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MODFLOW-2005 Ground-Water Model – User Guide to the

1. There are variations of the basic process that can significantly influence the efficiency of the calculations depending on the use of the sensitivities and on the structure of the flow simulation The highest efficiency is reached if the gradient of a weighted sum of square error is needed Only a single adjoint state calculation is performed in this case The gradient is used by truncated Newton variable metric conjugate gradient and quasi Newton optimization procedures The MODFLOWP Hill 1992 implemented parameter estimation using adjoint state based gradients with a conjugate gradient routine for an earlier version of MODFLOW Harbaugh and McDonald 1988 A different form of computational efficiency can be gained If the flow simulation has constant time steps Significant increases in efficiency in determining the full sensitivity matrix can then be gained by reusing the adjoint state calculation of an observation for other observations at the same observation location Introduction The adjoint state based sensitivity ADJ process for MODFLOW 2005 provides the sensitivity of calculated observations to model parameters Throughout this user s guide we assume that the reader is familiar with the use and terminology of MODFLOW 2005 Harbaugh 2005 In this document the terms observations and parameters have formal meanings Observations are defined for various MODFLOW packages and entered as such using the package input files For exa
2. Nument PARNAM OBSNAM Misfit Cutoff icol irow ilay Sensitivity Parameter _ Cell Value Definition of Variables Nument Number of entries to follow for that particular distributed parameter observation pair If sensitivities for an observation cannot be calculated because the cell in which the observation lies has gone dry then the NUMENT entry will be zero Misfit Difference between observed observation and calculated observation If the observation cannot be calculated because the cell in which the observation lies has gone dry then the Misfit entry will be 999 Cutoff The absolute value of the smallest sensitivity that would be printed for this observation parameter pair Cutoff is the product of THRESH and the largest absolute sensitivity of any cell Sensitivity Sensitivity of the observation to the value for the cell listed in the appropriate multiplication array defined in the LPF file Parameter Cell Value the value of the parameter in the cell This value is the product of the parameter value listed in the LPF file and the value for the cell listed in the appropriate multiplication array defined in the LPF file The parameter cell value is printed only f the PVALUEOUT variable is set to 1 4 3 4 Format Option 4 Option 4 compresses the output into distributed parameter matrices in way readable using JUPITER UTL_READMATRIKX The sensitivities of each parameter observation pair are written in compressed format To r
3. The observation head value As with OBSNAM the observation value is not used if multiple observations are entered for the location MLAY i The i layer used for a multi layer head observation PR i The proportion of the calculated head of layer MLAY i used to determine the calculated observation value The sum of all for an observation PR should equal 1 0 ITT Indicator of head observation type If ITT is 1 then the head values given as HOBS are used as the observation values If ITT is 2 then the first observation of the list is treated as a reference head value Subsequent observations are calculated as changes from the reference observation The HOBS values for these observations are still entered as head values The change in head from the reference observation is calculated within the code STATISTIC Observation STATISTIC If STAT FLAG is 3 then STATISTIC is used as the observation weighting factor For values of STAT FLAG less than 3 then STATISTIC interpreted as a measurement error statistic STAT FLAG STAT FLAG is used in the following manner to determine the observation weighting factor If STAT FLAG 0 the weight is STATISTIC is treated as a 1 EVF WEIGHT variance observation measure If STAT FLAG the weight is STATISTIC is treated as an EVF WEIGHT observation standard deviation measure 1 EVF WEIGHT Hoss STATISTIC is treated as an observation coefficient of
4. IADJSCL a value of 1 causes 1 scaled sensitivities to be output to file IADJSU If IADJSCL is set to 1 then the reported sensitivities are multiplied by the parameter value and divided by 100 If IADJSCL is not set to 1 then the sensitivities are not scaled They are not scaled by the parameter value IADJPC a non zero value causes observation and parameter names to be added to file IADJSU IUOBSMAP File unit number for mapping of observations of all types to output observation number File name should be supplied in the MODFLOW Name File using type DATA IUSOLV File unit number for adjoint calculation solver control File name should be supplied in the MODFLOW Name File using type DATA The file supplies separate solver control information for the adjoint solution The solver must be the same as used in the head calculations If IUSOLV is zero then the solver control used for the head calculations is used to control the adjoint state calculations It would be a rare situation where using the same control parameters for both the head solution and the adjoint calculations would be appropriate NPE Number of lumped parameters to be used to calculate sensitivity information Any distributed parameters must be included as lumped parameters also NPARDIS Number of distributed parameters to calculate sensitivities for Each distributed parameter must have a corresponding lumped parameter IADJXDU File unit number for distributed pa
5. scaled They are not scaled by the parameter value IADJPC a non zero value causes row measurement and column parameter numbers to be added to file IADJSU IUOBSMAP File unit number for mapping of observations of all types to output observation number File name should be supplied in the MODFLOW Name File using type DATA CFNAME File name for concentration location and time data This name is supplied only if velocity sensitivity data has been calculated for a transport model VSNAM File name from velocity sensitivity data This name is supplied only if velocity sensitivity data has been calculated for a transport model IUSOLV File unit number for adjoint calculation solver control File name should be supplied in the MODFLOW Name File using type DATA The file supplies separate solver control information for the adjoint solution The solver must be the same as used in the head calculations If IUSOLV is zero then the solver control used for the head calculations is used to control the adjoint state calculations It would be a rare situation where using the same control parameters for both the head solution and the adjoint calculations would be appropriate NPE Number of lumped parameters to be used to calculate sensitivity information Any distributed parameters must be included as lumped parameters also NPARDIS Number of distributed parameters to calculate sensitivities for Each distributed parameter must have
6. 2 9G13 6 14 10F6 1 3 15F7 1 15 10F6 2 4 15F7 2 16 10F6 3 5 15F7 3 17 10F6 4 6 15F7 4 18 10F6 5 7 20F5 0 19 5G12 5 8 20F5 1 20 6G11 4 9 20F5 2 21 7G9 2 10 20F5 3 11 20F5 4 12 10G11 4 3 5 Observation weights for gradient calculation There is no need for observation weights in the standard usage of MODFLOW 2005 However the calculation of the gradient of the sum of square observation error requires weights The adjoint code re implements the assignment of observation weights used in MODFLOW 2000 However observation weights are optional If weighting of the gradient calculation is not requested input files for standard MODFLOW 2005 simulations do not need to be modified The following description replaces the input instructions for observations Further detail regarding the concepts of observations in MODFLOW 2005 can be obtained from the file OBS pdf distributed with this software 3 5 1 Weighted Head Observations 1 Text Text is optional and can include as many text lines as desired as long as they begin with 2 NH MOBS MAXM IUHOBSV HOBDRY 3 TOMULT EVF 4 OBBSNAM LAYER ROW COLUMN IREFSP TOFFSET ROFF COFF HOBS STATISTIC STAT FLAG 5 MLAY 1 PR 1 MALY 2 PR 2 6 ITT 7 OBSNAM IREFSP TOFFSET HOBS STATISTIC STAT FLAG Definition of Variables Variables that are needed in addition to the standard MODFLOW 2005 head observation data are STATISTIC and STAT FLAG T
7. 56 Drain Conductivity Parameter The DRT parameter influences the boundary conditions of the model These influences are represented in matrices P and Q of Equation 2 16 Equation 6 11 of the MODFLOW 2005 user guide presents the drain boundary condition as QD CD HD 7 h a h 4 gt HD QD 0 h lt HD ijk 7 2 57 Where QD is the flow from the drain to the aquifer QD will always be less than or equal to zero CD is the drain conductance and HD is the drain elevation When the drain is active h jx gt HD matrices P and C have an element along the diagonal for cell i j k consisting of CD and CD HD respectively When CD is defined by a drain parameter it is composed of two terms multiplied together a multiplication factor RTF for each cell and the parameter value Thus for any cell where a drain defined by a parameter is active the derivative of A with respect to the parameter value is RTF and RTF HD in the case of matrix C A drain return reintroduces a fixed proportion RFPROP of the flow to a drain back into the aquifer at another cell location The returned flow is not dependent on the head at the other location For return flow the matrices P and C have an off diagonal element in the row of the return cell location aligned with location i j k of RFPROP RTF and RFPROP RTF HD respectively 2 10 2 Matrix Derivatives with Respect to Head Derivative of Conductance with Respect to Head B
8. dependence of J on h we choose aL XM A 2 19 oh Now we want to manipulate this equation to be the same form as the ground water flow equation aR Ch 2 20 i j k Where we have taken advantage of the symmetry of the discrete groundwater flow equations A A The line and subscript to the right of the last term indicates that the derivative is non zero only at the location of the observation MODFLOW can be used to solve this equation where the initial conditions for are zero and Oh is taken as the only external source term If the function L is the error of a calculated head measurement 4 located at the OL oe fen top center of a cell then 5p 00 1 j jpk k i j k Returning to the augmented performance measure dJ _ L H AA dp p p p dJ L 0 Bh C An A 2 21 dp Op Op dJ _ OL or OB yo OC A 1 dp Op Op Op Op If the observation does not depend directly on the parameter the first term in Equation 2 21 is zero If the parameter is not storage then the term involving the matrix B is zero If there are no source terms or head dependent boundary conditions that depend directly on the parameter then the derivative of the matrix C with respect to the parameter is zero For these conditions the sensitivity equation reduces to ie es J p 2 22 2 3 Multi step transient For a multi step transient we have a series of time step equations A h B k C 2 23 and a pe
