A High-Order, Data-Driven Framework for Joint Simulation of Categorical Variables

  • Ilnur MinniakhmetovEmail author
  • Roussos Dimitrakopoulos
Part of the Quantitative Geology and Geostatistics book series (QGAG, volume 19)


Relatively recent techniques for categorical simulations are based on multipoint statistical approaches where a training image (TI) is used to derive complex spatial relationships using patterns. In these cases, simulated realizations are driven by the TI utilized, while the spatial statistics of the hard data is not used. This paper presents a data-driven, high-order simulation approach based upon the approximation of high-order spatial indicator moments. The high-order spatial statistics are expressed as functions of spatial distances similar to variogram models for two-point methods. It is shown that the higher-order statistics are connected with lower orders via boundary conditions. Using an advanced recursive B-spline approximation algorithm, the high-order statistics are reconstructed from hard data. Finally, conditional distribution is calculated using Bayes rule and random values are simulated sequentially for all unsampled grid nodes. The main advantages of the proposed technique are its ability to simulate without a training image, which reproduces the high-order statistics of hard data, and to adopt the complexity of the model to the information available in the hard data. The approach is tested with a synthetic dataset and compared to a conventional second-order method, sisim, in terms of cross-correlations and high-order spatial statistics.


Training Image Hard Data Variogram Model Unsampled Location Sequential Indicator Simulation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



Funding was provided by the Natural Sciences and Engineering Research Council of Canada (NSERC) Discovery Grant 239019 and mining industry partners of the COSMO Lab (AngloGold Ashanti, Barrick Gold, BHP Billiton, De Beers Canada, Kinross Gold, Newmont Mining, and Vale).


