5 4 2 it will recommence execution at the beginning of the iteration during which it was interrupted PEST will also dump the Jacobian
488. screen by clicking on its respective tab Where a plot is unavailable for example a plot of optimisation results prior to the completion of a PEST run if the respective ICOV ICOR or IEIG variable in the PEST control file was set to 0 the pertinent window is left bare until the plot can be produced Each of WinPEST s graphical windows will now be briefly discussed 4 2 Phi The Phi window plots the objective function against iteration number As the optimisation process progresses the objective function should fall at first until it eventually flattens out to a level where PEST can reduce it no further Where observations and or prior information items are assigned to more than one observation group the contribution to the objective function made by each such group is also plotted in this window Plots of these objective function contributions can be selected deselected for display by clicking with the right mouse button anywhere within this window and then clicking on the Select groups item of the resulting pop up menu When PEST is running in regularisation mode only two variables are plotted in this window viz the measurement objective function and the regularisation objective function The latter can be selected de selected through the Select groups item on the pop up menu accessible by clicking anywhere on the graph with the right mouse button 4 3 Regularisation This window plots the value of the regular
489. se predictive analysis is a necessity Alternatively some kind of problem regularisation is required this being a mechanism by which parameter uncertainty can be reduced through the enforcement of known or suspected relationships between parameters such as a smoothing condition in the case of distributed parameter models Such techniques can be very effective in combating the deleterious effects of over parameterisation if properly designed they can be such as to ensure that departures from simplicity in parameterisation are limited to those that are just sufficient to reproduce the required level of detail in system behaviour See Chapter 7 for a description of PEST s uses in regularisation mode for more details 5 7 6 Covariance Matrix for Best Fit Parameters The user is reminded that once PEST has calculated an optimal set of parameter values and in accordance with one of its termination criteria finishes execution it calculates the covariance matrix and the statistics derived therefrom on the basis of the Jacobian matrix giving rise to the best set of parameter values unless these were achieved on the very last optimisation iteration If this is the case then PEST s termination criteria are such as to ensure that the statistics calculated on the basis of the not quite optimal parameter set will depart only minimally from those calculated on the basis of the optimal parameter set Use of the not quite optimal
490. sed as a result of the previous instruction Primary Marker Unless it is a continuation of a previous line each instruction line must begin with either of two instruction items viz a primary marker or a line advance item The primary marker has already been discussed briefly It is a string of characters bracketed at each end by a marker delimiter If a marker is the first item on an instruction line then it is a primary marker if it occurs later in the line following other instruction items it is a secondary marker the operation of which will be discussed below On encountering a primary marker in an instruction file PEST reads the model output file line by line searching for the string between the marker delimiter characters When it finds the string it places its cursor at the last character of the string Note that this cursor is never actually seen by the PEST user it simply marks the point where PEST is at in its processing of the model output file This means that if any further instructions on the same instruction line as the primary marker direct PEST to further processing of this line that processing must pertain to parts of the model output file line following the string identified as the primary marker Note that if there are blank characters in a primary or secondary marker exactly the same number of blank characters is expected in the matching string on the model output file Often as in Example 3 7 a primar
491. seeesseeeeeeee 3 20 SENSAN viniani a ces 1 13 11 1 SENSAN Control File ccccccecesseessseeesseeesseeeesees 11 4 SENSCHEK viscgcdsasscvctenscassccievecuvaviesivseteeconvavetvevesess 11 8 SENSFLE 11 6 11 10 DEMSILVILY e e Seasksuesteseviss soucetvressseatesadzasssausetesd 5 16 Sensitivity ANALYSIS aniue aa 5 13 Sharing violation sssseeeseeeeeeeseeereeeeeeeee 9 9 9 17 13 2 SLAVDIR riii irii iil iE i R 9 9 SLANG AEE E AEEA cabs aos ea ives AOR ek bs 9 3 9 4 Index SMOOUMIG ces oukentedseccapeiceronsactauneasnecsbeest ceipuvesseetoee 7 2 Standard deviation cccesccesscceseceesseeesseeesseeeeeee 5 14 MAD coc re OENE A E BTR aCe OTE Taylor s theorem TEMPCHEK scssscsovesssedevideosncinrgivegssveensee TEMPFLE Template files Terminal input Terminal output sensein eanan Tied parameters 0 eeeeeseeeeeneeeeees User interaction Whitespace eee ceesesecseeseceseeeeeetseeeeeeaeees WINDOWS oon iudessticsetiees haves terddovenn satin
492. ses involving spatially distributed parameters individual parameter values or the differences between individual parameter values may exhibit some degree of distance dependent correlation perhaps expressed by a variogram In this case it may be a good idea to include that correlation in the inversion process by assigning a covariance matrix to the prior information equations that reflects the observed parameter interdependence rather than a set of weights based on the false premise that the value of each parameter is independent of that of its neighbours Observation correlation may be important in other situations as well For example consider the case where for a particular ground water model the extent of outflow from the ground water domain into two neighbouring reaches of a stream is used in the model calibration process Consider also as often occurs in practice that the total outflow into both of the neighbouring reaches can be more accurately measured than the outflow into each individual reach However it may be considered desirable for a particular model application that the model be calibrated using both of the individual reach outflows rather than the total outflow Because the uncertainties associated with the individual reach outflow measurements will not be independent for a positive error in one is likely to be complemented by a negative error in the other it would be better to assign a covariance matrix to these obs
493. sis of the sensitivity of particular model outputs to particular model inputs must be performed Such an analysis may be required as part of an effort to increase a modeller s understanding of the processes simulated by the model Or it may be the first step in a model calibration exercise whereby key system parameters are identified SENSAN facilitates the sensitivity analysis process by allowing a modeller to automate the tedious task of adjusting certain model inputs running the model reading the outputs of interest recording their values and then commencing the whole cycle again Using SENSAN a modeller can prepare for an unlimited number of model runs and then let the computer undertake these runs overnight over a weekend or simply while he she is doing other things SENSAN reads user prepared parameter values and writes specified model output values to files which can easily be imported to a spreadsheet for further processing If requested a system command can be issued after each model run For example a user may wish to rename certain model output files after some model runs have been completed hence these model output files are not overwritten during subsequent model runs and are thus available for later inspection SENSAN is model independent This means that it can be used to conduct a sensitivity analysis in conjunction with any model It achieves this by communicating with a model through the model s own input and output files
494. sitive parameters that can cause most problems in the calibration of complex models especially where parameters are many and correlation is Preface high At any stage of the optimisation process a user can request that certain troublesome parameters be held at their current values Such parameters can be demarcated either individually or according to whether their sensitivity drops below a certain threshold in the course of the parameter estimation process Alternatively a user can request that the x least sensitive parameters within a certain group be held while the parameter upgrade vector is calculated where x is supplied by the user according to his her knowledge and intuition with regard to the current parameter estimation problem As well as this certain variables controlling how the parameter upgrade vector is calculated can now be altered during a PEST run Calculation of the parameter upgrade vector can now be repeated Thus if a user thinks that PEST could have done a better job of lowering the objective function during a certain optimisation iteration he she can halt PEST execution instruct PEST to hold certain parameters at current values and ask PEST to calculate the parameter upgrade vector again This can be done without the need to re calculated the Jacobian matrix the most time consuming part of PEST s operations because the latter is stored every time it is calculated in anticipation of just such a request As a
495. sly read as the current project If Validate then Instructions is selected all instruction files cited in the current PEST control file are checked using INSCHEK If Validate then Check all is selected program PESTCHEK is run to check the control file and all template and instruction files cited therein Alternatively checking of the input dataset using PESTCHEK can be initiated by clicking on the Validate PEST input files button v Preparing for a PEST Run 7 on the WinPEST toolbar Or you can click on the neighbouring arrow to select between activation of the template file instruction file or entire dataset checking procedures If any errors are detected through any of these checking mechanisms a number of things happen First WinPEST opens the file in which the error condition was encountered and the focus is shifted to its text editing window whereupon that file is displayed This window is then divided into two compartments with the file listed in the upper compartment and errors listed in the bottom compartment line numbers accompany most error messages so that the source of the error can be easily found By pressing the right mouse button while in the upper compartment and selecting Go to from the pop up menu you can move to the appropriate line within the offending file and rectify the error This file can then be saved by pressing the Save button kl on the toolbar It can be closed by clicking w
496. spect to a particular parameter can be ascertained by clicking on the pertinent histogram bar A bubble will then appear in which the numerical value of the sensitivity is printed remove it by clicking on it Note also that if the legend on the side of this plot does not supply all of the information on parameter or observation colour coding that it should move it to the bottom of the window by dragging it with the mouse If there are many parameters and or observations involved in the optimisation process plotting of the Jacobian elements can take a long time In these circumstances WinPEST warns the user of this when the Jacobian tab is selected and gives him her the option of selecting parameters and observations for display before the plot is actually generated 4 10 Histograms Ideally when the optimisation process is complete weighted residuals should have a mean of zero In many cases they should also be normally distributed about this mean Where the measurement dataset upon which the parameter estimation process is based contains data of more than one type or from more than one source this should apply both to the residuals WinPEST Graphical Windows 16 taken as a whole as well as to the residuals pertaining to each data type or source individually Through its observation group functionality PEST allows these different measurement types to be assigned to different groups Various aspects of the behaviour of each such g
497. st A number of PEST control variables can also be re set through the parameter hold file click on the Controls tab to view a dialogue box through which new values for these variables can be supplied The user is advised to exercise caution in altering these variables however This applies especially to the Marquardt lambda If you forget to clear this entry after the change to lambda has been effected then the parameter hold file will continue to instruct PEST to use the altered lambda during every subsequent optimisation iteration This could do irreparable damage to the parameter estimation process for it is an integral part of PEST s operations that it regularly adjusts the Marquardt lambda to its optimum value at all stages of this process After entries are made to the Group Eigenvectors or Control dialogue boxes of the WinPEST Hold Status window click on Apply to write a parameter hold file with the appropriate information However when holding parameters on an individual basis in the Parameters subwindow there is no need to press Apply for the new parameter hold file is written automatically each time a change is made Once the parameter hold file has been written PEST should be re started with the switch Actually the parameter hold file can be edited and saved with PEST still running if you wish it is read by PEST on the next occasion that it completes calculation of th
498. stem resources available for the running of the model on the same machine For such a case running PSLAVE from the model working directory thus designating it as the PSLAVE working directory is again entirely appropriate This directory can also be that from which PEST is run and may thus hold PEST control template and instruction files See Section 9 3 8 for a further discussion of this topic 9 2 5 The Parallel PEST Run Management File The purpose of the Parallel PEST run management file is to inform PEST of the working directory of each slave as seen through the network from the machine on which PEST is run and of the names and paths pertaining to each model input file which it must write and each model output file which it must read The run management file must possess the same filename base as the current PEST case its extension must be rmf Thus if Parallel PEST is run using the command ppest calib then PEST will look to file calib pst to read its control data ie calib pst is the PEST control file for the current case and file calib rmf to read data pertaining to the distribution of model Parallel PEST 9 8 runs across the network Example 9 1 shows the structure of a run management file while Example 9 2 shows an example of such a file for the case where there are three slaves prf NSLAVE IFLETYP WAIT PARLAM SLAVNAME SLAVDIR once for each slave RUNTIME I I 1 NSLAVE Any lines after this point are r
