BANG, A SYSTEM FOR SURVEYING THE
Contents
1. DewPDeg the decrease in temperature at weather point required for water vapour to condensate to water Dir direction of wind gradient at weather point Speed speed of the wind gradient at weather point m s Newton Raphson data maxiter maximal number of iterations resmin largest residual for convergence smallest residual for divergence dz integration step in height for interpolation of weather points inc step length for numerical derivation sf scale factor Output data KXtrue Y true Z true T true Position and time point of bang s Input file for Matlab with bang and microphone positions Graphical view of microphone and bang positions with Matlab graphics window Bachelor thesis Benjamin Donald Oakes 2011 07 05 Appendix 2 GUI 2 Tab for inputting vicinam dum When running the program the GUI will appear 2 Run Press to run the calculation with the currently selected input data A browse window will pop up Choose the path where you want to save the input file Input data tab Here the input data is applied Load default data Resets to the default input data so BANG can be tested Read input file For selecting an input data file A browse window pops up and an input data file can be selected from any path Under the input data tab there are several sub tabs Microphone data Select2. a Input data Output data Load default data Input file name C Users Benjamin Desktop BANG_FINAL Kopia bin ReleaseNala txt Microphone Bang Weather Newton Raphson data For first round For second round Number of iterations Number of iterations 500 1000 Minimum residual Minimum residual 1E 18 1E 18 Maximum residual Maximum residual 10000 10000 Increment Increment 0 0001 0 0001 Scale factor step size Scale factor step size 0 8 08 Integration step in altitude 1 5 Newton Raphson data tab after having read the input file Reada weather file Press Read weather file under the weather tab A weather file is selected in the same way as an input data file See the opened weather file 6 n 7 WeatherNewTest txt Anteckningar CK Arkiv Redigera Format Visa Hj lp Date 20070616 Time GMT 17 25 Sond B3931336 Station Name Bofors Station position Lat 61 59 Lon 13 63 Alt 100 Time Pressure Temp RH DewP Dir Speed s hPa degC 96 degc deg m s 0 0 937 5 10 35 0 00 0 15 50 931 4 10 31 0 55 90 20 100 926 0 15 32 0 25 90 25 6 Input weather file Because of limited time for this work only temperature and wind speed and direction are taken into account Once the weather file is read new data will appear in the Weather data tab 7 Bachelor thesis Benjamin Donald Oakes 201
3. number of bangs nmic number of microphones xmic n ymic n zmic n coordinates for the position of microphone m m tmic n k the scaled registration time of bang k at microphone n m x0 y0 zO start guess of the position of one bang m tO start guess of the first registration time m xb yb zb tb currently estimated position and first registration time for a bang m tbtrue true time of first microphone registration s inc step length for numerical differentiation 0 0001 good enough maxiter maximal number of iterations for Newton Raphson dz integration step in height from microphone to bang m ds integration step along true line from microphone to bang m zb zmic n DOREM int nstep Number of integration steps aspeed reference speed of sound 340 3 m s 55 1 local speed of sound at step i m s q cost function value m residual the smallest value of q found for a tried combination of times m Note all the listed times here are scaled to distances with the speed of sound initially aspeed to simplify calculations Markings figures expressions equations 1 references Bachelor thesis Benjamin Donald Oakes 2011 07 05 BANG Theory description method for the new BANG For re writing the old FORTRAN 77 code to an object oriented C application with optimal solutions an understanding of the structure
4. Table of contents i ig i Foreword Last tm dd ae oe oles dens soi he mn emer i Table or eontentssse atout e otio lou utto noti ido Deb Oba aus tatus iii 15 T 1 Objectives ofthis WO L PUE 2 Requirem enisu ME T washay aq RO RM 2 D finitio NS M 3 Theory description method for the new BANG rennen 4 OVBIVIBWE Loo RR apy eru EU I dpa QUU UR Qoid ut 4 How BANG WOLKE cans beeen rte us pae tis ED Eon Ee t dE PAIGE USO e pP Dag Nd duas 5 Surveying a single bang Newton Loop nenne 6 Calculating the speed of sound with weather distortion 8 Discerning multiple bangs and sonic boom from bang 11 Geometrical analysis of the problem L n trennen 14 Surveying in one dimension eren ennt 14 Surveying in two dimensions nana nsn 14 A new version of BANG using a graphical solution a 15 REilzu puce D cr MM 16 The Matlab pf gralfi A 17 20 DISCUSS yu c ce 21 How well does BANG
5. V s 2 Hy Ve Make a conversion matrix built up of the base vectors for the new system S Va Vor V F W V Vs V22 V23 V33 Also calculate the inverse of F for the next step Calculate the coordinates of the microphones in the new system We know that F S Xold Yold Zold The coordinates of the new system will be 5 Xnew Ynew Znew F 1 Xold Yold Zold When the position of the bang has been found in the new system convert back to the original system to get the true point Xold Y old 2014 F S F Xnew Ynew Znew 19 Bachelor thesis Benjamin Donald Oakes 2011 07 05 BANG Procedure Here is a short description of what was involved in the various steps of this work A short list of the executed steps in order Analysis the old BANG Fortran code Structure diagrams for the new BANG make skeleton and class diagram Make prototype code of the calculation modules and make sure the code can be compiled Make GUI so input data easily can be given for tests and output data read Test run the new BANG and compare results with old results When new results are identical to old make fine adjustments to functionality and GUI Integrate Matlab for graphical presentation of result Document code and write a user manual 96 OQ tA AWN PE The report writing began at step 6 and continued till the end of the work Thoughts of other methods of improvement where thou
6. LP Jj a j HS C MA E s 7 Shows the position of the bang b red point surrounded by S in relation to microphones blue circles surrounded by S1 2 and 5 17 Bachelor thesis Benjamin Donald Oakes 2011 07 05 BANG This program is merely a prototype of an alternative solution and is not complete Though an acceptably functional version would not require an exceeding amount of effort this solution was not the focus of this work It does however illustrate the problematic well and is thought to be used for quickly getting a rough result when only one bang Another use of this prototype is finding more accurate start values from measured microphone data before running the original optimization program The code of this program is much less and simpler than that of the original optimization program and a solution is very quickly reached This makes the program an attractive alternative for a quick solution and easier to troubleshoot Probably the biggest plus with this solution is that convergence is almost a certainty since the spheres are locked a solution is always held with four microphones whatever the time differences should be The Newton Raphson method can however spin off to infinity should the start values not be sufficiently accurate Newton Raphson may even co
