Multiple-point simulation of multiple categories Part 1. Testing against multiple truncation of a Gaussian field

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1 Multiple-point simulation of multiple categories Part 1. Testing against multiple truncation of a Gaussian field Tuanfeng Zhang November, 2001 Abstract Multiple-point simulation of multiple categories ( > 2 ) is implemented using program snesim. The training image was derived by truncating a Gaussian realization into 4 categories. Nonconditional and conditional simulations of these 4 categories are generated using snesim. In both cases, the reproduction of indicator variograms and multiple-point connectivity function are seen to be excellent, as good as a would be obtained by direct truncation of a Gaussian field. 1 Objective The objective of this first part is to test the ability of the newly developed multiplepoint simulation algorithm to reproduce multiple category indicator variograms and continuity functions, in case when these measures are analytically known. This congenial case is provided by the categories obtained by multiple truncation of a Gaussian field. The variograms of the resulting category indicators are known functions of the original Gaussian field variogram. Multiple-point (mp), multiple category simulation using program snesim (Strebelle 2000) is applied using for training image a 4-category realization obtained by truncation of a simulated realization of a continuous Gaussian field. As long as that training image does reflect accurately the Gaussian-related measures of continuity, it is expected that snesim realizations would reflect them. This would prove that the mp simulation algorithm performs as expected, reproducing measures of continuity as well as direct simulation algorithms calling for explicit (analytical) knowledge of the multiple-point distribution. The key to good snesim results is a training image reflecting the target continuity. 1

2 2 Standard Gaussian simulation Twelve continuous realizations of a standard Gaussian random function over an area of size are generated using the GSLIB sequential Gaussian simulation program sgsim. These are shown in Figures 1 and 2. The variogram model is spherical with zero nugget and anisotropic ranges: a x =30 in the E-W direction and a y =15 in the N-S direction: fl(h) =Sph s ( h x 30:0 )2 +( h y 15:0 )2 Each realization histogram is plotted (see Appendix Figures A1 and A2). As expected the realization means fluctuate around the zero value. The same ergodic fluctuations lead to non-symmetric lower and upper quartiles. Such ergodic fluctuations are to be expected from stochastic simulations, particularly when the field size is not very large with regard to the range of correlation. 3 Four categories training image By multiple truncation of the previous twelve Gaussian realizations, twelve realizations with 4-categories are created. The three threshold values used are specific to each realization, such that the 4 categories generated have equal proportions. These are the lower quartile, the median, and the upper quartile of each original Gaussian realization. To check reproduction of the 2-point statistics, the theoretical indicator variogram of the standard Gaussian model is calculated using the GSLIB program bigaus; that program utilizes the analytic formula linking the indicator variogram at threshold y p to the covariance function C Y (h) of the original continuous standard Gaussian random function (Deutsch and Journel, 1992, p.139): Z fl I (h;y p )=p(1 p) 1 arcsincy (h)» exp y 2 p d (1) 2ß 0 1+sin where p = G(y p ) 2 [0; 1], and y p is the standard Gaussian p quantile. Let S i denote the four cumulative categories i = 1; 2; 3; 4 with the following notations: S1 = fu : I(u; z1) =1g S1 + S2 = fu : I(u; z2) =1g S1 + S2 + S3 = fu : I(u; z3) =1g 2