499. struction files can read the same model output file PRECIS PRECIS is a character variable which must take either the value single or double It determines whether single or double precision protocol is used to represent a very large or very small number or a number in a wide parameter space see Section 3 2 6 for more details The value single is usually appropriate DPOINT DPOINT must be supplied as either point or nopoint In the latter case the decimal point is omitted if there is a tight squeeze of a parameter value into a parameter space Use point if at all possible for some models make assumptions regarding the location of a missing decimal point See Section 3 2 6 for more details 11 2 7 SENSAN Files VARFLE VAREFLE is the name of the parameter variation file for the current model run The number of parameters cited in this file must agree with the value of NPAR cited in the control data section of the SENSAN input file ABSFLE RELFLE and SENSFLE The names of the three SENSAN output files Contents of these files are discussed below 11 2 8 Model Command Line Provide the command that you would normally use to run the model Remember that you can enter the name of a batch file here to run a model consisting of multiple executables To prevent screen output from occurring during execution of batch file commands if desired you can disable echoing of each batch file command using th
500. such as these it is essential that PESTCHEK be used to check all input data prior to a PEST run PESTCHEK also carries out some of the tasks undertaken by programs TEMPCHEK and INSCHEK viz it checks all template and instruction files cited in the PEST control file for correct syntax Unlike TEMPCHEK and INSCHEK PESTCHEK cannot generate a model input file nor read a model output file nevertheless it does check that all parameters and observations cited in the PEST control file are also cited in the template and instruction files referenced in the PEST control file and that parameters and observations cited in template and instruction files are also listed in the PEST control file PESTCHEK is run using the command pestchek case where case is the filename base of a PEST control file PEST Utilities 10 5 No filename extension should be provided here an extension of pst is added automatically This is the same filename base which should be provided to PEST on its command line see Section 5 1 2 PESTCHEK reads an identical dataset to PEST PESTCHEK writes any errors it encounters to the screen If you wish error messages can be redirected to a file using the gt symbol on the PEST command line Thus to check the dataset contained in the PEST control file calib pst and the template and instruction files cited therein directing any error messages to the file errors chk invoke PESTCHEK using the command pestchek c
501. supplied threshold value of m while minimising the regularisation objective function r The PEST Algorithm 2 16 The remainder of this section describes the theory behind PEST s accommodation of the use of one or more observation covariance matrices in place of observation weights to characterise the uncertainty associated with groups of measurements and or prior information equations However as the implementation of this theory within PEST takes place behind the scenes it is not essential to the use of PEST that this theory be fully understood Let c be a vector whose elements are stochastic variables i e numbers with a random component For the purposes of the present discussion consider that the elements of are the set of measurements to be used in the calibration process Suppose that these measurements are statistically dependent on each other and thus that the uncertainties associated with them must be represented by a covariance matrix rather than by a set of individual variances Suppose that the covariance matrix associated with the elements of the vector is the matrix C Because C is a covariance matrix it must be positive definite which means that it is also symmetric Let R be a matrix whose columns are comprised of the normalised eigenvectors of C Itis easily shown that R R 2 34 ie that the transpose of R is also its inverse Now let us introduce another stochastic vector d calculable from the vec
502. t However in spite of the fact that g defines the direction of steepest descent of it can be shown that u is normally a far better parameter upgrade direction than g especially in situations where parameters are highly correlated In such situations iteratively following the direction of steepest descent leads to the phenomenon of hemstitching where the parameter set jumps from side to side of a valley in as parameters are upgraded on successive iterations The PEST Algorithm 2 9 convergence toward the global minimum is then extremely slow See Figure 2 2 wt Initial eS parameter t estimates Parameter 2 Contours of equal objective function value Parameter 1 Figure 2 2 The phenomenon of hemstitching Nevertheless most parameter estimation problems benefit from adjusting u such that it is a little closer to the direction of g in the initial stages of the estimation process Mathematically this can be achieved by including in equation 2 18 the so called Marquardt parameter named after Marquardt 1963 though the use of this parameter was in fact pioneered by Levenberg 1944 Equation 2 18 becomes u J QJ ol FQr 2 20 where q is the Marquardt parameter and I is the n x n identity matrix It can be shown that the gradient vector g can be expressed as g 2J Qr 2 21 It follows from equations 2 20 and 2 21 that when is very high the direction of u approaches tha
503. t between model outcomes and field measurements can be improved with relative ease by declaring more parameters as adjustable or by simply adding more parameters to the model This is particularly the case for distributed parameter models where it is an easy matter to undertake a finer subdivision of the model domain for parameterisation purposes thus endowing the model with a greater number of parameters that require estimation Ultimately through adding more and more parameters it may be possible to reduce the objective function to almost zero as every nuance of system behaviour is replicated by the model outputs It is very important to be aware of the fact that a good fit under calibration conditions does not guarantee an accurate model prediction In general the more parameters that are estimated the more highly are they correlated and the more likely it is that some of them are insensitive Both of these will contribute to a high degree of parameter uncertainty which may result in a high degree of uncertainty for at least some types of model predictions If the system under study is such that a high level of parameterisation is nevertheless required because it is necessary that the model be capable of reproducing the fine detail of system response then the user should be very aware of the uncertainty surrounding estimated parameters and of the uncertainty that is likely to accompany many of the predictions made by the model In this ca
504. t can almost result in a unique parameter set all by itself almost here implies that there may still be a degree of nonuniqueness but that this might only be in relation to a factor by which all parameters can be multiplied and still satisfy the regularisation conditions However if the regularisation scheme is such as to allow high and untrammelled variability of parameters PEST can encounter serious difficulties if the inversion problem to which the regularisation scheme refers is already unstable In the belief that if regularisation observations were to Regularisation 7 8 make a more substantial contribution to the objective function than measurement observations greater stability will ensue in accordance with the aims of introducing regularisation into the optimisation process in the first place PEST may try to assign greater weights to regularisation observations than it otherwise would when the inversion process runs into difficulties Where the number of adjustable parameters is high and numerical solution of the weight factor equation is thus very time consuming this may place an undue strain on PEST s operation in regularisation mode If the upper weight factor limit WFMAX is set at an appropriate value the user will be able to detect this aberrant behaviour sooner than he she otherwise would and take appropriate action if necessary WEMIN and WFMAX values of 10 and 10 respectively are suitable for most occasions
505. t changes to be made to the criteria that govern termination of a PEST run The PEST Algorithm 2 27 2 3 The Calculation of Derivatives 2 3 1 Forward and Central Differences The ability to calculate the derivatives of all observations with respect to all adjustable parameters is fundamental to the Gauss Marquardt Levenberg method of parameter estimation these derivatives are stored as the elements of the Jacobian matrix Because PEST is independent of any model of which it takes control it cannot calculate these derivatives using formulae specific to the model Hence it must evaluate the derivatives itself using model generated observations calculated on the basis of incrementally varied parameter values Note however that there may be occasions where a model can calculate derivatives of its outputs with respect to its adjustable parameters itself If this is the case PEST can make direct use of these derivatives if they can be provided to it in the correct format This is further described in Section 8 of this manual Most of the discussion in the remainder of this Chapter assumes that PEST must calculate parameter derivatives itself Accuracy in derivatives calculation is fundamental to PEST s success in optimising parameters Experience has shown that the most common cause of PEST s failure to find the global minimum of the objective function in parameter space is the presence of roundoff errors incurred in the calculation of derivativ
506. t data PEST will terminate execution with a run time error message Unlike PESTCHEK see Chapter 10 PEST will not continue reading its input data files in order to determine whether more errors are present so that it can list them as well rather it ceases execution as soon as it has noticed that something is wrong Other errors can arise later in the estimation process For example if the instruction set fails to locate a particular observation PEST will cease execution immediately with the appropriate run time error message This may happen if the model has varied the structure of its output file in response to a certain set of parameter values in a way that you did not anticipate when you wrote the instruction set It may also arise if the model terminated execution prematurely Hence if a run time error informs you that PEST was not able to read the model output file correctly you should check both the screen and the model output file for a model generated error message If there is a compiler generated error message on the screen informing you of a floating point or other error and this is followed by a PEST run time error message informing you that an observation could not be found then the model not PEST was responsible for the error You should then carefully inspect the model output file for clues as to why the error occurred In many cases you will find that one or a number of Running PEST 5 23 model parameters have transgressed thei
507. t objective function rises no higher than a specified level In all of these cases the extent to which model outputs are in agreement with their field measured counterparts is apparent from the value of the objective function or the measurement objective function when working in regularisation mode Another measure of goodness of fit is provided by the correlation coefficient as defined in Cooley and Naff 1990 Unlike the objective function the correlation coefficient is independent of the number of observations involved in the parameter estimation process and of the absolute levels of uncertainty associated with those observations Hence use of this measure of goodness of fit allows the results of different parameter estimation exercises to be directly compared The correlation coefficient R is calculated as wc m w c m w c m w c m wic m Ow c M r an where Ci is the i th observation value Coi is the model generated counterpart to the i th observation value m is the mean value of weighted observations Mo is the mean of weighted model generated counterparts to observations and Wi is the weight associated with the 7 th observation or rotated observation if a covariance matrix is used to specify observation uncertainty instead of individual observation weights Generally R should be above 0 9 for the fit between model outputs and observations to be acceptable Hill 1998
508. t of steam failing to lower it much further Make sure that model outcomes are being written to the model output file with the maximum precision which the model allows If the model places an upper limit on output precision ensure that the parameter increments used for derivatives calculation are large enough to cause a useable change in all model calculated observations given the number of significant digits in which they are expressed Try not to make parameter increments too large though or finite difference generated derivatives will be a poor approximation to the real thing However if they must be large use one of the three point methods of derivatives calculation Try the parabolic method first if that doesn t work use the best fit method Check that parameter values are written to model input files with enough significant digits to reflect parameter increments If some parameters become very low as the optimisation process progresses you may need to provide a suitable lower bound on derivative increments through the parameter group variable DERINCLB or calculate the increment using the largest member of a parameter group by denoting the group variable INCTYP as rel_to_max For a model which solves large matrix equations using an iterative method you should ensure that the model s solution convergence criterion is set very low so that model generated observations are calculated with a high degree of precision
509. t of the negative of the gradient vector when is zero equation 2 20 is equivalent to equation 2 18 Thus for the initial optimisation iterations it is often beneficial for q to assume a relatively high value decreasing as the estimation process progresses and the optimum value of is approached The manner in which PEST decides on a suitable value for q for each iteration is discussed in Section 2 1 7 2 1 6 Scaling For many problems especially those involving different types of observations and parameters whose magnitudes may differ greatly the elements of J can be vastly different in magnitude The PEST Algorithm 2 10 This can lead to roundoff errors as the upgrade vector is calculated through equation 2 20 Fortunately this can be circumvented to some extent through the use of a scaling matrix S Let S be a square n x n matrix with diagonal elements only the i th diagonal element of S being given by Si FQ 2 22 Introducing S into equation 2 20 the following equation can be obtained for S u S u JS QJS aS S JS Qr 2 23 It can be shown that although equation 2 23 is mathematically equivalent to equation 2 20 it is numerically far superior If is zero the matrix JS QJS aS S has all its diagonal elements equal to unity For a non zero the diagonal elements of JS QJS aS S will be greater than unity though in general they will not be equal Let the largest element of S S be denoted as referred to