7. t inc2 q q xb inc yb zb inc tb q xb inc zb tb q xb yb zb inc tb q xb zb tb z x 529 q xb yb inc zb inc tb q xb yb inc zb tb q xb yb zb inc tb q xb yb zb tb z y inc2 q q xb yb zb inc tb q xb yb zb tb q xb zb tb q xb yb zb inc tb z inc q q xb yb zb inc tb inc q xb yb zb inc tb q xb yb zb tb inc q xb yb zb tb z t inc q P q xb inc yb zb tb inc q xb inc yb zb tb q xb yb zb tb inc q xb yb zb tb t x inc Bachelor thesis Benjamin Donald Oakes 2011 07 05 Appendix 1 q _ q xb yb inc zb tb inc q xb yb inc zb tb q xb yb zb tb inc q xb yb zb tb t y 2 2q _ q xb yb zb inc tb inc q xb yb zb inc tb q xb yb zb tb inc q xb yb zb tb t z inc 67q q xb yb zb tb inc q xb yb zb tb q xb yb zb tb q xb yb zb tb inc t2 inc inc Bachelor thesis Benjamin Donald Oakes 2011 07 05 Appendix 2 Appendix 2 User manual Installation A list of all necessary complied and debug files for running BANG is shown in figure 1 To run the program double click on BANG NEW exe A doxy file is also provided for detailed documentation of the code This is not showing here but is inc luded in the program catalogue 1 and can be opened with the
8. 07 05 11 07 01 11 07 01 11 07 01 11 07 01 11 07 01 11 07 01 11 07 01 11 07 01 11 07 01 11 07 01 27 Bachelor thesis Benjamin Donald Oakes 2011 07 05 Appendix 1 Appendix Appendix 1 Derivative calculations The gradient g _ q b inc yb zb tb q xb yb zb tb inc q q xb yb inc zb tb q xb yb zb tb y inc q xb yb zb inc tb q xb yb zb tb z inc q _ q xb yb zb tb inc q xb yb zb tb t inc The hessian H q _ q xb inc yb zb tb q xb yb zb tb q xb yb zb tb q xb inc zb tb 829 q xb inc yb inc zb tb q xb inc yb zb tb q xb yb inc zb tb q xb yb zb tb x y inc 529 q xb inc yb zb inc tb q xb inc yb zb tb q xb yb zb inc tb q xb yb zb tb x z inc 529 q xb inc yb zb tb inc q xb inc yb 2 tb q xb yb zb tb inc q xb yb zb tb x t inc q _ q xb inc yb inc zb tb q xb inc yb zb tb q xb yb inc zb tb q xb yb zb tb inc2 q 2 q xb yb inc zb tb q xb yb zb tb q xb yb zb tb q xb yb inc zb tb by inc q 7 q xb yb inc zb inc tb q xb yb inc zb tb q xb yb zb inc tb q xb yb zb tb 6y z inc2 q _ q xb yb inc zb tb inc q xb yb inc zb tb q xb yb zb tb inc q xb yb zb tb 6y
9. The total calculation time will not necessarily be faster because of the calculation of the second derivatives and handling of the hessian matrix In the case of larger calculations with many combinations see page 11 and weather points see the next section the convergence may well be somewhat faster For an even faster convergence the Newton direction can be multiplied by a scale factor found with a backtracking line search method see 1 This is not yet implemented as it is not absolutely necessary for good results more a finesse Perhaps the greatest downside of the Newton Raphson is its poor global convergence properties The prerequisite is that the start values are relatively close to the true values Convergence however is still not a guarantee even if the start values are thought of as being qualified though the chances of convergence will be high with accurate start values Calculating the speed of sound with weather distortion As we have noted earlier the times in the cost function are scaled to distances with the speed of sound The speed of sound is not constant but varies with the air properties and weather distortion The following weather data is measured with a weather balloon at certain heights going up over time Pressure air pressure Temp temperature RH relative humidity DewPTemp Dewpoint temperature Dir wind direction Speed wind speed Each sample is saved in a text file which B
10. WOIKY pu ied 21 CORVENOEIGE n atta 21 alae 21 The effect of weather data i e qut cpm 21 The improvements made on BANG 22 GENS Fonran m s u a casa hen le au a TE 22 Graphical view with Matlab Ree uo Coetus tot teo ined e 22 All calculations for multiple bangs are executed automatically 22 Possibilities of further improvement n s 23 SUUS VS Clas OO ID CHF aceite cedet oy cee dtd eden dos tdt teda deret toda tdeo 23 Improvements in weather calculations eese 23 Frequency analysis for discerning sonic boom 23 iii Bachelor thesis Benjamin Donald Oakes 2011 07 05 BANG Is the dummy variable time known necessary 23 Discerning more than two bangs nsn 24 CHOICE OF microphoneS un ast E 26 la dc ig vee u uuruuu 27 fioe qme TP TETTE 1 Appendix 1 Derivative calculations I eene 1 B s xelr te ioa M 1 The SS SAN d oscar aea tcp iN RM a anes 1 Appendix 2 Usermini al R
11. an advanced method of optimisation with Newton Raphson at the bottom Calculations in BANG The engine or motor in BANG consists essentially of two parts 1 Discern the bangs if nbang gt 1 otherwise step 2 2 Accurate positioning of single bang s severalif nbang 1 Program structure Calculation 1 Discern the bangs if nbang gt 1 else if nbang 1 make accurate calculation For int i0 ixncomb i SetStartV alues x0 y0 z0 t0 NewtonLoop timecombination i j Calculation 2 if nbang gt 1 More accurate calculation for each bang separately If nbang gt 1 For int 10 ixnbang i SetStartValues x y z t of CorrectTimeComb i NewtonLoop timesforbang i Bachelor thesis Benjamin Donald Oakes 2011 07 05 BANG Surveying a single bang Newton Loop This calculation is the fundamental calculation n B ANG It is used for determining the near exact position and time point of one bang The cost function For positioning a point in R one must have n 1 microphones more theory on page 14 In the real 3D world at least four microphones are required to position one or more bangs For simplicity s sake a 2D system figure 1 with a minimum number of microphones is illustrated This simple example is sufficient to understand the theory of BANG The engine of BANG minimises the so called cost function q 1 The value of q also known as the residual is an indication of how close
12. second is executed a new window pops up 9 r Processing data Calculating bangs seperately plase be patient 9 Progress bar window while calculating each bang separately for more accurate results When window 9 is up a little patience is required It may take a while before the progress bar moves at all It will progress after each bang has been calculated When all calculations are complete a final window emerges 10 n Processing data 10 Final progress window If Cancel is pressed during the calculations the output data files currently being produced will be deleted A finesse to avoid useless text files taking up disk space and making a mess The content of the output data files are not shown here because they are long and take up much space This is however not necessary since the result can be viewed under the Output data tab in the GUI form window 11 The position and time points of the two bangs are shown in the datagridview 2011 07 05 k 20 BANG Deluxe Input data Output data Bang pos and time TrueTime X true Y true Z true Ss TENE 149 9683542074 249 8703326269 99 76961327608 9 999846009835 124 9941280954 299 9905986977 100 1004200474 11 Result window showing position and time point of bang s 2011 07 05 Benjamin Donald Oakes Bachelor thesis Ap
13. the number of microphones the microphones positions and the registration times for each microphone 2 Bachelor thesis Appendix 2 Benjamin Donald Oakes Bang data 2011 07 05 o s de Number of bangs 1 Calculate guess position using average of x y mic positions and 500m high Set start time as smallest mic time Guessed position and time of first explosion 500 YO 500 20 1000 3 Tab for inputting bang data Select the number of bangs start values and if the first registration time is known or unknown default 3 Weather data Read weather file Select a weather file from any path A browse window is opened The weather data is observed froma weather balloon always given as a function of height Included measurements from each weather point t me Z Pressure Temp RH DewPTemp Dir and Speed Weather data is edited in window 4 Bachelor thesis Benjamin Donald Oakes 2011 07 05 Appendix 2 4 Tab for inputting weather data Newon Raphson Set maxiter resmin resmax inc and sf Set these parameters for the second calculation when nbang gt 1 If nbang gt 1 set also dz for the second calculations 5 reati rane 5 Tab for inputting Newton Raphson data Bachelor thesis Ben