9. model cells along columns CC are conductance between model cells along rows and CV are conductance between model cells in adjacent layers The matrices P and some entries in Q are a result of head dependent flow boundary conditions to a cell such as flow from a river Q may also represent flow sources that are not head dependent such as pumping rate SS is the specific storage of a cell Ar Ac AV are the row column and layer thicknesses of the cell and t is the time step increment To simplify the development of the adjoint equations the hydraulic conductance matrices are assumed to be insensitive to changes the hydraulic head The assumption is relaxed in Section 2 6 If we collect all terms that include h on the left hand side of the equation and all terms that include 4 on the right hand side along with the Q terms we can write the flow equations in matrix form as Ah B C 2 17 If the equations have the correct solution f Bh C Ah 0 In this development A and B depend on model parameters p but not on 4 or h The model equivalent of the observation is a function of head at a single cell at the end of the time step which is represented as L Lis Augment L with f to form J L h p 4 Bho C Ah i j k H A Bho C J H h p A E h p 4 Ah 2 18 T T T T fp apt a a a 2 aN Alah oh Op Op Op oh For this one step example h is known and hence dh 0 To eliminate the
10. optimization of a function is described in Section 2 10 The concept of using adjoint states to calculate sensitivities is 2 1 Adjoint states for constrained optimization We now take a detour to provide intuition of why the adjoint state approach works Let L h m be a function A common function is the error in matching an observation L h p F p d 2 4 h is the hydraulic head solution throughout the model Clearly h is a function of p and often dj will be equal to h for a cell at some particular time Now take f to represent the set of discrete flow equations 2 2 of the MODFLOW 2005 manual rewritten as Ah fh p gt Q SE Fs aa 2 5 The set of equations f h p has one equation at each time step or steady state calculation for each active cell in the model domain Because f 0 the function L h p can be augmented with f without changing its value H h p L A p Af h p 2 6 At this point A is arbitrary Equation 2 6 is the equation formed if we wanted to constrain h to be the correct solution to the flow equations using as a Lagrange multiplier Because H L oH OH dH dh d X T op n and OH OL of 4 oh Oh oh ea oH o Now set such that ah 1 A a x 2 9 oh oh By doing so the dependence of H on A is eliminated from Equation 2 8 and H can be minimized with respect to m without worrying about the influence of m on h Minimizing H also minimizes L because H L 2 1 1 Simple e
11. s manual However not all MODFLOW 2005 extension packages are supported Most importantly the calculations are limited to parameters defined using the Layer Property Flow package or in one of the stress packages used with the Layer Property Flow package This report presents a theoretical development of the adjoint state sensitivity method The report documents input to the program to control the sensitivity calculations and the format of the sensitivity output files This document does not provide of description of the input files needed to define the ground water flow simulation The reader should refer to the MODFLOW 20005 User s Guide for instructions on controlling the groundwater simulation Harbaugh 2005 Acknowledgments I am thankful for the support given development of the adjoint sensitivity code by Warren Barrash Center for the Investigation of the Shallow Subsurface Boise State University and John Barich Region 10 Office U S Environmental Protection Agency The initial code development was supported by Army Research Office grants DAAD04 961 0318 and DAAD19 00 1 0454 The support for documentation and expansion of the code for general MODFLOW 2005 capabilities was provided by U S EPA grant X 970085 01 0 il Table of Contents PADS URAC siategiet hecat aa dtc cuat sheild SiMe alanentis ie alia a a a eae hotest Si sat 1 TRO CON oss fives aes sg enna Gap nec aa ea das a Sy lp oa as sn aaa Data des aay iV neo Shes LTS 2 1 1Ov
12. second data set IPER ISTEP TOFF IEND ISTART 1 lines per location Explanation of Variables NUMLOC Number of observation locations NUMMEAS Total number of observations COLUMN The column index of the observation location ROW The row index of the observation location LAYER The layer index for the location IEND Observation number for last observation at location ISTART Observation number for first observation at location IPER Stress period number when observation occurred ISTEP Time step in the stress period when observation occurred TOFF Fraction of time step completed when observation occurred File Listing Format for File VSNAM The sensitivity data file requires a very restrictive format The file has been created using FORTRAN DIRECT ACCESS format This is accomplished with an OPEN statement similar to OPEN UNIT Unit number NAME name FORM UNFORMATTED ACCESS DIRECT RECL Len Where Len is the record length of the sensitivity data Len 4 3 NCOL NROW NLAY The stored sensitivities are single precision e g 4 bytes long A single record contains the array VELSENS 3 NCOL NROW NLAY There are three directions with indicies column direction 1 row direction 2 vertical 3 This record contains the velocity sensitivity for a single observation at a one time step The data are stored with the direction index cycling first then column then row and finally layer This order i
13. variation measure If STAT FLAG 2 the weight is If STAT FLAG 3 the weight is STATISTIC EVF 3 5 2 Flow Observations Flow observations have similar input formats Each observation type requires a separate observation definition input file The specific meaning of the input variable FLOWOBS changes with observation type The other variables have consistent meanings 1 Text 2 NQ NQC NQT IUOBSV 3 TMULT EVF 4 NQOB NQCL Item 4 through 6 are listed NQC times 5 OBSNAM IREFSP TOFFSET FLOWOBS STATISTIC STAT FLAG Item 5 read NQOB times after item 4 6 LAYER ROW COLUMN FACTOR Item 6 is read NQCL times after the last item 5 Definition of Variables NQ The number of cell groups that define the observation locations NQC The total number of cells for all observations listed in the input file NQT The number of observations listed in the file IUOBSV a FORTRAN unit number defining the output file used to report the calculated observation values The file name of the unit must be specified in the MODFLOW Names File as type DATA TMULT used to convert the time scale of TOFFSET to the time scale used in the calculations The product TOMULTH TOFFSET must have time units specified in the DIS file EVF EVF works as a divisor of the observation weights It provides a convenient mechanism to change the relative weighting of all observations of this type to weighting of other observations with a sing
14. 05 Automatic history matching in a Bayesian framework example applications SPE Reservoir Evaluation amp Engineering 8 3 214 223 Zheng C and Wang P 1999 MT3DMS A modular three dimensional multispecies transport model for simulation of advection dispersion and chemical reactions of contaminants in groundwater systems Documentation and user s guide Contract Report SERDP 99 1 U S Army Engineer Research and Development Center Vicksburg MS Zheng C Hill M C and Hsieh P A 2001 MODFLOW 2000 The U S Geological Survey Modular Ground Water Model User Guide to the LMT6 Package The Linkage with MT3DMS for Multi Species Mass Transport Modeling Open File Report 01 82 U S Geological Survey
15. 17 DramC onductiwvity Parameter sccciccyisierenspadtensseauladeren tassel edi eacadta eae yied 18 2 10 2 Matrix Derivatives with Respect to Head ccccecccecssccsseceeeceeeeeeeecesecneeeeeeenseeeaeenes 19 Derivative of Conductance with Respect to Head cccceccecceeseeeseceteceeeeeeeeeeeeeeaeeees 19 lnpOt IMs iCtiOns 48 aah ease esata hcl etal E E incarnate Maca OKEE 19 3 1 ADJ file type in MODEL OW Names File s1 ss lt ctraksccaxt dniecanddivscaavhsgactesstvatiapicadaeneshasc 20 3 2 Type ADJ ANU Hesh aia erates sere n sell E cease E A E R tie 20 Explanation of WV abla CS nne A tad ela att a oes E EE 21 3 3 Format control of the lumped parameter sensitivity OUtPUt eee ceeceeeeeeeeeteeeeteeenseeees 24 3 5 Observation weights for gradient calculation ce cccescceseceeeeeesceececeseceeeeeeeeeeeeeeeseeesaeenes 25 3 5 1 Weighted Head Observations setecssvssaieunst ae cleieestidesesaceancassaawieloeesidyn cinta 25 Defimitnonof Varnables re nerne ee testes e E aa E east 25 39 2 PLOW ODSETVAtlONS eentenari 28 Definition oF Variables mi srronrcetinnnn eisin aa a a dene a a e Aaa 28 Output Eiles se a a a e a a a e EER 30 Aol Obs ryation Mapsin beaten seas E EARE R E EEA E E E hats 30 4 2 Lumped Parameter Sensitivity Output ccccccsceesseesseceteceseceecceeseecsaecnseeseceeeaeecsseceeesaes 30 Fil Listing Format so siscceus sents moneda ase ai E antl heat a thc Made metre 30 POSTINI OF Ve Ab DIC Soa cect ctin oa tea peca
16. ALUEOUT THRESH NPARDIS PARNAM One entry per line One line for each distributed Parameter NOBS OBSNAM One entry per line One line for each observation NCOL NROW NLAY Sensitivity Sensitivity Sensitivity One list per parameter observation pair 4 3 3 Format Option 3 Option 3 compresses the output to eliminate small sensitivity values After a header that lists each distributed parameter and each observation in the orders in which they are cycled each line of the output lists numbers for the distributed parameter identification column row layer observation and sensitivity only for sensitivity values larger than DISCUT Output for option 3 contains two possible extra data that the options 2 and 4 The misfit between the input observation value and the calculated observation value is reported The misfit can also obtained from the MOFLOW 2005 output files defined for observations in the MODFLOW Names File If the PYALUEOUT variable is set to 1 then the parameter value for each cell is also included These values are defined through the LPF and multiplication array input files File Listing Format IAFORM PVALUEOUT THRESH NPARDIS PARNAM One entry per line One line for each distributed Parameter NOBS OBSNAM One entry per line One line for each observation Nument PARNAM OBSNAM Misfit Cutoff One for each parameter observation pair icol irow ilay Sensitivity Parameter_Cell_ Value Nument lines are printed
17. IGHTS or IUPAROBS 3 PREFIX PREFIX is read only if OPERATION has a value of 3 or 4 4 IADJSU IADJFM IADJSCL IADJPC 5 IUOBSMAP 6 IUSOLV 7 NPE NPARDIS 8 IADJXDU IAFORM DISCUT PVALUEOUT Read only if NPARDIS is greater than zero 9 IPDISP 1 IPDISP 2 IPDISP is a NPARDIS long integer list read only if NPARDIS is greater than zero 10 PARNAM LN Read NPE times Explanation of Variables OPERATION Defines the mode of operation of the adjoint sensitivity code OPERATION is an integer value between 1 and 4 The following lists how OPERATION controls the program 1 Sensitivities will be calculated for each observation creating a Jacobian matrix of sensitivities of each observation to each parameter 2 Gradient of the weighted sum of squares errors of all observations is calculated 3 Only the solution for hydraulic heads will be performed Sensitivity calculations will not be done The resulting heads and necessary boundary condition information will be stored in a series of files starting with the PREFIX character string 4 Sensitivities will be calculated for a single observation that will be obtained from file unit IUPAROBS defined in the MODFLOW Names File Head and boundary condition information will be read from files starting with the PREFIX character string MODE Read only if OPERATION 1 If MODE is greater than zero then adjoint states are solved for the last observation at a location The adjoint states for earli