3 S4 = fu : I(u; z1) =0g I(u; z i )= ( 1 if z(u)» z i 0 if not where I(u; z i ) is the indicator variable corresponding to the threshold value z i. Each of the three indicator variables I(u; z i ) relates to an accumulation of the initial individual categories S i ; this is because program bigaus considers such cumulative indicators. Note: S1 ρ S2 ρ S3 ρ S4 For each of the initial 12 continuous Gaussian realizations, the three indicator variograms of I(u; z i ) standardized by their respective marginal variances are calculated. These realization variograms are compared to the theoretical expression (1) through fl fl plots. The fl fl scatter plots of each realization indicator variogram vs. its theoretical Gaussian model are displayed in Appendix Figures A3 to A14. This is done for the 2 directions in the E-W and N-S. Realization #6 presents the best fl fl fit repeated in Figure 3, and is chosen as the 4-category training image (TI). This TI will be used to generate realizations of these 4 categories using the multiple-point simulation program snesim, see parameter file in Figure 4. The resulting snesim realizations are then checked for reproduction of the theoretical Gaussian 2-point and multiple-point statistics. Realization #6 (bottom right of Figure 1) was truncated with 3 quartile thresholds to generate the training image with 4 equal frequency categories shown at the top left of Figure 5. 4 Nonconditional categorical simulation Using the previous categorical TI, program snesim was run to generate five nonconditional simulations over an area of size , see Figure 5. The corresponding snesim parameter file is shown in Figure 4. The resulting simulated categorical proportions are listed in Table 1. From Table 1, we find that the realization #4 has the best reproduction of the target :25 equal proportions and realization #3 is the worst. To check reproduction of the training image indicator variograms, fl fl scatterplots of the simulated indicator variograms are displayed in Appendix Figures A15 to A19. The reference fl model used for these scatterplots is now that of the training image; the horizontal value corresponds to the TI indicator variogram. When these scatterplots are compared with those derived by direct truncation of continuous realizations (recall Appendix Figures A3 to A14), the snesim reproduction of the training image indicator variograms 3

4 p1 p2 p3 p4 Training image Realization Realization Realization worst Realization best Realization Table 1: Categorical proportions of nonconditional simulation appears equivalently good. The best fit fl fl plot corresponding to realization #4 is reproduced as Figure 6 and should be compared to Figure 3. Reproduction of a multi-point ( > 2 ) statistics is now checked. We consider for such statistics the following multiple-point (mp) connectivity measure calculated along the E-W direction: Conn(k; h) =Prob(I(u) =1;I(u + h) =1; :::; I(u + kh) =1) (2) where Conn(k; h) is defined as the k step connectivity along vector h of the category associated to the binary indicator I(u). In an one dimensional space (x) with N values evenly spaced, the k step connectivity would be expressed as: Conn(k) = 1 N k X ky i(x + jk): N k x=1 j=1 For k =0, that connectivity expression yields the category proportion: Conn(N; 0) = 1 N P N x=1 i(x). In Appendix Figures A20 to A24, the solid line corresponds to the TI connectivity and the dashed line corresponds to the simulated realization connectivity. It appears from these figures that, in general, the TI connectivity is well reproduced by the snesim realizations. Note the symmetry induced by the Gaussian origin of the TI: categories 1 and 4 have similar connectivity, so do categories 2 and 3. Note also the greater connectivity of the two extreme categories 1 and 4 which are more clustered in space than categories 2 and 3, see Figure 5. Note that realization #4 noted as best in Table 1 for reproduction of proportions also provides good reproduction of 2-point and mp statistics, see Appendix Figure A18 and A 23. Conversely, realization #3 noted as worst in Table 1 also provides poor reproduction of 2-point and mp statistics, see Appendix Figure A17 and A22. 4

5 5 Conditional categorical simulation We proceed now to conditional simulation. We wish to check whether both the 2- point indicator variograms and the multiple-point connectivity function remain well reproduced after conditioning to hard data. To this goal, the previous nonconditional categorical realization #4 generated by snesim (bottom left of Figure 5) is chosen as the true image from which to take hard data. Four hard data for each of the 4 categories were selected, thus ensuring that each category has the same sample proportion, see the location map of the resulting 16 hard data at the top left of Figure 7. The same TI used previously (top left of Figure 5) was retained as input to program snesim. The same parameter file given in Figure 4 are retained for this snesim conditional simulation. Five conditional realizations are created, see Figure 7. The resulting simulated categorical proportions are listed in Table 2. p1 p2 p3 p4 Training image Realization worst Realization best Realization Realization Realization True image Table 2: Categorical proportions of conditional simulation fl fl scatterplots are used to evaluate the reproduction of the TI 2-point statistics (indicator variograms), see Appendix Figures A25 to A29. Reproduction of the mp connectivity functions are displayed in Appendix A30 to A34. From these figures, it appears that both the 2-point indicator variograms and the multiple-point connectivity functions remain equally well reproduced after conditioning the snesim realizations to hard data. Note that realization #2 noted as best in Table 2 for reproduction of proportions also provides good reproduction of 2-point and mp statistics, see Appendix Figures A26 and A31. Conversely, realization #1 noted as worst in Table 2 also provides poor reproduction of 2-point and mp statistics, see Appendix Figures A25 and A30. 5