14. the smallest registration time 714 to ti 4 The geometrical surveying 5 system in simplest form on line to mi bi m illustrate the problem Surveying in two dimensions 5 The 2D interpretation of the surveying system The position and time point of b correspond to the radius and centre point of the circle C 14 Bachelor thesis Benjamin Donald Oakes 2011 07 05 BANG The same problem can be envisioned on a two dimensional plane R2 5 The circle C represents the propagation of the sound wave from b at the time of the first microphone registration The radius r of C is directly proportional to the time of the first registration tbtrue The radii rj are directly proportional to tmic 2 and tmic 3 the time difference between the time of first registration and the time of registration at m and respectively The sought parameters are x y Known and fixed parameters x4 y4 X2 X 3 1 2 Unknown and sought parameters x y r Three parameters are required to define a circle the two coordinates for the centre point and the radius alt three of the rand points Figure 5 can be described with the equation system 14 consisting of three equations and three unknowns 1 y 2 r r 2 x x2 y y2 r r gt 2 x y r 14 An explicit solution to the system 14 exists but the path to the solution is messy Solving this
15. thesis Benjamin Donald Oakes 2011 07 05 BANG Objectives of this work The primary objective of this work was to make the software of B ANG more user friendly and document the code well Requirements Primary requirements 1 Rewrite the software program in a modern high level language preferably Cit 2 The program should be compatible with Windows XP Vista and Windows 7 Secondary requirements 1 Input data should be easy to feed in both manually and by reading text files 2 The result should be simple to interpret and be written to text files Stretch goals 1 Develop the functionality of the program A few points Can more than two bangs be discriminated at once 2 Develop weather data calculations so they are more accurate 2 Can separation be done with frequency analysis Smaller study of the hardware involved Do microphones need to be changed 4 Are there other possible algorithms better than the currently used method Analyse di The theory used in BANG is described in the report Documentation is important for continued development The code is documented with Doxygen an html documentation Bachelor thesis Benjamin Donald Oakes 2011 07 05 BANG Definitions Here are some definitions of terms frequently used in this report Note if the definition has a unit the unit is stated after the explaining text bang sound from the separation of a cargo grenade can also refer to a loud noise or sonic boom nbang
16. 1 07 05 Appendix 3 ui BANG Deluxe Input filename _C Users Benjamin Desktop BANG_FINAL Kopia bin Release Vala bt Weather Integration step in altitude Weather file name C Users Benjamin Downloads WeatherNew Test bt 7 Weather data tab after having read the weather file The program is now ready to be run once input data file and weather file have been read Press Run to execute BANG A browse window will come up asking where you want to save the input data Select a file or make a new one and click save The calculations will now begin Two output data files are produced the first part of their name being the name of the saved input data file The first file contains data from the first calculation regarding discerning the bangs This file will be named for instance somefilename_output0 txt with outputO txt on the end For each new file created the number at the end of the file name will increment A second file with the separated bang data will be created named somefilename output separatedO txt This contains the more accurate calculations for each bang Once Run is pressed and the input data file saved a progress window with Processing data will pop up 8 The first calculation discerns the bangs 8 Progress bar window while separating bangs Bachelor thesis Appendix 3 Benjamin Donald Oakes When the first calculation is complete and the
17. ANG reads Because of limited time for this work BANG only takes wind and temperature into consideration It is however these two factors that typically have the most effect on the speed of sound and the two that vary the most more in the next section Linear interpolation As the weather data is specified at a number of discrete heights linear interpolation is used for determining values at heights between the measured points The number of new points calculated is determined by the step length dz As we have discussed earlier the measured and calculated times for each microphone are compared when calculating the residual q The difference in height from the actual microphone to the bang is divided by the step length dz to get the number of points to be calculated The points of height are evenly spread from the bang to the microphone Points in between the measured points are calculated with linear interpolation This determines the local speed of sound at each point The distance from the bang to each microphone 7 ttot is calculated by adding the step length along the true line from microphone to bang multiplied by the reference speed of sound divided by the local speed of sound to 2 for each interpolation point ssi 7 Where the step length is 8 Bachelor thesis Benjamin Donald Oakes 2011 07 05 BANG ds s dz zb zmic n 8 Where s is the original ttot from 2 Temperature calculations Let s look at the Newton Lap
18. Bachelor thesis Benjamin Donald Oakes 2011 07 05 BANG Bachelor thesis 15 higher education credits C level BANG A SYSTEM FOR SURVEYING THE SEPARATION POINTS OF CARGO GRENADES Benjamin Donald Oakes Audio engineering programme 180 higher education credits rebro spring term 2011 Examiner Dag Stranneby rebro universitet 5 rebro University Akademin f r naturvetenskap och teknik me es School of Science and Technology 701 82 rebro e SE 701 82 rebro Sweden UNINSS Bachelor thesis Benjamin Donald Oakes 2011 07 05 BANG Summary This report describes the theory of the surveying system BANG which is used to determine the position and time point of one to two separations of cargo grenades From the time differences of microphone registrations and microphone positions the sought coordinates and time points of a separation can be determined The major aim of the current project was to recode an earlier version of the software program written in FORTRAN 77 to a modern high levellanguage This report describes the theory and calculations made by the new program BANG practical use of the new software program an alternative graphical solution which ensures convergence and in addition improvements and expansions of the software A few words on the choice of microphone is also included Summary Swedish Rapporten beskriver teorin om positioneringssystemet BANG som anv nds f r att avg ra positionen och ti