510. t will appear in the covariance window only after the parameter estimation process is complete or the user has brought the process to a premature end using the stop with statistics option The plot that is available in the covariance window is a graphical representation of the covariance matrix Elements of this matrix are represented by red circles for positive values and blue squares for negative values the sizes of these symbols being proportional to the sizes of the elements that they represent If a particular element appears to be absent this is because it is probably too small to see click where you think the element should be in order to see its magnitude displayed in a bubble or zoom in on it Only adjustable parameters are featured in this and other matrices fixed and tied parameters are not represented Recall that the covariance matrix is calculated on the basis of the transformation state of the various parameters involved in the parameter estimation process Thus if a parameter is log transformed the log to base 10 of the actual parameter is represented in the covariance matrix and the matrices derived from it Recall also that the diagonal elements of the covariance matrix represent the variances of the respective transformed parameters If PEST is running in regularisation mode calculation of the covariance matrix takes all of the regularisation information into account with weights assigned to this information
511. ta in the inversion process especially if these measurements are targeted at the offending parameters Extra information in the form of prior information can also be very effective in increasing parameter sensitivity and in reducing parameter correlation Use of PEST in regularisation mode can also be very effective in reducing the deleterious effects of parameter insensitivity 5 7 4 Residuals An analysis of the differences between model outcomes and corresponding field or laboratory data is an extremely important part of any parameter estimation application The mathematical basis of the parameter estimation algorithm used by PEST relies on the Running PEST 5 38 assumption that measurement uncertainties are uncorrelated ie that the uncertainty associated with any one measurement is unrelated to that associated with any other measurement If measurements do in fact exhibit correlation then an observation covariance matrix should be used in place of observation weights so that the rotated residuals are uncorrelated See Section 4 3 for details One of the first tasks that should be undertaken after the parameter estimation process is complete is to determine whether residuals or rotated residuals are in fact randomly distributed about a mean of zero with minimal correlation between them PEST provides some degree of assistance in making this determination As described in Section 5 2 8 and in Section 4 3 on its run record f
512. tant to note that a rotated observation whose name is formulated by adding the string _r to the name of the original observation has no more of a direct relationship to that original observation than it does to any other member of the original observation group this observation naming convention is just a convenience 4 3 3 4 Analysis of Residuals PEST calculates and records a number of basic statistics pertaining to optimised residuals to the end of its run record file Due to the fact that these statistics are calculated on the basis of weighted residuals rather than the residuals themselves PEST calculates them using rotated residuals rather than true residuals for those observation groups for which a covariance matrix is supplied The fact that rotated residuals rather than direct residuals are used in this calculation is recorded on the run record file Also where the names of any such rotated residuals are cited the _r suffix appended to a residual s name indicates its rotated status Running PEST 5 1 5 Running PEST 5 1 How to Run PEST 5 1 1 Checking PEST s Input Data PEST s input file requirements have been discussed in detail in the previous two chapters Before submitting these files to PEST for a parameter estimation run you should check that all information contained in them is syntactically correct and consistent This can be done using the utility programs PESTCHEK TEMPCHEK and INSCHEK described in Ch
513. tc subdirectory appropriate to that slave Each such subdirectory should also be cited as a slave subdirectory for the purposes of writing the signal files listed in Table 9 1 Once the Parallel PEST run management file has been constructed accordingly Parallel PEST can be started and should run without problems If the slave and model working directories are on a server s disk is not the server running the model No the model is run by the machine from which PSLAVE is launched irrespective of whether the current working directory is on the current machine s disk or on a disk belonging to another machine Similarly even if the model executable resides on another machine s disk the model is run by the machine from which the model run command is issued this is the machine running PSLAVE for it is PSLAVE which issues the command to run the model If the model executable resides on another machine then it has to be loaded into the current machine s memory across the network prior to execution Do I have to install the model executable on each slave machine Not at all If a single model executable is accessible through the network by all slave machines then it can be run from each slave machine without having to install it on any of them The disadvantage of this scheme however is that if the model executable is large reloading it to each slave machine every time that slave runs the model may be a slow process on heavily used network
514. te a model input file on the basis of the template file which you have just prepared You can then run your model making sure that it reads the input file correctly In this way you can be sure prior to running PEST that PEST will write a model input file that satisfies your model s requirements INSCHEK does for instruction files what TEMPCHEK does for template files INSCHEK checks that an instruction file is syntactically correct and consistent Then if you wish INSCHEK will read a model output file using the directives contained in the instruction file listing the values of all observations cited in the instruction file as read from the model output file In this way you can be sure prior to running PEST that PEST will read a model output file correctly Like template and instruction files the PEST control file can be prepared using a text editor However it is generally easier to prepare it using program PESTGEN PESTGEN generates a PEST control file using parameter and observation names cited in template and instruction files which have already been built However as it uses default values for all variables which control PEST execution you will probably need to make some changes to a PESTGEN generated PEST control file usually not too many in order to tune PEST to your current problem After all PEST input files have been prepared viz the PEST control file and all template and instruction files you can use program PESTCHEK to
515. ted Continuation You can break an instruction line between any two instructions by using the continuation character amp to inform PEST that a certain instruction line is actually a continuation of the previous line Thus the instruction file fragment 11 SRESULTS STIME 4 obs1 obs2 obs3 is equivalent to 11 amp ESULTS amp amp amp obsl amp amp For both these fragments the marker delimiter is assumed to be Note that the continuation character must be separated from the instruction which follows it by at least one space 3 3 7 Making an Instruction File An instruction file can be built using a text editor This is particularly easy in the WINDOWS environment where you can open two command line windows one to view a model output file and the other to write the instruction file If the viewing program is a text editor which displays cursor line and column numbers the job is even easier note that the text editor EDIT provides these numbers Furthermore with the help of the WINDOWS clipboard facility you can easily copy markers from a model output file to an instruction file using the mouse You must always exercise caution in building an instruction set to read a model output file especially if navigational instructions such as markers whitespace tabs and dummy observations are used PEST will always follow your instructions to the letter but it may not read the
516. ted measurements their rotated model generated counterparts residuals calculated therefrom and the same functions of these quantities Because through the use of rotated observations the observation covariance matrix is diagonalised weights can be used in the calculation of these various functions As is explained in Section 2 1 11 these weights are actually the reciprocals of the square roots of the eigenvalues of the original observation covariance matrix supplied by the user Where a covariance matrix is supplied for only a few of the many observations used in the parameter estimation process most of the entries in the rotated residuals file will be the same as those found in the normal residuals file However entries pertaining to observation groups for which a covariance matrix is supplied will be different Because a new set of rotated observations is calculated for members of this group the user assigned names for the original observations are no longer applicable Hence when PEST lists the names of the new observations to this file it formulates new observation names by adding the string _r to the names of the original observations Rotated observations are listed in order of decreasing observation weight ie in order of increasing eigenvalues of the original covariance matrix New names for these observations are formulated in the order in which these observations are supplied in the original PEST control file It is impor
517. ted residuals should have a mean of zero and be randomly distributed The information contained in this section of the run record file helps to assess whether this is the case It also allows the user to immediately identify outliers those Running PEST 5 14 observations for which the residuals are unusually high In calculating residual statistics observations with zero weight are ignored 5 2 9 The Parameter Covariance Matrix The covariance matrix is always a square symmetric matrix with as many rows and columns as there are adjustable parameters hence there is a row and column for every parameter which is neither fixed nor tied The order in which the rows and columns are arranged is the same as the order of occurrence of the adjustable parameters in the previous listing of the optimised parameter values This is the same as the order of occurrence of adjustable parameters in both the PEST control file and in the first section of the run record file Being a by product of the parameter estimation process see Chapter 2 the elements of the covariance matrix pertain to the parameters that PEST actually adjusts this means that where a parameter is log transformed the elements of the covariance matrix pertaining to that parameter actually pertain to the logarithm to base 10 of that parameter Note also that the variances and covariances occupying the elements of the covariance matrix are valid only in so far as the linearity assum
518. tegy e Each parameter difference comprising the set of regularisation observations is formed between a certain layer and the layer underneath it However for the bottommost layer the difference is formed between that layer and the topmost layer Experience has demonstrated that a regularisation scheme such as that depicted in Example 7 4 works very well However if an attempt is made to estimate layer thicknesses as well as resistivities then the scheme breaks down If layer thicknesses are estimated as well as resistivities then the regularisation observations on their own do not provide a sufficiently strong constraint on parameter values to constitute a suitable regularisation scheme The Regularisation 7 15 reason for this is obvious when it is realised that if all regularisation conditions were exactly obeyed and the subsurface was uniform layer thicknesses would be completely indeterminate for layer thicknesses have no meaning in a uniform half space 7 6 2 A Heterogeneous Aquifer Under steady state conditions the flow of ground water through the subsurface is mathematically described by Darcy s law and constrained by the conservation of mass These two laws can be combined into a single partial differential equation which can be used to calculate the distribution of hydraulic heads throughout a study area for different dispositions of sources and sinks of water and for different system boundary conditions Where the shape of the
519. tely the template concept used for model input files will not work for model output files as the latter may change from run to run depending on parameter values However if a person is capable of locating a pertinent model output amongst the other data on a model output file then so too is a computer As it turns out the instruction set by which this can be achieved is relatively simple involving only a handful of basic directives PEST requires then that for each model output file which must be opened and perused for observation values an instruction file be provided detailing how to find those observations This instruction file can be prepared using any text editor It can be checked for syntactical correctness and consistency using the utility programs PESTCHEK and INSCHEK Once interfaced with a model PEST s role is to minimise the weighted sum of squared differences between model generated observation values and those actually measured in the laboratory or field this sum of weighted squared model to measurement discrepancies is referred to as the objective function The fact that these discrepancies can be weighted makes some observations more important than others in determining the optimisation outcome Weights should be inversely proportional to the standard deviations of observations trustworthy observations having a greater weight than those which cannot be trusted as much Also if observations are of different type
520. teration NUMLAM This integer variable places an upper limit on the number of lambdas that PEST can test during any one optimisation iteration It should normally be set between 5 and 10 For cases where parameters are being adjusted near their upper or lower limits and for which some parameters are consequently being frozen thus reducing the dimension of the problem in parameter space experience has shown that a value closer to 10 may be more appropriate than one closer to 5 this gives PEST a greater chance of adjusting to the reduced problem dimension as parameters are frozen RELPARMAX and FACPARMAX As was explained in Section 2 2 5 there should be some limit placed on the amount by which parameter values are allowed to change in any one optimisation iteration If there is no limit parameter adjustments could regularly overshoot their optimal values causing a The PEST Control File 4 9 prolongation of the estimation process at best and instability with consequential estimation failure at worst the dangers are greatest for highly nonlinear problems PEST provides two input variables which can be used to limit parameter adjustments these are RELPARMAX and FACPARMAX both real variables RELPARMAX is the maximum relative change that a parameter is allowed to undergo between optimisation iterations whereas FACPARMAX is the maximum factor change that a parameter is allowed to undergo Any particular parameter can be subject to o