6 6 Conclusions This application of the multiple-point (mp) snesim code to simulation of multiple categories generated by truncation of a continuous Gaussian field shows that ffl snesim can be used to capture multiple category structures as well as would a direct Gaussian simulation algorithm in presence of Gaussian-type (high entropy) structures. ffl the categorical training image used was derived by truncation of a Gaussian field. The training image 2-point statistics and multiple-point connectivity function are well reproduced. ffl for categorical variables, a preliminary good reproduction of proportions (singlepoint statistics) is important since these proportions control the multiple-point statistics. Reference [1] S. Strebelle., Seqential simulation drawing structures from training images. SCRF report, 2000, and PhD thesis. [2] C.V.Deutsch and A.G.Journel., GSLIB - Geostatistical Software Library and User s Guide. Oxford University Press,

7 200 Realization Realization North North East 200 East Realization Realization North North East 200 East Realization Realization North North East 200 East 200 Figure 1: Standard Gaussian realizations 7

8 200 Realization Realization North North East 200 East Realization Realization North North East 200 East Realization Realization North North East 200 East 200 Figure 2: Standard Gaussian realizations 8

9 Gama-Gama plot (Real6, Cat1, E-W) Gama-Gama plot (Real6, Cat1, N-S) Gama-Gama plot (Real6, Cat1+2, E-W) Gama-Gama plot (Real6, Cat1+2, N-S) Gama-Gama plot (Real6, Cat1+2+3, E-W) Gama-Gama plot (Real6, Cat1+2+3, N-S) Figure 3: fl fl scatter plots of simulated indicator variograms vs. Gaussian models: 3 thresholds, 2 directions. Realization #6 9

10 Parameters for SNESIM ******************** START OF PARAMETERS: nondata.dat - file with conditioning data columns for x, y, z, variable 4 - number of categories category codes global (target) pdf 0 - use (target) vertical proportions (0=no, 1=yes) vertprop.dat - file with target vertical proportions target pdf repro. (0=no, 1=yes) cpdf correction factor 0 - debugging level: 0,1,2,3 snesim.dbg - debugging file snesim.out - file for simulation output 5 - number of realizations to generate nx,xmn,xsiz ny,ymn,ysiz nz,zmn,zsiz random number seed template48.dat - file for data template 25 - max number of conditioning data 0 - max number of data per octant (0=not used) 10 - min number of data events number of mult-grid, number with search gtsim.out - file for training image - Rotation angle for training image Compressed factors for data event training image dimensions: nxtr, nytr, nztr 1 - column for variable maximum search radii (hmax,hmin,vert) 9 - angles for search ellipsoid Figure 4: snesim parameters used to generate the 5 categorical realizations of Figure 5 10

200 Training Image 100 Realization 1 from Ti cat4 cat4 cat3 cat3 North North cat2 cat2 cat1 cat1 East 200 East 100 100 Realization 2 from Ti 100 Realization 3 from Ti cat4 cat4 cat3 cat3 North North