19. a button under the output tab Graphical view with Matlab The result can be viewed graphically with an integrated Matlab program The positions of microphones and bangs are viewed in a coordinate system that is rotatable zoomable and possible to view in 2D with any of the two axes These results are outputted from BANG to a text file read by the Matlab program when it is run See details on how Matlab was integrated 6 Why Matlab Matlab is used because good looking and well functional 3D graphics easily can be plotted The Matlab graphics window has built in functions such as rotate zoom and 2D view A good graphical solution is fully possible in C This would however demand much time in learning C graphics Functions such as rotation require extensive matrix operations Problems arise with these matrix operations such as gimbal lock 7 Gimbal lock is however avoidable with a quaternion 8 solution A nice graphical solution in C might be preferred in the long run as the complied Matlab program takes a while to start up Less debug files would then be required and Matlab text files would not have to be created All calculations for multiple bangs are executed automatically The old BANG had to be run several times in the case of more than one bang The first execution was to discern the bangs One extra execution was required for each bang to calculate a more accurate position for each bang once the correct combination order of time regist
20. amount of time that can be spent is the deciding factor of using this method B ANG is however not designed for more than two bangs since it is not used for more than two For more other methods should be examined 25 Bachelor thesis Benjamin Donald Oakes 2011 07 05 BANG Choice of microphones Condenser microphones should be used as the sound from the bangs is very short and sharp having a chaotic nature The capacitor effect will capture the steep slopes of the impulse from the bang The microphones should even be of a directed type not an isotropic microphone so there is a possibility of ruling out a faulty convergence see page 16 26 Bachelor thesis Benjamin Donald Oakes BANG References 1 http trond hjorteland com thesis node28 html 2 http en wikipedia org w iki Speed of sound 3 http translate google se translate hl sv amp lanepair enlsv amp u http en wikipedia org wiki Linear inte rpolation 4 http s6 aeromech usyd edu au aero atmospher e atmosphere pdf 5 http en wikipedia org w iki Trilateration 6 http pages stern nyu edu nwhite scrc Compilin gMatlab htm for the integration of Matlab in C 7 http en wikipedia org w ik Gimbal lock 8 http en wikipedia org w iki Quaternion 9 http www rane com pdf ranenotes Enviromen tal 20Effects 20 20the 20S peed 200f 20Sound pdf 10 programbeskrivning BANG separationspunktsbest mning mha ljud SIMBAL 2011
21. and theory of the old BANG program was required This part of the work goes beyond a straight translation from FORTRAN to C The C structure is very different to FORTRAN and improved In order to improve the efficiency and functionality of the program an understanding of the theory behind BANG was necessary This was needed not only to improve the program but also because vital code in the old BANG was written in machine code making it difficult to blindly copy the code Identifying and analysing the underlying mathematical methods was a major part of this work Overview A number of microphones are placed about the field of fire The microphones are sampled at a relatively high frequency 40 kHz The sample data from each microphone is saved in text files IRIG B GPS time code The registration times of each microphone are found by checking at which sample the voltage exceeds a certain threshold The sample number is multiplied by the sample time 1 40 k 25 us to obtain the time From the time data of each microphone and know ledge of the positions of the microphones the position and time point of a bang in the air can be determined The principle is analogous of GPS positioning only using sound instead of electromagnetic radiation Bachelor thesis Benjamin Donald Oakes 2011 07 05 BANG How BANG works Because of messy algebraic equations see page 12 imprecise measurements and weather distortion from the real world BANG uses
22. are many and tightly together With fewer data points with larger distance between them it may be rewarding to use more advanced methods of interpolation with use of derivatives This is more time and data consuming but the fewer weather points will hopefully even out the effort in exchange for a more accurate result Perhaps the degree of change in the measurements should be taken into account as the weather data will not remain constant 2 Bachelor thesis Benjamin Donald Oakes 2011 07 05 BANG The improvements made on BANG C VS Fortran The program in C is built up of classes in object oriented form The classes make objects that are natural for the scenario for instance Microphone Bang Weather etc Aneditor can quickly interpret the code and acquire an understanding of what s going should a new functionality be desired This makes the new BANG is a good platform for new functionalities and improvements The old Fortran code was substantially more difficult to edit as the entire code was stashed up in a single file The C version is faster and has a modern data structure Easy to use GUI The new BANG has a user friendly GUI Input data is easily fed in and changed Tabs make it easy to flick between different input data such as microphone bang and weather data The result of BANG is easy to interpret Positions and times of bangs can be viewed under the output tab A graphical view of a compiled Matlab file can be viewed by pressing
23. be of any size as long as l is chosen so that both C1 and CZ intersect each other The length cannot however be determined so that C tangents m at this stage Imagine instead a graphical program where the circle Ccan be blown up and shrunk down to what ever size providing it always tangents C1 and C2 The user changes l with a bar or textbox until C intersects the microphone of first registration m4 Trilateration The same thing can be done in 3D space with fixed spheres instead of circles The bang sphere S tangents three fixed spheres 51 2 and 53 and is blown up manually by the user until it intersects the reference microphone As with finding the intersection point of the two fixed circles C1 and 2 the intersection points of three spheres S1 S2 and 53 are found with trilateration 16 5 P r x 2X2 deo aus 2 Z r x2 Y 16 Bachelor thesis Benjamin Donald Oakes 2011 07 05 BANG Where X Y and Z are the coordinates for the two intersection points Two solutions exist with two different values of Z This is because the intersection of two spheres is a circle and a third sphere will intersect this circle in two points Which of these points is the correct one can never be known with this method alone A negative Z value can however be ruled out since it is not likely the bang occurred below ground level This is easily checked and the positive alternat
24. doxy wizard Doxygen n EE X BANG_FINAL Kopia bin Release 7 Arkiv Redigera Visa Verktyg Hj lp orte fil Ses Oppna stang PSD W Ordna 2 Visa i Namn Senast ndrad Typ 1 BANG NEW exe 2011 05 29 17 48 Program BANG NEW pdb 2011 05 29 17 48 PDB fil B Musik WJBANG NEW vshostexe 2011 05 2917 49 Program Mer _ BANG_NEW vshost exe 2007 07 21 02 33 MANIFEST fil EJ fee HLE Favoritl nkar E Dokument Mappar zd 1 Shows the project catalogue with program and debug files The output data files will be saved in this catalogue automatically while the input data files are saved in a chosen catalogue Bachelor thesis Benjamin Donald Oakes 2011 07 05 Appendix 2 Input and output data Input data Microphone data mmic number of microphones maximum 10 for the present xmic n ymic n zmic n the position of microphone n tmic n registration time at microphone n Bang data nbang number of bangs maximum two for the present x0 y0 z0 t0 Start values for position and time of one bang first registered bang time known If the time point for the first registration is known true false Weather data time time of measurement s 2 height of measurement m Pressure air pressure at weather point hPa Temp temperature at weather point C RH relative humidity at weather point