521. ters c cesseeeseeeeees 5 31 320 User Interven thon oeiee ie e are e e EET E e eE e EE ie E a EEEE 5 31 5 6 1 An Often Encountered Cause of Aberrant PEST Behaviour 5 31 5 6 2 Fixing the Problem posone sinnen tenie ereti erro eieko oi 5 32 5 6 3 The Parameter Hold File ee eeeeecceseceneeceeeceseceseeesceesaeesaeesseeesseesaeesseeeaes 5 32 5 6 4 Re calculating the Parameter Upgrade Vector ccceseesceeeceeeeceeeceneeeaeeenees 5 35 5 6 5 Maximum Parameter Change cccccscccesssecsesecesseeeesseceeseeceseeeseeeeeseeeeeaeeeses 5 35 927 PEST Postprocessin Sse ecvcaieidics ini E e ER iiebde tebe telat A EAR ones 5 36 5 1 1 Generalnie aan iaai a a E a a i a E aii a k 5 36 5 1 2 Parameter Valles neee sy caoasteorss i eoe ka ee AE a R E aa e eaa EE 5 36 5 753 Parameter Statistics senhores en eE AE TE shiek SEA RE a Ei 5 36 JA R sid alS onena ee e ea i a sv ao EREE eTe ERS a E E 5 37 5 7 5 Over Parameterisation eeseeeeseeseseesesresressssstssresrestesestrsteereesteseesresreseessessessee 5 39 5 7 6 Covariance Matrix for Best Fit Parameters eeseeeeneeeeeeserreerrerrerrererrerresse 5 39 5 7 7 Model Outputs based on Optimal Parameter Values ce eeceeeeeereeereeees 5 40 6 Predictive Analysis anayea he dedt eens E RER eee dn eset E AE os sel ER 6 1 6 1 The Concept ni nin settee eae i a a a ee oe este eae 6 1 6 1 1 What Predictive Analysis Means eseseeeeseesesesesresresressessrssresresressessrssresresese
522. ters to a manageable level a two or three dimensional model domain is often subdivided into a small number of zones of assumed parameter constancy zone boundaries can be chosen on the basis of geological and or other evidence where this evidence exists or inferred from the field data itself While this methodology works well there are many problems associated with it For example zonation of the model domain may be difficult as it may not be immediately apparent from the observation dataset where property contrasts exist or whether property transitions are smooth rather than abrupt In many cases the modeller may prefer to ask PEST to infer areas of high or low property value itself rather than attempting to construct a parameter zonation scheme based on an incomplete knowledge of the locations of property discontinuities Unfortunately a calibration and or data interpretation strategy which does not rely entirely on an externally supplied zonation will nearly always result in the necessity to estimate a large number of parameters Some of these parameters will invariably be more sensitive to the field data than others insensitive parameters are not only difficult to estimate themselves but may hamper PEST s ability to estimate other more sensitive parameters see Section 5 6 for details Furthermore parameter correlation is often very strong in highly parameterised systems this resulting in a high degree of nonuniqueness in parameter es
523. tes such a file Note that a PEST parameter value file can be used by program TEMPCHEK in building a model input file based on a template file by program PESTGEN in assigning initial parameter values to a PEST control file and by program PARREP in building a new PEST control file from an old PEST control file see Chapter 10 for further details single point rol 1 000000 1 000000 0 0000000 ro2 40 00090 1 000000 0 0000000 ro3 1 000000 1 000000 0 0000000 h1 1 000003 1 000000 0 0000000 h2 9 999799 1 000000 0 0000000 Example 5 2 A parameter value file The first line of a parameter value file cites the character variables PRECIS and DPOINT the values for which were provided in the PEST control file see Section 4 2 2 Then follows a line for each parameter each line containing a parameter name its current value and the values of the SCALE and OFFSET variables for that parameter 5 3 2 The Parameter Sensitivity File 5 3 2 1 The Composite Parameter Sensitivity Most of the time consumed during each PEST optimisation iteration is devoted to calculation of the Jacobian matrix During this process the model must be run at least NPAR times where NPAR is the number of adjustable parameters As is explained in Section 2 2 7 based on the contents of the Jacobian matrix PEST calculates a figure related to the sensitivity of each parameter with respect to all observations with the latter weighted as per user assigned weights The
524. tes derivatives of model outcomes with respect to adjustable parameters using finite parameter differences successive model runs are independent ie the parameters used for one particular model run do not depend on the results of a previous model run The complete independence of model runs undertaken as part of the process of filling the Jacobian Parallel PEST 9 2 matrix makes this process easily parallelised Under these circumstances Parallel PEST simply distributes model runs to different machines or processors as they become available and processes the outcomes of these runs as they are finished 9 1 3 Parallelisation of the Marquardt Lambda Testing Process Unlike the Jacobian calculation process the lambda search process see Section 2 1 7 is difficult to parallelise This is because with the exception of the first two model runs undertaken as part of the lambda search procedure during each optimisation iteration the Marquardt lambda used at subsequent stages of this procedure is dependent on the results of model runs undertaken on the basis of previous lambda values Hence it is necessary for PEST to wait until the results of a previous model run have been evaluated before it can undertake a further model run on the basis of a new parameter set calculated using a new Marquardt lambda However while the lambda search process is not immediately amenable to parallelisation it is not impossible to accelerate this process somewhat through
525. th respect to the incrementally varied parameter as the observation increment divided by the parameter value increment For log transformed parameters this quotient is then multiplied by the current parameter value Hence if derivatives with respect to all parameters are calculated by the method of forward differences the filling of the Jacobian matrix requires that a number of model runs be carried out equal to the number of adjustable parameters as the Jacobian matrix must be re calculated for every optimisation iteration each optimisation iteration requires at least as many model runs as there are adjustable parameters plus at least another one to test parameter upgrades The calculation of derivatives is by far the most time consuming part of PEST s parameter estimation procedure The PEST Algorithm 2 28 If the parameter increment is properly chosen see below this method can work well However it is often found that as the objective function minimum is approached attainment of this minimum requires that parameters be calculated with greater accuracy than that afforded by the method of forward differences Thus PEST also allows for derivatives to be calculated using three parameter values and corresponding observation values rather than two as are used in the method of forward differences Experience shows that derivatives calculated on this basis are accurate enough for most occasions provided once again that parameter increments are
526. th such a TEMPCHEK prepared input file you can verify that the model will have no difficulties in reading input files prepared by PEST 3 3 Instruction Files Of the possibly voluminous amounts of information that a model may write to its output file s PEST is interested in only a few numbers viz those numbers for which corresponding field or laboratory data are available and for which the discrepancy between model output and measured values must be reduced to a minimum in the weighted least squares sense These particular model generated numbers are referred to as observations or model generated observations in the discussion which follows in order to distinguish them from those model outcomes which are not used in the parameter estimation process while the field or laboratory data to which they are individually matched are referred to as measurements For every model output file containing observations you must provide an instruction file The Model PEST Interface 3 10 containing the directions which PEST must follow in order to read that file Note that if a model output file is more than 2000 characters in width PEST will be unable to read it however a model output file can be of any length Some models write some or all of their output data to the terminal You can redirect this screen output to a file using the gt symbol and teach PEST how to read this file using a matching instruction file in the usual man
527. that more model runs will be required to fill the Jacobian matrix than are really needed at that stage of the estimation process If PHIREDSWH is set too low PEST may waste an optimisation iteration or two in lowering the objective function to a smaller extent than would have been possible if it had made an earlier switch to central derivatives calculation Note that PHIREDSWH should be set considerably higher than the input variable PHIREDSTP which sets one of the termination criteria on the basis of the relative objective function reduction between optimisation iterations NOPTMAX The input variables on the ninth line of the PEST control file set the termination criteria for the parameter estimation process These are the criteria by which PEST judges that the optimisation process has been taken as far as it can go These should be set such that either parameter convergence to the optimal parameter set has been achieved or it has become obvious that continued PEST execution will not bear any fruits The first number required on this line is the integer variable NOPTMAX This sets the maximum number of optimisation iterations that PEST is permitted to undertake on a particular parameter estimation run If you want to ensure that PEST termination is triggered by other criteria more indicative of parameter convergence to an optimal set or of the futility of further processing you should set this variable very high A value of 20 to 30 is often appr
528. that under which it will operate to make predictions in other cases the stress regimes will be similar In some cases model predictions will be of the same type as those that were used for calibration in other cases a model will be required to generate predictions of a very different type from those used in the calibration process In some cases predictions will be required at the same locations as those for which data was available to assist in the calibration process in other cases predictions will be required at very different Predictive Analysis 6 4 places In general the more similarity that model predictions bear to the type of data used in calibrating the model and the more similar is the stress regime in which predictions are made to those prevailing at the time of calibration the greater is the likelihood that parameter uncertainty arising as an outcome of the calibration process will not result in a large degree of predictive uncertainty If parameter uncertainty doesn t matter in terms of the model s ability to replicate historical system conditions then it will quite possibly not matter in terms of its ability to predict future conditions provided things are not too different in the future However where conditions are different and where predictions are of a different type from those used in the calibration process it is possible that a large degree of predictive uncertainty may be associated with model parameter uncerta
529. the parameter estimation process If they are outside of the o contour PEST automatically works in parameter estimation mode until it is within reach of o at which stage it modifies its operations to search for the critical point When run in predictive analysis mode PEST still needs to calculate a Jacobian matrix so derivatives of model outcomes with respect to adjustable parameters are still required Derivatives can be calculated using two or three points PEST can switch from one to the other as the solution process progresses Parameters must still be assigned to groups for the purpose of assigning variables which govern derivatives calculation Parameters can be log transformed linked to one another or fixed in predictive analysis mode just as in parameter estimation mode In fact parameter transformations and linkages must be the same for a predictive analysis run as they were for the preceding parameter estimation run in which min was determined for the value supplied to PEST for o must be consistent with the previously determined value of min Just as in parameter estimation mode a Marquardt lambda is used to assist PEST in coping with model nonlinearities when it is run in predictive analysis mode this lambda is adjusted by PEST as the optimisation process progresses The same user supplied control variables affect PEST s lambda adjustment procedure as when it is used in parameter estimation mode plus a couple more see be
530. the run record file the best objective function calculated during the whole parameter estimation process is used in computing the reference variance If the best objective function was computed on the final parameter upgrade this will differ slightly from that calculated at the beginning of the last optimisation iteration resulting in slight differences between the matrices recorded on the final matrix file and those recorded on the run record file 5 3 6 Other Files If requested through the PEST control variable RSTFLE PEST intermittently stores its data arrays and last two Jacobian matrices in binary files named case rst case jac and case jst If PEST execution is re commenced using the r switch it reads the first of these binary files in addition to its normal input files if it is re started with the j switch it reads all of them As is explained in Section 10 6 PEST records the Jacobian matrix corresponding to optimised parameter values to a file named case jco This is accessible by the JACWRIT utility for recording of the Jacobian matrix in text format Parallel PEST uses a number of files for communication between PEST and its various Running PEST 5 22 slaves It also writes a run management file documenting the communications history between the various programs taking part in the optimisation process All of these files are described in detail in Chapter 9 5 3 7 PEST s Screen Output As well as record
