Home

EllipseFit 3 User Manual

1. am T pg E ipt a kair 01 LI ei be a s CRE wi af TT FR i 2 Figure 15 The Fry graph without normalizing using the settings displayed in Figure 14 To normalize the graph select Normalize as shown in SAC EllipseFit Preferences Figure 16 Note that the Normalized radius is now used Digitize E E due to the normalization to a unit circle the default oe FESTER sk erred value is 1 5 as shown an _ Enhance Graph radius 50 00 Stretch M Show all points Nadai Fit ellipse Selection factor 1 050 Ad Map Mx right Normalized radius M Y down _ Auto scale Figure 16 Settings to display a normalized Fry graph Note that the Normalized radius is now used due to the normalization to a unit circle EllipseFit User Manual Page 18 The resulting normalized graph is shown in Figure 17 Note the clear sharpening of the central void The final question addressed in his section is how to find the ellipse corresponding to the central void The enhanced normalized Fry method Erslev and Ge 1990 uses a user specified cutoff radius to exclude particles beyond the a certain distance from the void center This is a subjective value chosen here with a default value of 1 05 In the Preferences dialog check Normalize and uncheck Show all points EllipseFit calculates the best fit ellipse through the cloud of points using the least squares method described in Sect
2. For the purposes of this section it will be assumed that the void has been defined well enough to pick out the void by eye which may be a close enough estimate and also makes a good exercise for introductory students Click on the Centered Ellipse Icon Digitize gt Centered Ellipse and click at the edge of the void An orange circle marks the starting point subsequent points are marked by a yellow circle When finished click on the orange circle and the ellipse will be calculated and displayed in the Log Window EllipseFit User Manual Page 11 Figure 8 Digitizing the central void The orange point is the start point the yellow are subsequent points Click on the orange point when finished and the ellipse is calculated The point size is set larger than the default size for the illustration For this sample the calculated results are reported by EllipseFit as N 60 Pairs 1770 Best Fit Ellipse Manual n 1 7 R 1 758 25 45 RMS 0 0583 A centered ellipse was calculated from the 17 digitized points The calculation is rotationally invariant and the best fit found by minimizing the sum of the squares of the distance of the points from the ellipse i e the residuals The minimization is solved from the linear equations using a LU decomposition The RMS value is the root mean square measure of the variation of the residuals from the ellipse that is the square root of the sum of the squares of the residua
3. Hossack J R 1968 Pebble deformation and thrusting in the Bygdin area Southern Norway Tectonophysics 5 315 339 Yamaji A 2008 Theories of strain analysis from shape fabrics A perspective using hyperbolic geometry Journal of Structural Geology 30 1451 1465 EllipseFit User Manual Page 55 History 3 1 0 Added bootstrap error analysis to ellipsoid calculations This has some similarities to the kernel density estimation approach of Mookerjee and Nickleach 2011 Added saving of the ellipsoid axes orientations for plotting on spherical projections in Orient Changed column headers A B C to Max Int Min to clarify the axial lengths EllipseFit will open files with the old headers but will save them using the new headers Removed option to save files as Space Delimited This format potentially causes issues parsing files with spaces in the header column EllipseFit will still open space delimited files with recognizable headers Added 95 confidence regions to Nadai graph Added 95 confidence regions to Flinn graph Added option to save bootstrap ellipsoid axes Added numerous options to Synthesize Data command These include generating the strain ratio from a Gaussian normal distribution generating particle size from a Gaussian normal distribution generating a preferred orientation from a Von Mises circular distribution generating centers at a truncated Poisson distribution The latter is performed
4. Figure 48 Lower hemisphere equal area projection of the strain ellipsoid axes Circles are the axes calculated from all 8 sections diamonds with section 6 removed Red Smax green Rm blue Rmn EllipseFit User Manual Page 47 Documentation in preparation Axis bT14 bT24 bT34 bT45 bT46 bT47 bT48 bT56 bT57 bT58 S1 2 569 25 0 2 570 2570 2569 2 569 2570 2568 2568 2 5 0 t1 35 060 35 100 35 010 35 180 35 030 35 030 35 010 35 230 35 220 35 010 p1 10 900 10 910 10 890 10 930 10 900 10 900 10 890 10 940 10 940 10 890 S2 1 130 1131 1130 1132 1130 1130 1130 1133 1133 1 130 t2 127 450 127 490 127 400 127 570 127 420 127 420 127 400 127 620 127 610 127 400 p2 12 210 12 200 12 220 12 190 12 220 12 220 12 220 12 180 12 180 12 220 S3 0 344 0 344 0 344 0 344 0 344 0 344 0 344 0 344 0 344 0 344 t3 264 450 264 440 264 450 264 440 264 450 264 450 264 450 264 440 264 440 264 450 p3 73 510 73 520 73 510 73 520 73 510 73 510 73 510 73 520 73 520 73 510 Table 5 Results of test of ellipsoid fitting using two ellipses and six lineations from synthetic section ellipses calculated from b Table 4 For ten tests six of the eight RTj values were omitted Subscripts indicate the sections with RTj data Results are all identical down to round off error Axis b14 b24 b34 b45 b46 b47 b48 b56 b57 b58 S1 nan 3 422 4379 3 196 3389 3 371 3 469 3126 3301 3 127 t1 nan 41 760 47 150 43 140 20 310 20 330 20 320 42 680 45 960 37 500 pl nan 11 690
5. and the potential to port to other platforms I simultaneously develop several programs that use common graphics and matrix libraries that I have written 1 1 Installation On Macintosh OS X double click the disk image file dmg and drag the EllipseFit application to your Applications folder or other desired location On Windows unzip the zip file zip using the Extract All option and drag the EllipseFit folder to any desired location The EllipseFit folder contains the EllipseFit application EllipseFit exe and a Resources folder which is required Please make sure to entirely extract the EllipseFit folder from the zip file this is the most common installation problem EllipseFit User Manual Page 2 On Linux unpack the gzip file tar gz and copy the EllipseFit folder to any desired location The EllipseFit folder contains the EllipseFit application ellipsefit and a Resources folder which is required An application icon ellipsefit png is included in the Resources folder if desired for installation There is also a folder of example data and images to show how data is formatted these are referred to in this guide After installing a new version it is recommended that you reset the preferences using the Reset Preferences command in the Help menu This will clear any options that may have changed and set them to default values The preferences are stored in the file EllipseFit3 xml which is located in the fol
6. 12 580 12 310 8 060 8 060 8 060 12 240 12 790 11 280 S2 nan 0 902 0 836 1 052 0 584 0 585 0578 0 301 1054 1 021 t2 nan 133 950 139 230 135 430 235 100 234 930 235 570 264 470 138 190 129 850 p2 nan 10 430 9 240 10 370 80 220 80 240 80 160 73 780 9 730 11 610 S3 nan 0 323 0273 0 297 0505 0 507 0499 0 561 0 287 0 313 t3 nan 264 630 264 590 264 450 111 090 111 110 111 110 264 470 264 450 264 490 p3 nan 74 230 74 300 73 800 5 510 5 470 5 600 73 780 73 830 73 700 Table 6 Test of ellipsoid fitting using two ellipses and six lineations from eight measured section ellipses Table 5 For ten tests six of the eight Rj values were omitted Subscripts indicate the sections with Rj data Results are highly variable especially as axial ratios which are plotted in Fig 8 EllipseFit User Manual Page 48 1 T 00 BC Figure 49 Test of ellipsoid fitting using two ellipses and six lineations from eight measured section ellipses Table 5 For ten tests six of the eight Rj values were omitted Subscripts indicate the sections with Rj data Results are highly variable especially as axial ratios EllipseFit User Manual Page 49 9 Ellipsoid Data Graphs Documentation in preparation 9 1 Flinn Graphs Documentation in preparation ANA Flinn Graph A Flinn graph is a graph of the ratios A B Smax Sm versus B C Sm Smin and is commonly used for displaying strain ellipsoid data e g Ramsay and Huber As with the ellipse graphs the Fl
7. 97716 9 0 31454 1 80 23 00 87 24708 0 69591 0 99044 10 0 31451 1 90 25 00 86 98953 0 69099 1 01624 MLLF Results Point statistics Number 52 EllipseFit User Manual Page 23 Calculated density 0 00004 Real density 0 00004 Results Mean log likelihood 0 31327 R strain ratio 1 90000 Phi angle of max strain axis 25 00000 Cutoff radius 86 98953 Finished 2014 05 31 22 49 58 The results of pass 0 are identical to the previous result however the cross validation procedure located a slightly better solution in pass 8 the mean log likelihood is 0 31327 instead of 0 31829 The resulting Fry graph with 8 less neighbor points is shown in Figure 25 Figure 25 Fry graph of the results using the cross validation option for mean log likelihood maximization EllipseFit User Manual Page 24 4 Strain from Lines Documentation in preparation 4 1 Automated Wellman Analysis The Wellman method can be applied to objects in which two lines can be identified that have constant initial angles such as brachiopod hinge and medial lines which are initially perpendicular Wellman 1962 Ramsay 1967 For brachiopods not parallel to a principal strain this angle will be distorted by shear strain Wellman s graphical technique is illustrated in many structural geology laboratory manuals e g Ragan 2009 An analytical solution to the problem was given by Vollmer 2011 which is implemented in implemented in