25. dpunkten hos en till tv separationer av cargogranater Utifr n tidsskillnader hos mikrofonregistreringar samt k nnedom om mikrofonernas positioner kan de s kta koordinaterna och tidpunkterna ber knas M let med examensarbetet var att koda om den befintliga mjukvaran av programmet skriven i FORTRAN 77 till ett modernt h gniv spr k Rapporten beskriver teorin och ber kningar utf rda av det nya programmet praktisk anv ndning av det nya mjukvarupro grammet en ny alternativ metod som anv nder sig av en grafisk l sning f r garanterad konvergens samt m jligheter till f rb ttringar och p byggnader N gra ord om mikrofonval yttras i slutet av rapporten Bachelor thesis Benjamin Donald Oakes 2011 07 05 BANG Foreword I would like to acknowledge Dag Stranneby U For helping me find and realise this fantastic graduate job He is an inspiration and a fountain of knowledge Carl Arnesson and Tord Kemi Bofors Test Center For the graduate job Niklas Gillstr m Student OU For helping me with bug management and integrating Matlab in C for graphical view of result Timmy Jahrl Student OU For helping me with matrix methods using base vectors for converting to another coordinate system And finally I d like to thank my supervisor at OU Kjell Mardensj OU Worthy of a gold medal for his good advice inspiration and guidance into the world programming technology ii Bachelor thesis Benjamin Donald Oakes 2011 07 05 BANG
26. e is the fraction of water molecules in the air can be determined with 20 0L RH 020386 20 9 Where is the air pressure The humidity has a smaller effect on the speed of sound The question is how much more accurate would the results from BANG be if RH was taken into account Is the influence of RH greater than the deviation in weather observations and results from BANG Research is needed to answer this question Frequency analysis for discerning sonic boom The possibility of using frequency analysis for discerning the sonic boom should be examined This will most likely be more efficient than the combination method described on page 11 Is the dummy variable time known necessary The question arose if the possibility of setting the time of first registration to known is necessary This means that the guess time remains unchanged throughout the entire calculation and is not updated with a new approximation Knowing the time means minimising the calculation time substantially by roughly a quarter This is more noticeable in the case of larger calculations In a test scenario measuring the flight time can be worth the effort and having the option of changing time known does not hurt 23 Bachelor thesis Benjamin Donald Oakes 2011 07 05 BANG Discerning more than two bangs If the need of positioning more than two bangs should arise an expansion of the current method used is not that difficulty implem
27. e oen ete entente pu edet 1 lucir 1 Input dala oper tne ebat ore ettet 2 RI cl cult ta 3 Appendix 3 Example of a test rn iter read cud de 1 Iv Bachelor thesis Benjamin Donald Oakes 2011 07 05 BANG Background BANG developed by Bofors Test Center Is a system for determining the position and time point of one or two separations of certain projectiles The purpose of this system is to survey the separation point of cargo grenades In the situation of one bang the coordinates and time point of separation are calculated directly with BANG s main algorithm based on minimisation of a cost function When more than one bang is present each bang is discriminated first by finding the smallest sum of residuals out of all possible combinations of registration times Once the microphone registrations have been assigned to each bang more accurate approximations are then obtained for each bang using cost function minimisation as in the case of a single bang This algorithm is also required to distinguish a sonic boom which can be generated by the shock wave as the grenades travel at supersonic veloc ity The previous software program BANG was last updated in 1991 At the time this project was started the software program BANG was an MS DOS application coded in FORTRAN 77 It was difficult to use and poorly documented Bachelor
28. ented For details on how this method works se page 11 Earlier we have seen that each possible combination is given a time array This is the case even here with more than two bangs An example is illustrated below with three bangs and three microphones 111 211 311 121 221 321 131 231 331 112 212 312 122 222 322 132 232 332 113 213 313 123 223 323 133 233 333 Naturally at least four microphones must be present for measurements in 3D space This simple example is merely to show the principle and does not need to be more complicated to understand the idea As in the case of two bangs all combinations are matched with their opposite compliment The difference is that more than one combination will match each combination Examples of matches 111 matches with the following 222 322 232 332 223 323 233 and 333 If 111 and 222 333 If 111 and 322 ET 233 etc The next combination 211 matches the following 122 322 132 332 123 323 133 and 333 If 211 and 122 333 If 211 and 322 133 As for two bangs the residuals are added for each possible set of matched combinations Restotl res 111 res 222 res 333 Restot2 res 111 res 322 res 233 etc The method can theoretically be used for an unlimited number of bangs 24 Bachelor thesis Benjamin Donald Oakes 2011 07 05 BANG This method may not be actual for larger calculations and more than two bangs The
29. exactly zero with more than four microphones However the more microphones used the more accurate the result will be ml T tmic 1 1 Shows the problem of BANG in R with three microphones and one bang When xb yb zb and tb are chosen correctly the puzzle pieces will fit together and the correct position and time point of the bang is found ttot 2 Bachelor thesis Benjamin Donald Oakes 2011 07 05 BANG Minimising the cost function using Newton Raphson The cost function is minimised iteratively with a refined Newton Raphson method For a more thorough description see 1 Start values An initial guessed position and time point of the bang is required to start the convergence process If an accurate guess cannot be made there is an acceptable alternative The guessed position s x and y coordinates x0 y0 are set to the average of the x and y coordinates of all microphones The height 20 is just set to a height of 500 m tO is set to the smallest microphone registration time Calculating the next approximation On each iteration an improved estimate is calculated using the previous approximation the first being the start values This is done with a Newton Raphson method with knowledge of the second derivative The value of the cost function q 1 at the current approximation point is calculated The first and second derivatives of q with respect to x y z and t are then calculated by using the step length for n
