Mathematical Geosciences

, Volume 42, Issue 5, pp 487–517 | Cite as

Stochastic Simulation of Patterns Using Distance-Based Pattern Modeling

  • Mehrdad HonarkhahEmail author
  • Jef Caers
Special Issue


The advent of multiple-point geostatistics (MPS) gave rise to the integration of complex subsurface geological structures and features into the model by the concept of training images. Initial algorithms generate geologically realistic realizations by using these training images to obtain conditional probabilities needed in a stochastic simulation framework. More recent pattern-based geostatistical algorithms attempt to improve the accuracy of the training image pattern reproduction. In these approaches, the training image is used to construct a pattern database. Consequently, sequential simulation will be carried out by selecting a pattern from the database and pasting it onto the simulation grid. One of the shortcomings of the present algorithms is the lack of a unifying framework for classifying and modeling the patterns from the training image. In this paper, an entirely different approach will be taken toward geostatistical modeling. A novel, principled and unified technique for pattern analysis and generation that ensures computational efficiency and enables a straightforward incorporation of domain knowledge will be presented.

In the developed methodology, patterns scanned from the training image are represented as points in a Cartesian space using multidimensional scaling. The idea behind this mapping is to use distance functions as a tool for analyzing variability between all the patterns in a training image. These distance functions can be tailored to the application at hand. Next, by significantly reducing the dimensionality of the problem and using kernel space mapping, an improved pattern classification algorithm is obtained. This paper discusses the various implementation details to accomplish these ideas. Several examples are presented and a qualitative comparison is made with previous methods. An improved pattern continuity and data-conditioning capability is observed in the generated realizations for both continuous and categorical variables. We show how the proposed methodology is much less sensitive to the user-provided parameters, and at the same time has the potential to reduce computational time significantly.


Geostatistics Multiple point statistics Distance-based method Kernel Mapping Pattern classification Training image 