531. third line of the PEST control file must be set to prediction Also the PEST control file must contain a predictive analysis section which controls PEST s operations in this mode Note the following e if PEST is run in parameter estimation mode the predictive analysis section of the PEST control file is ignored and can in fact be omitted Predictive Analysis 6 12 e if there is no prior information the prior information section of the PEST control file should either be omitted or simply left empty 6 2 2 PEST Variables used for Predictive Analysis The role of those PEST variables which govern its operation in predictive analysis mode will now be discussed in detail Although PESTMODE was discussed in Section 4 2 2 specifications for this variable will now be repeated All other variables discussed below reside in the predictive analysis section of the PEST control file and hence pertain only to PEST s operation in predictive analysis mode PESTMODE This is a character variable that appears on the third line of the PEST control file just after the RSTFLE variable It must be supplied as either estimation prediction or regularisation In the first case PEST will run in parameter estimation mode its traditional mode of operation in the second case PEST will run in predictive analysis mode in the third case PEST will run in regularisation mode see Chapter 7 As mentioned above
532. this comprised of the entries out2 ins and out2 dat 10 Add a predictive analysis section to file twofit pst Because we wish to maximise the prediction NPREDMAXMIN is assigned the value of 1 As mentioned above PDO is 5 0E 3 Set PD1 to 5 2E 3 and set PD2 to be twice as high as PDO ie 1 0E 2 Variables governing operation of the Marquardt lambda the switching from two point to three point derivatives calculation and the termination of execution will be set in relative rather than absolute terms so set ABSPREDLAM ABSPREDSWH and ABSPREDSTP to 0 0 RELPREDLAM RELPREDSWH and RELPREDSTP should be set to their recommended values of 005 05 and 005 respectively NPREDNORED and NPREDSTP should be set at their recommended values of 4 and 4 Contrary to the advice provided in Chapter 6 we will not conduct a line search for each Marquardt lambda so set the control variables INITSCHFAC MULSHFAC and NSEARCH to 1 0 2 0 and 1 respectively 12 2 4 Template and Instruction Files Inspect files in2 tpl and out2 ins provided with the example files These are respectively a template file for in2 dat and an instruction file to read the single prediction observation from file out2 dat Notice how parameter spaces for each of the four parameters involved in the predictive analysis process appear in both of files in tpl and in2 tpl This is because these parameter values are used by the model under both calibration and prediction con
533. tical for the two cases Also identical for the two cases are the template and instructions files however the instruction file for the SENSAN example is named out ins instead of out ins in order to avoid out obf used in the PEST example of Chapter 12 being overwritten when SENSAN runs INSCHEK Five parameter sets are provided in parvar dat requiring that five model runs be undertaken After the third model run has been completed the model output file out dat is copied to file out kp for safekeeping until later inspection Before running SENSAN make sure that the PEST directory is cited in the PATH environment variable so that SENSAN can run TEMPCHEK and INSCHEK Run SENSAN 11 14 SENSCHEK using the command senschek twofit After verifying that there are no errors or inconsistencies in the SENSAN input dataset run SENSAN using the command sensan twofit After SENSAN has completed execution inspect files out txt out2 txt and out3 txt the three SENSAN output files You may also wish to verify that file out kp exists this being a record of out dat generated on the third model run An Example 12 12 An Example 12 1 Parameter Estimation 12 1 1 Laboratory Data This section takes you step by step through an example which demonstrates the application of PEST to a practical problem Once PEST has been installed on your computer the files cited in this chapter can be found in the pestex subdirectory of the main PEST dir
534. timates Hence even if a good fit is obtained between model outputs and field data the parameters giving rise to this fit will probably be just one set out of a virtually infinite number of sets that will also result in a good fit between model outputs and field data A related problem in working with highly parameterised systems is that unless some constraints are imposed on parameter values or on relationships between parameter values individual parameter estimates tend to show a high degree of spatial variability and can even take on extreme values as PEST tries to use every parameter at its disposal in order to accommodate every nuance of the observation dataset upon which the calibration or data interpretation process is based In many cases these nuances are better considered as system noise Hence an appropriate level of misfit between model outputs and field data can be tolerated on this basis for it is well known that a parameterisation which attempts to fit all of the fine detail of a measurement dataset is often a poor representation of true system properties Use of an appropriate regularisation procedure in the parameter estimation process Regularisation 7 2 overcomes this problem by allowing the modeller to inject some sanity into the process by limiting the degree of spatial variability that optimised parameter values are allowed to show While this may be done at the cost of obtaining a perfect fit between model o
535. tion 2 4 is the best linear unbiased estimator of the set of true system parameters appearing in equation 2 1 As an estimator it is one The PEST Algorithm 2 3 particular realisation of the n dimensional random vector b calculated through equation 2 4 from the m dimensional random vector of experimental observations of which the actual observations are but one particular realisation If o represents the variance of each of the elements of c the elements of being assumed to be independent of each other then o can be calculated as o m n 2 5 where m n the difference between the number of observations and the number of parameters to be estimated represents the number of degrees of freedom of the parameter estimation problem Equation 2 5 shows that o is directly proportional to the objective function and thus varies inversely with the goodness of fit between experimental data and the model generated observations calculated on the basis of the optimal parameter set It can further be shown that C b the covariance matrix of b is given by C b o X X 2 6 Notice that even though the elements of c are assumed to be independent so that the covariance matrix of contains only diagonal elements all equal to g in the present case C b is not necessarily a diagonal matrix In fact in many parameter estimation problems parameters are strongly correlated the estimation process being better able to estimat
536. tion problems can be undertaken using the parameter preprocessor PAR2PAR see Chapter 10 for details 5 5 5 Highly Nonlinear Problems If the relationship between parameters and observations is highly nonlinear the optimisation process will proceed only with difficulty As discussed above such nonlinearity may sometimes be circumvented through appropriate transformation of some parameters However in other cases this will make little difference In such cases the Gauss Marquardt Levenberg method of parameter estimation on which PEST is based may not be the most appropriate method to use Sometimes the use of a high initial Marquardt lambda is helpful in cases of this type Also the relative and absolute parameter change limits RELPARMAX and FACPARMAX on the PEST control file may need to be set lower than normal a careful inspection of the PEST run record file may suggest suitable values for these variables and indeed which parameters should be relative limited and which should be factor limited Parameter increments for derivatives calculation should be set as low as possible without incurring roundoff errors The three point parabolic method may be the most appropriate method for calculating derivatives because of its quadratic approximation to the relationship between observations and parameters The incorporation of prior information into the parameter estimation process with a suitably high weight assigned to each prior information
537. tive parameter change 2 000 h2 OPTIMISATION ITERATION NO 3 Model calls so far T3 Starting phi for this iteration 230 1 Contribution to phi from observation group group_1 29 54 Contribution to phi from observation group group_2 84 81 Contribution to phi from observation group group_3 91 57 Contribution to phi from observation group group_4 24 17 All frozen parameters freed Lambda 7 81 parameter phi 89 25E 02 ro2 frozen 49 gradient and update vectors out of bounds 0 39 of starting phi No more lambdas phi is now less than 0 4000 of starting phi Lowest phi this iteration 89 49 Current parameter values rol ro2 ro3 hl h2 Maximum fac Maximum relative parameter change OPTIMISATION ITE Model calls s Starting phi Contribution Contribution Contribution Contribution All Lambda 3 90 parameter phi 19 Lambda 12 95 0 500000 10 0000 1 00000 0 472096 34 3039 tor paramete RATION NO o far for this ite to phi from to phi from to phi from to phi from 63E 02 ro2 frozen 20 31E 02 3 0 r change 4 17 ration 89 observation observation observation observation frozen parameters freed 000 6667 49 group group group group Previous parameter values rol ro2 ro3 hl h2 group_1 group_2 group_3 group_4 0 500000 10 0000 1 00000 1 41629 3143239 9 345 34 88 2857
538. to operators brackets and functions However they cannot appear within numbers parameter names or function names Some examples of allowable mathematical expressions follow trans5 k5 top5 bottom pi 3 14159 par3 3 4 4 5 trans5 3 sin 0 6 par4 par3 pi exp 5 0 par3 trans5 par5 parl par2 cosh pi trans5 If an expression is long it may be continued onto the next line by placing the amp character at the beginning of that line Thus the expression parS parl par2 cosh pi trans5 is equivalent to pars amp parl par2 amp cosh amp pi trans5 10 7 2 4 Generation of Model Input Files Once it has calculated values for all parameters PAR2PAR writes these values to one or more model input files using templates of these files to govern parameter value placement Use of template files for writing model input files is fully discussed in Chapter 3 of this manual As is described in that chapter slight variations of the way in which numbers representing parameter values are written to model input files can be effected through use of the PRECIS and DPOINT variables values for these variables are supplied in the optional control data section of a PAR2PAR input file If PRECIS is set to single numbers are written to model input files using the E character for exponentiation However if it is set to double the D character is used Furthe
539. to phi from observation group group_4 27 24 All frozen parameters freed Lambda 4 8828E 03 gt parameter ro2 frozen gradient and update vectors out of bounds phi 63 59 1 00 of starting phi Lambda 2 4414E 03 gt phi 63 59 1 00 of starting phi Lambda 9 7656E 03 gt phi 63 59 1 00 of starting phi No more lambdas relative phi reduction between lambdas less than 0 0300 Lowest phi this iteration 63 59 Current parameter values Previous parameter values rol 0 500000 rol 0 500000 ro2 10 0000 ro2 10 0000 ro3 1 00000 ro3 1 00000 hl 0 261177 h1 0 265320 h2 42 2006 h2 42 2249 Maximum factor parameter change 1 016 h1 Maximum relative parameter change 1 5615E 02 h1 Optimisation complete the 3 lowest phi s are within a relative distance of eachother of 1 000E 02 Total model calls 42 OPTIMISATION RESULTS Adjustable parameters gt Parameter Estimated 95 percent confidence limits value lower limit upper limit ro2 10 0000 0 665815 150 192 hl 0 261177 1 00256 1 52491 h2 42 2006 0 467914 3806 02 Note confidence limits provide only an indication of parameter uncertainty They rely on a linearity assumption which may not extend as far in parameter space as the confidence limits themselves see PEST manual Tied parameters gt Parameter Estimated value ro3 1 00000 Fixed parameters gt Parameter Fixed value rol 0 500000 Observations gt Obs
540. tor using the relationship d R c 2 35 It is easy to show that the covariance matrix of d which will be named D can be calculated from the covariance matrix of c ie C using the relationship D R CR 2 36 and that D is a diagonal matrix whose elements are equal to the eigenvalues of C This is a very important relationship for it expresses the fact that through a simple rotational transformation of the original stochastic vector c another stochastic vector d can be obtained whose elements are statistically independent Hence if the rotated observation vector d is used as a basis for parameter estimation instead of the original observation vector c weights can be used in the inversion equations instead of a covariance matrix Equation 2 11 describes how optimised parameter values as encapsulated in the vector b are calculated from measurements ie the vector c for a linear model For the sake of simplicity the present discussion is restricted to a linear model the theory is easily extended to a nonlinear model using the methodology presented above The equation repeated from equation 2 11 is b X QX X Qe 2 37 If the elements of the measurement vector are not statistically independent then the cofactor matrix Q of equation 2 37 has non diagonal elements and as is explained above is proportional to the inverse of the covariance matrix of c It is not too difficult to show that the vector of optimised para
541. trol data restart estimation 4 13 4 0 1 1 1 single point 1 0 0 5 0 2 0 Oes 05 03 10 3 0 3 0 0 001 Oud 30 0 01 3 3 0 01 3 1 1 1 parameter groups sl relative 0 01 0 0 switch 2 0 parabolic 2 relative 0 01 0 0 switch 2 0 parabolic y1 relative 0 01 0 0 switch 2 0 parabolic xC relative 0 01 0 0 switch 2 0 parabolic parameter data sl none relative 0 300000 1 00000E 10 1 00000E 10 s1 1 0000 0 000 1 52 none relative 0 800000 1 00000E 10 1 00000E 10 s2 1 0000 0 000 1 y1 none relative 0 400000 1 00000E 10 1 00000E 10 y1 1 0000 0 000 1 xc none relative 0 300000 1 00000E 10 1 00000E 10 xc 1 0000 0 000 1 observation groups obsgroup observation data ol 0 501000 0 obsgroup o2 0 521000 0 obsgroup o3 0 520000 0 obsgroup o4 0 531000 0 obsgroup o5 0 534000 0 obsgroup o6 0 548000 0 obsgroup o7 0 601000 0 obsgroup 08 0 626000 0 obsgroup o9 0 684000 0 obsgroup 010 0 696000 0 obsgroup oll 0 706000 0 obsgroup 012 0 783000 0 obsgroup 013 0 832000 0 obsgroup model command line model model input output model tpl model inp model ins model out prior information Example 12 10 The PESTGEN generated control file twofit pst twoline model input output in tpl in dat out ins out dat model command line Example 12 11 Altered section of twofit pst As a final check that the entire PEST input dataset is complete correct and consistent you should run program
542. tural oO hs oO tg m z o bs Oo ono a oes B f Do D D o Oo n Oo c rot ro2 h1 Parameter Parameter n a Weighted Sensitivity n a Not Running Figure 3 The Jacobian window displays the sensitivity of each observation with respect to each parameter WinPEST Graphical Windows 15 In the plots available through this window elements of the Jacobian matrix are referred to as sensitivities The user has the option of multiplying these by observation weights to obtain weighted sensitivities if desired Sensitivities can be natural scaled or unscaled Natural parameter sensitivities represent the elements of the Jacobian matrix as seen by PEST Thus if a parameter is untransformed then elements of the Jacobian matrix pertaining to that parameter represent the derivatives of various observations with respect to that parameter However if a parameter is log transformed then elements of the Jacobian matrix pertaining to that parameter represent derivatives of respective observations with respect to the log to base 10 of that parameter Unscaled sensitivities are simply derivatives with respect to parameter values whether they are log transformed or not Scaled sensitivities are computed by multiplying unscaled sensitivities by respective parameter values thus they represent sensitivities of observations to changes in relative parameter values They are related to the natur