18. MODFLOW 2005 Ground Water Model User Guide to the Adjoint State Based Sensitivity Process ADJ Tom Clemo Center for the Geophysical Investigation of the Shallow Subsurface Boise State University Technical Report BSU CGISS 07 01 November 2007 Preface This report describes a version of the MODFLOW 2005 modular ground water model that uses the adjoint state method to calculate the sensitivity of observations to parameters The sensitivity information provided by this program may be useful in many applications including parameter estimation However the program itself does not use the sensitivity in a parameter estimation process or for any other end use application The program provides an alternative to calculating sensitivity information using the direct sensitivity approach implemented in MODFLOW 2000 or using a parameter perturbation method The most popular optimization procedures require the Jacobian matrix of sensitivities with elements consisting of the sensitivity of each observation to each parameter Other procedures use the gradient of an objective function with respect to the parameters This program will calculate sensitivities needed to form the Jacobian matrix or the gradient of a weighted least square error objective function The resulting sensitivity information is reported in the JUPITER API format or other optional formats The program is based on the basic features of MODFLOW 2005 described in the MODFLOW 2000 user
19. RDIS 11 IADJXDU IAFORM DISCUT PVALUEOUT Read only if NPARDIS is greater than zero 12 IPDISP 1 IPDISP 2 IPDISP is a NPARDIS long integer list read only if NPARDIS is greater than zero 13 PARNAM LN Read NPE times Explanation of Variables OPERATION Defines the mode of operation of the adjoint sensitivity code OPERATION is an integer value between 1 and 4 The following lists how OPERATION controls the program 1 Sensitivities will be calculated for each observation creating a Jacobian matrix of sensitivities of each observation to each parameter 2 Gradient of the weighted sum of squares errors of all observations is calculated 3 Only the solution for hydraulic heads will be performed Sensitivity calculations will not be done The resulting heads and necessary boundary condition information will be stored in a series of files starting with the PREFIX character string 4 Sensitivities will be calculated for a single observation that will be obtained from file unit IUPAROBS defined in the MODFLOW Names File Head and boundary condition information will be read from files starting with the PREFIX character string MODE Read only if OPERATION 1 If MODE is greater than zero then adjoint states are solved for the last observation at a location The adjoint states for earlier observations are calculated from this solution If the simulation does not have constant time steps then these calculated adjoint states are an appr
20. TAT FLAG 3 the weight is STATISTIC EVF LAYER the layer index for the location ROW the row index of the observation location COLUMN the column index of the observation location FACTOR The proportion of this cell s contribution to the total calculation of the simulated FLOWOBS equivalent Usually the sum of FACTORs for each cell group should sum to 1 0 However if the observation results from measuring only a portion of the flow then the FACTORs may sum to less than 1 0 See Equation 2 34 or Equation 5 of the file OBS pdf Output Files 4 1 Observation Map The observation map relates individual observations to a global observation list An ordered listing of the observations is also included in the header of both the lumped and distributed parameter sensitivity files The unit number of the observation map is specified by input variable IUOBSMAP If this value is less than or equal to 0 then the map will not be printed The file name should be defined in the MODFLOW Names File as type DATA The observation map will contain a line for each observation type such as Head observations of grid 1 are global observations 1 to 3 River flow observations of grid 1 are global observations 4 to 6 The file anticipates support for the local grid refinement LGR package However the LGR package cannot be used with this version of the adjoint sensitivity code 4 2 Lumped Parameter Sensitivity Output The unit number for the lumped param
21. a corresponding lumped parameter IADJXDU File unit number for distributed parameter sensitivity output File name should be supplied in the MODFLOW Name File Type of file DATA or DATA BINARY controls whether the file is formatted or binary The type of binary file can be controlled using the openspec inc file prior to compilation of the program IAFORM controls the format of the distributed parameter sensitivity output Briefly the value must be 1 4 See Section 3 5 Distributed Sensitivity Output for more detail 1 Option 1 causes single column of sensitivities to be output with the following cycle pattern The columns of the grid are the inner most cycle then rows layers parameters with observations as the outer cycle Comment lines starting with are written prior to each new distributed parameter All columns rows and layers are included which means that both active and inactive cells are included Inactive cells are listed with a sensitivity of zero 2 Option 2 creates a row of sensitivities for each distributed parameter observation pair using the same cycling as for option 1 Formatted output file type DATA uses 20 1PG16 7 for each line No text lines are included Each new parameter will cause a new line in the output 3 Option 3 compresses the output to eliminate small sensitivity values After a header that lists each distributed parameter and each observation in the orders in which they are cycled each line of t
22. anual have been implemented Unsupported packages include FHB RES STR IBS LAK SFR GAGE SUB UZF We hope to add SFR2 LAK GAGE and UZF to the supported list in the near future The adjoint code does not support local grid refinement However it has been written in anticipation of support for the LGR package The code can be used to calculate the sensitivity of each observation to each parameter creating a sensitivity Jacobian matrix Alternately the code can be used to calculate the gradient of a weighted sum of square error objective function To calculate the sensitivity matrix the adjoint state equations need to be solved for each observation Only a single adjoint state calculation is needed for the gradient calculation Weighting of the objective function requires assignment of weighting coefficients for the observations The MODFLOW 2005 code has been extended to make it backward compatible with the observation weighting used in MODFLOW 2000 Section 3 5 describes how to add observation weights to the standard MODFLOW 2005 observation input files The sensitivity of an observation to a lumped parameter is a single number whereas the sensitivity of an observation to a distributed parameter has a value for each cell where the parameter is defined The sensitivity matrix for lumped parameters is reported in a form similar to the sc and _sc1 files of MODFLOW 2000 The user has the option of reporting unscaled or 1 scaled sensitivi
23. certainty in model prediction Tonkin and others submitted Jyrkama and Sykes 2006 used sensitivities calculated by the adjoint state method in a first order reliability analysis method to estimate the reliability of a pumping well to recharge uncertainty 1 1 Overview of Capabilities The current implementation of the adjoint sensitivity code has limited compatibility with respect to the MODFLOW 2005 packages The LPF package must be used which excludes use of the BCF6 and HUF2 packages All MODFLOW 2005 observation types are supported These are HOB GBOB DRNOB RVOB and CHOB The following parameter types are supported HK VK VKCB HANI VANI SS SY CHD GHB RIV RCH WEL DRN EVT and ETS In addition to overcome a difficulty in calculating the sensitivity to log hydraulic conductivity for distributed parameters two new parameter types have been introduced These are YHK log horizontal conductivity and Y VK log vertical conductivity The HFB parameter is not on this list The HFB6 package is supported but not the HFB parameters We expect to include the HFB parameter in a future version The HK parameter is limited to harmonic mean averaging The alternate approaches for calculating horizontal conductance have not yet been implemented All of the basic stress packages are supported The stress packages are WEL RCH GHB RIV DRN DRT and EVT Few of the MODFLOW extension packages packages not described in the MODFLOW 2005 m
24. ded with the end of time steps MODFLOW 2005 uses linear interpolation of calculated observations at the time steps to determine the observation at the proper time Interpolation is in effect a weighted sum of calculated observations before and after the time step that bounds the observation time The two weighting factors are given by F WyF wF N t t Woran N 1 1 ape 2 30 o 4t N N N 1 Where N references the time at the end of the time step that bounds the observation time The performance measure for an observation that does not fall on a time step is L W F WyF d The following equations result from using this performance measure to calculating the adjoint states for Equation 2 26 EN A AN Wy L 2 2 31 PE OF AN VAN 1 BN YN W _ Pe and for each time step before N A A Brym 2 32 2 5 Multi cell observations If a head observation is offset from the center of the specified cell and or composed from more than one layer then the simulated head is the weighted sum of multiple calculated head values Calculated observations offset from the center of a cell use weighting factors 8 gt which are calculated by linear interpolation between the cell and it neighbors See file OBS pdf The factors R are non zero for at most four head values and sum to 1 0 Multi layer head observations which might represent a piezometer open over multiple layers also have weighting factors P k
25. del User guide to the Observation Sensitivity and Parameter Estimation Processes and Three Post Processing Programs Open File Report 00 184 U S Geological Survey Hill M C 1992 A computer program MODFLOWP for estimating parameters of a transient three dimensional ground water flow model using nonlinear regression Open File Report 91 484 U S Geological Survey Jyrkama M and Sykes J 2006 Sensitivity and uncertainty analysis of the recharge boundary conditions Water Resources Research 42 W01404 doi 1029 2005WR004408 McDonald M G and Harbaugh A W 1988 A modular three dimensional finite difference ground water flow model U S Geological Survey Techniques and Methods book 6 chapter Al Poeter E Hill M Mehl S and Christensen S 2000 UCODE_2005 and six other computer codes for universal sensitivity analysis calibration and uncertainty evaluation U S Geological Survey Techniques and Methods 6 A11 U S Geological Survey RamaRao B LaVenue A de Marsily G and Marietta M 1995 Pilot point methodology for automated calibration of an ensemble of conditionally simulated transmissivity fields 1 Theory and computational experiments Water Resources Research 31 3 475 494 Sykes J Wilson J and Andrews R 1985 Sensitivity analysis for steady state groundwater flow using adjoint operators Water Resources Research 21 3 359 371 Zhang F Skjerheim J Reynolds A and Oliver D 20