11 200 Training Image 100 Realization 1 from Ti cat4 cat4 cat3 cat3 North North cat2 cat2 cat1 cat1 East 200 East Realization 2 from Ti 100 Realization 3 from Ti cat4 cat4 cat3 cat3 North North cat2 cat2 cat1 cat1 East 100 East Realization 4 from Ti 100 Realization 5 from Ti cat4 cat4 cat3 cat3 North North cat2 cat2 cat1 cat1 East 100 East 100 Figure 5: Training image (top left) and five nonconditional realizations using program snesim 11

12 Gama-Gama plot (x-ti, Cat1, E-W) Gama-Gama plot (x-ti, Cat1, N-S) Gama-Gama plot (x-ti, Cat1+2, E-W) Gama-Gama plot (x-ti, Cat1+2, N-S) Gama-Gama plot (x-ti, Cat1+2+3, E-W) Gama-Gama plot (x-ti, Cat1+2+3, N-S) Figure 6: fl fl scatter plots of snesim nonconditional simulated indicator variograms vs. that of the training image: 3 threshholds, 2 directions. Realization #4. 12

13 100. Locations of Clustered Data 100 Realization 1 from Ti cat cat3 North cat2 20. cat East Realization 2 from Ti 100 Realization 3 from Ti cat4 cat4 cat3 cat3 North North cat2 cat2 cat1 cat1 East 100 East Realization 4 from Ti 100 Realization 5 from Ti cat4 cat4 cat3 cat3 North North cat2 cat2 cat1 cat1 East 100 East 100 Figure 7: Hard data locations (top left) and 5 conditional realizations from snesim using the same TI in figure 5 13

14 Appendix A: Figures A1 to A34 showing histograms of standard Gaussian realizations and reproduction of spatial statistics by multiple-point categorical simulation using program snesim 14

15 Frequency Histogram of realization 1 Number of Data mean 0.21 std. dev coef. of var undefined maximum 4.72 upper quartile 0.95 median 0.21 lower quartile minimum Frequency Histogram of realization 2 Number of Data mean -9 std. dev coef. of var undefined maximum 3.64 upper quartile 0.59 median -9 lower quartile minimum value value Frequency Histogram of realization 3 Number of Data mean 0.10 std. dev coef. of var undefined maximum 3.58 upper quartile median 0.13 lower quartile minimum Frequency Histogram of realization 4 Number of Data mean 2 std. dev coef. of var undefined maximum 3.60 upper quartile 0.68 median 3 lower quartile minimum value value Frequency Histogram of realization 5 Number of Data mean 2 std. dev coef. of var undefined maximum 3.76 upper quartile 0.73 median lower quartile minimum Frequency Histogram of realization 6 Number of Data mean std. dev coef. of var undefined maximum 3.31 upper quartile 0.48 median - lower quartile minimum value value Figure A1: Histograms from standard Gaussian realizations 15

16 Frequency Histogram of realization 7 Number of Data mean std. dev coef. of var undefined maximum 3.60 upper quartile 0.51 median lower quartile minimum Frequency Histogram of realization 8 Number of Data mean 1 std. dev coef. of var undefined maximum 4.02 upper quartile 0.65 median lower quartile minimum value value Frequency Histogram of realization 9 Number of Data mean 5 std. dev coef. of var undefined maximum 4.05 upper quartile 0.77 median 4 lower quartile minimum Frequency Histogram of realization 10 Number of Data mean -3 std. dev coef. of var undefined maximum 4.22 upper quartile 0.63 median -1 lower quartile minimum value value Frequency Histogram of realization 11 Number of Data mean std. dev coef. of var undefined maximum 3.53 upper quartile 0.50 median lower quartile minimum Frequency Histogram of realization 12 Number of Data mean 9 std. dev. coef. of var undefined maximum 3.55 upper quartile 0.81 median 9 lower quartile minimum value value Figure A2: Histograms from standard Gaussian realizations 16

