Approximations of High-Order Spatial Statistics Through Decomposition

  • Ryan GoodfellowEmail author
  • Hussein Mustapha
  • Roussos Dimitrakopoulos
Part of the Quantitative Geology and Geostatistics book series (QGAG, volume 17)


Many of the existing multiple-point and high-order geostatistical algorithms depend on training images as a source of patterns or statistics. Generating these training images, particularly for continuous variables, can be a labor-intensive process without any guarantee that the true high-order statistics of the deposit are accurately represented. This work proposes a decomposition of a high-order statistic (moment) into a set of weighted sums that can be used for approximating spatial statistics on sparse data sets, which could lead to new data-driven simulation algorithms that forgo the use of training images. Using this decomposition, it is possible to approximate the n-point moment by searching for pairs of points and combining the pairs in the various directions at a later step, rather than searching for replicates of the n-point template, which is often unreliable for sparse data sets. Experimental results on sparse data sets indicate that the approximations perform much better than using the actual n-point moments on the same data. Additionally, the quality of the approximation does not appear to degrade significantly for higher-orders because it is able to use more information from pairs of data, which is generally not true for the moments from the sample data, which requires all n points in a template.


Training Image Stochastic Simulation Algorithm Unknown Point Spatial Moment Spatial Random Field 
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.


  1. 1.
    Arpat GB, Caers J (2007) Conditional simulation with patterns. Math Geosci. doi: 10.1007/s11004-006-9075-3 Google Scholar
  2. 2.
    Chilès JP, Delfiner P (1999) Geostatistics: modeling spatial uncertainty. Wiley, New York CrossRefGoogle Scholar
  3. 3.
    David M (1988) Handbook of applied advanced geostatistical ore reserve estimation. Elsevier, Amsterdam Google Scholar
  4. 4.
    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. doi: 10.1007/s11004-009-9258-9 Google Scholar
  5. 5.
    Goodfellow R, Albor F, Dimitrakopoulos R, Lloyd T (2010) Quantifying multi-element and volumetric uncertainty in a mineral deposit. In: MININ 2010: proceedings of the 4th international conference on mining innovation, Santiago, Chile, pp 149–158 Google Scholar
  6. 6.
    Goodfellow R, Mustapha H, Dimitrakopoulos R (2011) Approximations of high-order spatial statistics through decomposition. COSMO research report, No 5, vol 2, Montreal, QC Google Scholar
  7. 7.
    Goovaerts P (1998) Geostatistics for natural resources evaluation. Cambridge University Press, Cambridge Google Scholar
  8. 8.
    Guardiano J, Srivastava RM (1993) Multivariate geostatistics: beyond bivariate moments. In: Soares A (ed) Geostatistics Tróia ’92, vol 1. Kluwer, Dordrecht, pp 133–144 CrossRefGoogle Scholar
  9. 9.
    Journel AG, Huijbregts ChJ (1978) Mining geostatistics. Academic Press, San Diego Google Scholar
  10. 10.
    Lebedev NN, Silverman R (1972) Special functions and their applications. Dover, New York Google Scholar
  11. 11.
    Mao S, Journel AG (1999) Generation of a reference petrophysical/seismic data set: the Stanford V reservoir. Report 12, Stanford Center for Reservoir Forecasting, Stanford, CA Google Scholar
  12. 12.
    Mustapha H, Dimitrakopoulos R (2010) A new approach for geological pattern recognition using high-order spatial cumulants. Comput Geosci. doi: 10.1016/j.cageo.2009.04.015 Google Scholar
  13. 13.
    Mustapha H, Dimitrakopoulos R (2010) High-order stochastic simulations for complex non-Gaussian and non-linear geological patterns. Math Geosci. doi: 10.1007/s11004-010-9291-8 Google Scholar
  14. 14.
    Mustapha H, Dimitrakopoulos R (2011) HOSIM: a high-order stochastic simulation algorithm for generating three-dimensional complex geological patterns. Comput Geosci. doi: 10.1016/j.cageo.2010.09.007 Google Scholar
  15. 15.
    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
  16. 16.
    Strebelle S (2002) Conditional simulation of complex geological structures using multiple point statistics. Math Geosci. doi: 10.1023/A:1014009426274 Google Scholar
  17. 17.
    Zhang T, Switzer P, Journel AG (2006) Filter-based classification of training image patterns for spatial simulation. Math Geosci. doi: 10.1007/s11004-005-9004-x Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2012

Authors and Affiliations

  • Ryan Goodfellow
    • 1
    Email author
  • Hussein Mustapha
    • 2
  • Roussos Dimitrakopoulos
    • 1
  1. 1.McGill UniversityMontrealCanada
  2. 2.Schlumberger Abingdon Technology CenterAbingdonUK

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