26. dp Op ma Op Op op Op Op The steady state adjoint Ay can be defined equivalently by either of the following equations Carrera and Medina 1994 N Ay gt 2 2 40 m 0 or aL A A 7 2 41 The steady state adjoint is not the same as the adjoint for a steady state period prior to a transient For a single steady state stress period then the second form defines the adjoint state equation ar L ac aA e Ap 2 42 dp op a p 2 7 Location based adjoint state calculations Carrera and Medina 1994 and Carrera and others 1997 present a technique that reduces the number of adjoint solutions from the number of observations to the number of measurement locations if a transient simulation has constant A and B matrices Constant A and B matrices require 1 constant time steps 2 insensitivity of A and B to head changes and 3 no head dependent boundary conditions that change with time We introduce a series of objective functions L h corresponding to observations at times all at the same location Corresponding adjoint states 4 are for time step m of observation The A are zero if time step m is after time step m gt The adjoint states Ax are for the last observation at the location For each other observation the solutions to Equation 2 28 result m m l N 255 ind Ay This equivalence does not apply to the time zero adjoint because the time zero adjoint equation differs from t
27. e adjoint state of location i j k multiplies derivatives of A with respect to the head at locations i j k but does not multiply the derivatives of A with respect to any other head 2 9 Gradient of a weighted sum of squares objective function MODFLOWP Hill 1992 provides two options for parameter estimation One option is to calculate the parameter sensitivity matrix using parameter perturbations for use in the Levenburg Marquart non linear optimization procedure The objective function used to define the parameter estimation problem is the weighted sum of squares of mismatch of the observations The second option is to calculate the gradient of the objective function using the adjoint state method for use in a conjugate gradient solution procedure A very attractive feature of this second approach is that the gradient of the objective function can be calculated using a single adjoint state solution A problem with the approach is that the conjugate gradient procedure requires many more iterations than the Levenburg Marquart procedure to converge Carrera and Medina found that the long convergence sequence for the conjugate gradient procedure made the approach inferior to Levenburg Marquart for typical ground water problems Carrera and Medina 1994 However Zhang and others 2005 found that preconditioning of the limited memory Broyden Fletcher Goldfarb Shanno LBFGS algorithm significantly reduces the convergence sequence making the LBFGS a
28. ead the file created using option 4 the sensitivity data for each distributed parameter observation pair would be read using sequential calls to UTL_READMATRIKX The sequence would be one call for each distributed parameter in order of parameters listed in the top of the file which is repeated for each observation listed File Listing Format IAFORM PVALUEOUT THRESH NPARDIS PARNAM One entry per line One line for each distributed Parameter NOBS OBSNAM One entry per line One line for each observation COMPRESSEDMATRIX PARNAM_OBSNAM One per parameter observation pair NNZ NROW NCOL NLAY IPOS Sensitivity repeated NNZ times COMPRESSEDMATRIX PARNAM_ OBSNAM Nnz NCOL NROW NLAY Ipos Sensitivity Definition of Variables COMPRESSEDMATRIX an ASCII string PARNAM_OBSNAM a concatenation of the parameter name and the observation name connected by an underscore _ Nnz The number of sensitivity values printed for the parameter observation pair Only sensitivities that are equal or greater than Threshold are printed Ipos The location index for the associated sensitivity Ipos identifies the location of the associated sensitivity in the model domain The counting cycles columns first then rows then layers For example Ipos 6 in a 5 column by 5 row by 5 layer model would indicate the first column of the second row in the top layer Usage Notes 5 1 WARNING RATE DISCREPANCIES Some error checking of the adjoi
29. er observations are calculated from this solution If the simulation does not have constant time steps then these calculated adjoint states are an approximation Care is required to ensure that observation times correspond to the time step times Observation time data needs to be within 1 of the time step duration WEIGHTS Read only if OPERATION 2 If WEIGHTS is non zero then observations are weighted in the calculation of the sum of squares gradient Otherwise all observations are assigned a weight of 1 Setting WEIGHTS to a non zero value changes the input requirements for observations so that observation data must be entered using the instructions indicated in Section 3 5 for head observations and 3 6 for flow observations If WEIGHTS is zero then the standard MODFLOW 2005 input instructions apply IUPAROBS Read only if OPERATION 4 IUPAROBS defines the unit number from which to read the active observation This unit number must be defined as a DATA file type in the MODFLOW Names File PREFIX A character string without blanks used to define the names of head solution and boundary condition files created using OPERATION 3 or read using OPERATION 4 Only the first 70 characters will be used IADJSU File unit number of lumped parameter sensitivities File name should be supplied in the MODFLOW Name File The file must be type DATA IADJFM Print Format for IADJSU Section 3 4 describes the influence of IADJFM on the output format
30. ered using item 5 ROW the row index of the observation location COLUMN the column index of the observation location IREFSP is the stress period to which the observation time is referenced The reference point is the beginning of the stress period If the value of IREFSP is negative then there are multiple observations at the observation location If IREFSP is negative then IREFSP observations must be using the item 7 format TOFFSET the observation time measured from the beginning of the reference stress period The reference stress period is defined by IREFSP TOFFSET can be specified using time scale units that differ from the simulation time scale TOMULT is used to convert TOFFSET time units to the simulation time scale The product of TOMULT TOFFSET must have the same units as specified in the DIS file ROFF This is the row offset used to locate the observation location relative to the center of the ROW Row offset may vary from 0 5 to 0 5 If ROFF is not zero then multiple calculated head values contribute to the calculated observation See the file OBS PDF for a more detailed explanation of ROFF COFF This is the column offset used to locate the observation location relative to the center of the COLUMN Column offset may vary from 0 5 to 0 5 If COFF is not zero then multiple calculated head values contribute to the calculated observation See the file OBS PDF for a more detailed explanation of COFF HOBS
31. essessrssressresrssseeseesresseessessessesse 36 6A NEO escort ccc tactics a cu n E a A a E A E Siac A A SAS 37 6 2 ADJ input file for velocity sensitivity data lt jsic4esa kes ishcduchoatisssrseeseadiae le Guiae aluadnudianeced 39 File Listing Formatera tananana e e A K EE AR eae a ee 39 Explanation of Varia E634 ais nace uusictel es styceasnie wactencecacettecans antes oneaaecausestneatee tad aeidetaoeaeantces 39 6 3 Input Instructions for velocity sensitivity data eccccscesceseeeceescecececseceeeeeeeeeeeeesseenes 43 File Listing Format for File CENAME 3 visscs scecseceiieeiatstiesceeessthtetasetsidertsieteascaetieess 43 Explanation of Variables 205 acca cass bochscareveater ovaves ae ict edusans coves cine Siucrauseeeus eatocuatens as ane 43 File Listing Format for File VSNAM c 4 2 tesces Baan tea ata can 44 References Cited cscs cassicsesstesngusncesctissgeives sens dias ssews eed TESE EEEE a EEEE EEEE nae 44 MODFLOW 2005 Ground Water Model User Guide to the Adjoint State based Sensitivity Process ADJ By Tom Clemo Abstract This report describes the adjoint state based sensitivity process for MODFLOW 2005 that calculates the sensitivity of observations to parameters The process is composed of three basic components described here for one observation and one parameter first is the solution of the ground water flow problem next this solution is used to calculate the adjoint state for the observation and finally the sensitiv
32. eter output file is defined by input variable IADJSU This file should be defined in the MODFLOW Names file as type DATA File Listing Format NOBS Obsnum OBSNAM Misfit One line for each observation PARNAM PARNAM PARNAM PARNAM PARNAM OBSNAM Sensitivity Sensitivity Sensitivity Sensitivity Sensitivity Sensitivity The printing of observation and parameter names is optional and are included if the input variable ADJPC is not zero The format of the sensitivity values is controlled by input variable IADJFM using the specifications listed in Section 3 4 Definition of Variables NPE Number of parameters NOBS Number of observations represented in this file PARNAM Name of the parameter PARNAM corresponds to PARNAM in the LPF file OBSNAM Name of observation Misfit Input observation value Calculated Observation value Sensitivity The sensitivity of observations to lumped parameters The sensitivity of distributed parameters is listed as zero in this file 4 3 Distributed Parameter Sensitivity Output The unit number for distributed parameter sensitivity files is defined by input variable IADJXDU This unit number can be defined as either type DATA or DATA BINARY in the MODFLOW Names File All format options us a common header which lists the distributed parameters and observations represented in the file Common Header Format IAFORM PVALUEOUT THRESH NPARDIS PARNAM One entry per line One l
33. he output lists numbers for the distributed parameter identification column row layer observation and sensitivity only for sensitivity values larger than DISCUT See below 4 Option 4 compresses the output into distributed parameter matrices in way readable using JUPITER UTL_READMATRIX The sensitivities of each parameter observation pair are written in compressed format For each pair a line is written with the text COMPRESSEDMATRIX and a name composed of the parameter name and the observation name separated with and underscore _ The next line lists the total number of printed elements followed by the product of NCOL NROW and then the number of layers After this each line contains a pointer to the column row and layer plus the sensitivity value The pointer uses the column row layer cycling as described for option 1 To read the file created using option 4 the sensitivity data for each distributed parameter observation pair would be read using sequential calls to UTL_READMATRIKX The sequence would be one call for each distributed parameter in order parameters listed in the top of the file which is repeated for each observation listed DISCUT DISCUT is a relative cutoff criterion for compressed output options 3 and 4 Only sensitivities with relative sensitivities greater the DISCUT are written Relative sensitivity is in comparison to the largest cell sensitivity of an observation to a parameter PVALUEOUT If PYVALUEOUT i
