Mathematical Geosciences

, Volume 46, Issue 1, pp 95–123 | Cite as

CDFSIM: Efficient Stochastic Simulation Through Decomposition of Cumulative Distribution Functions of Transformed Spatial Patterns

  • Hussein MustaphaEmail author
  • Snehamoy Chatterjee
  • Roussos Dimitrakopoulos


Simulation of categorical and continuous variables is performed using a new pattern-based simulation method founded upon coding spatial patterns in one dimension. The method consists of, first, using a spatial template to extract information in the form of patterns from a training image. Patterns are grouped into a pattern database and, then, mapped to one dimension. Cumulative distribution functions of the one-dimensional patterns are built. Patterns are then classified by decomposing the cumulative distribution functions, and calculating class or cluster prototypes. During the simulation process, a conditioning data event is compared to the class prototype, and a pattern is randomly drawn from the best matched class. Several examples are presented so as to assess the performance of the proposed method, including conditional and unconditional simulations of categorical and continuous data sets. Results show that the proposed method is efficient and very well performing in both two and three dimensions. Comparison of the proposed method to the filtersim algorithm suggests that it is better at reproducing the multi-point configurations and main characteristics of the reference images, while less sensitive to the number of classes and spatial templates used in the simulations.


Pattern-based simulation Clustering Patterns coding Conditional simulation Training image 



We thank the Associate Editor of Mathematical Geosciences handling our manuscript and the anonymous reviewers for their detailed comments that have helped improve the manuscript. The work in this paper was funded by Natural Science and Engineering Research Council of Canada CRDPJ 411270-10, Discovery Grant 239019, and the industry members of the COSMO Stochastic Mine Planning Laboratory: AngloGold Ashanti, Barrick Gold, BHP Billiton, De Beers, Newmont Mining and Vale.


  1. Allard D, Froidevaux R, Biver P (2006) Conditional simulation of multi-type non stationary Markov object models respecting specified proportions. Math Geol 38(8):959–986CrossRefGoogle Scholar
  2. Arpat GB (2004) Sequential simulation with patterns. PhD thesis, Stanford UniversityGoogle Scholar
  3. Arpat G, Caers J (2007) Conditional simulation with patterns. Math Geol 39(2):177–203CrossRefGoogle Scholar
  4. Chatterjee S, Dimitrakopoulos R (2012) Multi-scale stochastic simulation with a wavelet-based approach. Comput Geosci 45:177–189CrossRefGoogle Scholar
  5. Chatterjee S, Dimitrakopoulo R, Mustapha H (2012) Dimensional reduction of pattern-based simulation using wavelet analysis. Math Geosci 44(3):343–374CrossRefGoogle Scholar
  6. Chilès JP, Delfiner P (1999) Geostatistics—modeling spatial uncertainty. Wiley, New YorkCrossRefGoogle Scholar
  7. Comunian A, Renard P, Straubhaar J (2012) 3D multiple-point statistics simulation using 2D training images. Comput Geosci 40:49–65CrossRefGoogle Scholar
  8. Daly C (2004) Higher order models using entropy, Markov random fields and sequential simulation. In: Leuangthong O, Deutsch CV (eds) Geostatistics Banff. Kluwer, Dordrecht, pp 215–225Google Scholar
  9. Deutsch CV (2002) Geostatistical reservoir modeling. Oxford University Press, New YorkGoogle Scholar
  10. Dimitrakopoulos R, Mustapha H, Gloaguen E (2010) High-order statistics of spatial random fields: exploring spatial cumulants for modeling complex non-Gaussian and non-linear phenomena. Math Geosci 42(1):65–99CrossRefGoogle Scholar
  11. Gloaguen E, Dimitrakopoulos R (2009) Two-dimensional conditional simulation based on the wavelet decomposition of training images. Math Geosci 41(7):679–701CrossRefGoogle Scholar
  12. Goovaerts P (1998) Geostatistics for natural resources evaluation. Oxford University Press, New YorkGoogle Scholar
  13. Guardiano FB, Srivastava RM (1993) Multivariate geostatistics: beyond bivariate moments. In: Soares (ed) Geostatistics Troia ‘92. Kluwer, Dordrecht, pp 133–144Google Scholar
  14. Honarkhah M, Caers J (2010) Stochastic simulation of patterns using distance-based pattern modelling. Math Geosci 42:487–517CrossRefGoogle Scholar
  15. Huysmans M, Dassargues A (2011) Direct multiple-point geostatistical simulation of edge properties for modeling thin irregularly shaped surfaces. Math Geosci 43(5):521–536. doi: 10.1007/s11004-011-9336-7 CrossRefGoogle Scholar
  16. Journel AG (1997) Deterministic geostatistics: a new visit. In: Baafi E, Schofield N (eds) Geostatistics Woolongong ‘96. Kluwer, Dordrecht, pp 213–224Google Scholar
  17. Liu Y (2006) Using the Snesim program for multiple-point statistical simulation. Comput Geosci 23(2006):1544–1563CrossRefGoogle Scholar
  18. Mao S, Journel AG (1999) Generation of a reference petrophysical and seismic 3D data set: the Stanford V reservoir. In: Stanford center for reservoir forecasting annual meeting. Available at: Google Scholar
  19. Mariethoz G, Renard P (2010) Reconstruction of incomplete data sets or images using direct sampling. Math Geosci 42(3):245–268CrossRefGoogle Scholar
  20. Mariethoz G, Renard P, Straubhaar J (2010) The direct sampling method to perform multiple-point simulation. Water Resour Res. doi: 10.1029/2008WR007621 Google Scholar
  21. Mustapha H, Dimitrakopoulos R (2010) High-order stochastic simulation of complex spatially distributed natural phenomena. Math Geosci 42(5):455–473CrossRefGoogle Scholar
  22. Mustapha H, Dimitrakopoulos R, Chatterjee S (2011) Geologic heterogeneity representation using high-order spatial cumulants for subsurface flow and transport simulations. Water Resour Res. doi: 10.1029/2010WR009515 Google Scholar
  23. Ortiz JM, Deutsh CV (2004) Indicator simulation accounting for multiple-point statistics. Math Geol 36(5):545–565CrossRefGoogle Scholar
  24. Remy N, Boucher A, Wu J (2008) Applied geostatistics with SGeMS: a user’s guide. Cambridge University Press, CambridgeGoogle Scholar
  25. Sarma P, Durlofsky L, Aziz K (2008) Kernel principal component analysis for efficient, differentiable parameterization of multipoint geostatistics. Math Geosci 40(1):3–32CrossRefGoogle Scholar
  26. Scheidt C, Caers J (2009) Representing spatial uncertainty using distances and kernels. Math Geosci 41:397–419CrossRefGoogle Scholar
  27. Straubhaar J, Renard P, Mariethoz G, Froidevaux R, Besson O (2011) An improved parallel multiple-point algorithm using a list approach. Math Geosci 43(3):305–328CrossRefGoogle Scholar
  28. Strebelle S (2000) Sequential simulation drawing structures from training images. PhD thesis, Stanford UniversityGoogle Scholar
  29. Strebelle S (2002) Conditional simulation of complex geological structures using multiple-point statistics. Math Geol 34(1):1–21CrossRefGoogle Scholar
  30. Tjelmeland H (1998) Markov random fields with higher order interactions. Scand J Stat 25:415–433CrossRefGoogle Scholar
  31. Tjelmeland H, Eidsvik J (2004) Directional Metropolis: hastings updates for conditionals with nonlinear likelihoods. In: Geostatistics Banff 2004, vol 1. Springer, Berlin, pp 95–104Google Scholar
  32. Wu J, Zhang T, Journel A (2008) Fast FILTERSIM simulation with score-based distance. Math Geosci 40(7):773–788CrossRefGoogle Scholar
  33. Zhang T, Switzer P, Journel A (2006) Filter-based classification of training image patterns for spatial simulation. Math Geol 38(1):63–80CrossRefGoogle Scholar
  34. Zhang T, Stein Inge Pedersen SI, Christen Knudby C, McCormick D (2012) Memory-efficient categorical multi-point statistics algorithms based on compact search trees. Math Geosci 44(7):863–879CrossRefGoogle Scholar

Copyright information

© International Association for Mathematical Geosciences 2013

Authors and Affiliations

  • Hussein Mustapha
    • 1
    Email author
  • Snehamoy Chatterjee
    • 1
  • Roussos Dimitrakopoulos
    • 1
  1. 1.COSMO—Stochastic Mine Planning Laboratory, Department of Mining and Materials EngineeringMcGill UniversityMontrealCanada

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