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  1. Arpat GB (2005) Sequential simulation with patterns. PhD thesis, Stanford University Google Scholar
  2. Arpat BG, Caers J (2007) Stochastic simulation with patterns. Math Geol 39(202):177–203 CrossRefGoogle Scholar
  3. Borg I, Groenen PJF (1997) Modern multidimensional scaling. Springer, New York Google Scholar
  4. Buja A, Swayne DF, Littman M, Dean N, Hofmann H, Chen L (2008) Data visualization with multidimensional scaling. J Comput Graph Stat 17(2):444–472 CrossRefGoogle Scholar
  5. Caers J (2008) Distance-based random field models and their applications. In: Proceedings of the eighth international geostatistics congress, Santiago, Chile, vol 1, Plenary Google Scholar
  6. Caers J, Park, KA (2008) Distance-based representation of reservoir uncertainty: the Metric EnKF. In: Proceedings to the 11th European conference on the mathematics of oil recovery, Bergen, Norway Google Scholar
  7. Chiles JP, Delfiner P (1999) Geostatistics: modeling spatial uncertainty. Wiley, New York Google Scholar
  8. Cover TM, Thomas JA (1991) Elements of information theory. Wiley, New York CrossRefGoogle Scholar
  9. Cox TF, Cox MAA (2001) Multidimensional scaling. Chapman & Hall, London Google Scholar
  10. Daly C (2004) Higher order models using entropy, Markov random fields and sequential simulation. In: Leuangthong O, Deutsch CV (eds) Geostatistics, Banff 2004. Springer, Dordrecht, pp 215–224 Google Scholar
  11. Daly C, Knudby C (2007) Multipoint statistics in reservoir modelling and in computer vision. In: Petroleum Geostatistics 2007. Cascais, Portugal Google Scholar
  12. Deutsch CV, Gringarten E (2000). Accounting for multiple-point continuity in geostatistical modeling. In: 6th International Geostatistics Congress, Geostatistics Association of Southern Africa, vol 1, pp 156–165 Google Scholar
  13. Deutsch CV, Wang L (1996). Hierarchical object-based geostatistical modeling of fluvial reservoirs. Paper SPE 36514 presented at the 1996 SPE Annual Technical Conference and Exhibition, Denver, Oct 6–9 Google Scholar
  14. Dimitrakopoulos R, Mustapha H, Gloaguen E (2010) High-order statistics of spatial random fields: Exploring spatial cumulants for modelling complex, non-Gaussian and non-linear phenomena. Math Geosci 42(1):65–100 CrossRefGoogle Scholar
  15. Dujardin B, Wu J, Journel A (2006), Sensitivity analysis on filtersim. In: 19th SCRF affiliate meeting, Stanford University Google Scholar
  16. Gloaguen E, Dimitrakopoulos R (2009) Two-dimensional conditional simulations based on the wavelet decomposition of training images. Math Geosci 41(6):679–701 CrossRefGoogle Scholar
  17. Goovaerts P (1997) Geostatistics for natural resources evaluation. Oxford University Press, New York Google Scholar
  18. Guardiano F, Srivastava RM (1993) Multivariate, geostatistics: beyond bivariate moments. In: Soares A (ed) Geostatistics Troia. Kluwer Academic, Dordrecht, pp 133–144 Google Scholar
  19. Haldorsen HH, Lake LW (1984) A new approach to shale management in field-scale models. Soc Pet Eng J 24(8):447–452 Google Scholar
  20. Holden L, Hauge R, Skare O, Skorstad A (1998) Modeling of fluvial reservoirs with object models. Math Geol 30(5):473–496 CrossRefGoogle Scholar
  21. Honarkhah M, Caers J (2008) Classifying existing and generating new training image patterns in kernel space. In: 21st SCRF affiliate meeting, Stanford University Google Scholar
  22. Honarkhah M, Caers J (2009) Stochastic simulation of patterns using distance-based pattern modeling. In: 22nd SCRF affiliate meeting, Stanford University Google Scholar
  23. Isaaks E (1990) The application of Monte Carlo methods to the analysis of spatially correlated data. PhD thesis, Stanford University Google Scholar
  24. Journel AG (1983) Non-parametric estimation of spatial distributions. Math Geol 15(3):445–468 CrossRefGoogle Scholar
  25. Kjønsberg H, Kolbjørnsen O (2008) Markov mesh simulations with data conditioning through indicator kriging. In: Proceedings of the Eighth International Geostatistics Congress, Santiago, Chile Google Scholar
  26. Lyster S, Deutsch CV (2008) MPS simulation in a Gibbs sampler algorithm. In: Proceedings of the eighth international geostatistics congress, Santiago, Chile, vol 1, Plenary Google Scholar
  27. MacKay DJC (2003) Information theory, inference, and learning algorithms. Cambridge University Press, Cambridge Google Scholar
  28. MacQueen JB (1967) Some Methods for classification and analysis of multivariate observations. In: Proceedings of 5-th Berkeley symposium on mathematical statistics and probability, Berkeley, University of California Press, vol 1, pp 281–297 Google Scholar
  29. Maitre H, Campedel M, Moulines E, Datcu M (2005) Feature selection for satellite image indexing. In: ESA-EUSC: image information mining, Frascati, Italy Google Scholar
  30. Ortiz JM, Deutsch CV (2004) Indicator simulation accounting for multiple-Point statistics. Math Geol 36(5):545–565 CrossRefGoogle Scholar
  31. Park KA, Schiedt C, Caers J (2008) Simultaneous conditioning of multiple non-Gaussian geostatistical models to highly nonlinear data using distances in kernel space. In: Proceedings of the eighth international geostatistical congress, Santiago, vol 1, Plenary Google Scholar
  32. Parra A, Ortiz JM (2009) Conditional multiple-point simulation with a texture synthesis algorithm. In: IAMG 2009 Conference, Stanford University Google Scholar
  33. Remy N, Boucher A, Wu J (2008) Applied geostatistics with SGeMS: a user’s guide. Cambridge University Press, Cambridge Google Scholar
  34. Russ JC (1995) Image processing handbook, 2nd edn. CRC Press, Boca Raton Google Scholar
  35. Scheidt C, Caers J (2008) Representing spatial uncertainty using distances and kernels. Math Geosci 41(4):397–419 CrossRefGoogle Scholar
  36. Scheidt C, Caers J (2009) A new method for uncertainty quantification using distances and kernel methods. Application to a deepwater turbidite reservoir. In: SPEJ, SPE-118740-PA Google Scholar
  37. Scholkopf B, Smola AJ (2001) Learning with kernels: support vector machines, regularization, optimization, and beyond. MIT Press, Cambridge Google Scholar
  38. Shannon CE (1948) A mathematical theory of communication. Bell Syst Tech J 27:379–423 Google Scholar
  39. Shawe-Taylor J, Cristianini N (2004) Kernel methods for pattern analysis. Cambridge University Press, Cambridge Google Scholar
  40. Srivastava RM (1992) Reservoir characterization with probability field simulation. SPE paper no. 24753 Google Scholar
  41. Stoyan D, Kendall WS, Mecke J (1987) Stochastic geometry and its applications. Wiley, New York Google Scholar
  42. Strebelle S (2000) Sequential simulation drawing structures from training images. PhD thesis, Stanford University Google Scholar
  43. Strebelle S (2002) Conditional simulation of complex geological structures using multiple-point geostatistics. Math Geol 34(1):1–22 CrossRefGoogle Scholar
  44. Suzuki S, Caers J (2006) History matching with an uncertain geological scenario. In: SPE Annual Technical Conference and Exhibition, SPE 102154 Google Scholar
  45. Suzuki S, Caers J (2008) A distance-based prior model parameterization for constraining solutions of spatial inverse problems. Math Geosci 40(4):445–469 CrossRefGoogle Scholar
  46. Suzuki S, Caumon G, Caers J (2008) Dynamic data integration for structural modeling: model screening approach using a distance-based model parameterization. Comput Geosci 12(1):105–119 CrossRefGoogle Scholar
  47. Tjelmeland H (1996) Stochastic Models in reservoir characterization and Markov random fields for compact objects. Doctoral Dissertation, Norwegian University of Science and Technology. Trondheim, Norway Google Scholar
  48. Tjelmeland H, Eidsvik J (2004) Directional metropolis-Hastings updates for posteriors with non linear likelihood. In: Leuangthong O, Deutsch CV (eds) Geostatistics, Banff. Springer, Dordrecht, pp 95–104 Google Scholar
  49. Wu J (2007) 4D seismic and multiple-point pattern data integration using geostatistics. Phd thesis, Stanford University Google Scholar
  50. Zhang T (2006) Filter-based training pattern classification for spatial pattern simulation. PhD thesis, Stanford University, Stanford, CA Google Scholar
  51. Zhu M, Ghodsi A (2006) Automatic dimensionality selection from the scree plot via the use of profile likelihood. Comput Stat Data Anal 51(2):918–930 CrossRefGoogle Scholar

Copyright information

© International Association for Mathematical Geosciences 2010

Authors and Affiliations

  1. 1.Department of Energy Resources EngineeringStanford UniversityStanfordUSA

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