34. he transient adjoint calculations To find the time zero adjoint we make use of Equation 2 33 Once 4 has been defined using Equation 2 33 with the 2y then A is found by L eae ae A m 1 2 43 _ X gee m l 2 8 Multi step transient for head dependent flow equations If the A B and C matrices of the flow equations are head dependent then the full derivative of Equation 2 25 becomes N 1 m m aJ Af Bian F are Bm a ae am a ban PTET oh oh T N m m m a y AN OA hY dh i gt git OB ml C 2 1 ph dp oh on vap Op 2 44 m The B matrix does not depend on h therefore ah 0 resulting in adjoint update equations of the form a aL AY _fA a z and for each time step before N T k i fr peje 2 46 Primarily head dependent conditions arise from variable saturated thickness of a cell Details for oh makes the adjoint state equation asymmetric To avoid solving unsymmetric equations T a m m m the derivative of A with respect to head are given in Section 2 10 The term h A T oA m m se h A is shifted to the right hand side producing a system of equations that is solved iteratively a m i A A Brym h Ar 2 47 E on T A T A A Bym E h Aa 2 48 oh m Where m indicates the adjoint state solution in the last iteration The effect of transposition in the final term on the right hand side is that th
35. hese are listed in bold type NH Number of head observations MOBS Number of head observations that are multilayer MAXM Maximum number of layers use for any multilayer observation IUHOBSV The file unit number for reporting calculated equivalents to the observations The file type must be listed as DATA HOBDRY The value written to file UHOBSV when a cell needed to calculate the observation is dry at the time of observation TOMULT TOMULT is used to convert the time scale of TOFFSET to the time scale used in the calculations The product TOMULT TOFFSET must have time units specified in the DIS file EVF EVF works as a divisor of the observation weights It provides a convenient mechanism to change the relative weighting of all head observations to weighting of flow observations with a single change to the input OBSNAM Name of the observation OBSNAM is a string of up to 12 characters After the OSNAM entry enough blank characters must be used to fill the first twelve spaces of the line OBSNAM should be unique However the OBSNAM supplied for input item 4 is not used if the IRESP of that entry is negative which indicates multiple observations at the specified location The OBSNAM for the observations at this location are entered using item 7 LAYER the layer index for the location If LAYER is less than zero then multiple layers are used to calculate the observation The indexes for these layers are ent
36. ine for each distributed Parameter NOBS OBSNAM One entry per line One line for each observation Definition of Variables IAFORM An IAFORM value of three indicates that Format Option 3 is being used to report the distributed parameter sensitivities PVALUEOUT If PVALUEOUT is 1 then Parameter_Cell_ Values are printed Otherwise no Parameter_Cell_ Values are printed Threshold The sensitivity threshold below which sensitivity is not printed THRESH is the product of DISCUT times the largest sensitivity for the parameter observation pair NPARDIS Number of distributed parameters NOBS Number of observations represented in this file PARNAM Name of the distributed parameter PARNAM corresponds to PARNAM in the LPF file OBSNAM Name of observation 4 3 1 Format Option 1 Option causes single column of sensitivities to be output with the following cycle pattern The columns of the grid are the inner most cycle then rows layers parameters with observations as the outer cycle Comment lines starting with are written prior to each new distributed parameter if the file is type DATA but not if the file type is DATA BINARY All columns rows and layer are included which means that both active and inactive cells are included Inactive cells are listed with a sensitivity of zero If the PYVALUEOUT variable is set to 1 then the parameter value for each cell is also included These values are defined through the LPF and
37. interested in using the code to calculate sensitivities from previously determined observation sensitivities to Darcy velocity then this section can be safely ignored Transport codes such as MT3DMS Zheng and Wang 1999 and RT3D Clement 1997 can use the flow results from MODFLOW simulations to define the flow velocities for transport simulations Velocity sensitivity refers to the sensitivity of transport related calculation to the flow results from MODFLOW transformed Darcy velocity Since porosity is not a property that is represented in a MODFLOW calculation the sensitivity to fluid velocity can not be used directly How the sensitivity to Darcy velocity is calculated is beyond the scope of this document We simply assume that the sensitivity to Darcy velocity is available in the form of an input file A paper by Samper and Neuman 1986 has been used as the basis of the development of the velocity sensitivity capability The presentation below however uses the conventions adopted in Section 2 6 1 Theory In effect the Darcy velocity sensitivity becomes a pseudo observation that is distributed throughout the model domain and may also be distributed in time Henceforth Darcy velocity will be shortened to velocity Applying the chain rule to the sensitivity of an observation C to a MODFLOW parameter p a representation similar to Equation 2 2 can be formed dC dC dV_ dC dV df dp dV dp dV df dp 6 1 where V is velocity and as in sect
38. ion 2 frepresents the flow equations The velocity has components defined across each cell boundary V jix would be the velocity along a row across the face common to cell j i k and cell j i k Equation 6 1 shows that is a scaling factor dV df that multiplies the sensitivity of velocity to the parameter As before the factor dp is applied aoe ac _ W dC after solving for adjoint states A av df By multiplying df by dV advantage of the fact that the adjoint state equations are linear even for non linear flow problems The equations are linear because the A and B matrices are fixed once the head solution is known we are taking Using this definition of the adjoint states we can solve for dp with a single adjoint state solution Following the conventions used in Section 2 we define the function L as gt dv V V represents every velocity in the model domain calculated throughout the flow solution including the possibility of a steady state velocity Thus each element of the five dimensional matrix V contributes to the source term in the adjoint state equations except where WV is zero The five dimensions are the three spatial coordinates that define a cell location plus a dimension to represent the three directions of flow and a time dimension The function L augmented by the flow equations becomes J L h p 2 c An e er c ah 6 2 m 1 eran eS ae Seite Ste ma p amp p 6 3 v ZJ uye a
39. itivity and Uncertaintlysis Theory volume 1 Chapman amp Hall CRC Boca Raton Cacuci D Ionescu Bujor M and Navon I 2003 Sensitivity and Uncertainty Analysis Applications to Large Scale Systems volume 2 Chapman amp Hall CRC Boca Raton Carrera J and Medina A 1994 An improved form of adjoint state equations for transient problems In Peters A Wittum G Herrling B Meissner U Brebbia C Gray W and Pinder F editors Computational Methods in Water Resources X pages 199 206 Dordrecht Netherlands Kluwer Carrera J Medina A Axness C and Zimmerman T 1997 Formulations and computational issues of the inversion of random fields In Dagan G and Neuman S editors Subsurface Flow and Transport A Stochastic Approach International Hydrology Series chapter II 3 pages 62 79 Cambridge University Press Clement T P RTSD A Modular Computer Code for Simulating Reactive Multi Species Transport in 3 Dimensional Goundwater Aquifers PNNL 11720 Pacific Northwest National Laboratory Richland Washington Doherty J 2004 PEST Model Independent Parameter Estimation Users manual Fifth Edition Watermark Numerical Computing Harbaugh A 2005 MODFLOW 2005 The U S Geological Survey Modular Ground Water Flow Process Techniques and Methods 6 A16 U S Geological Survey Hill M Banta E Harbaugh A and Anderman E 2000 MODFLOW 2000 the U S Geological Survey Modular Ground Water Mo
40. ity of the of the observation to the parameter is determined by summing the product of the adjoint state with the derivative of the ground water flow equations with respect to the parameter for each time step of the flow simulation The theoretical development presents the mathematical basis for the second two steps in the process Sensitivity information is useful as part of a parameter estimation process for reliability analysis in uncertainty analysis and to describe error propagation The program described herein only determines the sensitivities To implement any of the above analyses the program must be used in conjunction with other software Sensitivity information can be determined using other approaches such as the direct sensitivity calculations of MODFLOW 2000 Hill and Others 2000 and the parameter perturbation method implemented in UCODE_2005 Poeter and others 2005 and PEST Doherty 2004 This report describes when the adjoint sensitivity process can be expected to be more efficient than these other methods As a general rule but not always adjoint sensitivities require the equivalent computational effort of a head solution simulation for each observation The other approaches require the computational effort of a head solution for each parameter If a full matrix of sensitivities is needed the adjoint state based sensitivities is expected to be more efficient that the other approaches if there are more parameters than observations
41. le change to the input NQOB The number of observations listed for this cell group NQCL The number of cells in this cell group OBSNAM the observation name The observation name can have up to 12 characters After the OBSNAM entry enough blank characters must be used to fill the first twelve spaces of the line TOFFSET the observation time measured from the beginning of the reference stress period The reference stress period is defined by IREFSP TOFFSET can be specified using time scale units that differ from the simulation time scale TOMULT to convert TOFFSET time units to the simulation time scale The product of TOMULT TOFFSET must have the same units as specified in the DIS file FLOWOBS The value of the observation STATISTIC Observation STATISTIC If STAT FLAG is 3 then STATISTIC is used as the observation weighting factor For values of STAT FLAG less than 3 then STATISTIC interpreted as a measurement error statistic STAT FLAG STAT FLAG is used in the following manner to determine the observation weighting factor 1 If STAT FLAG 0 the weight is EVF WEIGHT STATISTIC is treated as a variance observation measure 1 If STAT FLAG 1 the weight is EVE WEIGHT STATISTIC is treated as an observation standard deviation measure 1 If STAT FLAG 2 the weight is EVF WEIGHT HOBS treated as an observation coefficient of variation measure STATISTIC is If S