543. tween these various sub models with maximum precision Thus every sub model comprising the composite model should record numbers to those of its output files which are read by other sub models with maximum numerical precision Many models calculate their outcomes using one or a combination of numerical approximations to differential equations for example the finite difference method finite element method boundary element method etc Problems which are continuous in space and or time are approximated by discrete representations of the same problem in order that the partial differential equation s describing the original problem can be cast as a matrix equation of high order The matrix equation is often solved by an iterative technique for example preconditioned conjugate gradient alternating direction implicit etc in which the solution vector is successively approximated the approximation being fine tuned until it is judged that convergence has been attained Most such iterative matrix solution schemes judge that the solution is acceptable when no element of the solution vector varies by more than a user specified tolerance between successive iterations If this threshold is set too large model precision is reduced If it is set too small solution convergence may not be attainable in any case the smaller it is set the greater will be the model computation time If a numerical model of this type is to be used with PEST it is essential
544. uch an analysis for example Monte Carlo methods and linear uncertainty propagation However unlike most other methods the PEST predictive analysis algorithm relies on no assumptions concerning the linearity of a model or the probability distribution associated with its parameters furthermore notwithstanding the fact that calculation of model predictive uncertainty is a numerically laborious procedure PEST s predictive analysis algorithm is less numerically intensive than any other method of nonlinear predictive analysis It is hoped that the use of PEST s predictive analyser will allow modellers from all fields of science and engineering to make yet another quantum leap in the productive use of computer simulation models in whatever field of study they are currently engaged John Doherty October 1999 Preface to the Fourth Edition Production of the fourth edition of the PEST manual marks two important milestones in the development of PEST The first of these is the addition of advanced and powerful regularisation functionality underpinning the release of version 5 0 of PEST otherwise known as PEST ASP ASP stands for Advanced Spatial Parameterisation The second is PEST s change in status from that of a commercial product to that of a public domain package Over the last few years the continued development of PEST has focussed on its ability to work successfully with complex highly parameterised models First there was
545. ue to each prior information article Following the prior information label is the prior information equation To the left of the sign there are one or more combinations of a factor PIFAC plus parameter name PARNMB with a log prefix to the parameter name if appropriate PIFAC and PARNME are separated by a character which must be separated from PIFAC and PARNME by at least one space signifying multiplication All parameters referenced in a prior information equation must be adjustable parameters ie you must not include any fixed or tied parameters in an article of prior information Furthermore any particular parameter can be referenced only once in any one prior information equation however it can be referenced in more than one equation The parameter factor must never be omitted Suppose for example that a prior information equation consists of only a single term viz that an untransformed adjustable parameter named parl has a preferred value of 2 305 and that you would like PEST to include this information in the optimisation process with a weight of 1 0 If this article of prior information is given the label pil the pertinent prior information line can be written as pil 1 0 parl 2 305 1 0 pr_info If you had simply written pil parl 2 305 1 0 pr_info PEST would have objected with a syntax error If a parameter is log transformed you must provide prior information pertinent to the log o
546. ues resulting in very slow convergence to an objective function minimum Furthermore for the case of relative limited parameters which are permitted to change sign it is possible that the denominator of equation 4 3 could become zero To obviate these possibilities choose a suitable value for the real variable FACORIG If the absolute value of a parameter falls below FACORIG times its original value then FACORIG times its original value is substituted for the denominator of equation 4 3 For factor limited parameters a similar modification to equation 4 4 applies Thus the constraints that apply to a growth in absolute value of a parameter are lifted when its absolute value has become less than FACORIG times its original absolute value However where PEST wishes to reduce the parameter s absolute value even further factor limitations are not lifted nor are relative limitations lifted if RELPARMAX is less than 1 FACORIG is not used to adjust limits for log transformed parameters FACORIG must be greater than zero A value of 0 001 is often suitable PHIREDSWH The derivatives of observations with respect to parameters can be calculated using either forward differences involving two parameter observation pairs or one of the variants of the central method involving three parameter observation pairs described in Section 2 3 As discussed below you must inform PEST through the variables FORCEN and DERMTHD which method is to be used for th
547. ues for 5 parameters the parameter names appear in the top row As usual a parameter name must be twelve characters or less in length The same parameter names must be cited on template files provided to SENSAN In fact if there is a naming discrepancy between the parameters cited in the parameter variation file and those cited in the template files supplied to SENSAN parameters cited in the parameter variation file which are absent from any template file s will not be provided with updated values from model run to model run This will manifest itself on the SENSAN output files as a total lack of sensitivity for some parameters named in the parameter variation file A comprehensive checking program named SENSCHEK see below has the ability to detect such inconsistencies in the SENSAN input dataset Hence SENSCHEK should always be run prior to running SENSAN The second and subsequent rows of a parameter variation file contain parameter values for SENSAN to use on successive model runs A separate model run will be undertaken for each such row A parameter variation file can possess as many rows as a user desires hence SENSAN can be set up to undertake thousands of model runs if this is considered necessary as it may be in Monte Carlo simulation In many sensitivity analyses a user is interested in the effect of varying parameters either individually or in groups from certain base values In such cases parameter base values should appea
548. uld be pointed out that PEST writes a parameter value to a model input file with the maximum possible precision given the parameter field width provided in the pertinent template file Also for the purposes of derivatives calculation PEST adjusts a parameter increment to be exactly equal to the difference between a current parameter value and the incremented value of that parameter as represented possibly with limited precision in the model input file as read by the model 2 3 3 How to Obtain Derivatives You Can Trust The PEST Algorithm 2 31 Reliability of derivatives calculation can suffer if the model which you are trying to parameterise does not write its outcomes to its output file using many significant figures If you have any control over the precision with which a model writes its output data you should request that the maximum possible precision of representation be used Although PEST will happily attempt an optimisation on the basis of limited precision model generated observations its ability to find an objective function minimum decreases as the precision of these model generated observations decreases Furthermore the greater the number of parameters which you are simultaneously trying to estimate the greater will be the deleterious effects of limited precision of model output If a model is comprised of multiple sub model executables run by PEST through a batch file then you should also ensure that numbers are transferred be
549. ulties in finding parameter sets which can raise or lower the prediction whichever is appropriate while keeping the model calibrated it may be advisable to raise PD1 to 10 higher than PDO PD2 When used in predictive analysis mode the solution procedure used by PEST is very similar to that used in parameter estimation mode During each optimisation iteration PEST first fills the Jacobian matrix then it calculates some trial parameter upgrade vectors on the basis of a number of different values of the Marquardt lambda The latter is automatically altered by PEST during the course of the solution process using a complex adjustment procedure If the current value of the objective function is above PD1 PEST adjusts the Marquardt lambda in such a manner as to lower the objective function if the objective function is below PD1 PEST s primary concern is to raise lower the model prediction After PEST has tested a few different Marquardt lambdas it must make the decision as to whether to continue calculating parameter upgrade vectors based on new lambdas or whether it should move on to the next optimisation iteration If the objective function is above PD1 this decision is made using the same criteria as in normal PEST operation these criteria are based on the efficacy of new lambdas in lowering the objective function An important variable in this regard is PHIREDLAM As is explained in Section 4 2 2 of this manual if PEST fails to lower the ob
550. up_2 32 58 Contribution to phi from observation group group_3 0 1115 Contribution to phi from observation group group_4 27 21 Correlation Coefficient gt Correlation coefficient 0 9086 Analysis of residuals gt All residuals Number of residuals with non zero weight 21 Mean value of non zero weighted residuals 0 4399 Maximum weighted residual observation ar13 1 260 Minimum weighted residual observation pil 5 216 Standard variance of weighted residuals 3 933 Standard error of weighted residuals 1 880 Note the above variance was obtained by dividing the objective function by the number of system degrees of freedom ie number of observations with non zero weight plus number of prior information articles with non zero weight minus the number of adjustable parameters If the degrees of freedom is negative the divisor becomes the number of observations with non zero weight plus the number of prior information items with non zero weight Residuals for observation group group_1 Number of residuals with non zero weight 6 Mean value of non zero weighted residuals 0 7424 Maximum weighted residual observation ar4 1 038 Minimum weighted residual observation ar6 0 3916 Variance of weighted residuals 0 6144 Standard error of weighted residuals 0 7838 Note the above variance was obtained by dividing the sum of squared residuals by the number of items with non zero
551. uperimposed on a map of the area proportional posting and or contouring of these residuals may also be useful Such a two dimensional graphical portrayal of residuals will immediately indicate any spatial correlation between them If such spatial correlation exists this is an indication that the manner in which the model domain has been subdivided for parameterisation purposes could do with some improvement If measurements used in the calibration process represent the variation of some quantity over time at one or a number of measurement sites then superimposed plots of model generated and measured quantities against time or of the residuals themselves against time at all measurement sites should be inspected Any tendency of the model to overpredict or underpredict over extended periods of time or over certain segments of the graphs should be noted for this may indicate an inadequacy in the model s ability to represent facets of real world behaviour such an inadequacy may have unwanted repercussions when the model is used for predictive purposes If time varying measurements are made at different geographical locations plots of the spatial distribution of residuals at different times may also reveal worrying departures from independent behaviour spatial correlation possibly indicating once again certain inadequacies on the part of the model Running PEST 5 39 5 7 5 Over Parameterisation In many cases of model deployment the fi
552. upplied in the control data section of the PEST control file Experience has shown that in most instances the most appropriate value for FORCEN is switch This allows speed to take precedence over accuracy in the early stages of the optimisation process when accuracy is not critical to objective function improvement and accuracy to take precedence over speed later in the process when realisation of a normally smaller objective function improvement requires that derivatives be calculated with as much accuracy as possible especially if parameters are highly correlated and the normal matrix thus approaches singularity DERINCMUL If derivatives are calculated using one of the three point methods the parameter increment is first added to the current parameter value prior to a model run and then subtracted prior to another model run In some cases it may be desirable to increase the value of the increment for this process over that used for forward difference derivatives calculation The real variable DERINCMUL allows you to achieve this If three point derivatives calculation is employed the value of DERINC is multiplied by DERINCMUL this applies whether DERINC holds the increment factor as it does for relative or rel_to_max increment types or holds the parameter increment itself as it does for absolute increment types As discussed in Section 2 3 3 for many models the relationship between observations and The