17 Gama-Gama plot (Real1, Cat1, E-W) Gama-Gama plot (Real1, Cat1, N-S) Gama-Gama plot (Real1, Cat1+2, E-W) Gama-Gama plot (Real1, Cat1+2,N-S) Gama-Gama plot (Real1, Cat1+2+3, E-W) Gama-Gama plot (Real1, Cat1+2+3, N-S) Figure A3: fl fl scatter plots of simulated indicator variograms vs. Gaussian models: 3 thresholds, 2 directions. Realization #1 17

18 Gama-Gama plot (Real2, Cat1, E-W) Gama-Gama plot (Real2, Cat1, N-S) Gama-Gama plot (Real2, Cat1+2, E-W) Gama-Gama plot (Real2, Cat1+2, N-S) Gama-Gama plot (Real2, Cat1+2+3, E-W) Gama-Gama plot (Real2, Cat1+2+3, N-S) Figure A4: fl fl scatter plots of simulated indicator variograms vs. Gaussian models: 3 thresholds, 2 directions. Realization #2 18

19 Gama-Gama plot (Real3, Cat1, E-W) Gama-Gama plot (Real3, Cat1, N-S) Gama-Gama plot (Real3, Cat1+2, E-W) Gama-Gama plot (Real3, Cat1+2, N-S) Gama-Gama plot (Real3, Cat1+2+3, E-W) Gama-Gama plot (Real3, Cat1+2+3, N-S) Figure A5: fl fl scatter plots of simulated indicator variograms vs. Gaussian models: 3 thresholds, 2 directions. Realization #3 19

20 Gama-Gama plot (Real4, Cat1, E-W) Gama-Gama plot (Real4, Cat1, N-S) Gama-Gama plot (Real4, Cat1+2, E-W) Gama-Gama plot (Real4, Cat1+2, N-S) Gama-Gama plot (Real4, Cat1+2+3, E-W) Gama-Gama plot (Real4, Cat1+2+3, N-S) Figure A6: fl fl scatter plots of simulated indicator variograms vs. Gaussian models: 3 thresholds, 2 directions. Realization #4 20

21 Gama-Gama plot (Real5, Cat1, E-W) Gama-Gama plot (Real5, Cat1, N-S) Gama-Gama plot (Real5, Cat1+2, E-W) Gama-Gama plot (Real5, Cat1+2, N-S) Gama-Gama plot (Real5, Cat1+2+3, E-W) Gama-Gama plot (Real4, Cat1+2+3, N-S) Figure A7: fl fl scatter plots of simulated indicator variograms vs. Gaussian models: 3 thresholds, 2 directions. Realization #5 21

22 Gama-Gama plot (Real6, Cat1, E-W) Gama-Gama plot (Real6, Cat1, N-S) Gama-Gama plot (Real6, Cat1+2, E-W) Gama-Gama plot (Real6, Cat1+2, N-S) Gama-Gama plot (Real6, Cat1+2+3, E-W) Gama-Gama plot (Real6, Cat1+2+3, N-S) Figure A8: fl fl scatter plots of simulated indicator variograms vs. Gaussian models: 3 thresholds, 2 directions. Realization #6 22

23 Gama-Gama plot (Real7, Cat1, E-W) Gama-Gama plot (Real7, Cat1, N-S) Gama-Gama plot (Real7, Cat1+2, E-W) Gama-Gama plot (Real7, Cat1+2, N-S) Gama-Gama plot (Real7, Cat1+2+3, E-W) Gama-Gama plot (Real7, Cat1+2+3, N-S) Figure A9: fl fl scatter plots of simulated indicator variograms vs. Gaussian models: 3 thresholds, 2 directions. Realization #7 23

24 Gama-Gama plot (Real8, Cat1, E-W) Gama-Gama plot (Real8, Cat1, N-S) Gama-Gama plot (Real8, Cat1+2, E-W) Gama-Gama plot (Real8, Cat1+2, N-S) Gama-Gama plot (Real8, Cat1+2+3, E-W) Gama-Gama plot (Real8, Cat1+2+3, N-S) Figure A10: fl fl scatter plots of simulated indicator variograms vs. Gaussian models: 3 thresholds, 2 directions. Realization #8 24