42. lgorithm more efficient than Levenburg Marquart for their highly non linear two phase flow problem Rama Rao and others 1995 produced an efficient method of estimating transmissivity using both transmissivity estimates from pump tests and temporal head measurements Their approach used the gradient of the objective function to select optimal pilot point locations The weighted sum of square error objective function contains measurements from possibly many times L h m gt W d F m 2 49 Following the development of a multi step transient L in the augmented objective function similar to equation 2 25 becomes a function of h rather than a single measurement time h J L h p 5 2 Bh a Arn 2 50 m 1 Collecting terms and multiplying the heads at each time step results in m 1 2 oS Fi E ly aJ 2 Bian F ai An B A A dh S AN ala op p amp p 2 51 with adjoint state equations L A A 2 52 and for each time step before N A A Brym Z 2 53 OL Z has non zero terms at the location of each measurement made during the time step OL If the observations represent cell centered head measurements then becomes a series of terms V F d 0 i Si Seka There is one term for each observation at time step m 2 10 Evaluation of matrix derivatives Evaluation of the matrix derivative with respect to parameters is presented by two examples rather tha
43. mple observed seepage below a river specified as a FLWOBS value in the river package is a formal observation MODFLOW 2005 will calculate a simulated value for the seepage Parameters are defined either using an LPF input file or a within another package such as the WEL package Sensitivity then is defined as the ratio of changes of the calculated observations with respect to small changes in the parameter values Computational demands of the adjoint state based sensitivity process are influenced less by the number of parameters than the number of observations To take advantage of this aspect parameters have two different interpretations The first is consistent with the description of parameters in the MODFLOW 2005 manual We refer to this interpretation as umped parameters The value of a lumped parameter is determined by the value supplied in the LPF file or in a specific package input These values may be manipulated using multiplication matrices to specify different properties for different cells in the model If the value of a lumped parameter is changed then the change influences all cells where that parameter is defined The sensitivity to these parameters reflects the fact that the parameters influence many cells We use the term distributed parameter to refer to a different interpretation of these same parameters The key difference is that distributed parameters are defined on a cell by cell basis by using multiplication matrices Typicall
44. multiplication array input files File Listing Format IAFORM PVALUEOUT THRESH NPARDIS PARNAM One entry per line One line for each distributed Parameter NOBS OBSNAM One entry per line One line for each observation NCOL NROW NLAY SENSITIVITY MATRIX FOR OBSERVATION OBSNAM PARAMETER PARNAM These two lines are listed for each parameter observation pair If DATA BINARY is used to define the file type the lines will not be printed Sensitivity Parameter Cell Value Sensitivities for all NCOL NROW NLAY cells are written SENSITIVITY MATRIX FOR OBSERVATION OBSNAM PARAMETER PARNAM Sensitivity Parameter _Cell Value Definition of Variables Sensitivity Sensitivity of the observation to the value for the cell listed in the appropriate multiplication array defined in the LPF file Parameter Cell Value the value of the parameter in the cell This value is the product of the parameter value listed in the LPF file and the value for the cell listed in the appropriate multiplication array defined in the LPF file The parameter cell value is printed only f the PVALUEOUT variable is set to 1 4 3 2 Format Option 2 Option 2 creates a row of sensitivities for each distributed parameter observation pair using the same cycling as for option 1 Formatted output uses 20 1PG16 7 for each line No text lines are included Each new parameter will cause a new line in the output File Listing Format IAFORM PV
45. n being inclusive of all cases For simplicity the presentation is made with respect a single cell of a distributed parameter The sensitivity to lumped parameters is just the sum of sensitivities to each cell where the parameter is defined so the extension from a single distributed parameter to a lumped parameter is trivial 2 10 1 Matrix Derivatives with Respect to Parameters Hydraulic Conductivity Parameter An interesting example is horizontal conductivity For harmonic mean averaging of conductivity the only averaging method supported by the adjoint code the MODFLOW 2005 manual Equation 5 16 defines the horizontal conductance terms as TR TR j l k CR jaz 2DELC ds Jt TR DELR TR pa PDELR a TE TE g 2 54 CCira 2DELR a ES de TC DELC TC j DELC where TR jx AV jx HK 2 55 TC jx Av HK HANI z 2 55 Av is the saturated thickness for a cell HK is the horizontal hydraulic conductivity and HANI is the ratio of conductivity in the column direction to conductivity in the row direction The derivatives of the CRi je and CCit2 terms of Equation 2 15 with respect to HK j k are OCR j 1 2 k TR i l k TR IR j 1 DELR j tt _ 2 Av DELC ue A OHK Lj k A TR DELR TR pi PDELR pa TR DELR TR pa PDELR a OC Girne C24 TC TC DELC i 1 2 j k 2Av HANI DELR i j k Es i j k i l j k i OHK ij k r w TC DELC TC 41 PELC a TC DELC F TC a a DELC 2
46. ns eran sheng A a aueaeems Rave ate ties 31 4 3 Distributed Parameter Sensitivity Output eee cece ceeeceeceeesceceeceeeeneeeeeaeecaecnseeeeeeenstees 31 Common Header Fornat sss eneit neas tiean aae a a sabe eai i ERa a 31 Definition of Variables sas exis ces fusessctvsedy airen chdeintedanawaia nie ceeisariaceaesaetaesneanna 31 4 3 1 Format Optiot li 15 ca crete ssczeedl e a a wade e a a meen indoor oes a 32 File bisting POT dl cas ctensue a uasi nnan n a tea SL a E AN 32 Definition OF Variables vis ccs iss cncscsssesa veareesccinccadcdaecuavest4doasvasassvedeis seeacaed saaveddasuacsatasnagtes 33 432 F rmat QPO 2 hress etniei an A a E E RSA RRA E teen R Eia 33 File Listing Formaten a a e EE E E E a E AOA EE 33 433 ok 916 401 160 08 0 8 810 J Ren Ree an er irr a e Er E R A EE CEREA 34 ill Fil Listing 0 101 ieee ara een en ome roe RTM Nan E a Ree i nT ete Re DER ae nee 34 D finition GE Variables recs each gae encase neu teed eacwlee masa neci sande giaa a a Aai ee 34 4 34 Format ODON oc a hasase aa a Medea a Ava eetiuaaa Avensis lh alae 35 IAA Sae OUT VAL 2 scandy E uc cae E TE A ws esas eee aoa acne Baise 35 Definitonof V ania Diese cassie curses E a eee a a tesa 36 Usage NOES ccc sss chcsacadccas sh pecs eie ila tyakteaveds Sediciedatecnlg a a a a haan nsec aece cates 36 51 WARNING RATE DISCREPANCIES cire oirnne idani en Aa EAE ia CTA oa ia 36 Parameter Sensitivities from Velocity Sensitivity s sssessessessssss
47. nt state calculations is done by calculating a mass balance of net incoming flow and outgoing flow during a simulation This is similar to the familiar flow budget of MODFLOW An imbalance is referred to as a rate discrepancy If any rate discrepancy is larger than 1x10 percent of the influx to the cell then a warning message is printed to the GLOBAL file at the end of the time step that reports the discrepancy calculated during the time step A typical warning message looks like SUM OF POSITIVE RATES 2 89639E 10 SUM OF NEGATIVE RATES 2 89639E 10 PERCENT DISCREPANCY 9 85E 05 At the end of a simulation in which any discrepancy was more than 1x10 a final message listing the largest discrepancy encountered is printed to both the GLOBAL file and the screen standard out A discrepancy of 1x10 percent would rarely indicate a significant problem with the simulation A discrepancy of 1x10 percent may be significant Significance depends on the use of the sensitivity information and on when the discrepancy occurs If the positive and negative rates are small compared to the adjoint states then a large percentage error in the rates is not significant Rate discrepencies may be controlled by solver control variables for the adjoint state calculations Parameter Sensitivities from Velocity Sensitivity This section describes a capability that is a significant variation from other capabilities of the code If the reader is not
48. ociated velocity sensitivity array Presumably the sensitivity to Darcy velocity will have been calculated using a transport code However it may be possible to define sensitivities along specific path lines analytically or with simple numerical calculations The Darcy velocity sensitivity does not need to be determined using the adjoint state approach A paper by Samper and Nueman 1986 provided the theoretical basis for our implementation of the transformation Use of the code for transforming Darcy velocity sensitivities requires significantly more information than other uses of the code Therefore we have relegated description of this aspect of the code to Section 6 This section is essentially a mini manual about using the code for transport problems Section 6 is strictly limited to the transformation of Darcy velocity sensitivities The issue of calculating velocity sensitivities is not addressed Theory Sykes Wilson and Andrews 1985 introduced the use of adjoint calculations the use of adjoint calculations to the groundwater literature Their development of the equations remains an excellent source and is recommended as a complement to the development presented here A more in depth development in a non groundwater context is presented in Cacuci 2003 Cacuci and others 2003 Cacuci derives the adjoint equations for general physical systems His book also covers uncertainty analysis using sensitivity information The development of
49. or some applications Parallelization is another way to speed computation The code supports a simple method of parallelization An optional use of the code is to solve the flow problem without calculating sensitivities This will produce a number of files that are needed in subsequent adjoint state calculations The files resulting from the flow solution can be used for calculations of sensitivities of individual observations on separate processors The current release of the adjoint sensitivity code is a compromise between providing sensitivity calculations for the basic MODFLOW 2005 program and supporting the large number of processes that extend the capabilities of MODFLOW 2005 The intent to expand support to some packages has already been mentioned In addition some work to implementing adjoint sensitivity calculations in the transport code MT3DMS Zheng and Wang 1999 has been done Support for adjoint calculations in MT3DMS has not been developed to a condition where formal release of a program is possible However the linkage to MODFLOW 2005 is complete and may be useful to some researchers The linkage has the capability of transforming sensitivity of observations to Darcy velocity to sensitivity of the observations to MODFLOW parameters Darcy velocity sensitivity must be supplied as a set of arrays which contain the sensitivity of each component of velocity for each cell in the domain Each time step of the simulation should have an ass