8. Vollmer 2011 It is therefore generally preferred over the R graph of the next section Most of the graphs in EllipseFit are interactive When the Binoculars Icon is selected points can be selected and the selection will automatically update on other graphs and in the Data Window To illustrate Figure 33 shows a Fry graph with the points generated by the outlier selected in Figure 32 Ann Fry Graph ne He 6 FG Figure 32 Polar Elliot graph with digitized data from the oolith photomicrograph in Figure 1 One outlier is selected TLI 63 0 Figure 33 Fry graph with data generated from the oolith photomicrograp in Figure 1 The selected points are those generated by the outlier selected in the polar graph of Figure 32 EllipseFit User Manual Page 31 This outlier falls well inside the central void and probably does not meet the assumptions necessary for a Fry analysis i e a truncated Poisson distribution 6 2 Rob Documentation in preparation CRE Ri Phi Graph The R 4 graph Dunnet 1969 is probably more widely recognized and used than the polar Elliott graph e g Lisle 1985 however it distorts the strain space especially at low strains and a polar graph is generally preferred Vollmer 2011 4 70 8 67 4 Figure 34 R graph with digitized data from the oolith photomicrograph in Figure 1 One outlier is selected the same as in Figures 32 and 33 all of
9. Wellman M Show pe Graph radius 180 00 nan ie x Stretch P MES AE PA age T RS ne Nadai Fit ellipse selection factor 1 050 EL Flinn AR X right etes t Strain Map Ex rig Normalized rad 1 50 M Y down _ Auto scale 4 gt c Preview Cancel OK Se 1 Shen EN a 172 2 96 6 Figure 6 Set the graph radius to display the central void by unchecking Auto scale and entering a smaller radius EllipseFit User Manual Page 10 ae fe N it CE Li AC et Ph z Sy A Fei iy ey ae E s F Pa Sia Figure 7 Close up of the central voids for the two data examples of 60 and 252 points Figure 7 shows the zoomed in central voids for the two examples The next step is to determine the best fit ellipse for the central void displayed in Figure 7 This can be a subjective process and objectively choosing this ellipse is the subject of a number of papers e g Erslev 1988 Erslev and Ge 1990 Shan and Xiao 2011 Waldron and Wallace 2011 Mulchrone K F 2013 The normalized Fry method Erslev 1988 Erslev and Ge 1990 is one that is commonly employed but requires the digitized ellipses of each particle The normalized Fry method is the subject of Section 32 Ideally a method should require only the point data e g Shan and Xiao 2011 Waldron and Wallace 2011 Mulchrone K F 2013 Currently EllipseFit implements the algorithm of Shan and Xiao 2011 discussed in Section 2 3
10. by randomizing the location in x y and discarding collisions Added an option to the Strain Map command to either plot scaled strain ellipses or particle axes Implemented the maximum mean log likelihood function MLLF search procedure of Shan and Xiao 2011 This gives a high accuracy strain estimate from Fry type data that is data from truncated Poisson distributions It does not require ellipse data and it is not subjective and is reproducible Fixed auto scaling on Fry graphs Significant progress on the User Manual 3 0 3 13 May 2014 Added transforms to image to rotate flip strain unstrain etc To strain or unstrain both image and data transform the image first This calculates the origin offset in the new bitmap Then transform the data at X0 YO 0 0 0 0 with Rectify checked Added transform data to Wellman type data Changed default bootstrap resamples from 300 to 5000 Rewrote ellipse standard error and confidence interval methods Changed from using resample trials to calculate standard error and Student T for confidence interval to use resampled data for both Non bootstrap MRL uses analytical error and Student T following Mulchrone 2005 Added option to save bootstrap resample ellipses Added option to plot 95 confidence regions on Polar and Rf Phi graphs using analytical error Fixed bug that was swapping A and B radii while digitizing polygons 3 0 2 21 April 2014 F
11. data Before continuing open the Fry graph Analyze gt Fry Graph You should have something similar to Figure 2 I in TH 20 127 79 234 36 Figure 1 EllipseFit s Image Window used for digitizing with photomicrograph of a deformed oolite from Ramsay and Huber 1983 Continue digitizing point centers you should ideally work out from one point digitizing adjacent points keeping a roughly circular area The Fry graph will start to develop as you digitize with each new set of generated points highlighted Figure 3 Use the Hand Tool Digitize gt Hand Tool or the Hand Icon to scroll and the Zoom Tool to zoom Digitize gt Zoom or the Magnifying Glass Icon You can also use the Command Mac or Control Windows and Linux and keys to zoom in and out Holding down the Shift key allows scrolling with the cursor Points can be deleted by using the Find Tool Digitize gt Find Tool or the Binoculars Icon to highlight a point and delete it using the Cut command Edit gt Cut or red X icon A point can also be deleted by selecting it in the Data Window and deleting it there EllipseFit User Manual Page 7 4 El os felicia Li himin m ee LE Dali CE TELE 0 ECOLE sh oo Her se FL en 7 Figure 2 EllipseFit s Image Window Data Window and Fry Graph displaying a single data point It is important to be objective and you may wish to digitize all ava
12. file that can be opened in Orient 2 Vollmer 2010 for plotting the axes on spherical projections The Bootstrap option performs a bootstrap type error analysis using the number of resamples specified in the Resamples edit box 5000 is the default value Finally the Save bootstrap will save the 5000 results of the resampling which is normally unnecessary Press OK to start the calculation You will be prompted to save the orientation data files and shortly the results appear in the Data Window Figure 43 and the Log Window 800 ES Owens 1984 csv ay xs N Max Min R Phi Strike Dip 1 1 16 5000 4 5000 3 6667 165 00 302 00 78 00 2 2 9 5000 3 5000 2 7143 166 00 301 00 77 00 3 3 20 5000 6 8000 3 0147 166 00 302 00 75 00 4 4 37 0000 6 0000 6 1667 173 00 201 00 71 00 5 5 7 5000 1 5000 5 0000 0 00 178 00 71 00 6 6 16 7000 3 0000 5 5667 10 00 18 00 73 00 7 7 22 0000 4 0000 5 5000 8 00 17 00 78 00 8 8 18 0000 3 0000 6 0000 7 00 19 00 78 00 gt Figure 41 The section data from a sample of slate from Dinorwic North Wales from Owens 1984 displayed in the EllipseFit Data Window AN Calculate Ellipsoid Options M Append results IM Save orientations Error Analysis Mi Bootstrap Save bootstrap Resamples 5000 Figure 42 EllipseFit s Calculate Ellipsoid Dialog The Data Window now displays the ellipsoid principal axes Max Int Min as stretches Smax Sm Smin and 95 confidence intervals calcula
13. v Eigenvector M Mean log likelihood Mean radial length Hyperbolic mean Centroids Simple means Max R 20 Calculate Ellipse Error Analysis vV Bootstrap Distance Save bootstrap Resamples 5000 Cross validate Passes 10 Caneel Figure 20 The Calculate Ellipse dialog showing the MLLF options available for a set of point only data off 953 60 00004 00000 31829 90000 00000 98953 Stat 0 67361 Density 0 84687 EllipseFit User Manual Page 21 Figure 22 Fry graph with results of the mean log likelihood function MLLF maximization search The ellipse is the result of the MLLF grid search The green markers highlight the Fry neighbor points The Fry graph of the mean log likelihood function MLLF maximization search results is shown in Figure 22 The ellipse is the result of the MLLF grid search The green markers highlight the Fry neighbor points those that maximize the MLLF Note the ellipse is the result of the intensive grid search and is not simply a linear least squares fit as used in Sections 3 1 and 3 2 EllipseFit User Manual Page 22 To test the cross validation procedure go back and check the Cross validation option in the Preferences dialog The progress dialog now is displayed as in Figure 23 There are now three iteration passes displayed the first is O to 10 where 0 is the first calculation as done above Passes 1 to 10 are the coss validation iterati
14. which are automatically updated interactively EllipseFit User Manual Page 32 6 3 Hyperboloidal Projections Documentation in preparation C 10 07 Figure 35 The unit hyperboloid H showing cartesian axes x0 x1 x2 and point C 1 0 0 which corresponds to the circle R 1 The plane x x is the projection plane for azimuthal projections the polar strain graph Points on H are x Xo x1 X2 with origin C If strain is represented by p W log R 26 then an ellipse is x cosh p sinh p cos w sinh p sin w EllipseFit User Manual Page 33 Figure 36 The unit hyperboloid with superimposed cylinder with axis x0 The cylinder is the projection surface for cylindrical projections as the R amp graph EllipseFit User Manual Page 34 Figure 37 Synthetic data of 300 ellipses strained to values of R 2 and R 4 displayed on hyperboloidal azimuthal projections a equidistant b stereographic c equal area d orthographic and e gnomic The best fit ellipse is plotted as a white circle the centroid of the projected data is plotted as a gray circle 7 EllipseFit User Manual Page 35 Mean Ellipse Calculation Documentation in preparation Data Set Imposed R D Eigenvector Mean Radial Hyperbolic Oolith 1 0 1 628 25 74 1 628 25 74 1 628 25 74 n 252 0 018 0 73 0 018 0 62 0 013 0 614 25 74 1 000 113 32 1 000 113 32 1 000 113 32 0 007 55 27 0 011 633 74 0 0
15. 13 Synth 1 1 0 1 031 40 20 1 031 40 20 1 031 40 20 n 300 0 021 33 24 0 025 22 81 0 030 2 0 2 012 1 16 2 012 1 16 2 012 1 16 0 048 1 16 0 050 0 92 0 032 4 0 4 023 0 46 4 023 0 46 4 023 0 46 0 101 0 53 0 099 0 37 0 031 Synth 2 1 0 1 016 146 03 1 016 146 03 1 016 146 03 n 1000 0 012 35 35 0 014 24 51 0 016 2 0 2 012 179 46 2 012 179 46 2 012 179 46 0 026 0 71 0 27 0 51 0 016 4 0 4 024 179 78 4 024 179 78 4 024 179 78 0 052 0 30 0 053 0 21 0 017 Table 1 Comparative results for ellipse fitting techniques implemented in EllipseFit Eigenvector Shape matrix eigenvectors Shimamoto and Ikeda 1976 Radial Mean radial length Mulchrone et al 2003 Mulchrone 2005 Hyperboloidal Hyperboloidal vector mean Yamaji 2008 From Vollmer 2010 Shape matrix eigenvector Shimamoto and Ikeda 1976 mean radial length Mulchrone et al 2003 and hyberbolic vector mean Yamaji 2008 ellipse fitting methods give precisely identical results 7 1 Shape Matrix Eigenvectors Documentation in preparation 7 2 Mean Radial Length MRL Documentation in preparation 7 3 Hyperbolic Vector Mean Documentation in preparation EllipseFit User Manual Page 36 7 4 Bootstrap Error Analysis Documentation in preparation 7 5 Simple Means and Centroids Documentation in preparation 90 Figure 38 The best fit strain ellipse is simply the hyperboloidal
16. 291 305 Fossen H 2010 Structural geology Cambridge University Press 463 p Fry N 1979 Random point distributions and strain measurement in rocks Tectonophysics 60 806 807 Hanna S S Fry N 1979 A comparison of methods of strain determination in rocks from southwest Dyfed Pembrokeshire and adjacent areas Journal of Structural Geology 2 155 162 EllipseFit User Manual Page 53 Hobbs B E Means W D and Williams P F 1976 An outline of structural geology Wiley New York 571 pp Hossack J R 1968 Pebble deformation and thrusting in the Bygdin area Southern Norway Tectonophysics 5 315 339 Jensen 1981 On the hyperboloid distribution Scandinavian Journal of Statistics 8 193 206 Launeau L and Pierre Yves F Robin P F 2005 Determination of fabric and strain ellipsoids from measured sectional ellipses implementation and applications Journal of Structural Geology 27 2223 2233 Lisle R J 1985 Geological Strain Analysis A Manual for the Rf o Technique Pergamon Press Oxford Mardia K V 1972 Statistics of Directional Data Academic Press 329 p Marshak S and Mitra G 1988 Basic methods of structural geology Prentice Hall 446 p McNaught M A 1994 Modifying the normalized Fry method for aggregates of non elliptical grains Journal of Structural Geology 16 493 503 McNaught M A 2002 Estimating uncertainty in normalized Fry plots using a bootstrap approach Journal of Structu
17. 4 Pollard and Fletcher 2005 Twiss and Moores 2007 Ragan 2009 Fossen 2010 Ragan 2009 and Ramsay and Huber 1983 provide excellent overviews of techniques for the analysis of strain in deformed rocks The literature is vast but in the references I have tried to included many of the papers that are relevant to using EllipseFit The following chapters discuss techniques of strain analysis that are implemented in EllipseFit in terms of the type of data collected points lines ellipses and polygons Points are the simplest type of data collected however as discussed in Chapter 3 Strain from Points it can be difficult to objectively extract strain from point distributions A new objective procedure mean log likelihood function or MLLF maximization Shan and Xiao 2011 has been added in version 3 1 The analysis of line data depends on the known initial lengths of or angles between lines and has important applications for some data as discussed in Chapter 4 Strain from Lines Chapter 5 Strain from Ellipses and Polygons covers ellipse data which is collected assuming that particles such as sand grains initially approximated a collection of random spheres or ellipsoids It turns out however that ellipse data is a subcategory of polygon data An important mathematical proof Mulchrone and Choudhury 2004 shows that all particles of any shape that can be assumed to have been initially randomly oriented can be used to calculate st
18. EllipseFit To try the method open the file LA Ragan 1985 F10_la png as an image This is from Ragan 1985 and is used in many structural geology classes as an exercise To begin click on the digitizing icon at the second from the left until the LinePair Icon is displayed or from the menu choose Digitize Line Pair Click on the endpoints of each of the two lines in turn When done the lines appear in red and the yellow cursor appears at the intersection Mistakes can be corrected by using the red X Cut Icon or by deleting the line pair in the Data Window M OLA Ragan 1985 F10_1a png BARBARA aaa Figure 26 The Image Window after opening the example data for the analytical Wellman method from Ragan 1985 The hinge and medial lines are assumed initially perpendicular One line pair has been digitized Note the Line Pair Icon is visible EllipseFit User Manual Page 25 After digitizing one line pair open the Wellman Graph Spas tt using the menu command Analyze gt Wellman Graph The graph shows the parallelogram corresponding to the brachiopod Figure 27 The parallelogram sides parallel the line pair Note the two additional points used for the construction 1 8 LS No Figure 27 The analytical Wellman graph diplayed in a graph window after digitizing one line pair as in Figure 26 Note the Binoculars Icon is selected and that the parrallelogram and corresponding brachiopod are selected with
19. EllipseFit 3 User Manual Version 3 1 0 June 3 2014 Copyright Frederick W Vollmer 2014 Table of Contents License and Citation 1 Introduction 1 1 Installation 1 2 Example Data Files 2 Overview of Strain Analysis 3 Strain from Points 3 1 Fry Analysis 3 2 Normalized Fry Analysis 3 3 Mean Log Likelihood Function MLLF 4 Strain from Lines 4 1 Automated Wellman Analysis 4 2 Line Stretch Analysis 5 Strain from Ellipses and Polygons 5 1 Polygon Moment Equivalent Ellipses 5 2 Digitizing Ellipses 6 Ellipse Data Graphs 7 1 Elliott Polar Graph 7 2 Rd 7 3 Hyperboloidal Projections 7 Mean Ellipse Calculations 5 1 Shape Matrix Eigenvectors 5 2 Mean Radial Length MRL 5 3 Hyperbolic Vector Mean 5 4 Bootstrap Error Analysis 5 5 Simple Means and Centroids 8 Ellipsoid Calculation 8 1 Global Coordinates 8 2 Shan Ellipsoid Calculation 8 3 Error Analysis 9 Ellipsoid Data Graphs 9 1 Flinn Graphs 9 2 Nadai Graphs 10 Data Transformation 11 Data Synthesis 12 Image Analysis 12 1 Filtering 12 2 Edge Detection Acknowledgements References History License and Citation License EllipseFit 3 software and accompanying documentation are Copyright Frederick W Vollmer They come with no warrantees or guarantees of any kind The software is free and may be downloaded and used without cost however the author retains all rights to the source binary code and accompanying files It may not be redistributed or posted online
20. EllipseFit is the first available implementation of Shan s method Before giving an example calculation it is useful to compare it with some other methods Shan s method has been tested on synthetic and natural samples the following are some of the results of Vollmer 2010 Owens 1984 tested his method on a sample of slate from Dinorwic North Wales for which the strains had been calculated from reduction spots on 8 sections His data was also used by Launeau and Robin 2005 to test Robin s 2002 method Table 3 shows results of Vollmer s 2010 tests on Shan s method using Owen s data 6 5 A B R R AR Ad RT 6T ART MT 302 78 165 45 3670 165 3 083 165 700 0 587 0 700 3 082 165 700 0 002 0 000 301 7 95 35 2710 166 3 076 165 380 0 366 0 620 3 075 165 380 0 005 0 000 302 75 205 68 3010 166 3 024 165 310 0 014 0 690 3 023 165 310 0 003 0 010 201 71 370 60 6 170 173 6 418 172 780 0 248 0 220 6 420 172 780 0 001 0 000 178 71 75 15 5 000 0 4 618 179 090 0 382 0 910 4 618 179 090 0 002 0 000 18 79 16 7 30 5 570 10 593 7 870 0 353 2130 5 924 7 870 0 004 0 000 17 78 22 0 40 5 500 8 5 792 7710 0 292 0 290 5 793 7 710 0 003 0 000 19 78 18 0 30 6 000 7 5 987 8 200 0 013 1 200 5 989 8 200 0 001 0 000 Table 3 Results of test of Shan s 2008 method using data from Owens 1984 R o are the calculated b Table 4 section ellipses Misfits AR Ad indicate the error between calculated and measured ellipses Calculated section ellipses were used to bac
21. It is requested that acknowledgment and citation be given for any usage that leads to publication This software and any related documentation are provided as is without warranty of any kind either express or implied including without limitation the implied warranties or merchantability fitness for a particular purpose or non infringement The entire risk arising out of use or performance of the software remains with you Citation EllipseFit is the result of many hours of work over several decades Algorithms used in the program come from numerous sources however many have been developed by the author some of which have not yet been published and are the subject of papers in preparations I have released the program publicly with the hope that the structure and tectonics community will find it useful and ask forgiveness for the limited documentation as well as respect for publication priority In return for free use I request that any significant use of the software in analyzing data or preparing diagrams be cited and acknowledged in publications presentations or other works An acknowledgement could be I thank Frederick W Vollmer for the use of his EllipseFit 3 software Appropriate references include see References Vollmer 2010 discusses ellipse and ellipse fitting techniques including Shan s method and their implementation in EllipseFit Vollmer 2011a discusses methods for contouring finite strain on the unit h
22. R 0 333 Phi 0 85 Distance 0 126 See data grid for section residuals Bootstrap confidence intervals 5000 resamples Maximum A Stretch 0 973 Stretch 95 1 385 Stretch 99 3 603 Trend 0 186 Trend 95 0 269 Trend 99 0 369 Plunge 0 037 Plunge 95 0 058 Plunge 99 0 083 Intermediate B Stretch 0 106 Stretch 95 0 234 Stretch 9 0 415 Trend 0 187 Trend 95 0 273 Trend 99 0 382 Plunge 0 041 Plunge 95 0 057 Plunge 99 0 073 Minimum C Stretch 0 030 Stretch 95 0 063 Stretch 99 Trend Trend 95 Trend 99 Plunge Plunge 95 Plunge 99 Oo Oo OOOOOCO 117 031 043 056 014 020 026 EllipseFit User Manual Page 43 EllipseFit User Manual Page 44 This includes all 3 principal stretches and their trends and plunges with measures of error To view the results graphically first select Analyze gt Flinn Graph A Flinn graph Section 9 1 is a graph of the ratios A B Smax Sin Versus B C Sm Smn and is commonly used for displaying strain ellipsoid data e g Ramsay and Huber Now select Analyse gt Nadia Graph to display the results on a Nadai graph A Nadai graph Nadia 1950 Hossack 1968 Section 9 2 is based on natural or logarithmic strain which is also the basis for the hyberboldal projections discussed in Section 6 3 This provides an undistorted representation of the deviatoric strains a
23. Unstrain W M Strain ratio M Rectify SetMean kiz of5 Si 2 00 Sz 0 50 Yx 1 00 Yy 0 00 Transform X 0 00 Ya 0 00 Cancel Figure 11 The Transform Data dialog with values entered to unstrain the data Set Mean is only used with ellipse data Rectify resolves the offsets caused by the image transformation Figure 12 Fry graph of the unstrained 60 point data after using the Transform Data command to unstrain retrodeform the data using the calculated values EllipseFit User Manual Page 15 3 2 Normalized Fry Analysis As discussed in Section 3 1 the Fry analysis is a two dimensional solution to a three dimensional problem since initial particles are assumed circular instead of spherical Even if the particles have a uniform size a section through a sample will show them as different size particles One solution developed to overcome this is the normalized Fry analysis Erslev 1988 Erslev and Ge 1990 McNaught 1994 McNaught 2002 The distances between particles are normalized to account for the difference in the sizes of the particles which can greatly improve the sharpness of the central void Unfortunately the ellipse sizes and orientations are required for this and in most cases if the ellipse data is available it should used for the strain analysis following techniques in Chapter 7 Strain from Polygons However as mentioned in Section 3 1 a Fry analysis can provide different information regarding part
24. ain from two angulars of shear Journal of Structural Geology v 15 p 1359 1360 Ragan D M 2009 Structural Geology An Introduction to Geometrical Techniques 4th Ed John Wiley and Sons Inc 393 p Ramsay J G and Huber M I 1983 The Techniques of Modern Structural Geology Volume 1 Strain Analysis Academic Press London 307 p Ramsay J G 1967 Folding and Fracturing of Rocks McGraw Hill 568 p Robin P F 2002 Determination of fabric and strain ellipsoids from measured sectional ellipses theory Journal of Structural Geology 24 531 544 Rogers D F And Adams J A 1976 Mathematical Elements for Computer Graphics McGraw Hill New York 239 p EllipseFit User Manual Page 54 Shan Y 2008 An analytical approach for determining strain ellipsoids from measurements on planar surfaces Journal of Structural Geology 30 539 546 Shan Y and Xiao W 2011 A statistical examination of the Fry method of strain analysis Journal of Structural Geology v 33 p 1000 1009 Shimamoto T Ikeda Y 1976 A simple algebraic method for strain estimation from ellipsoidal objects Tectonophysics 36 315 337 Steger C 1996 On the Calculation of Arbitrary Moments of Polygons Technical Report FGBV 96 05 Forschungsgruppe Bildverstehen FG BV Informatik IX Technische Universitat Munchen Germany 18 p Twiss R J and Moores E 2007 Structural geology 2nd edition W H Freeman New York 736 pp Van der Pluij
25. ata synthesis for making artificial samples from random populations Chapter 12 Image Analysis discusses image analysis techniques including filtering and edge finding that can aid in highlighting particle edges prior to digitizing It is essential to be aware of the assumptions involved in strain analysis Refer to the referenced texts for a complete discussion An important consideration is whether the particles such as fossils or clasts record the same deformation as the rock In general this means whether there was a viscosity contrast between the particles and the matrix that encloses them This is discussed briefly in Chapter 3 EllipseFit User Manual Page 5 A second problem to consider is whether there was an initial preferred orientation of the particles this can be related to an initial sedimentary fabric or compaction Unimodal or orthogonal sedimentary fabrics and compaction essentially apply a deformation that is indistinguishable from a tectonic deformation without additional information Detection of initial fabrics is discussed briefly in Chapter 7 Similarly volume change is difficult to quantify and strain is generally calculated with volume equivalent to an initial unit sphere This User Manual is written in a tutorial fashion in order to become acquainted with the program it is a good idea to work through the examples provided This User Manual is also not yet finished itis a work in progress 3 Strain from P
26. ces can then be photographed or thin sections made and photographed Keeping thin sections correctly oriented is challenging keep the strike arrow parallel to one side and pointing right To minimize confusion make sure each photograph is oriented with the section strike to the right and with the dip line down Careful photography is best but EllipseFit can rotate an image an arbitrary amount if necessary see Chapter 12 Image Analysis It is better to do it now than after digitizing the data although EllipseFit can rotate the data if needed see Chapter 11 Data Transformation One last important detail is to keep track of the viewing direction The strike arrow must point to the right in the section image This means it is dipping towards you If the strike arrow points left you are looking at the underside of the section and it is dipping away from you If so you need to flip the image horizontally about a vertical axis EllipseFit can do this Edit gt Rotate Image gt Flip Horizontal and it is better to fix the image before digitizing Vertical sections are not a problem if the recorded strike is kept to the right in the images If one is lucky to have outcrops with well exposed sections the process is greatly simplified but the same principles apply Fields Alternate Symbol Definition N Datum number KY eZ Global coordinates North East Down X Y Local coordinates normally strike and dip line Strike Theta 0 Strike of secti
27. der EllipseFit in your operating system s application preferences folder To deinstall simply delete the EllipseFit application folder and optionally delete the preference folder No other files are installed on your computer No administrative permissions are required to install EllipseFit and it is possible to keep a copy on a thumb drive to run on any computer 1 2 Example Data Files The included example files and images can be used to determine input data formats These are simple files that can be generated using a text editor or spreadsheet EllipseFit 3 will read comma separated csv tab separated tsv and Open Document ods formats The header line indicates the type of data required in each column The included example files are named to indicate their contents this is not required EllipseFit will examine the headers to determine the available data and extra columns are ignored E2 Ramsay and Huber 1983 small csv E2 Ramsay and Huber 1983 small jpg E2 Ramsay and Huber 1983 large jpg Example ellipse data and thin section photomicrograph from Ramsay and Huber 1983 This data type can contain X Y coordinates for Fry type analyses or complete ellipse data including X Y A B R Phi axes data Note that there are small and large versions I use the large version which does not include a data file for teaching E3 Hossack 1968 csv Example ellipsoid data from Hossack 1968 with A B C axes data for Flin
28. determining two and three dimensional strain from oriented photographs and is designed for field and laboratory based structural geology studies The graphical interface and multi platform deployment also make it ideal for introductory or advanced structural geology laboratories I use the software to teach structural geology at SUNY New Paltz where hundreds of students have used it in laboratory and field studies EllipseFit is currently implemented for Windows 32 Macintosh 10 5 and Linux Ubuntu 64 bit platforms EllipseFit is suitable for determining two and three dimensional strain using various objects including center points Fry analysis lines ellipses and polygons Polygons include ooids pebbles fossils or particles of any initial shape The analysis of strain from polygons is widely applicable to many rocks in thin section hand sample or suitable outcrops EllipseFit allows digitizing polygons directly or indirectly by using a flood fill method EllipseFit converts them to moment equivalent ellipses and the mean ellipse is equivalent to the strain Mulchrone and Choudhury 2004 Given three or more oriented sections EllipseFit can calculate the three dimensional strain using the method of Shan 2008 This User Manual was prepared for the strain workshop at the 2014 Structural Geology and Tectonics Forum at the Colorado School of Mines with Paul Karabinos and Matty Mookerjee and is not however complete EllipseFit 3 has numero
29. ea Norway where the data graphed in Figures 50 and 51 was collected by Hossack 1968 Photograph by F W Vollmer EllipseFit User Manual Page 52 Acknowledgements I thank Y Shan K Burmeister S Treagus G Mitra S Wojtal H Fossen P Karabinos M Mookerjee J Davis W Dunn E Erslev Y Kuiper R Bauer D Wise D Czeck N Mancktelow J M Crespi B M Klemm S Dirringer and others for suggestions comments discussions and encouragement Y Shan kindly provided Fortran code for his MLLF calculation I especially thank R Twiss W Means and P Hudleston mentors whose clear thinking and quantitative approaches inspired me as a student References Brandon M T 1995 Analysis of geological strain data in strain magnitude space Journal of Structural Geology 17 1375 1385 Cloos E 1947 Oolite deformation in the South Mountain Fold Maryland Geological Society of America Bulletin 58 843 918 Cloos E 1971 Microtectonics Along the Western Edge of the Blue Ridge Maryland and Virginia The Johns Hopkins Press Baltimore and London 234 p Crespi J M 1986 Some guidelines for the practical application of Fry s method of strain analysis Journal of Structural Geology 16 p 1327 1330 Davis J C 1986 Statistics and Data Analysis in Geology Wiley 646 p Dirringer S and Vollmer F W 2013 A test of the analytical Wellman and mean polygon moment ellipse methods of strain analysis using a sample of defo
30. he process was subjective and single outliers significantly effected the result The result for 43 data points was R 2 761 0 50 RMS 0 294 parallel to cleavage They concluded that the necessary assumptions about initial geometry for the analytical Wellman method were not met and the polygon method with no such required assumptions about initial geometry was preferred EllipseFit User Manual Page 28 Figure 30 Sample of deformed graptoliferous slate used by Dirringer an Vollmer 2013 for comparison of the automated Wellman and mean polygon moment ellipse methods Figure 31 The graptoliferous slate sample of Figure 24 after retrodeforming to remove the strain calculated by the mean polygon moment ellipse method R 2 079 177 48 4 2 9 1 5 2 EllipseFit User Manual Page 29 Line Stretch Analysis Documentation in preparation Strain from Ellipses and Polygons Documentation in preparation Digitizing Ellipses Documentation in preparation Moment Equivalent Polygons Documentation in preparation EllipseFit User Manual Page 30 6 Ellipse Data Graphs Documentation in preparation 6 1 Elliott Polar Graph Documentation in preparation The polar Elliott graph Elliott 1970 is a polar plot of the natural log R and 26 This is a natural parameter EEE space for strain and the grapg is a simple hyperboloidal projection that gives an undistorted representation Yamaji 2008
31. he right and Y is down the image These coordinate axes are indicated by the blue lines on the top and left of the Image Window The angle 0 is the positive angle clockwise from X This coordinate system was chosen to simplify the relationship to the global coordinates referred to here as X Y Z and to simplify the calculation of the three dimensional strain ellipsoid The global coordinates are equivalent to North East Down NED In Figure 39 the gray plane is a section plane that corresponds to an image analyzed for two dimensional strain as discussed in earlier chapters The X axis is parallel to the strike of the plane using the standard right hand rule e g Pollard and Fletcher 2005 as shown in Figure 37 The strike is given by 6 the clockwise angle from North the standard azimuth in degrees The dip of the plane is the angle 6 The calculated strain ellipse is given by R A B Lmax Lmin and the angle from X which is its pitch in global coordinates This is referred to here as a section ellipse In order to calculate the strain ellipsoid from the section ellipses each section ellipse must undergo a coordinate transformation from local X Y coordinates to global X Y Z coordinates This is done automatically by EllipseFit but the user must take great care to properly prepare samples Time taken at this stage will save much aggravation later on A sample collected in the field must be carefully oriented rec
32. icle versus matrix strains The digitizing of ellipses is discussed in Chapter 5 Strain from Ellipses so for an example of this analysis open the image file E2 Ramsay and Huber 1983 small jpg and the data file E2 Ramsay and Huber 1983 small This is the 252 point data set used in Section 3 1 The data is overlain on the image and if the Binoculars Icon is selected you can select individual particles that are highlighted in the Data Window and the Fry Graph This selection method is implemented for most of the graphs discussed in subsequent chapters The Fry graph will look like Figure 5B EllipseFit User Manual Page 16 Figure FE EllipseFit Image Window with ellipse data overlain Selecting the Binoculars Icon as shown allows interactive selectiin of particles that are highlighted in the Data Window as well as on data graphs including the Fry Graph To zoom in on the central void open the Preferences dialog Gear Icon deselect Auto scale and enter 50 EP PR for the Graph radius as shown in Figure 14 Bohr RR ENTER Ratio _ Enhance or Graph radius 50 00 re M Show all points Nadai _ Fit ellipse Selection factor 1 050 Are Map Mx right Normalized radius 1 50 My down _ Auto scale 4 CD Figure 14 Settings to display the central void without normalizing EllipseFit User Manual Page 17 The unnormalized graph is displayed in Figure 15 k a i Are 2 1 Pe 8 het ae
33. ilable points however note that some particles may not meet the required assumptions In particular note that the centers of the particles in two dimensions do not generally correspond to their three dimensional centers as they lie on an arbitrary plane cutting through the rock so the assumption of of a uniform cutoff is weakened This is discussed further in Section 3 2 Normalized Fry Analysis Additionally it is desirable to select approximately equal numbers of particles in all directions so the point density is not biased by direction This is one reason to maintain a uniform point density in a circular area while digitizing and why having the interactive Fry graph open can assist in particle selection This is discussed further in Section 3 2 Mean Log Likelihood Function MLLF EllipseFit User Manual Page 8 N m Est Be BAAS Be ARCS 285 8 51 4 259 7 145 5 Figure 3 Fry graph after digitizing 20 adjacent particle centers The generated points are highlighted On the right note the presence of the spurious data point each point is mirrored about the center generated by clicking too close to an existing point i e an operator error which can be deleted EllipseFit Preferences If you wish to change the size of the AC digitized points click the Gear Icon or ne Ft at from which you can set most of the EllipseFit preferences Note some selections have multiple pages use the left right arrows Command lt gt to g
34. inn and Nadia graphs are interactive selecting a point in one will automatically select the corresponding data point on the other graph and in the Data Window Figure 50 Log Flinn graph displaying deformed pebble ellipsoids Bygdin area Norway from Hossack 1968 This graph is interactive with the Binoculars Icon selected data points can be selected and will be simultaneously updated on the Nadai graph and in the Data Window the selected data point is also displayed in Figure 51 EllipseFit User Manual Page 50 9 2 Nadai Graphs Documentation in preparation The Nadai graph Nadia 1950 Hossack 1968 Section 9 2 is based on natural or logarithmic strain which is also the basis for the hyberboldal projections discussed in Section 6 3 This provides an undistorted representation of the deviatoric strains and is preferred by many for that reason Brandon 1995 ADO Nadai Graph Be He Strain Symmetry Nu 0 0 Figure 51 Nadia graph displaying deformed pebble ellipsoids Bygdin area Norway from Hossack 1968 This graph is interactive with the Binoculars Icon selected data points can be selected and will be simultaneously updated on the Flinn graph and in the Data Window the selected data point is also displayed in Figure 48 EllipseFit User Manual Page 51 m Zr y 2 P ge A re ut mL a ya u ad p WG FREE nen me gt Figure 52 Deformed pebble conglomerate Bygdin ar
35. ion 3 1 The results from the Log Window are N 252 Pairs 31626 Normalized Enhanced Selection factor Enhanced pairs Best Fit Ellipse 1 050 142 Automatic n 142 R 1 581 D 24 46 RMS 0 1383 Figure 17 Graph of the normalized data Note the better resolution of the central void Figure 19 Fry graph with ellipse fitted to the enhanced normalzed points Again the RMS is a measure of the deviations of the residuals and can be used to refine the selection factor However note that smaller number of points will generally have a smaller RMS For example three points give RMS 0 so finding the minimum RMS is not a valid strategy EllipseFit User Manual Page 19 Section 3 3 Mean Log Likelihood Function MLLF Calculating the strain from a sample of points should ideally require no additional information about the particle s shapes and there are a number of methods that have been developed for this purpose e g Shan and Xiao 2011 Waldron and Wallace 2011 Mulchrone 2013 EllipseFit implements the mean log likelihood function MLLF method of Shan and Xiao 2011 They examine the statistics of a truncated Poisson distribution and define the MLLF as the average sum of the log probability distribution function PDF of each individual point in the deformed state This is related to the density distribution around each point The PDF in the deformed state is related to the pre deformation PDF by the sha
36. ixed bug in fill ellipse routine causing hangs at high thresholds Fixed bug causing crash when opening page size dialog Added strain map Added synthesize data to create data sets Added transform data to strain unstrain shear etc data Changed names of digitize routines to reflect the objects e g center points ellipses polygons EllipseFit User Manual Page 56 instead of the results e g polygon moment ellipse Changed names of graphs to more common specific names attr buting authors Fry Flinn etc instead of generic names Internal change in form communication from flags and timers to messages Numerous additional fixes and changes 3 0 1 6 April 2014 Fixed bug effecting symbol colors n svg graphics Cleaned up the polar graph Fixed cursor status strings on graphs Fixed up contouring preferences Added axial ratio Flinn type graph Added octahedral Nadai Hsu type strain graph Added ellipse digitizing with polygon fill and moments Fixed file save warning Numerous internal changes 3 0 0 24 March 2014 First public release 3 0 0 28 August 1 2012 Initial prerelease version
37. k calculate bT Table 4 and RT oT Misfits ART AoT indicate that the method does retrieve b From Vollmer 2010 CON MD OF FP N FF e The test involves calculating the strain ellipsoid from the section ellipses then from the calculated ellipsoid determining the two dimensional sections corresponding to the input data These are reported as R o in the table The difference is a residual These are reported as AR A9 in the table An additional result is shown by using the calculated section ellipses to calculate an ellipsoid These are reported as ART AQT and are negligible indicating success in retrieving the ellipsoid Table 4 shows the results of the ellipsoid calculation from this sample as calculated using the methods of Owens 1984 Robin 2002 and Shan 2008 The results are compared graphically in Figure 40 The calculations and graphs were done in EllipseFit 2 Vollmer 2011 and Orient 2 Vollmer 2012 There negligible differences between the results using the methods of Robin and Shan the results using the method of Owen deviate a small amount from them EllipseFit User Manual Page 40 Axis Owens Robin Shan b b S1 2 340 2 626 2 565 2 567 t1 29 000 37 100 34 960 34 970 p1 10 000 11 300 10 890 10 890 S2 1 197 1 112 1 131 1 131 t2 122 000 129 500 127 350 127 360 p2 14 000 11 700 12 230 12 230 S3 0 357 0 343 0 345 0 345 t3 265 000 264 500 264 440 264 440 p3 73 000 73 600 73 510 73 510 Table 4 Comparison
38. ls of the data from the fitted ellipse RMS is a common way to express goodness of fit of least squares solutions It is not a measure of the error in the strain calculation and is not technically an error It is however a measure of how closely the digitized points fit the ellipse A small RMS means that the entered points lie close to an ellipse It makes a good class exercise for students to solve and compare their results and RMS EllipseFit User Manual Page 12 Asa final step in this analysis select the Edit gt Transform Image command and enter the results into the dialog as in Figure 9 The image will be unstrained to remove the calculated strain as shown in Figure 10 an Transform Image Unstrain BE M Strain ratio Set Mean R 1 758 d 25 45 S 1 00 S 1 00 Preview Cancel ok Figure 9 The Transform Image dialog with values entered to unstrain the mage EllipseFit User Manual Page 13 Figure 10 The oolith photomicrograph after being unstrained using EllipseFit s Image Transform command EllipseFit User Manual Page 14 Next select the Analyze gt Transform Data command and enter the calculated values as shown in Figure 11 Press Transform and then Accept The data is unstrained using the calculated values as shown by the Fry graph in Figure 12 The Rectify option resolves the offsets caused by the image transformation so the data points remain centered over the particle centers or Transform Data
39. m B A and Marshak S 2004 Earth structure 2nd edition W W Norton New York 656 p Vollmer F W 1995 C program for automatic for automatic contouring of spherical orientation data using a modified Kamb method Computers amp Geosciences 21 31 49 Vollmer F W 2010 A comparison of ellipse fitting techniques for two and three dimensional strain analysis and their implementation in an integrated computer program designed for field based studies Abstract T21B 2166 Fall Meeting American Geophysical Union San Francisco California 1 Vollmer F W 2011a Automatic contouring of two dimensional finite strain data on the unit hyperboloid and the use of hyperboloidal stereographic equal area and other projections for strain analysis Geological Society of America Abstracts with Programs v 43 n 5 p 605 2 Vollmer F W 2011b Best fit strain from multiple angles of shear and implementation in a computer program for geological strain analysis Geological Society of America Abstracts with Programs v 43 3 Waldron J W F and Wallace K D 2011 Objective fitting of ellipses in the centre to centre Fry method of strain analysis Journal of Structural Geology 29 p 1430 1444 Wellman H G 1962 A graphic method for analyzing fossil distortion caused by tectonic deformation Geological Magazine 99 384 352 Wheeler J 1984 A new plot to display the strain of elliptical markers Journal of Structural Geology 6 417 423
40. n and Nadai graphs ES Owens 1984 csv Example ellipse section data fron Owens 1984 for calculating the three dimensional strain ellipsoid from three or more faces using Shan s 2008 method The strikes and dips of each section must be included LA Ragan 1985 F10 1a csv LA Ragan 1985 F10 1a png Example line angular shear data and image from Ragan 1985 for analytical Wellman type analysis Vollmer 2011 Each data point requires the endpoints of two lines that originally had a constant angle This is an analytical solution to the classic multiple brachiopod problem illustrated in a number of structural geology texts LS Ragan 2009 T14 9 csv Example line stretch data for lines with known initial and final lengths such as boudins and folds EllipseFit does not yet provide digitizing of this type of data Please contact me if this would be of interest Note that the LS data is from fold flattening index example Ragan 2009 which is EllipseFit User Manual Page 3 mathematically related MLLEF Test 60 csv Sample of 60 points used to test the maximum mean log likelihood function MLLF method of Shan and Xiao 2011 EllipseFit User Manual Page 4 2 Overview of Strain Analysis Geological strain analysis and theory is an important aspect of structural geology that is covered in numerous textbooks e g Means 1976 Hobbs Means and Williams 1976 Ragan 1985 Marshak and Mitra 1988 Van der Pluijm and Marshak 200
41. nd is preferred by many for that reason Brandon 1995 A B B C Figure 44 Flinn graph of the ellipsoid axial ratios determined from the Shan calculation with a 95 confidence region strain Symmetry Mu 0 0 0 0 Figure 45 Nadai graph of the ellipsoid axial ratios determined from the Shan calculation with a 95 confidence region EllipseFit User Manual Page 45 The calculated strain has large 95 error region as shown in both graphs Examining the data Figure 43 shows that section 6 has the largest distance residual Select it delete it and preform the ellipsoid calculation again Figure 46 shows the updated Flinn graph which now shows both solutions Similarly the Nadia graph has been updated to reflect the newly calculated results 5 E B C Figure 46 Flinn graph of the ellipsoid axial ratios determined from the Shan calculation with 95 confidence regions after deleting section 6 Strain Symmetry Nu 0 0 00 Figure 47 Nadia graph of the ellipsoid axial ratios determined from the Shan calculation with 95 confidence regions after deleting section 6 EllipseFit User Manual Page 46 Finally the resulting axes are plotted on a lower 0 hemisphere equal area projection using Orient 2 1 2 Vollmer 2010 The strain axes calculated from all 8 sections are plotted as circles and the axes section 6 removed are plotted as diamonds Red Smax green Rn blue Rmin
42. o through them You can preview the effect of preference changes before setting then with the OK button Fry Graph Symbols 0O Data symbol Mi MLLF points _ best fit ellipse JM MLLF ellipse CI W Frame Start point ol Digitize point To view the data as a Strain Map select BB Cursor poiri Analyze gt Strain Map This displays the data as particle centers this population oun Gar can be strained and unstrained as SEC BR 5 Preview Cancel F OK described in Chapter 10 Data Transformation Figure 4 The EllipseFit Preferences Dialog where most preferences are set Note the left right arrows used to scroll to additional pages if present EllipseFit User Manual Page 9 Figure 5 is the graph after carefully selecting 60 particle centers a probable minimum number for analysis Shan and Xiao 2011 and after digitizing 252 points essentially all of them Figure 5 Fry graphs after digitizing 60 carefully selected points and after digitizing 252 points essentially all of them These images are PNG files as saved from EllipseFit To zoom in for a better image of the central void open the Preferences Dialog Gear Icon uncheck Auto scale and enter a number smaller than the displayed Data radius Figure 6 ANA Fry Graph ale da A A E 2 BO EllipseFit Preferences Digitize 5 2 Fry Graph Settings Polar A Rf d Normalize Data radius 617 25 i 2 Ratio Enhance oe i Me a Se a
43. of calculated strain ellipsoids Owens from Owens 1984 Robin from Launeau and Robin 2005 unweighted method of Robin 2002 Shan b from Vollmer 2010 Shan s 2008 method b is a test to retrieve b The data is graphed in Figure 38 From Vollmer 2010 a C K R a AB R Bs e a0 BIC Figure 40 Comparison of calculated strain ellipsoids O Owens 1984 R Launeau and Robin 2005 using unweighted method of Robin 2002 S EllipseFit using Shan s 2008 method From Vollmer 2010 The file ES Owens 1984 csv contains the 8 section ellipse data from Owens 1984 Open this file in EllipseFit The data as displayed in the Data Window is shown in Figure 41 There are 8 section ellipses for each there is the Max and Min the axial lengths Lmax Lin the strain ratio R Max Min Phi 6 the pitch of R from the X axis X strike the strike angle 0 and the dip angle see Figure 39 This is data then that in EllipseFit would be determined from oriented photographs of each of the 8 sections EllipseFit User Manual Page 41 Select the command Analyze gt Calculate Ellipsoid and the Calculate Ellipsoid Dialog is displayed as in Figure 42 The results will be written to the Log Window Checking Append results will append the ellipsoid results to the open Data Window so it can be plotted on Hsu and Nadia graphs Check Save orientations to save the trends and plunges of the principal axes to a
44. oints It is common in nature for objects to be distributed randomly but with some minimum cutoff distance between them A random distribution in space follows a Poisson distribution see for example Davis 1986 basically a distribution gotten by throwing pingpong balls randomly into an empty room However the centers of the pingpong balls can never touch giving a cutoff distance of twice the radius of the balls This distribution is called a truncated Poisson distribution e g Shana and Xiao 2011 Examples of this type of data include the centers of clasts in many sedimentary rocks such as sandstones and conglomerates The centers of phenocrysts in igneous rocks where nucleation of new crystals is prevented in proximity to existing crystals due to the chemical gradient is another example Note that it the particles have a different viscosity than the enclosing matrix even if they are perfectly rigid it is possible to get an estimate of the strain of the rock Thus it is possible to extract different information than by an analysis of the particle shapes 3 1 Fry Analysis The basic idea for methods utilizing point distributions e g Ramsay and Huber 1983 is that the distance between the initial object centers is the same in all directions and after a deformation the particles are closer in some directions and further in others This new distribution will be elliptical in two dimensions or ellipsoidal in three dimensions A Fry analy
45. on following right hand rule Dip Delta Dip of section plane from horizontal Max Int Min A B C Axes of an ellipsoid Max Min A B Axes of a sectional ellipse R Strain ratio Max Min Phi Pitch amp Angle in XY from X to ellipse axis Max R Best fit estimate of R Phi D Best fit estimate of D Delta R AR Misfit between R and R Delta Phi Ab Misfit between b and amp S1 S2 S3 S1 S2 S3 Principal stretches Trend ti t2 t3 Trend of ellipsoid axis Plunge p1 p2 p3 Plunge of ellipsoid axis Table 2 Data file field headers and corresponding symbols The headers define columns in data files read and written by EllipseFit EllipseFit User Manual Page 39 8 2 Shan Ellipsoid Calculation Shan s method for determining the strain ellipsoid from section ellipses has similarities to the methods of Owens 1984 and Robin 2002 as they are all direct non iterative calculations Shan s method however also allows the inclusion of stretching lineation data so has additional flexibility Ellipsoids can be represented by shape matrixes and the solution desired is the optimal shape matrix Each section ellipse or section lineation adds one or two linear equations describing the shape matrix which can be solved as an eigenvalue problem Shan solved the problem by assuming the matrix can be located on a six dimensional hypersphere centered at the origin and recognized that the smallest eigenvector of the data matrix is an optimal solution
46. ons 1 to 60 are the data points and 1 to 180 are the grid search in degrees The R grid search values 0 1 to 20 0 by default and the 1 to 50 distance search loops are not displayed The MLLEF search is computationally intensive especially for cross validation during some test runs I set my laptop on marble coasters to keep it from overheating After about 6 hours on a 3 06 GHz Intel Core 2 Duo iMac the process completes and the dialog displays OK You can cancel the run at any time and the results of the completed passes will be displayed Mean Ellipse Calculations MLLF Test 60 tTSV 2014 05 31 16 30 46 N 60 a 7 Log Likelihood Progress Pass 0 1 69 of 10 60 180 Time 14 seconds Cancel Figure 23 Progress dialog for the MLLF grid search with cross validation Log Likelihood Progress Pass 10 51 180 of 10 51 180 Time 6 hours 19 minutes Figure 24 Progress dialog for the MLLF grid search with cross validation when complete Pass Mean LL R Phi Cutoff Stat Density 0 0 31829 1 90 25 00 86 98953 0 67361 0 84687 1 0 31610 1 90 25 00 86 98953 0 68773 0 86122 2 0 31603 1 90 25 00 86 98953 0 69522 0 87607 3 0 31882 1 90 25 00 86 98953 0 67496 0 89144 4 0 31651 1 90 25 00 86 98953 0 68968 0 90736 5 0 31536 1 90 25 00 86 98953 0 70494 0 92386 6 0 32428 1 80 23 00 87 24708 0 68393 0 93542 7 0 31554 1 90 25 00 86 98953 0 69945 0 95872 8 0 31327 1 90 25 00 86 98953 0 71578 0
47. ording its strike and dip other conventions are fine but the strike is the X coordinate axis so is used here A suitable marking is a strike arrow and a dip tick Figure 39 if possible on a surface that is not Figure 39 Coordinate system for section ellipses The global coordinates are X North Y East and Z Down NED The plane with the section ellipse has a strike 8 using the right hand rule and dip 6 The section ellipse has a pitch p and R A B where A and B are the maximum and minimum axes A suggested strike arrow and dip tick marking is shown EllipseFit User Manual Page 38 overhanging A minimum of three sections must be made through the sample although more is preferred Shan s method Section 8 2 relaxes this requirement if lineation data is used as well but Vollmer 2010 showed that the error range in natural samples can be large so a minimum of three sections is recommended If available lineation data can supplement the section ellipses Section 8 2 The sections should be made at high angles to each other but it does not need to be 90 a restriction of some methods e g Shimamoto and Ikeda 1976 In making the sections be careful not to destroy the strike arrow and dip tick it happens The sample can then be taken outside away from magnetic fields and reoriented The strikes and dips of the section planes can then be measured and a strike arrow and dip tick marked on each face The fa
48. pe and orientation of the central void giving as parameters a cutoff distance the ratio R and the orientation The function is complex however and is solved using a gird search to locate the maximum MLLEF The search is over the range 0 to 179 in steps of 1 and R 1 to 20 in steps of 0 1 The latter value is the default that can be changed if desired a smaller value will speed up the search Once R and are determined the sample is retro deformed and a 50 step search is done to locate the cutoff radius Shan and Xiao 2011 further suggest an approach to improve the results using a cross validation technique for detecting spurious points by sequentially removing up to 10 points the default value in EllipseFit and repeating the search These algorithms were implemented by Y Shan in a Fortran program which he provided EllipseFit has been carefully tested to insure that identical results are obtained The result are the best estimates values of R initial cutoff distance and a set of neighborhood points This method has advantages in that it is a robust numerical solution and one that uses all of the points to define the central void In comparison the enhanced normalized Fry method that only examines the points close to the void A disadvantage of the method is the computing time required to calculate the solution In particular the cross validation can take several hours Shan and Xiao 2011 also note wryly that it i
49. rain This allows numerous geological objects to be used for strain analysis using objective calculations developed for ellipse analysis Chapter 6 Ellipse Data Graphs covers graphical techniques for two dimensional strain plots including R graphs and polar Elliott graphs which are types of hyperboloidal projections Hyperboloidal projections are analogous to spherical projections such as the stereographic and equal area projections that are used to create stereonets and Schmidt nets respectively familiar to students of structural geology Chapter 7 Mean Ellipse Calculation discusses the calculation of a mean ellipse from a sample of ellipses As discussed in Chapter 5 these calculations apply to polygons as well as ellipses as the use of polygon moment equivalent it ellipses removes the requirement that particles were initially elliptical The techniques mentioned thus far are related to two dimensional strain analysis Chapter 8 Ellipsoid Calculation covers the more complex steps involved in determining three dimensional strain ellipsoids from oriented sections for which the two dimensional strain ellipse has been determined Chapter 9 Ellipsoid Graphs covers strain graphs used to display this type of data Flinn and Nadia graphs Chapter 10 Data Transformation discuses methods for transforming data sets including unstraining or retrodeforming data sets and images to their pre deformation state Chapter 11 Data Synthesis covers d
50. ral Geology 24 p 311 322 Mitra S 1978 Microscopic deformation mechanisms and flow laws in quartzites within the South Mountain anticline Journal of Geology 86 p 129 152 Mookerjee M and Nickleach S 2011 Three dimensional strain analysis using Mathematica Journal of Structural Geology 33 p 1467 1476 Mulchrone K F O Sullivan F Meere P A 2003 Finite strain estimation using the mean radial length of elliptical objects with bootstrap confidence intervals Journal of Structural Geology 25 529 539 Mulchrone K F 2005 An analytical error for the mean radial length method strain analysis Journal of Structural Geology 27 1658 1665 Mulchrone K F 2013 Fitting the void Data boundaries point distributions and strain analysis Journal of Structural Geology 46 22 33 Nadai A 1950 Theory of Flow and Fracture of Solids McGraw Hill New York 572 p Onasch C M 1986 Ability of the Fry method to characterize pressure solution deformation Tectonophysics 122 187 193 Owens W H 1984 The calculation of a best fit ellipsoid from elliptical sections on arbitrarily orientated planes Journal of Structural Geology 6 571 578 Pollard D D and Fletcher R C 2005 Fundamentals of structural geology Cambridge University Press Cambridge 463 500 p Ragan D M 1985 Structural Geology An Introduction to Geometrical Techniques 3rd Ed John Wiley and Sons Inc 393 p Ragan D M and Groshong R H 1993 Str
51. rmed Ordovician graptoliferous slate from the Taconic orogen New York Geological Society of America Abstracts with Programs 247 52 Dunne W M Onasch C M Williams R T 1990 The problem of strain marker centers and the Fry method Journal of Structural Geology 12 p 933 1990 Dunnet D 1969 A technique of finite strain analysis using elliptical particle Tectonophysics 7 117 136 Dunnet D and Siddans A W B 1971 Non random sedimentary fabrics and their modification by strain Tectonophysics 12 307 325 Efron B 1979 Bootstrap methods Another look at the jackknife Annals of Statistics 7 1 26 Elliott D 1970 Determination of finite strain and initial shape from deformed elliptical objects Geological Society of America Bulletin 81 2221 2236 Erslev E A 1988 Normalized center to center strain analysis of packed aggregates Journal of Structural Geology 10 201 209 Erslev E A Ge H 1990 Least squares center to center and mean object ellipse fabric analysis Journal of Structural Geology 8 1047 1059 Fisher N I Lewis T and Embleton B J J 1987 Statistical Analysis of Spherical Data Cambridge University Press 329 p Flinn D 1962 On folding during three dimensional progressive deformation Quarterly Journal of the Geological Society of London 118 p 385 433 Flinn D 1978 Construction and computation of three dimensional deformations Journal of the Geological Society of London 135 p
52. s a pity that the treatment does not require the Fry plot which will disappoint structural geologists who prefer manual manipulation and visual appreciation To assist I have tried to make the output plot as visually pleasing as possible To run a test sample open the file MLLF Test 60 csv This data is the 60 point oolth sample used in section 3 1 and was carefully selected to avoid spurious points and to avoid a directional bias EllipseFit User Manual Page 20 Select the command Analyze gt Calculate Ellipse Note that the only available options are the MLLF options the other options all require ellipse data Select Mean log likelihood leave Cross validate off as in Figure 20 and press OK A progress dialog will appear as in Figure 21 the display shows the search iteration passes in degrees and is done at 180 The process should complete in less than a minute and the results displayed in the Log Window and on the Fry graph Figure 22 a 7 Log Likelihood Progress Pass 52 of 180 Time 12 seconds Cancel Figure 21 Progress dialog for the MLLF grid search without cross validation The results reported in the log file are N 60 MLLF Calculations Pass Mean LL R Phi Cut 6 0 31829 1 90 25 00 86 98 MLLF Results Point statistics Number Calculated density 0 Real density 0 Results Mean log likelihood 0 R strain ratio 1 Phi angle of max strain axis 25 Cutoff rad us 86 Method
53. s of the residuals of the data from the fitted ellipse It is a measure of goodness of fit of the ellipse but is not technically an error The RMS will be zero for two line pairs The calculation includes the constriction line so the ellipse has 9 point pairs including the 8 data points In theory objects like graptolites that have a constant non perpendicular angle between stipe and thecae can be treated in the same fashion Ramsay 1967 Dirringer and Vollmer 2013 compared the automated Wellman method and the mean polygon moment ellipse method Section 5 1 using a sample of slate with deformed Ordovician graptolites The sample was oriented with the slaty cleavage as the X axis The center lines and lower thecae lines were digitized in 120 locations for the Wellman test only one species had clearly defined thecae lines The outlines of 31 whole graptolites and 38 partial graptolites were digitized for the polygon method test The mean polygon moment ellipse was R 2 079 0 122 177 48 4 57 parallel to the slaty cleavage The polygon method does not require assumptions about initial shapes only that they are initially random Interpreting the data for the analytical Wellman method was problematic as it many outliers around a central ellipse Removal of 77 outliers believed to be due to initial variations in thecae angle was required before the ellipse could be clearly resolved While most outliers could be clearly identified t
54. sis Fry 1979 is an important and widely used technique for analyzing this type of data and there is an extensive literature on it and its variations e g Hanna and Fry 1979 Crespi 1986 Onasch 1986 Erslev 1988 Erslev and Ge 1990 Dunne Onasch and Williams 1990 McNaught 1994 McNaught 2002 Shan and Xiao 2011 Waldron and Wallace 2011 Mulchrone 2013 A Fry analysis can be simply done with two pieces of tracing paper by tracing all of the particle centers on one sheet then drawing a center point on a second sheet overlain on the first and then sequentially moving the center point to each point and trace each point For n initial points this generates ny n 2 n 2 points which is a lot of points to draw by hand To illustrate the use of the method in EllipseFit start EllipseFit and open the file File gt Open E2 Ramsay and Huber 1983 large jpg This is a photograph of a deformed ironstone oolith in thin section from Ramsay and Huber 1983 that is widely used as a test image for strain analysis For point digitizing make sure the red Point Icon EllipseFit User Manual Page 6 second from left is displayed Digitize gt Center Point and the green Plus Icon is selected Digitize gt Add Tool as in Figures 1 and 2 ri TEN z H i 1 Use the zoom tools to enlarge the image and click on one particle center The Data Window will open and display a highlighted line of
55. ted by the bootstrap The section ellipses show the back calculated values for R and 6 and the corresponding residuals The last columns the distance residuals which are the hyperbolic distance residuals Dist Res Max 95 Int 95 Min 95 0 1766 0 1291 0 0328 0 0462 0 1081 0 2158 400 ES Owens 1984 csv a x H 2 N Max Int Min R Phi Strike Dip R Calc PhiCalc RRes Phi Res 1 1 16 5000 4 5000 3 6667 165 00 302 00 78 00 3 0856 165 70 0 5810 0 70 2 2 9 5000 3 5000 2 7143 166 00 301 00 77 00 3 0791 165 38 0 3648 0 62 3 3 20 5000 6 8000 3 0147 166 00 302 00 75 00 3 0266 165 31 0 0119 0 69 4 4 37 0000 6 0000 6 1667 173 00 201 00 71 00 6 4159 172 78 0 2492 0 22 5 5 7 5000 1 5000 5 0000 0 00 178 00 71 00 4 6179 179 09 0 3821 0 91 6 6 16 7000 3 0000 5 5667 10 00 18 00 79 00 5 9208 7 87 0 3541 2 13 i 7 22 0000 4 0000 5 5000 8 00 17 00 78 00 5 7901 7 71 0 2901 0 29 8 8 18 0000 3 0000 6 0000 7 00 19 00 78 00 5 9854 8 20 0 0146 1 20 gt a iach RE TCC a Oe ie Se el a ee dC COT Figure 43 The Data Window after calculating the optimal ellipse using Shan s method EllipseFit User Manual Page 42 The Log Window reports the following Best Fit Ellipsoid Calculations ES Owens 1984 2014 06 02 19 51 39 N 8 Ellipsoid axes as stretches Maximum A 2 565 Trend 35 02 Plunge 10 90 Intermediate B 1 132 Trend 127 41 Plunge 12 22 Minimum C 0 344 Trend 264 44 Plunge 73 51 Root mean square of section residuals
56. the yellow cursor Continue digitizing the remaining line pairs Figure 28 zarea s shows the graph after three line pairs The yellow cross cursor highlights the corresponding data point intersection and parallelogram and the data is selected in the Data Window If the Binoculars Icon is pressed as in Figure 28 you can search on the graph to locate the corresponding data As in digitizing points this allows the identification of outliers or spurious data 2 6 2 4 hel Figure 28 The analytical Wellman graph after three line pairs have been digitized EllipseFit User Manual Page 26 Figure 29 The final analytical Wellman graph after all 8 line pairs from the brachiopods in Figure 27 have been digitized Figure 29 shows the final analytical Wellman graph after all 8 line pairs have been digitized Examine the Log Window Window gt Log and note that at each step EllipseFit calculated the best fit ellipse EllipseFit User Manual Page 27 Analytical Wellman Ellipse Results Wellman Data tsv 2014 06 01 21 39 47 N 8 Point pairs 9 symmetric R 1 773 p 96 10 n 9 RMS 0 025 The calculation is the same as described in Sections 3 1 and 3 2 minimizing the sum of the squares of the residuals the points from the ellipse using a LU decomposition Similarly the RMS value is the root mean square measure of the variation of the residuals from the ellipse that is the square root of the sum of the square
57. us improvements over version 2 but has had more limited testing Additional releases are planned in the near future Version 2 is stable and has been widely used including for a strain workshop at the 2012 Structural Geology and Tectonics Forum at Williams College No updates are planned for EllipseFit 2 I am a professor of structural geology and have taught for over 30 years at SUNY New Paltz I had the luck to be introduced to analytical structural geology as a student and am particularly grateful to my mentors Rob Twiss at UC Davis Win Means at SUNY Albany and Peter Hudleston at U Minnesota whose clear thinking inspired me I was introduced to programming as a grade school student when my dear mother forced me to take a summer school course I subsequently joined the Computer Club as the third member and spent countless hours on the terminal connected remotely to a mainframe Writing code is still an obsession Version 1 of EllipseFit was written in the early 1980s in C for Macintosh in part based on code from a Fortran program written on punch cards for Win Means Version 2 was written in cross platform RealBasic however issues with licensing cost performance and the closed source led me to abandon that language Version 3 is fully rewritten with tens of thousands of lines of code in Free Pascal a professional open source compiler that runs on over 40 operating systems This allows improved code with better speed and extensibility
58. vector mean which gives identical values to other methods Yamaji 2008 Vollmer 2010 Error analysis is shown by an equidistant azimuthal graph of bootstrap results of 1000 resamples from oolite data The mean vector of the bootstrap mean vectors is rotated to C The dispersion of the points is a measure of the error in the best fit ellipse EllipseFit User Manual Page 37 8 Ellipsoid Calculation For regional strain studies it is generally necessary to determine the three dimensional strain ellipsoid with three stretches and their orientations normally expressed as trends and plunges This can be simplified if assumptions can be made about the relationship between foliations and strain for example slaty cleavage is commonly assumed perpendicular to the minimum stretch However in the general case it is necessary to determine the two dimensional strain on a number of different planes through a sample or outcrop where it can be considered homogeneous and combine them to determine the strain ellipsoid in three dimensions This is a difficult mathematical problem and numerous solutions have been proposed e g Shimamoto and Ikeda 1976 Owens 1984 Robin 2002 Shan 2008 Mookerjee and Nickleach 2011 EllipseFit implements the method of Shan 2008 as discussed in Section 8 2 8 1 Global Coordinates and Sample Collection The two dimensional strain ellipses considered thus far have been referred to X Y coordinates where X is to t
59. yperboloid and the use of hyperboloidal stereographic equal area and other projections for strain analysis Vollmer 2011b discusses best fit strain from multiple angles of shear and an analytical solution to the Wellman diagram A suitable references to the software and this documentation are Vollmer F W 2014 EllipseFit 3 1 0 http www frederickvollmer com ellipsefit Vollmer F W 2014 EllipseFit 3 1 0 User Manual http www frederickvollmer com ellipsefit Registration Please consider registering the software registration is free and helps me determine the software usage and justify the time spent in it s upkeep To register simply send an email to me at vollmerf gmail com with your user name affiliation and usage I will send you one email in reply with my thanks and will not place you on a mailing list For example send me an email with something like User Frederick Vollmer Affiliation SUNY New Paltz Geology Department Usage Undergraduate structural geology course and research I am happy to take emails with questions and suggestions either at the university SUNY New Paltz or at the gmail address used on my website However I am not reliable about checking email so please forgive me if I am slow in answering I will try to respond in as timely a fashion as possible EllipseFit User Manual Page 1 1 Introduction EllipseFit is an integrated program for geological finite strain analysis It is used for

EllipseFit 3 User Manual

Contents

Download Pdf Manuals

Related Search

Related Contents