Stochastic Simulation of Patterns using Distance-Based Pattern Modeling

2. Multiple-Point Geostatistics Search Template 9x9 Inner Patch 5x5 Multiple-Grids 3 Number of Clusters 100 Proposed Method Filtersim Stochastic Simulation of Patterns using Distance-Based Pattern Modeling Mehrdad Honarkhah

3. Motivation • Improving pattern reproduction • Less parameters – less user interaction • Reduce simulation time • Checking pattern/MPS reproduction

4. Introduction • better, more realistic models should not require an increase in user-set parameters (Filtersim) • Why use a complexmethod when a much simpler one works just as well ?

5. Distance-Based Methods:(what will be demonstrated) In distance-based modeling, many of the tasks usually performed in multiple-point geostatistical algorithms can be carried out in a surprisingly simple yet powerful way • Second • First • Answer queries through several inference mechanisms • Store and organize a domain of knowledge about the TI MPS Realization Distance-Based Outline

6. Broad Outlook on the Workflow

7. Implementation details(1) Pattern Database Pattern Database 3×3 SearchTemplate Training Image

8. Implementation details(2) MDS mapping

9. Implementation details(3) Kernel Mapping Cartesian Space Feature Space Projection

10. Implementation details(4) Kernel K-means Clustering Cartesian Space Feature Space Projection

11. Implementation detailsSummary Example δ12 Training Image Multi Dimensional Scaling δ13 δ14 δ23 δ24 δ34 Kernel Space Mapping Kernel K-Means

12. WorkFlow Summary ?

13. Implementation details(5) Simulation Recall: Simulation Algorithm in Filtersim • Classify training image patterns into clusters using filter scores and partition • Loop through all nodes of the simulation grid • Retrieve the data event at that node • Find the most similar cluster prototype to that data event • Randomly pick a pattern from that cluster • Paste it on the simulation grid • end

14. Implementation details(5) Simulation Simulation Algorithm in the Proposed Method • Classify training image patterns into clusters using Kernel k-means • Loop through all nodes of the simulation grid • Retrieve that data event at that node • Find the most similar cluster prototype to Data Event • Randomly pick a pattern from that cluster • Paste it on the simulation grid • end Note: The basic Filtersim algorithm is maintained. only the modeling of patterns changes

15. MPS Simulation Examples (1/2) Search Template 9x9 Inner Patch 5x5 Multiple-Grids 3 Number of Clusters 100

16. MPS Simulation Examples (2/2) Search Template 11x11 Inner Patch 5x5 Multiple-Grids 3 Number of Clusters 100

17. WorkFlow Summary

18. Reducing user interaction (1) Automatic Template Size Selection: • Calculating mean Entropy at different template Dimensions • Calculating second derivative of the mean Entropy Curve • Calculating Profile Log-Likelihood of the resulting Curve • Template Size = Maximum in the Profile

19. Reducing user interaction (1) Automatic Template Size Selection: 5 x 5 Training image Mean Entropy Log-Likelihood Template Size Template Size 1 2 3 4 5 1 2 3 The actual smallest template size is 5 x 5 4 5

20. Reducing user interaction (1) Automatic Template Size Selection: 13 x 13 Training image Mean Entropy Log-Likelihood Template Size Template Size

21. Reducing user interaction (2) Find Number of Clusters: • Eigenvalue decomposition of K = • Plot • Calculate Profile Log-Likelihood Number of Clusters = Maximum in the profile

22. Comparison with Filtersim Find Number of Clusters: • Eigenvalue decomposition of K = • Plot • Calculate Profile Log-Likelihood Number of Clusters = Maximum in the profile SIMULATION EXAMPLES

23. Comparison with Filtersim (1/6) 101 x 101 Training Image Proposed Methodology Realizations FilterSim Realizations Search Template 9x9 Inner Patch 5x5 Multiple-Grids 3 Number of Clusters 100

24. Comparison with Filtersim (1/6) 101 x 101 Training Image Proposed Methodology Realizations FilterSim Realizations Search Template 13x13 Inner Patch 5x5 Multiple-Grids 3 Number of Clusters 100

25. Comparison with Filtersim (2/6) 101 x 101 Training Image Proposed Methodology Realizations FilterSim Realizations Search Template 9x9 Inner Patch 5x5 Multiple-Grids 3 Number of Clusters 100

26. Comparison with Filtersim (2/6) 101 x 101 Training Image Proposed Methodology Realizations FilterSim Realizations K = 100 K = 100 Search Template 13x13 Inner Patch 7x7 Multiple-Grids 3 Number of Clusters ----- K = 400

27. Comparison with Filtersim (3/6) 111 x 111 Training Image FilterSim Realizations Proposed Methodology Realizations Search Template 15x15 Inner Patch 9x9 Multiple-Grids 3 Number of Clusters 1000

28. Comparison with Filtersim (4/6) 101 x 101 Training Image FilterSim Realizations Proposed Methodology Realizations Search Template 11x11 Inner Patch 5x5 Multiple-Grids 3 Number of Clusters 100

29. Comparison with Filtersim (5/6) 169 x 169 Training Image FilterSim Realizations Proposed Methodology Realizations Search Template 15x15 Inner Patch 11x11 Multiple-Grids 3 Number of Clusters 200

30. Comparison with Filtersim (6/6) 69 x 69x39 Training Image Proposed Methodology Realizations FilterSim Realizations Search Template 15x15x9 Inner Patch 9 x 9 x 5 Multiple-Grids 3 Number of Clusters 200

31. Studying variability between realizations How can we check the pattern/multiple point statisticalreproduction of the realizations with respect to the training image ?

32. Studying variability between realizations Variability between realizations correspond to the variability obtained while honoring the multiple-point information of the training image. Variability Quantification • Generate N realizations • Find a distance function that characterizes the variability between any two realizations • Map the N realizations into a Cartesian space using MDS • The cloud of points in that space represents variability

33. Studying variability between realizations Measure of Variability between two realizations : l1and l2 f ( l1 ,l2 ) =JSD{ MPH(l1),MPH ( l2) } Multiple-Point Histogram (MPH) Using a multiple-point template, scan the realizations Store the frequency of a specific configuration of outcomes Results in the multiple-point histogram Jensen Shannon Divergence (JSD) A measure of similarity between two probability distributions: pandq

34. Studying variability between realizations Red points : Filtersim realizations Black points : Proposed Method realizations Training Image Used Filtersim Method Proposed Method

35. Studying variability between realizations Training Image Used CONCLUSION Filtersim has less pattern reproduction than our method, and therefore increased artificial variability. Great way to diagnose pattern reproduction (i.e. by looking at ALL realizations at the same time).

36. Conclusion • Distance method: easy to implement, very few user-set parameters • Distance methods also allow easy evaluation of MPS reproduction • Future work: use model expansion techniques to generate additional patterns to further increase geological realism and data conditioning

37. Conclusion