50. oximation WEIGHTS Read only if OPERATION 2 If WEIGHTS is non zero then observations are weights in the calculation of the sum of squares gradient Otherwise all observations are assigned a weight of 1 Setting WEIGHTS to a non zero value changes the input requirements for observations In this case observation data must be entered using the instructions indicated in Section 3 5 for head observations and 3 6 for flow observations IUPAROBS Read only if OPERATION 4 IUPAROBS defines the unit number from which to read the active observation This unit number must be defined as a DATA file type in the MODFLOW Names File NCONCMX Maximum number of concentration measurements used to calculate velocity sensitivities in transport code PREFIX A character string without blanks used to define the names of head solution and boundary condition files created using OPERATION 3 or read using OPERATION 4 Only the first 70 characters will be used IADJSU File unit number of lumped parameter sensitivities File name should be supplied in the MODFLOW Name File The file must be type DATA IADJFM Print Format for IADJSU Section 3 4 describes the influence of IADJFM on the output format IADJSCL a value of 1 causes 1 scaled sensitivities to be output to file IADJSU If IADJSCL is set to 1 then the reported sensitivities are multiplied by the parameter value and divided by 100 If IADJSCL is not set to 1 then the sensitivities are not
51. rameter sensitivity output File name should be supplied in the MODFLOW Name File Type of file DATA or DATA BINARY controls whether the file is formatted or binary The type of binary file can be controlled using the openspec inc file prior to compilation of the program IAFORM controls the format of the distributed parameter sensitivity output Briefly the value must be 1 4 See Section 4 3 Distributed Sensitivity Output for more detail 1 Option 1 causes single column of sensitivities to be output with the following cycle pattern The columns of the grid are the inner most cycle then rows layers parameters with observations as the outer cycle Comment lines starting with are written prior to each new distributed parameter All columns rows and layer are included which means that both active and inactive cells are included Inactive cells are listed with a sensitivity of zero 2 Option 2 creates a row of sensitivities for each distributed parameter observation pair using the same cycling as for option 1 Formatted output file type DATA uses 20 1PG16 7 for each line No text lines are included Each new parameter will cause a new line in the output 3 Option 3 compresses the output to eliminate small sensitivity values After a header that lists each distributed parameter and each observation in the orders in which they are cycled each line of the output lists numbers for the distributed parameter identification column row la
52. rformance measure composed of matching a single observation made at time step N The performance measure is augmented with each of the time step equations J L h p 2 2 y Brp C Ann 2 24 m 1 Collecting terms of similar h together N I J L h p ar Y n sS e gt Chae a a a la a Bn 2 25 dJ 2 Ban a B a A dh 2 J Jaa l oh 2 26 T L N m m m 2 oS a CH ly p m l Op Op Op To eliminate the dependency of J on A the are constrained such that OL A a 3 and for each time step before N A A Brym 2 28 These equations can be solved by MODFLOW using the A and B matrices of the head solution and with boundary conditions set to 0 except for possible matrix P terms The equation for the adjoint state after time step N A is solved first and the adjoint steps for each other time step are solved in order of decreasing time steps The adjoint equations are solved backwards in time For specified initial conditions the resulting sensitivity is given by dJ L SOP OMN OB 3 C A sc 4 4 jl gare h 2 29 dp Op ma Op p p The calculation of the adjoint state does not depend on what parameter is involved Therefore once the head and adjoint state solutions are calculated the sensitivity of J to each parameter can be calculated relatively quickly 2 4 Observations between time steps Some observations may occur at times that do not coinci
53. s 1 then the parameter value of a cell is added after the cell sensitivity is listed in the distributed parameter sensitivity output This option is not allowed for IAFORM formats 2 or 4 The value listed is the product of the value listed in the LPF file and the multiplication matrix whereas the sensitivity is to the value in the sensitivity matrix If the value in the LPF file is 1 0 then the sensitivity and parameter value are consistent IPDISP 1 IPDISP 2 is an integer list all on one line corresponding to the PARNAM names read next 2 would correspond to the second PARNAM entry PARNAM read NPE times PARNAM must be identical to a parameter name supplied in the LPF input file PARNAM identifies which parameters are used to calculate lumped parameter sensitivities IPDISP identifies which lumped parameters are also distributed parameters LN If LN is 1 the sensitivities are calculated with respect to the natural log of the parameter Use YHK and YVK for natural log of conductivity For YHK and YVK parameters only an LN value of 0 in the ADJ file indicates dimensional values in the multiplier arrays and an LN value of 1 indicates log parameter values in the multiplier arrays 6 3 Input Instructions for velocity sensitivity data File Listing Format for File CFNAME NUMLOC NUMMEAS COLUMN ROW LAYER IEND ISTART NUMLOC sets of data IPER ISTEP TOFF IEND ISTART 1 lines per location COLUMN ROW LAYER IEND ISTART
54. s consistent with the default order when an entire array is written Note the all indices are included even inactive cells The openspec inc file can be used to make the UNFORMATTED style compatible with the code See Chapter 9 of the MODFLOW 2005 manual and the file openspec inc for more details The MODFLOW 2000 guide to linking MT3DMS to MODFLOW 2000 Zheng and others 2001 may also provide insight into adjusting the opensec inc file and writing the velocity sensitivity file Direct access files are written and read by record number A unique record number for a velocity sensitivity array is given by Record number IOBS 1 NTIMES KSTP 6 8 where IOBS is the index of the observations in the order listed in file CFNAME NTIMES is the total number of time steps in the simulation NTIMES is the sum of the time steps for each period and KSTP is the current time step of the adjoint state calculation This time step is the time step of the head solution Both NTIMES and KSTP include the steady state period if applicable References Cited Banta E R Poeter E P Doherty J E and Hill M C 2006 JUPITER Joint Parameter IdenTification and Evaluation of Reliability An Application Programming Interface API for model analysis U S Geological Survey Techniques and Methods 6 E1 U S Geological Survey Bryson A and Ho Y C 1975 Applied Optimal Control Halsted Press Revised printing edition Cacuci D 2003 Sens
55. special use of the adjoint sensitivity code If the ADJ file specification is missing then the program will behave very similarly to Version 1 1 of MODFLOW 2005 Released May 18 2006 Four other files for the adjoint code can be defined in the names file The optional observation map Section 4 1 assigned unit number IUVOBSMAP in the ADJ file must be defined as type DATA The optional solver control for the solution to the adjoint states must also be type DATA The same solver used for the head solution must be used for adjoint states solution but use of this file provides separate control of the solver specifically for the adjoint state solution The unit number for the solver control file is specified using IUSOLV in the ADJ file Sensitivities to lumped and distributed parameters will be written to unit numbers IADJSU and IADJXDU respectively File IADJSU must be defined as type DATA File IADJXDU can be defined either as type DATA or DATA BINARY 3 2 Type ADJ input file These input instructions assume that a calculation of parameter sensitivities from velocity sensitivities is not being requested See Section 6 if such a calculation is desired Input for the adjoint sensitivity calculations is read from a file that is specified with ADJ as the file type in the MODFLOW Name File indicate optional input data 1 Text Text is optional and can include as many text lines as desired as long as they begin with 2 OPERATION MODE or WE
56. the distributed parameter sensitivity output This option is not allowed for IAFORM formats 2 or 4 The value listed is the product of the value listed in the LPF file and the multiplication matrix whereas the sensitivity is to the value in the sensitivity matrix If the value in the LPF file is 1 0 then the sensitivity and parameter value are consistent IPDISP 1 IPDISP 2 is an integer list all on one line if desired corresponding to the PARNAM names read next 2 would correspond to the second PARNAM entry PARNAM read NPE times PARNAM must be identical to a parameter name supplied in the LPF input file PARNAM identifies which parameters are used to calculate lumped parameter sensitivities IPDISP identifies which lumped parameters are also distributed parameters LN If LN is 1 the sensitivities are calculated with respect to the natural log of the parameter Use YHK and YVK for natural log of conductivity For YHK and YVK parameters only an LN value of 0 in the ADJ file indicates dimensional values in the multiplier arrays and an LN value of 1 indicates log parameter values in the multiplier arrays 3 3 Format control of the lumped parameter sensitivity output IADJFM controls the length of each line and format of the reported sensitivities using the following corresponding formats Output longer than 80 characters Output less than 80 characters IADJFM FORMAT JADJFM FORMAT 1 11G10 3 13 10F6 0
57. the equations presented here follows Bryson and Ho 1976 We start with an abstraction of MODFLOW 2005 as a non linear operator F that calculates vector of observations d as using a vector of parameters p property values that are not formal parameters are embedded in F d F p 2 1 The first step is to expand the operator in a Taylor series about nominal parameter values po and then truncate it at the first term to get d F p Gdp 2 2 G is the sensitivity matrix composed of elements dp where indices and n identify particular elements of the observation and parameter vectors The parameter p could be either a lumped parameter or a single cell of a distributed parameter Using the chain rule of calculus we now decompose G into the product of two terms _ ad df df dp The variable f represents the discrete flow equations solved by MODFLOW 2005 See equations 2 1 2 2 and 2 24 of the MODFLOW 2005 manual Harbaugh 2005 The meaning of fis made dd more explicit in Equation 2 4 below We will show later that at are the adjoint states and 2 3 that they can be calculated using the computational equivalent of a simulation of the original flow problem dp is the derivative of the flow equations with respect to parameter n The calculation of dp is relatively fast compared to calculation of the adjoint states Calculation of df dp not intuitively obvious Our description begins with a related problem