25 Gama-Gama plot (Real9, Cat1, E-W) Gama-Gama plot (Real9, Cat1, N-S) Gama-Gama plot (Real9, Cat1+2, E-W) Gama-Gama plot (Real9, Cat1+2, N-S) Gama-Gama plot (Real9, Cat1+2+3, E-W) Gama-Gama plot (Real9, Cat1+2+3, N-S) Figure A11: fl fl scatter plots of simulated indicator variograms vs. Gaussian models: 3 thresholds, 2 directions. Realization #9 25

26 Gama-Gama plot (Real10, Cat1, E-W) Gama-Gama plot (Real10, Cat1, N-S) Gama-Gama plot (Real10, Cat1+2, E-W) Gama-Gama plot (Real10, Cat1+2, N-S) Gama-Gama plot (Real10, Cat1+2+3, E-W) Gama-Gama plot (Real9, Cat1+2+3, N-S) Figure A12: fl fl scatter plots of simulated indicator variograms vs. Gaussian models: 3 thresholds, 2 directions. Realization #10 26

27 Gama-Gama plot (Real11, Cat1, E-W) Gama-Gama plot (Real11, Cat1, N-S) Gama-Gama plot (Real11, Cat1+2, E-W) Gama-Gama plot (Real11, Cat1+2, N-S) Gama-Gama plot (Real11, Cat1+2+3, E-W) Gama-Gama plot (Real11, Cat1+2+3, N-S) Figure A13: fl fl scatter plots of simulated indicator variograms vs. Gaussian models: 3 thresholds, 2 directions. Realization #11 27

28 Gama-Gama plot (Real12, Cat1, E-W) Gama-Gama plot (Real12, Cat1, N-S) Gama-Gama plot (Real12, Cat1+2, E-W) Gama-Gama plot (Real12, Cat1+2, N-S) Gama-Gama plot (Real12, Cat1+2+3, E-W) Gama-Gama plot (Real12, Cat1+2+3, N-S) Figure A14: fl fl scatter plots of simulated indicator variograms vs. Gaussian models: 3 thresholds, 2 directions. Realization #12 28

29 Gama-Gama plot (Cat1,x-Ti,E-W ) Gama-Gama plot (x-ti,cat1,n-s) Gama-Gama plot (x-ti, Cat1+2, E-W) Gama-Gama plot (x-ti, Cat1+2, N-S) Gama-Gama plot (x-ti, Cat1+2+3, E-W) Gama-Gama plot (x-ti, Cat1+2+3, N-S) Figure A15: fl fl scatter plots of snesim nonconditional simulated indicator variograms vs. that of the training image: 3 threshholds, 2 directions. Realization #1. 29

30 Gama-Gama plot (x-ti, Cat1, E-W) Gama-Gama plot (x-ti, Cat1, N-S) Gama-Gama plot (x-ti, Cat1+2, E-W) Gama-Gama plot (x-ti, Cat1+2, N-S) Gama-Gama plot (x-ti, Cat1+2+3, E-W) Gama-Gama plot (x-ti, Cat1+2+3, N-S) Figure A16: fl fl scatter plots of snesim nonconditional simulated indicator variograms vs. that of the training image: 3 threshholds, 2 directions. Realization #2. 30

31 Gama-Gama plot (x-ti, Cat1, E-W) Gama-Gama plot (x-ti, Cat1, N-S) Gama-Gama plot (x-ti, Cat1+2, E-W) Gama-Gama plot (x-ti, Cat1+2, N-S) Gama-Gama plot (x-ti, Cat1+2+3, E-W) Gama-Gama plot (x-ti, Cat1+2+3, N-S) Figure A17: fl fl scatter plots of snesim nonconditional simulated indicator variograms vs. that of the training image: 3 threshholds, 2 directions. Realization #3. 31