553. urther details 2 1 2 Observation Weights The PEST Algorithm 2 4 The discussion so far presupposes that all observations carry equal weight in the parameter estimation process However this will not always be the case as some measurements may be more prone to experimental error than others Another problem arises where observations are of more than one type For example equation 2 1 may represent a plant growth model you may have a set of biomass and soil moisture content measurements which you would like to use to estimate some parameters for the model However the units for these two quantities are different kg ha and dimensionless respectively hence the numbers used to represent them may be of vastly different magnitude Under these circumstances the quantity represented by the larger numbers will take undue precedence in the estimation process if the objective function is defined by equation 2 3 this will be especially unfortunate if the quantity represented by the smaller numbers is in fact measured with greater reliability than that represented by the larger numbers This problem can be overcome if a weight is supplied with each observation the larger the weight pertaining to a particular observation the greater the contribution that the observation makes to the objective function If the observation weights are housed in an m dimensional square diagonal matrix Q whose i th diagonal element qi is the square of the weight w atta
554. ution PSTOP writes a 1 to tell PEST to cease execution while PSTOPSTP instructs PEST to cease execution with a full parameter statistics printout Running PEST 5 24 by writing a 2 to file pest stp Note that if the single window version of PEST is running PEST will not respond to the presence of file pest stp until the current model run is complete Parallel PEST will respond immediately however the various incidences of the model will continue to run to completion in their own windows after PEST execution has ceased While PEST execution is paused the run record file can be inspected by viewing it using a text editor or viewer from another window 5 4 2 Restarting PEST with the r Switch As was discussed in Section 4 2 2 you can instruct PEST to periodically dump the contents of its memory to a number of binary files so that if its execution is terminated at any stage it can later be restarted taking advantage of the work which it has already done Thus for example if you had been using the single window version of PEST and you had previously terminated its execution before the inversion process was complete you could restart it using the command pest case r where case is the filename base of the PEST control file The restart option is invoked in an identical manner for Parallel PEST however it may be necessary to restart the slaves first When PEST is restarted in this manner it will not resume executi
555. ution which calibrates the transient model also calibrates a steady state model There is no limit to the range of possibilities The main criteria for a regularisation methodology are that 1 it includes a substantial number of relationships between most or all of the parameters involved in the parameterisation process and 2 it encapsulates some preferred state of the system deviations from this preferred state are tolerated only to the extent that they allow the model to provide an acceptable fit to field measurements Because regularisation should attempt to impose some preferred state on system parameters and because it should involve as many of the model parameters as possible it constitutes a mechanism for making insensitive parameters sensitive Without the use of a suitable regularisation strategy a parameter pertaining to a part of the model domain far removed from locations where field measurements were made may have a sensitivity of almost zero Thus if that parameter is estimated together with other model parameters on the basis of field data alone its estimated value will be subject to a very large margin of uncertainty However if a regularisation condition is imposed the value estimated for such an ill determined parameter will most likely be the natural value of the parameter as defined through the regularisation process Similarly if when estimated on the basis of field measurements alone certa
556. utputs and field data the estimated parameters are often far more believable as result of this misfit Furthermore as will be described below when run in regularisation mode the user is able to inform PEST of the maximum cost that he she is prepared to tolerate in terms of model to measurement misfit when implementing a regularisation constraint The power of a properly implemented regularisation procedure is that it allows the modeller access to the benefits of using a large number of parameters in terms of the spatial resolution required to define the location of important property contrasts at the same time as it allows him her access to the benefits of a more sparingly parameterised system in terms of stability of the inversion process reduced parameter correlation and a reduced propensity for estimated parameter values to be wild and unbelievable It also allows the user to inject his her judgement into the parameterisation process by selecting a regularisation scheme that reflects his her current state of knowledge of that system 7 1 2 Smoothing as a Regularisation Methodology The most commonly used regularisation methodology is the imposition of a smoothing constraint on parameter values In most cases this is achieved by taking differences between neighbouring pairs of parameter values or between functions of these parameter values and requesting that each such difference be zero if possible This is done by supplying e
557. value of composite observation sensitivity would at first sight indicate that an observation is particularly crucial to the inversion process because of its high information content this may Running PEST 5 20 not necessarily be the case Another observation made at nearly the same time and or place as the first observation may carry nearly the same information content In this case it may be possible to omit one of these observations from the parameter estimation process with impunity for the information which it carries is redundant as long as the other observation is included in the process Thus while a high value of composite observation sensitivity does indeed mean that the observation to which it pertains is possibly sensitive to many parameters it does not indicate that the observation is particularly indispensable to the parameter estimation process for this can only be decided in the context of the presence or absence of other observations with similar sensitivities Example 5 4 shows part of an observation sensitivity file Observation Group Measured Modelled Sensitivity arl group_1 1 210380 1 639640 0 5221959 ar2 group_1 1 512080 2 254750 0 6824375 ar3 group_l 2 072040 3 035590 0 8591846 ar4 group_l 2 940560 3 978450 1 0338167 ar5 group_l 4 157870 5 047430 1 1915223 ar6 group_1l 5 776200 6 167830 1 3226952 ar7 group_2 7 789400 7 232960 1 4450249 ar8 group_2 9 997430 8 124100 1 5881968 ar9 group_2 11 83070 8 724950 1 750
558. variability that occurs as an outcome of parameter uncertainty As a general rule if predictions that are to be made by the model are of the same type as the measurements used in the calibration process and if the stress regime imposed on the system represented by the model under predictive conditions is not too different from that prevailing under calibration conditions then the effect of correlation induced parameter uncertainty on predictive uncertainty may not be too large However if any of these conditions are violated predictive uncertainty may be very large indeed Parameter uncertainty caused by parameter insensitivity is often difficult to rectify Recall that composite parameter sensitivities are listed in the parameter sensitivity file produced by PEST Parameter insensitivity results from the fact that the dataset used in the optimisation process simply does not possess the information content that is required to resolve the values of offending parameters Thus these parameters can assume a range of values with minimal effect on model outcomes Once again parameter uncertainty arising out of parameter insensitivity may or may not result in high levels of predictive uncertainty As always the extent of predictive uncertainty can be estimated using PEST s predictive analyser Parameter uncertainty resulting from either excessive correlation or from insensitivity of parameters can often be reduced by including more measurement da
559. veretcts tek aeons cent cies cea coms S E A SEE 4 2 2 WiInPES TV Windows sasaaa eaa aT ET EAE bee ca ies dele eek stake oes aakeaeles 4 3 Preparing for a PEST RUM c c c ccccssesedessuseccvessonsetosonnerooennceseecdoetesnavsccrorenededescsnenonnaes 6 3 1 Reading a Casen nah iiia ees ga nae ve EEA ta oe aas Beebe ned a E EA E 6 B D Brom CHeCkang vos esis ST T E epiee sgsbhcesunasnakavessoubestentees 6 3 3 Running PEST ssh en eaaa cde aa aatan e esis bens env AE Ea i aa e aaa aa anan EN T 4 WinPEST Graphical WindowS ccccccesssssceceeeeseeeneneeceeeeceesesnneceeeceseeesnaaaeeeeeeseeetnneeeess 10 4 1 Accessing the Windows eesseesesesesssessssresssressssreessretssesresesreeessretsstetesssressssreessrrees 10 A A 0 TAEAE EEEN EEE AE O EEA 10 4 3 Regu larisatio Nien aeii E EE T e Ta p EEA aE i 10 4 4 Parameter Sensitivity seina unites eeang ede sede dosbe sabes UnteeMensorsvo a e 10 4S5 Parameter Soa a n n ce I a EEA a cape aa a i n ia 12 AO De a 01010 Fe AONE AT T EEE 12 ANT EATCLEAN S 12 A SsRESIGUAIS aane e e e a a E E a a a seus e Ee a a aeei 13 49 Jacobian eeren aiea axl aa at a a daa a ces e a aa eds i eaPed 14 A Al D S EIKON E10 1 SEA E NEN 15 4l T Covanance eeen eE AEE AT E T EE aves deh EEA EAA 17 4 12 Correlation uainne Ge syed ine E EEEE N E E Relate ete E 17 EE AO E T S E 18 ASTA Eigenvalues iis eea ae ae E EEEE AE E ETEEN O Eet 19 4 15 UMCertainties iivcssc shscceeccaecastansesdstusishantaaes E eE EEA A EA O EEAS E
560. vided in Example 10 6 ptf parameter data infiltri SH Santi red filtrat2 Sinfiltrat2 filtrat3 Sinfiltrat3 F infilt1 infiltrat2 INEVLEZ anti lterat3 te and model input files l tpl model in trol data le point Example 10 6 A template for the PAR2PAR input file of Example 10 5 Based on the template file of Example 10 6 before PEST runs the model it will replace the strings infiltl S infiltrat2 and infiltrat3 with the current values of these parameters Note that these parameters do not need to be named the same as the PAR2PAR parameters to which values are assigned in the pertinent expressions in the PAR2PAR input file They could have been given any name at all the same parameter names are used by both PEST and PAR2PAR in this example simply as a matter of convenience Furthermore parameter spaces in the template of a PAR2PAR input file do not need to be restricted in their location to the right side of expressions comprised of a symbol followed by a single number See for example the PAR2PAR input file and corresponding template file depicted in Examples 10 7 and 10 8 These accomplish the same task as the files depicted in Examples 10 5 and 10 6 PEST Utilities 10 17 parameter data infiltl 0 3456 infilt2 infiltl 1 23983 infilt3 infilt2 1 53953 template and model input files model tpl model in Example 10 7 A PAR2PAR input file ptf
561. vity 5 7 Pest suspended Figure 1 For a nonlinear model parameter sensitivity varies with parameter value If a particular parameter is tied to another parameter or is held fixed during the inversion process then a composite sensitivity cannot be calculated for that parameter such parameters are thus not featured in composite sensitivity plots Composite sensitivities can be plotted as natural scaled or unscaled with the default being natural select the desired option from the pop up menu that appears if you click anywhere within this window with the right mouse button Natural sensitivities are sensitivities as PEST sees them and are calculated in the manner discussed in Section 5 3 2 of the PEST manual on the basis of the last computed Jacobian matrix the untransformed or logarithmically transformed status of each parameter is automatically taken into account when calculating its natural composite sensitivity If the Unscaled option is selected composite sensitivities are calculated with respect to the parameters themselves not their log transformed counterparts even for parameters that are log transformed throughout the parameter estimation process for untransformed parameters natural and unscaled composite sensitivities are thus identical The composite scaled sensitivity of a parameter is defined as the composite unscaled sensitivity of that parameter multiplied by the param
562. w parameters tested by running the model again By comparing parameter changes and objective function improvement achieved through the current iteration with those achieved in previous iterations PEST can tell whether it is worth undertaking another optimisation iteration if so the whole process is repeated At the beginning of a PEST run you must supply a set of initial parameter values these are the values that PEST uses at the start of its first optimisation iteration For many problems only five or six optimisation iterations will be required for model calibration or data interpretation In other cases convergence will be slow requiring many more optimisation iterations Often a proper choice of whether and what parameters should be logarithmically Introduction 1 8 transformed has a pronounced effect on optimisation efficiency the transformation of some parameters may turn a highly nonlinear problem into a reasonably linear one Derivatives of observations with respect to parameters are calculated using finite differences During every optimisation iteration the model is run once for each adjustable parameter a small user supplied increment being added to the parameter value prior to the run The resulting observation changes are divided by this increment in order to calculate their derivatives with respect to the parameter This is repeated for each parameter This technique of derivatives calculation is referred to as the method of forw
563. ween observation values on second and subsequent data lines of the ABSFLE output file and observation values cited on the first data line Hence if the first data line ie the line following the parameter name line of the parameter variation file lists parameter base values the second SENSAN output file lists the variations of model outcome values relative to model outcome base values If for a particular model outcome Op represents the base value and Op represents the value for a certain set of alternative parameter values then the value written to the RELFLE output file for that model outcome and parameter set is OO ui 0 11 1 Note that if Op is zero a value of 10 is written to RELFLE as an indicator of this condition The third SENSAN output file SENSFLE provides model outcome sensitivities with respect to parameter variations from their base values As usual parameter base values are assumed to reside on the first data line of the parameter variation file Sensitivity for a particular outcome is calculated as the difference between that model outcome and the pertinent model outcome base value divided by the difference between the current parameter set and the parameter base values The latter is calculated as the L2 norm ie the square root of the sum of squared differences between a current parameter set and the base parameter set Thus if only a single parameter p differs from the base set the sensitivity for a particular