58. ties _ scl The unscaled option does not apply any scaling to the sensitivities and thus are not the same as the sensitivities reported in the MODFLOW 2000 _sc files There are four choices for reporting distributed parameter sensitivities The formats are described in Section 4 3 Support programs to integrate the adjoint sensitivity code with the PEST program Doherty 2006 are included with the latest PEST distributions Format 4 is intended for use with the JUPITER API Banta and others 2006 If the flow problem can be simulated using constant time steps then a variation in the method of calculating the adjoint states can be used to significantly reduce the computational effort The underlying method is described in Carrera and Medina 1994 Carrera and others 1997 and in Section 2 7 With constant time steps and measurements that correspond to these time steps only a single adjoint calculation is needed for all the observations made at a single location The adjoint states for each observation can be inferred from this calculation It should be evident that the location based adjoint state calculations can significantly reduce the computational effort to determine a sensitivity matrix Constant time steps are not an absolute requirement Temporally varying time steps will introduce an error into this method However the computational advantage of the location based method may make an approximate calculation of the sensitivity matrix useful f
59. to represent the fraction contribution of each layer to the observation The factors P are obtained from the HOB file describing the observations In the case of multi cell head observations the adjoint state source equation source term becomes OL ee ant 22 DRIP 2 33 J i Flow observations may represent the flow from many cells Flow observations can be defined by where n is the index of the cells that are used to define the observation and fn are the appropriate flows for each cell The f factors are defined in the input file describing the observations The f factors are not proportionality factors which represent fractional contributions of the cells to the observation The fractional contribution of the cells is Aa 2 35 n The adjoint state equation source term as can be written as OL a ae ap Doo a fast hy 2 36 w 7 i 2 6 Multi step transient after a steady state calculation If a steady state stress period proceeds a transient stress period then h may depend on the parameters In this case we add an additional adjoint state to remove the dependence in Equation 2 21 on the steady state head solution The constraint is C A n 0 2 37 where A does not include the storage term on the right hand side of Equation 2 15 This creates an additional equation to solve for the time zero adjoint state AA By 2 38 And results in i 2 yi m 0 0 af Say OB jm OC _ 0A ym Y oC _ OA o 2 39
60. wya a laes S E am Be Jar ja L The terms ap are evaluated for each component of velocity as follows OL 4 7 dC On dC cen 4 dC cae Oh dv Oh AV Oh dV oh S where xe CR SG Sot ik 5G f e 6 5 ov P i CO OG f ER Hey FF 11 8 6 6 at CV SG Sa Hk 0G e SL e 6 7 OL Similarly if the parameter is a lumped hydraulic conductivity parameter the direct term ap contributes to the sensitivity for each non zero term in 7 gt Sensitivity of a cell for a distributed dV hydraulic conductivity parameter will have a direct term only when the cell is used to calculate one of the conductance terms That is only for the cell or neighboring cells 6 2 ADJ input file for velocity sensitivity data Specification of an input file of type COBS in the MODFLOW Names File signals the calculation of parameter sensitivities from velocity sensitivity data Input for the adjoint sensitivity calculations is read from a file that is specified with ADJ as the file type in the MODFLOW Name File The use of bracket indicates optional input data File Listing Format 1 Text Text is optional and can include as many text lines as desired as long as they begin with 2 OPERATION MODE or WEIGHTS or IUPAROBS 3 NCONCMX 4 PREFIX PREFIX is read only if OPERATION has a value of 3 or 4 5 IADJSU IADJFM IADJSCL IADJPC 6 IUOBSMAP 7 CFNAME 8 VSNAM 9 IUSOLV 10 NPE NPA
61. xample of constrained optimization To reinforce the concept of how the adjoint state can be used to simplify a problem we present an example introduced in Bryson and Ho 1976 Define L as I x m Subject to I x m x gm c 0 2 11 As shown in Figure 2 1 Bel gt Increasing L iy x gm c Figure 2 1 Minimization example Define I x m a i 5 m ACs emo 2 12 The resulting equations to solve are f 0 gt x gm c 0 OH Ox OH m 0 gt Ag 0 oa go The solution to these three equations with three unknowns is 0 gt 2 0 a 2 13 c A at gh b gc a g b 5 2 14 ac pel E 2a gb o Note that CA and oc q lo _ alg a Of 2 15 2 2 A one time step transient for a constant property model The adjoint sensitivity calculation for a one step transient solution is presented in this section The MODFLOW 2005 manual presents a development of the set of equations that are used to calculate the head distribution in MODFLOW Equation 2 25 of the MODFLOW 2005 manual is 1 1 1 1 1 1 CR ition a hi ia CR jzk n h x CC hiz g hy i i j lk 1 1 1 1 CCy 1125 4 ee a h sy CV pean ae AN hy x 2 16 hy in hy ix F Q jx SS jk ce ere ea CV jar i hi jx P h i ij k l i j k ti j k Where h is head in the cell before the time step and h are the head values after the time step The CR matrices are hydraulic conductance between
62. y a value of one would be supplied as the parameter value in the LPF file so that the parameter value of a cell is the same as the value in the multiplication matrix For distributed parameters sensitivities are calculated with respect the multiplication matrix value for each cell in the model not the value listed in the parameter definition statement Thus a distributed MODFLOW parameter actually represents a set of parameters one for each cell where the parameter is defined Only parameters that can be manipulated using multiplication arrays can also be distributed parameters The two parameter interpretations can be used to define identical flow simulations The adjoint state based sensitivity process is in large part a modification of the previous MODFLOW codes MODFLOWP Hill 1992 and MODFLOW 2000 Hill and Others 2000 In these codes the sensitivity of observations to parameters could be used to estimate parameters The current code is does not have a parameter estimation capability The ADJ process only supplies sensitivities There are many uses of the sensitivity information Parameter estimation programs UCODE_2005 Poeter and others 2005 and PEST Doherty 2000 require sensitivity information to define parameter update directions The MODFLOWP user s guide Hill 1992 describes use of the objective function gradient in conjugate gradient optimization Sensitivities can also be used to propagate uncertainty in parameter values to un
63. y inspection of Equation 2 55 we can see that the derivation of the conductance terms with respect to head are very similar to the conductance terms with respect to hydraulic conductivity if the saturated thickness of a cell is dependent on the hydraulic head in the cell If OAv the head in a cell is between the top and bottom elevation of the cell then Oh 1 For this circumstance the derivatives of CR and CC with respect to head are OCR jaiak 2HK DELC TR jst p TR TR ja DELR i j k i 2 Oh TR DELR TR a DELR TR DELR TR p DELR a OCC si TC TC TC a DELC i l1 2 j k 2HK HANI DELR i l j k i j k i l j k Oh k l TC DELC a TCi ja DELC TC DELC TC jaDELC a 2 58 with similar terms for the derivatives with respect to hj 7 k and hj j 1 k Input Instructions 3 1 ADJ file type in MODFLOW Names File The input file for MODFLOW that assigns file names to FORTRAN unit numbers is referred to in this report as the MODFLOW Names File This file is used to identify which packages are to be used in the MODFLOW simulation Two new file types have been added to the MODFLOW Names File These are type ADJ which signals the use of the adjoint based sensitivity package and file type COBS which signals that the adjoint package will be used to calculate parameter sensitivities from Darcy velocity sensitivities that presumably have been calculated using a transport code Section 6 describes this
64. yer observation and sensitivity only for sensitivity values larger than DISCUT See below 4 Option 4 compresses the output into distributed parameter matrices in way readable using JUPITER UTL_READMATRIX The sensitivities of each parameter observation pair are written in compressed format For each pair a line is written with the text COMPRESSEDMATRIX and a name composed of the parameter name and the observation name separated with an underscore _ The next line lists the total number of printed elements followed by the product of NCOL NROW and then the number of layers After this each line contains a pointer to the column row and layer plus the sensitivity value The pointer uses the column row layer cycling as described for option 1 To read the file created using option 4 the sensitivity data for each distributed parameter observation pair would be read using sequential calls to UTL_READMATRIKX The sequence would be one call for each distributed parameter in order parameters listed in the top of the file which is repeated for each observation listed DISCUT DISCUT is a relative cutoff criterion for compressed output options 3 and 4 Only sensitivities with relative sensitivities greater the DISCUT are written Relative sensitivity is in comparison to the largest cell sensitivity of an observation to a parameter PVALUEOUT If PYVALUEOUT is 1 then the parameter value of a cell is added after the cell sensitivity is listed in
65. yeryview of Capabilities siess is inina iieii iteresas seedee it tasetten edera taa niatan 3 THEO Yis r E eau Ei E E N R oh A E E R te cae ce 5 2 1 Adjoint states for constrained optimization 0 0 ccc ceeecceseceeeeeesceceeeceeceeeeeeseecsaeenseeeeeeenaes 6 2 2 A one time step transient for a constant property model sssssessessssseessessreseesseesesresseeseese 8 2 3 Mu lti step transiente erni Gin ta ach R E A E E E Sola an nad 10 2 4 Observations between time steps sc nic s0c8 sad cdecacas seaune cosh oveatisss Uadussicoadnnsessuvac snes cinces saveaunsepvadeass 11 230 M lti cell Observations issin istriani aas ai e aa aeai aas iniaa 12 2 6 Multi step transient after a steady state calculation s sssseseesesseesesseseesesseseesessessesseseesesse 13 2 7 Location based adjoint state calculations ccccccceseceseceeseeeseeceseceeeeeeeeeeseecaecneeeseeenseees 14 2 8 Multi step transient for head dependent flow equations ccceeeceeceeseeeseeceeceteeeeeeeeseees 14 2 9 Gradient of a weighted sum of squares objective fUNCTION cc eeceesceeseceteceteeeeeeeeseees 16 2 10 Evaluation of matrix derivatives assesses doasazaddeaaneaubetuntessievisasbaadeeasszta backs eovnastaoaidvtacdeaaeaied 17 2 10 1 Matrix Derivatives with Respect to Parameters cccccccceccessceesseceteceeeeeeeeenseeeaeenes 17 Hydraulic Conductivity Parameter sc ciscsccnsecssccdstaudedesnisd eseedasaesansvensoancesuiccvsndy ecandabaessnaroes

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