30. ght of from the beginning of the work 20 Bachelor thesis Benjamin Donald Oakes 2011 07 05 BANG Discussion How well does BANG work The study of how close the results of BANG are to the true results was not part of this work This study would require much research and a whole new thesis The goal of this work was to see that the new BANG gives identical results as the old BANG given identical input data For the tested examples this seems to be the case A smaller test of how well the algorithm works could however be done without much difficulty by firing a gun from a measured position so the position and time of the bang are known from the beginning Convergence Unfortunately BANG does not always converge because of Newton Raphson s poor global convergence properties Qualified start values are required and even then convergence is not a guarantee If the start values are sufficient however convergence is fast The graphical solution with spheres described on page 15 should be further examined to the fully as a solution is guaranteed to be reached As described on page 16 we have the problem of two possible solutions BANG can converge to the wrong solution This must be checked somehow The only check for the time being is if the height Z of the bang is below ground level In this case Z is simply set to its equivalent positive value This may not be the correct value of Z so a few more iterations should be made before claiming the
31. he speed of sound is even calculated the x and y rectangular form components of the wind gradient as a function of height When the effect of temperature has been calculated the effect of wind is added on to the temperature influenced speed The local x and y components of the wind are simply added on to the local speed of sound The points between measured points are calculated in the same way as 11 with linear interpolation The x and y components are however calculated separately 12 Bachelor thesis Benjamin Donald Oakes 2011 07 05 BANG Wy h1 Wx ho W h W ho hy Em ho ho 12 3 Wy hi T W ho hh W h Wy ho Other about weather data ISA stands for International Standard Atmosphere means the temperature vairies linearly with the height The temperature is in fact often linear as a function of height up to around 11000 m altitude Beyond this altitude the temperature remains roughly constant for a while 56 6 C 4 The old BANG had the option of selecting this temperature type This option is now however removed since weather data is always measured as a function of height and never assumed to vary linearly The idea of this weather type should however not be forgotten as it may be desired in future use for time and data saving 10 Bachelor thesis Benjamin Donald Oakes 2011 07 05 BANG Discerning multiple bangs and sonic boom from bang 2 Shows a system with two bangs and J
32. input file here Choose and input data file Here the file is called InputDataBangs txt we have two bangs 2 shows the input data file opened E InputDataBangs txt Anteckningar Arkiv Redigera Format Visa Hj lp 2 False 2 Input data file 1 1 We have 9 microphones 2 bangs the first registration time is unknown the 9 microphone positions time 1 and 2 for each microphone the start guess x0 175 y0 175 20 500 and t0 10 366 Because nbang gt 1 we have two rows of Newton Raphson parameters one for the first calculation discerning the bangs and one for the second more accurate calculations for each bang When selecting this file we can see that the tabs microphone data 3 bang data 4 and Newton Raphson 5 show the data from the selected file Bachelor thesis Benjamin Donald Oakes 2011 07 05 Appendix 3 Microphone data tmic5 10 553 10314 3 Microphone data tab after having read the input file Guessed position and time of first explosion 175 m 175 20 Calculate guess position using 500 average of x y mic positions and 500m high 4 Bang data tab after having read the input file Bachelor thesis Benjamin Donald Oakes 2011 07 05 Appendix 3 Newton Raphson data
33. ive is assumed the true one One possibility is to use directed microphones see page 25 or have a rough clue of where the bang is so the other alternative can be ruled out The Matlab program Input data microphone positions and the time data to the microphones The radius of the bang sphere can be fed in The radius should be changeable by pulling a bar up and down on the graphics window Output data graphical view of spheres and bang and microphone points A dummy variable checking whether the reference microphone is inside the bang sphere S With the input data the program generates three fixed spheres S1 52 and 53 around the microphones m and m and the bang sphere S which starts as its smallest possible size so its centre is positioned on the plane outlined by the points of mz and The user then increases its radius r until S tangents the reference microphone m4 Each time r is changed trilateration is used to calculate the new intersection points of 51 2 and 53 The new 5 is drawn out and a check on whether the reference microphone is inside S is made 7 Figure 1 o x File Edit View Insert Tools Desktop Window Help mz am je 2m Lr cA EM 7 i E q a LI 3 MU Tt m f 10 E f M LT I
34. jamin Donald Oakes 2011 07 05 Appendix 2 Output data tab The results of BANG are shown in the datagridview in the output tab 6 Goce TL UA Show 3D view click this button for 3D viewing of result A Matlab graphics window with graphical view will appear 7 after a minute This can be rotated with the built in Rotate 3D button by dragging the image with the mouse It s also possible to zoom in and out and set 2D views 6 Tab for viewing output data The position and time point of bang s are shown in the datagridview The graphical result from running the default input data 7 El Figure 1 7 Graphical result of BANG The blue circles represent the microphones and the red stars the bangs Bachelor thesis Benjamin Donald Oakes 2011 07 05 Appendix 3 Appendix 3 Example of a test run An example of running BANG This is the same test data as given in 10 The new BANG gives the same results as the old BANG for this input data Press Read input file window 1 will appear B 29 V lj en indatafil x QOO i Hamtadefiler BANG NEW Files 45 Sat P oS Typ Storlek Favoritlankar Dokument Z Nyligen besokta plat Skrivbord Mer Mappar v E Bilder E Dokument E Favoriter 8 Filmer I H mtade filer J BANG_NEW Finamn InputDataBangs bt Tex files ba 1 Select an
35. lace equation for determining the speed of sound in a gas air 9 PERS 9 2 Where P is the air pressure p is the air density and y the adiabatic index of air assumed 7 5 1 4 for dry air 2 We assume the air to be an ideal gas thus the formula 9 can be rewritten as 10 y RT M 10 2 Where R is the gas constant T is the absolute temperature in Kelvin and M the molar mass of air The air pressure and air density are now not required We have an expression with one dominating variable the temperature T The rest of the factors remain relatively constant R the molar gas constant is always the same M and y vary slightly with the relative humidity of air For dry air y 1 4 and 0 0289645 kg molt R 8 314510 J mol K 1 Re writing 10 gives 107 331 32 14 10 2 273 15 Where 2 is the air temperature in C The temperature could earlier be given as a constant as ISA profile or as a function of height in the old BANG Now the weather data can only be given as a function of height since Bofors own weather system takes measurements as a function of height and writes them to a text file The temperature at a certain height between measured points is calculated with linear interpolation 11 T hi T ho T T h a 11 3 Where ho and are the heights of the two weather points closest to Wind calculations The effect of the wind gradient on t
36. nverge to the wrong solution as described on page 16 With the new graphical program two solutions will be found with each system of four microphones The correct solution is easily found if more than four microphones are used as one of the solutions in each system will be close to the others The faulty Z value can be ruled out Note that such a sphere system can only be constructed with four microphones spread out in 3D space Real measurements will most probably be made with more than four microphones for more accurate results In this case three microphones are chosen at a time together with the reference microphone with smallest registration time Several of these systems can be made For a more accurate positioning the interesting result is the one given by the system of microphones with the smallest times For an even more accurate result the average of results of all systems can be made The total number of systems that can be made 17 nmic 17 4 is the number of microphones over four Different methods using standard deviation can be used for a finer result than the average of all results nsystems Another alternative to changing r manually is by using a simple bisection method where the radius increases in equally sized steps until the reference microphone is inside S The radius is then decreased with half the size of the original step When the reference point is outside S again the step size is halved again and radiu
37. our microphones in The first step is to determine which registration belongs to which bang at each microphone A more accurate position and time point of each bang can thereafter be calculated according to the previous section When more than one bang is present more than one registration will be detected at each microphone This is the case both for distinguishing several separations as well as a sonic boom from a separation The question is which of the registrations belong to which bang This is initially unknown See figure 2 with two bangs present Before an accurate position of each bang can be determined the bangs must be discerned by finding the matching registrations one at each microphone This is done by calculating every combination separately with the refined Newton Raphson method described in the previous section without the use of weather data One registration at each microphone is tried at a time The two matching combinations with the smallest summed residual q will most likely be the correct combinations These two combinations are thereafter calculated individually as if they were two single bangs so their position can be calculated more precisely with weather data and adapted Newton Raphson data Weather data is too time and data consuming the speed of sound is therefore assumed to be constant everywhere for the discerning calculation The previous example is of simplest form with multiple bangs We have two bang
38. pendix 3 For a 3D view of the result press Show input data A Matlab graphics window will appear 12 T D 1 1 12 Graphical view of the result 2 bangs red 9 microphones blue
39. rations is known Weather data was taken into account on these second calculations This meant having to feed in the results of the first calculation as start values for the second calculations The new BANG makes all these calculations automatically Weather data is given from the beginning and the program only has to be executed once 22 Bachelor thesis Benjamin Donald Oakes 2011 07 05 BANG Possibilities of further improvement Here are some ideas of possible improvement Structs VS classes in C For larger more time demanding calculations it may be worth implementing structs instead of classes This is especially worthwhile for the class handling the cost function Cost Improvements in weather calculations The wind gradient should be able to be given in cardinal coordinates These coordinates are converted to the reference coordinate system used This is simply done by knowing the phase difference between the cardinal and reference coordinate systems Relative humidity in the weather calculations The speed of sound increases with increased humidity The relative humidity influences two variables in equation 10 namely y and M The ratio between the speed of sound in dry air and in wet air will be 18 y My 18 9 Ya Ma Where y and M are y and M in wet air and yq and Mg are y and M in dry air The wet variables can be calculated 19 7 h Y gt o gt gt s Y S h 19 9 My Ma Wher
40. result reached The new graphical algorithm will however converge to the correct solution if more than four microphones are used see page 18 Calculation time The calculation time seems to vary greatly depending on the complexity of the system applied Determining factors are the number of microphones number of bangs number of weather points and interpolation points of weather data depending on the integration step length dz The more of any of these factors applied the longer the calculation time will be In the case of two or more in future bangs the time of the first half of the calculation is dependent on the number of microphones and bangs only The time difference between using a few microphones and around 10 microphones is relatively large but this calculation hardly takes one minute with 10 microphones and two bangs Compared to the next calculation after the bangs have been separated the first has little significance The number of weather points and interpolation steps have the heaviest influence on the time especially with a maximal number of bangs and microphones The effect of weather data With a certain input data the results of BANG with accounted weather data differ substantially from results without What needs to be researched on is the reliability of the weather observations and weather calculations Linear interpolation may give sufficiently accurate values of the points between measured points if the measured points
41. s and four microphones The total number of combinations will be nbang ic 24 16 Each combination has a time array 1111 2111 1211 2211 1121 2121 1221 2221 1112 2112 1212 2212 1122 2122 1222 2222 11 Bachelor thesis Benjamin Donald Oakes 2011 07 05 BANG The number of digits in each time array corresponds to the number of microphones In this case four The number 1 represents the first registration and 2 the second at the microphone with the index of the digit s place So 1111 means trying the combination of all the first microphone registrations 1122 means trying the first registrations at microphone 1 and 2 and the second registrations at microphone 3 and 4 The time arrays are matched together in the pairs that are logically possible 3 The same registration cannot occur twice at the same microphone The time arrays with opposite digits are therefore matched So 1111 and 2222 is one possible outcome 2121 and 1212 is another etc 1111 2111 1211 2211 1121 2121 1221 2221 1112 2112 1212 2212 1122 2122 1222 2222 3 The time arrays are matched with their compliments the arrows indicating each matched pair K A The residual for each pair is added 13 ResTot 1 res 1111 res 2222 ResTot 2 res 2111 res 1222 ResTot 3 res 1211 res 2122 ResTot 4 res 2211 res 1122 ResTot 5 res 1121 res 2212 13 ResTot 6 res 2121 res 1212 ResTo
42. s increased until the tolerance is reached Note that this can only be done with systems of four microphones ata time In addition the microphones must be placed in an origin system with one microphone in origin one on the X axis one in the XY plane and one in XYZ space Any set of four microphones can be converted to such a system and thus converted back This operation requires only a few simple matrix operations explained in the next section The example in figure 7 is of simplest form where m is in origin 73 on the X axes and m on the Y axes merely to illustrate that and how the system works Transition between coordinate systems in 3D space The following explains how a new coordinate system containing four microphones is constructed so that one microphone is placed in origin one on the X axes one on the XY plane and the third in XYZ space The condition is that the microphones are spread out in space in this way in the original system Let s start by defining a new system Xz Ys Zs in 3D space A B and C are the positions of three microphones in the original system 18 Bachelor thesis Benjamin Donald Oakes 2011 07 05 BANG T Find the base vectors of the new system S W AB B A W AC wWw Wed V AC Projy AC AC V V Where V V and V are the base vectors for new coordinate system S Normalise the base vectors vi V2 Vs V
43. system for two dimensions is a race with little prise since the equivalent 3D system still has to be solved In the case of three dimensions another term z and equation are added on The thought of trying to solve this system 15 is horrid x x 2 2 21 x x2 y yo z 22 n x x3 y z z rs 2 x x4 y 2 z 2 r2 15 Good luck in solving x y z and r A new version of BANG using a graphical solution There is a graphical way around solving 15 This is done with a prototype Matlab program illustrated a little further on For simplicity think back to the 2D system 5 with the circles The middle point of a circle with any chosen radius l tangenting the two fixed circles C1 and C2 can be found This is done by finding the intersection point of the circles C1 and 2 with their centre points in m and and radii rj 1 See figure 6 15 Bachelor thesis Benjamin Donald Oakes 2011 07 05 BANG 4 4 Lodo N 2 2 q 6 The principle of the new alternative graphical solution R The position and time point of b is found by blowing u p grap P P 1 8 up the circle C until it intersects the microphone of first registration m4 The bang circle is a circle with a chosen radius l that tangents C1 and C2 This circle can be chosen to
44. t 7 res 1221 res 2112 ResTot 8 res 2221 res 1112 The pair yielding the smallest residual is most likely the true combination order of times When the true combination is found only a rough approximation of an acceptable result will have been obtained The resulting position and time for each time array in the correct pair will be set as the start values for the separate second calculation finding a more exact position and time for each bang 12 Bachelor thesis Benjamin Donald Oakes 2011 07 05 BANG Note that we can discern multiple bangs but not determine m which order they occur Because of long distances between microphones the cargo grenades travel at the speed of sound or faster and that the second bang can occur much closer to a microphone than the first means the second bang can be registered first The order is determined with a video camera For the present BANG can discern no more than two bangs Surveying more than two bangs at a time is not presently necessary however it may well be possible as described on page 24 13 Bachelor thesis Benjamin Donald Oakes 2011 07 05 BANG Geometrical analysis of the problem In this section the problem is analysed geometrically A new graphical solution to the problem is discussed with an illustrative diagram from a new prototype Matlab program This is an alternative solution with more or less guaranteed convergence It however only works for one bang at
45. the approximated position and time are to the true position and time It is the distance from the estimated position to the true position squared nmic q gt tmic n tb ttot n gt 1 Where tmic n is the difference between the smallest registered time and the time of registration at microphone n tb is the current approximation of the time of the first registration and ttot n is the distance from microphone n to the approximated position xb yb zb Note that tmic n and tb are always scaled to distances with the reference speed of sound aspeed If weather distortion is taken into account the total distance ttot n will alter as the speed of sound will vary at different heights See page 8 for details on calculating ttot n with weather distortion 2 shows ttot n without weather taken into account ttot n xmic n xb ymic n yb zmic n zb 2 If the time and position are chosen correctly tmic n tb ttot n will be zero or close to zero from the bang to each microphone so q will be minimal ideally zero With four microphones spread out in the x y z space any values of tmic n for each microphone will lead to a fixed point In other words q is bound to converge to exactly zero With more than four microphones the measurements of each tmic n must be precisely accurate in order for q 0 In the real world with imperfections and weather distortion q is not likely to reduce to
46. the present and weather data is not taken into account though it could easily be The messy algebra the inexact measurements and possibility of discerning multiple bangs are however strong arguments for the use of optimization as described earlier The theory of the new method is explained in this section 1 microphones are required for positioning m One of them takes the first registration and the others hold the time differences between the first registration and the time of registration at each microphone In R the time difference is represented as a line In R the time differences are represented as circles and in R as spheres The other criterium for positioning in R is the microphones have to be spread out in R So in R the two microphones have to be placed on different points on the line In R and R all microphones cannot be on the same line and for R all microphones cannot be on the same plane No microphones can offcourse be placed on the same point in all cases Surveying in one dimension Let s start with imagining surveying a bang on a one dimensional line R1 4 The difference in registration time at microphone 1 m4 and microphone 2 mz is easily determined b s distance from the middle point of the microphones is found by multiplying the difference in the times to and t4 with the speed of sound b s position will be this distance from the middle point of the microphones in the direction the microphone holding
47. umerical differentiation inc See attachment 1 for details on the derivative calculations The first and second derivatives of q q q q l x z t 3 8 529 q q x x y x z x t 2q 8q 8q 2620 y x y y z y t 4 2q 529 8q 624 z x z y 622 zt q 2q 2q q l t x t y t z st Where 3 is the gradient g with partial derivatives and 4 is the symmetrical hessian matrix H with the partial second derivatives By rewriting a truncated Taylor series 5 the next approximation can be found by subtracting the Newton direction g H 1 from the current approximation qk 1 dk gk Hy 5 The new estimate of xb yb zb and tb is found with 6 xby44 xby gx 1 Hz 0 0 Hye 1 2 Hr 1G 3 Hz 4 ybya4 gx 22 2 0 Ht 22 H 1 2 3 H 2 4 2 zb gx 3 lt Hz 1 3 1 Hz 1 2322 H 3 3 H gt 3 4 1 thy gy 4 Hy 4 1 Hg 4 2 Wy 4 3 H 4 4 Where k 0 1 maxiter 6 Note that the numbers in brackets in 6 represent each index of the gradient and inverted hessian The inverted hessian is calculated by using Gauss elimination Bachelor thesis Benjamin Donald Oakes 2011 07 05 BANG Comments The reason for the use of second partial derivatives in the Newton Raphson calculation is that convergence is reached with fewer iterations
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