564. which is neither fixed nor tied must belong to a parameter group the group to which each such parameter belongs is supplied through the parameter input variable PARGP see below Each parameter group must possess a unique name of twelve characters or less The PEST input variables that define how derivatives are calculated pertain to parameter groups rather than to individual parameters Thus derivative data does not need to be entered individually for each parameter however if you wish you can define a group for every parameter and set the derivative variables for each parameter separately In many cases parameters fall neatly into separate groups which can be treated similarly in terms of calculating derivatives For example in Example 4 2 which is a PEST control file for a case involving the calculation of surface measured apparent resistivities from layered half space properties the layer resistivities can be assembled into a single group as can the layer thicknesses A tied or fixed parameter can be a member of a group however as derivatives are not calculated with respect to such parameters the group to which these parameters belong is of no significance except perhaps in calculating the derivative increment for adjustable group members if the increment type is rel_to_max see below Alternatively fixed or tied parameters can be assigned to the dummy group none If any group name other than none is cited for any
565. will result in nothing more than a message generated by the operating system to the PEST or PSLAVE screen this matters little as both PEST and PSLAVE check that their instructions have been obeyed re issuing them if they have not However if a model rather than PEST or PSLAVE encounters such an error through reading from or writing to a file which has not been closed by another process such as PEST or even the previous model run then the operating system will issue a message such as Sharing violation reading drive C Abort Retry Fail y n The slave running that particular model will drop out of the Parallel PEST optimisation process until this question is answered It would obviously be unfortunate if the question is asked at midnight when no one is around to answer it with a simple r to send the model on its way again Fortunately if WAIT is set high enough this should not happen If PARLAM the fourth variable appearing on the second line of the run management file is set to 1 partial parallelisation of the lambda search is undertaken However if it is set to 0 then the lambda search is conducted in serial fashion using just one processor Partial parallelisation is further discussed in Section 9 2 6 below Lines 3 to NSLAVE 2 ie NSLAVE lines in all of the run management file should contain two entries each The first is the name of each slave any name of up to 30 characters in length can be provided The name should b
566. working in parameter estimation mode the optimal set of parameters is that set for which the objective function is at its minimum PEST uses a number of different criteria to determine when to halt this iterative process Note that only one of them zero valued objective function is a guarantee that the objective function minimum has been obtained In difficult circumstances any of the other termination criteria could be satisfied when the objective function is well above its minimum and parameters are far from optimal Nevertheless in most cases these termination criteria do indeed signify convergence of the adjustable parameters to their optimal values In any case PEST has to stop executing sometime and each of the termination criteria described in this section provide as good a reason to stop as any If these criteria are properly set through user provided PEST input variables you can be reasonably assured that when PEST terminates the parameter estimation process either the optimal parameter set has been found or further PEST execution will not find it There are two indicators that either the objective function is at or very close to its minimum or that further PEST execution is unlikely to get it there The first is the behaviour of the objective function itself If it has been reduced very little or not at all over a number of successive iterations the time has come to cease execution The exact criteria determining this kind of te
567. y statistical calculation in this section of the PEST run record file the weights pertaining to regularisation observations are multiplied by the optimised regularisation weight factor ie by the regularisation weight factor used in the calculation of optimised parameter values Note that these optimised parameter values are also listed on the run record file and of course in the parameter value file Regularisation 7 12 A similar consideration applies to information written to the residuals file at the end of the optimisation process That is where weights are used in the calculation of any quantities pertaining to regularisation observations listed in this file the regularisation weights supplied by the user are multiplied by the optimised regularisation weight factor Note also that measurement standard deviations and natural weights are not calculated for regularisation residuals as these have no meaning Information recorded on the observation sensitivity file produced by PEST at the end of its run is also weight dependent see Section 5 3 3 for details In this case just as in the cases discussed already weights used for regularisation observations are equal to user supplied weights multiplied by the optimal regularisation weight factor 7 5 Other Considerations Related to Regularisation 7 5 1 Using PEST in Two Different Modes As is explained in Section 7 3 a PEST control file suitable for use by PEST in regularisat
568. y TEMPCHEK in which all parameters cited in file in tpl are listed By copying file in pmt to in par and adding parameter values scales and offsets to the listed parameter names as well as values for the character variables PRECIS and DPOINT we can create a PEST parameter value file Example 12 6 shows such a file because this file will shortly be used with program PESTGEN to generate a PEST control file the values supplied for each of the parameters are the initial parameter values to be used in the optimisation process sl s2 y1 xC Example 12 5 File in pmt single point sl 0 3 1 0 0 0 S206 8s sO 10470 yi Ori 1 0 0 0 xc 0 3 1 0 0 0 Example 12 6 File in par At this stage TEMPCHEK should be run again using the command tempchek in tpl in dat in par An Example 12 7 in par can be omitted if you wish for this is the default parameter value filename generated automatically by TEMPCHEK from the template filename When invoked with this command TEMPCHEK generates file in dat the TWOLINE input file using the parameter values provided in file in par you should then run TWOLINE making sure that it reads this file correctly 12 1 4 Preparing the Instruction File Next the instruction file should be prepared This can be easily accomplished by writing the instructions shown in Example 12 7 to file out ins using a text editor Using this instruction set all model generated observations are read as semi fixed
569. y aspects of it can be varied to suite the problem at hand allowing you to optimise PEST s performance for your particular model How you do this is described later in this manual In the course of the estimation process PEST writes what it is doing to the screen it simultaneously writes a more detailed run record to a file You can pause PEST execution at any time to inspect its outputs in detail when you have finished looking at these PEST will recommence execution exactly where it was interrupted Alternatively you can shut down PEST completely at any stage It can then be restarted later you can direct it to recommence execution either at the beginning of the optimisation iteration in which it was interrupted or at that point within the current or previous iteration at which it last attempted to upgrade parameter values As it calculates derivatives PEST records the sensitivity of each parameter with respect to the observation dataset to a file which is continuously available for inspection If it is judged that PEST s performance is being inhibited by the behaviour of certain parameters normally the most insensitive ones during the optimisation process these parameters can be temporarily held at their current values while PEST calculates a suitable upgrade for the rest of the parameters If desired PEST can be requested to repeat its determination of the parameter upgrade vector with further parameters held fixed Certain variables govern
570. y marker will be part or all of some kind of header or label such a header or label often precedes a model s listing of the outcomes of its calculations and thus makes a convenient reference point from which to search for the latter It should be noted however that the search for a primary marker is a time consuming process as each line of the model output file must be individually read and scanned for the marker Hence if the same observations are always written to the same lines of a model output file these lines being invariant from model run to model run you should use the line advance item in preference to a primary marker A primary marker may be the only item on a PEST instruction line or it may precede a number of other items directing further processing of the line containing the marker In the former case the purpose of the primary marker is simply to establish a reference point for further downward movement within the model output file as set out in subsequent instruction lines The Model PEST Interface 3 15 Primary markers can provide a useful means of navigating a model output file Consider the extract from a model output file shown in Example 3 8 the dots replace one or a number of lines not shown in the example in order to conserve space The instruction file extract shown in Example 3 9 provides a means to read the numbers comprising the third solution vector Notice how the SOLUTION VECTOR primary marker is preceded b
571. y may prevent parameter adjustment overshoot You should exercise caution in choosing which parameters are relative limited and which are factor limited Remember that if a parameter is factor limited or if it is relative limited with a limit of less than 1 the parameter can never change sign Conversely if a parameter is relative limited with a limit of 1 or greater it can be reduced right down to zero in a single step without transgressing the limit this may cause parameter overshoot problems for some nonlinear models and a factor limit may need to be considered However the latter cannot be used if the parameter can change sign Faced with quandaries of this type the parameter OFFSET variable may be useful in shifting the parameter domain such that it does not include Zero As described in Section 5 6 RELPARMAX and FACPARMAX can be altered midway through a PEST optimisation run Furthermore if the parameter adjustment vector is dominated by a particular insensitive parameter such that the change to that parameter is at its RELPARMAX or FACPARMAX limit and the changes to other parameters are minimal then the offending parameter can be held at its current value through the user intervention process described in Section 5 6 5 5 8 Poor Choice of Initial Parameter Values In general the closer are the initial parameter values to optimal ie the values for which the objective function is at its global minimum the faster will PE
572. y the PERIOD NO 3 primary marker The latter marker is used purely to establish a reference point from which a search can be made for the SOLUTION VECTOR marker if this reference point were not established using either a primary marker or line advance item PEST would read the solution vector pertaining to a previous time period TIME PERIOD NO 1 gt SOLUTION VECTOR 1 43253 6 43235 7 44532 4 23443 91 3425 3 39872 TIME PERIOD NO 2 gt SOLUTION VECTOR 1 34356 7 59892 8 54195 5 32094 80 9443 5 49399 TIME PERIOD NO 3 gt SOLUTION VECTOR 2 09485 8 49021 9 39382 6 39920 79 9482 6 20983 Example 3 8 Extract from a model output file pif PERIOD NO 3 SOLUTION VECTOR 11 obs1 5 10 obs2 12 17 obs3 21 28 obs4 32 37 obs5 41 45 amp obs6 50 55 Example 3 9 Extract from an instruction file Line Advance The syntax for the line advance item is ln where n is the number of lines to advance note that I is el the twelfth letter of the alphabet not one The line advance item must be the first item of an instruction line it and the primary marker are the only two instruction items which can occupy this initial spot As was explained above the initial item in an instruction line is always a directive to PEST to move at least one line further in its perusal of The Model PEST Interface 3 16 the mod
573. y varied parameter If the increment type for a group is rel_to_max the increment for all members of that group is calculated as DERINC times the absolute value of the group member of currently greatest absolute value This can be a useful means by which to calculate increments for parameters whose values can vary widely including down to zero The relative aspect of the rel_to_max option may maintain model outcome difference significance as described above however because the increment is calculated as a fraction of the maximum absolute value occurring within a group rather than as a fraction of each parameter an individual parameter can attain near zero values without its increment simultaneously dropping to zero A further measure to protect against the occurrence of near zero increments for relative and rel_to_max increment types is provided through the PEST group input variable DERINCLB This variable contains a standby absolute increment which can be used in place of the relative or rel_to_max increment if the increment calculated for a particular parameter using either of these latter methods falls below the absolute increment value contained in DERINCLB The group input variable FORCEN determines whether derivatives for the parameters of a particular group are calculated using the forward difference method the central difference method or both FORCEN can be designated as always_2 always_3
574. yond the original number making sure to leave whitespace or a comma between successive spaces or between a parameter space and a neighbouring character or number Similarly numbers read through field specifying format statements may not occupy the full field width in a model input file from which a template file is being constructed eg variable A in Example 3 4 In such a case you should again expand the parameter space beyond the extent of the number normally to the left of the number only until the space coincides with the field defined in the format specifier with which the model reads the number If you are not sure of this field because the model manual does not inform you of it or you do not have the model s source code you will often nevertheless be able to make a pretty good guess as to what the field width is As long as the parameter space you define does not transgress the bounds of the format specified field and the space is wide enough to allow discrimination between a parameter value and an incrementally varied parameter value this is good enough 3 2 6 How PEST Fills a Parameter Space with a Number PEST writes as many significant figures to a parameter space as it can It does this so that even if a parameter space must be small in order to satisfy the input field requirements of a model there is still every chance that a parameter value can be distinguished from its incrementally varied counterpart so as to allow prop

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