  1. Arndt C (2004) Information measures: information and its description in science and engineering. Springer, AmsterdamGoogle Scholar
  2. Babenko K (1986) Fundamentals of numerical analysis. Nauka, MoscowGoogle Scholar
  3. Caers J (2005) Petroleum geostatistics. SPE–Pennwell Books, HoustonGoogle Scholar
  4. Chilès JP, Delfiner P (2012) Geostatistics: modeling spatial uncertainty, 2nd edn. Wiley, New YorkCrossRefGoogle Scholar
  5. Chugunova TL, Hu LY (2008) Multiple-point simulations constrained by continuous auxiliary data. Math Geosci 40:133–146CrossRefGoogle Scholar
  6. Cressie NA (1993) Statistics for spatial data. Wiley, New YorkGoogle Scholar
  7. David M (1977) Geostatistical ore reserve estimation. Elsevier, AmsterdamGoogle Scholar
  8. David M (1988) Handbook of applied advanced geostatistical ore reserve estimation. Elsevier, AmsterdamGoogle Scholar
  9. Deutsch C, Journel A (1998) GSLIB: geostatistical software library and user’s guide, 2nd edn. 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 42:65–99CrossRefGoogle Scholar
  11. Evans J, Bazilevs Y, Babuška I, Hughes T (2009) N-widths, sup–infs, and optimality ratios for the k-version of the isogeometric finite element method. Comput Methods Appl Mech Engrg 198:1726–1741CrossRefGoogle Scholar
  12. Goovaerts P (1998) Geostatistics for natural resources evaluation. Cambridge University Press, CambridgeGoogle Scholar
  13. Guardiano J, Srivastava RM (1993) Multivariate geostatistics: beyond bivariate moments. Geosatistics Tróia’92:133–144Google Scholar
  14. Journel AG (1993) Geostatistics: roadblocks and challenges. Stanford Center for Reservoir ForecastingGoogle Scholar
  15. Journel AG (2003) Multiple-point Geostatistics: a State of the Art. Stanford Center for Reservoir ForecastingGoogle Scholar
  16. Journel AG, Alabert F (1990) New method for reservoir mapping. Pet Technol 42(2):212–218CrossRefGoogle Scholar
  17. Journel AG, Huijbregts CJ (1978) Mining geostatistics. Academic, San DiegoGoogle Scholar
  18. Kitanidis PK (1997) Introduction to geostatistics—applications in hydrogeology. Cambridge Univ Press, CambridgeCrossRefGoogle Scholar
  19. Mao S, Journel A (1999) Generation of a reference petrophysical/seismic data set: the Stanford V reservoir. 12th Annual Report. Stanford Center for Reservoir Forecasting, StanfordGoogle Scholar
  20. Mariethoz G, Kelly B (2011) Modeling complex geological structures withelementary training images and transform-invariant distances. Water Resour Res 47:1–2CrossRefGoogle Scholar
  21. Mariethoz G, Renard P (2010) Reconstruction of incomplete data sets or images using direct sampling. Math Geosci 42:245–268CrossRefGoogle Scholar
  22. Matheron G (1971) The theory of regionalized variables and its applications. Cahier du Centre de Morphologie Mathematique, No 5Google Scholar
  23. Mustapha H, Dimitrakopoulos R (2010a) A new approach for geological pattern recognition using high-order spatial cumulants. Comput Geosci 36(3):313–334CrossRefGoogle Scholar
  24. Mustapha H, Dimitrakopoulos R (2010b) High-order stochastic simulations for complex non-Gaussian and non-linear geological patterns. Math Geosci 42:455–473CrossRefGoogle Scholar
  25. Pyrcz M, Deutsch C (2014) Geostatistical reservoir modeling, 2nd edn. Oxford University Press, New YorkGoogle Scholar
  26. Remy N, Boucher A, Wu J (2009) Applied geostatistics with SGeMS: a user’s guide. Cambridge University Press, CambridgeCrossRefGoogle Scholar
  27. Straubhaar JRP, Renard P, Mariethoz G, Froidevaux R, Besson O (2011) An improved parallel multiple-point algorithm using a list approach. Math Geosci 43:305–328CrossRefGoogle Scholar
  28. Strebelle S (2002) Conditional simulation of complex geological structures using multiple point stastics. Math Geosci 34:1–22Google Scholar
  29. Strebelle S, Cavelius C (2014) Solving speed and memory issues in multiple-point statistics simulation program SNESIM. Math Geosci 46:171–186CrossRefGoogle Scholar
  30. Tikhonov AN, Arsenin VY (1977) Solution of Ill-posed problems. Winston & Sons, Washington, DCGoogle Scholar
  31. Tjelmeland H, Besag J (1998) Markov random fields with higherorder interactions. Scand J Stat 25(3):415–433Google Scholar
  32. Toftaker H, Tjelmeland H (2013) Construction of binary multi-grid Markov random field prior models from training images. Math Geosci 45:383–409CrossRefGoogle Scholar
  33. Vargas-Guzmán J (2011) The Kappa model of probability and higher-order rock sequences. Comput Geosci 15:661–671CrossRefGoogle Scholar
  34. Vargas-Guzmán J, Qassab H (2006) Spatial conditional simulation of facies objects for modeling complex clastic reservoirs. J Petrol Sci Eng 54:1–9CrossRefGoogle Scholar
  35. Wang W, Pottmann H, Liu Y (2006) Fitting B-spline curves to point clouds by curvature-based squared distance minimization. ACM Transactions on Graphics 25(2):214–238CrossRefGoogle Scholar
  36. Webster R, Oliver MA (2007) Geostatistics for environmental scientists. Wiley, New YorkCrossRefGoogle Scholar
  37. Yahya WJ (2011) Image reconstruction from a limited number of samples: a matrix-completion-based approach. Mater Thesis, McGill University, MontrealGoogle Scholar
  38. Zhang T, Switzer P, Journel A (2006) Filter-based classification of training image patterns for spatial simulation. Math Geosci 38(1):63–80Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.COSMO, Stochastic Mine Planning LaboratoryMcGill UniversityMontrealCanada

Personalised recommendations