32 Gama-Gama plot (x-ti, Cat1, E-W) Gama-Gama plot (x-ti, Cat1, N-S) Gama-Gama plot (x-ti, Cat1+2, E-W) Gama-Gama plot (x-ti, Cat1+2, N-S) Gama-Gama plot (x-ti, Cat1+2+3, E-W) Gama-Gama plot (x-ti, Cat1+2+3, N-S) Figure A18: fl fl scatter plots of snesim nonconditional simulated indicator variograms vs. that of the training image: 3 threshholds, 2 directions. Realization #4. 32

33 Gama-Gama plot (x-ti, Cat1, E-W) Gama-Gama plot (x-ti, Cat1, N-S) Gama-Gama plot (x-ti, Cat1+2, E-W) Gama-Gama plot (x-ti, Cat1+2, N-S) Gama-Gama plot (x-ti, Cat1+2+3, E-W) Gama-Gama plot (x-ti, Cat1+2+3, N-S) Figure A19: fl fl scatter plots of snesim nonconditional simulated indicator variograms vs. that of the training image: 3 threshholds, 2 directions. Realization #5. 33

34 Connectivity for cat1 (Rel1, Cat1) Connectivity for cat2 (Rel1, Cat2) Connectivity for cat3 (Rel1, Cat3) Connectivity for cat4 (Rel1, Cat4) Figure A20: Multiple-point connectivity along EW direction of snesim nonconditional simulation (dotted line) vs. that of the training image. Realization #1. 34

35 Connectivity for cat1 (Rel2, Cat1) Connectivity for cat2 (Rel2, Cat2) Connectivity for cat3 (Rel2, Cat3) Connectivity for cat4 (Rel2, Cat4) Figure A21: Multiple-point connectivity along EW direction of snesim nonconditional simulation (dotted line) vs. that of the training image. Realization #2. 35

36 Connectivity for cat1 (Rel3, Cat1) Connectivity for cat2 (Rel3, Cat2) Connectivity for cat3 (Rel3, Cat3) Connectivity for cat4 (Rel3, Cat4) Figure A22: Multiple-point connectivity along EW direction of snesim nonconditional simulation (dotted line) vs. that of the training image. Realization #3. 36

37 Connectivity for cat1 (Rel4, Cat1) Connectivity for cat2 (Rel4, Cat2) Connectivity for cat3 (Rel4, Cat3) Connectivity for cat4 (Rel4, Cat4) Figure A23: Multiple-point connectivity along EW direction of snesim nonconditional simulation (dotted line) vs. that of the training image. Realization #4. 37

38 Connectivity for cat1 (Rel5, Cat1) Connectivity for cat2 (Rel5, Cat2) Connectivity for cat3 (Rel5, Cat3) Connectivity for cat4 (Rel5, Cat4) Figure A24: Multiple-point connectivity along EW direction of snesim nonconditional simulation (dotted line) vs. that of the training image. Realization #5. 38

39 Gama-Gama plot (x-ti, Cat, E-W) Gama-Gama plot (x-ti, Cat1, N-S) Gama-Gama plot (x-ti, Cat1+2, E-W) Gama-Gama plot (x-ti, Cat1+2, N-S) Gama-Gama plot (x-ti, Cat1+2+3, E-W) Gama-Gama plot (x-ti, Cat1+2+3, N-S) Figure A25: fl fl scatter plots of snesim conditional simulated indicator variograms vs. that of the training image: 3 thresholds, 2 directions. Realization #1. 39

40 Gama-Gama plot (x-ti, Cat1, E-W) Gama-Gama plot (x-ti, Cat1, N-S) Gama-Gama plot (x-ti, Cat1+2, E-W) Gama-Gama plot (x-ti, Cat1+2, N-S) Gama-Gama plot (x-ti, Cat1+2+3, E-W) Gama-Gama plot (x-ti, Cat1+2+3, N-S) Figure A26: fl fl scatter plots of snesim conditional simulated indicator variograms vs. that of the training image: 3 thresholds, 2 directions. Realization #2. 40

41 Gama-Gama plot (x-ti, Cat1, E-W) Gama-Gama plot (x-ti, Cat1, N-S) Gama-Gama plot (x-ti, Cat1+2, E-W) Gama-Gama plot (x-ti, Cat1+2, N-S) Gama-Gama plot (x-ti, Cat1+2+3, E-W) Gama-Gama plot (x-ti, Cat1+2+3, N-S) Figure A27: fl fl scatter plots of snesim conditional simulated indicator variograms vs. that of the training image: 3 thresholds, 2 directions. Realization #3. 41

42 Gama-Gama plot (x-ti, Cat1, E-W) Gama-Gama plot (x-ti, Cat1, N-S) Gama-Gama plot (x-ti, Cat1+2, E-W) Gama-Gama plot (x-ti, Cat1+2, N-S) Gama-Gama plot (x-ti, Cat1+2+3, E-W) Gama-Gama plot (x-ti, Cat1+2+3, N-S) Figure A28: fl fl scatter plots of snesim conditional simulated indicator variograms vs. that of the training image: 3 thresholds, 2 directions. Realization #4. 42

43 Gama-Gama plot (x-ti, Cat1, E-W) Gama-Gama plot (x-ti, Cat1, N-S) Gama-Gama plot (x-ti, Cat1+2, E-W) Gama-Gama plot (x-ti, Cat1+2, N-S) Gama-Gama plot (x-ti, Cat1+2+3, E-W) Gama-Gama plot (x-ti, Cat1+2+3, N-S) Figure A29: fl fl scatter plots of snesim conditional simulated indicator variograms vs. that of the training image: 3 thresholds, 2 directions. Realization #5. 43

44 Connectivity for cat1 (Rel1, Cat1) Connectivity for cat2 (Rel1, Cat2) Connectivity for cat3 (Rel1, Cat3) Connectivity for cat4 (Rel1, Cat4) Figure A30: Multiple-point connectivity along EW direction of snesim conditional simulation (dotted line) vs. that of the training image. Realization #1 44

45 Connectivity for cat1 (Rel2, Cat1) Connectivity for cat2 (Rel2, Cat2) Connectivity for cat3 (Rel2, Cat3) Connectivity for cat4 (Rel2, Cat4) Figure A31: Multiple-point connectivity along EW direction of snesim conditional simulation (dotted line) vs. that of the training image. Realization #2 45

46 Connectivity for cat1 (Rel3, Cat1) Connectivity for cat2 (Rel3, Cat2) Connectivity for cat3 (Rel3, Cat3) Connectivity for cat4 (Rel3, Cat4) Figure A32: Multiple-point connectivity along EW direction of snesim conditional simulation (dotted line) vs. that of the training image. Realization #3 46

47 Connectivity for cat1 (Rel4, Cat1) Connectivity for cat2 (Rel4, Cat2) Connectivity for cat3 (Rel4, Cat3) Connectivity for cat4 (Rel4, Cat4) Figure A33: Multiple-point connectivity along EW direction of snesim conditional simulation (dotted line) vs. that of the training image. Realization #4 47

48 Connectivity for cat1 (Rel5, Cat1) Connectivity for cat2 (Rel5, Cat2) Connectivity for cat3 (Rel5, cat3) Connectivity for cat4 (Rel5, Cat4) Figure A34: Multiple-point connectivity along EW direction of snesim conditional simulation (dotted line) vs. that of the training image. Realization #5 48

Chapter 5. Describing Distributions Numerically. Finding the Center: The Median. Spread: Home on the Range. Finding the Center: The Median (cont.

Chapter 5. Describing Distributions Numerically. Finding the Center: The Median. Spread: Home on the Range. Finding the Center: The Median (cont. Chapter 5 Describing Distributions Numerically Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Copyright 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide