Mathematical Geosciences

, 42:65 | Cite as

High-order Statistics of Spatial Random Fields: Exploring Spatial Cumulants for Modeling Complex Non-Gaussian and Non-linear Phenomena

  • Roussos Dimitrakopoulos
  • Hussein Mustapha
  • Erwan Gloaguen


The spatial distributions of earth science and engineering phenomena under study are currently predicted from finite measurements and second-order geostatistical models. The latter models can be limiting, as geological systems are highly complex, non-Gaussian, and exhibit non-linear patterns of spatial connectivity. Non-linear and non-Gaussian high-order geostatistics based on spatial connectivity measures, namely spatial cumulants, are proposed as a new alternative modeling framework for spatial data. This framework has two parts. The first part is the definition, properties, and inference of spatial cumulants—including understanding the interrelation of cumulant characteristics with the in-situ behavior of geological entities or processes, as examined in this paper. The second part is the research on a random field model for simulation based on its high-order spatial cumulants.

Mathematical definitions of non-Gaussian spatial random functions and their high-order spatial statistics are presented herein, stressing the notion of spatial cumulants. The calculation of spatial cumulants with spatial templates follows, including anisotropic experimental cumulants. Several examples of two- and three-dimensional images, including a diamond bearing kimberlite pipe from the Ekati Mine in Canada, are analyzed to assess the relations between cumulants and the spatial behavior of geological processes. Spatial cumulants of orders three to five are shown to capture directional multiple-point periodicity, connectivity including connectivity of extreme values, and spatial architecture. In addition, they provide substantial information on geometric characteristics and anisotropy of geological patterns. It is further shown that effects of complex spatial patterns are seen even if only subsets of all cumulant templates are computed. Compared to second-order statistics, cumulant maps are found to include a wealth of additional information from underlying geological patterns. Further work seeks to integrate this information in the predictive capabilities of a random field model.

High-order statistics Non-Gaussian spatial random functions Spatial cumulants Complex geology 


  1. Arpat GB, Caers J (2007) Conditional simulation with patterns. Math Geosci 39(2):177–203 Google Scholar
  2. Billinger DR, Rosenblatt M (1966) Asymptotic theory of kth-order spectra. In: Harris B (ed) Spectral analysis of time series. Wiley, New York, pp 189–232 Google Scholar
  3. Boucher A (2009) Considering complex training images with search tree partitioning. Comput Geosci 35:1151–1158 CrossRefGoogle Scholar
  4. Caers J (2005) Petroleum geostatistics. SPE–Pennwell Books, Houston Google Scholar
  5. Chilès JP, Delfiner P (1999) Geostatistics: modeling spatial uncertainty. Wiley, New York Google Scholar
  6. Chugunova TL, Hu Ly (2008) Multiple-point simulations constrained by continuous auxiliary data. Math Geosci 40:133–146 CrossRefGoogle Scholar
  7. Cox DR (1972) Regression models and life-tables. J R Stat Soc B 34:45–58 Google Scholar
  8. Cressie NA (1993) Statistics for spatial data. Wiley, New York Google Scholar
  9. Daly C (2004) Higher order models using entropy, Markov random fields and sequential simulation. In: Geostatistics Banff 2004. Springer, Berlin, pp 215–225 Google Scholar
  10. David M (1977) Geostatistical ore reserve estimation. Elsevier, Amsterdam Google Scholar
  11. David M (1988) Handbook of applied advanced geostatistical ore reserve estimation. Elsevier, Amsterdam Google Scholar
  12. Dreiss N, Johnson S (1989) Hydrostratigraphic interpretation using indicator geostatistics. Water Resour Res 25(12):2501–2510 CrossRefGoogle Scholar
  13. Gaztanaga EP, Fosalba P, Elizalde E (2000) Gravitational evolution of the large-scale probability density distribution. Astrophys J 539:522–531 CrossRefGoogle Scholar
  14. 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
  15. Goovaerts P (1998) Geostatistics for natural resources evaluation. Cambridge University Press, Cambridge Google Scholar
  16. Guardiano J, Srivastava RM (1993) Multivariate geostatistics: Beyond bivariate moments. In: Soares A (ed) Geosatistics Tróia ’92, vol 1. Kluwer, Dordrecht, pp 133–144 Google Scholar
  17. Journel AG (1997) Deterministic geostatistics: A new visit. In: Baafy E, Shofield N (eds) Geostatistics Woolongong 1996. Kluwer, Dordrecht, pp 213–224 Google Scholar
  18. Journel AG, Huijbregts ChJ (1978) Mining geostatistics. Academic Press, San Diego Google Scholar
  19. Kitanidis PK (1997) Introduction to geostatistics—Applications in hydrogeology. Cambridge Univ Press, Cambridge Google Scholar
  20. Krishnan S, Journel AG (2003) Spatial connectivity: From variograms to multiple-point measures. Math Geol 35:915–925 CrossRefGoogle Scholar
  21. Mao S, Journel A (1999) Generation of a reference petrophysical/seismic data set: the Stanford v reservoir. Report 12, Stanford Center for Reservoir Forecasting, Stanford, CA Google Scholar
  22. Matérn B (1960) Spatial variation—Stochastic models and their application to some problems in forest surveys and other sampling investigations. Meddelanden fran Statens Skogsforskningsinstitud 49(5) Almaenna, Stockholm Google Scholar
  23. Matheron G (1971) The theory of regionalized variables and its applications. Cahier du Centre de Morphologie Mathematique, No 5, Fontainebleau Google Scholar
  24. Mendel JM (1991) Use of high-order statistics (spectra) in signal processing and systems theory: Theoretical results and some applications. IEEE Proc 79:279–305 CrossRefGoogle Scholar
  25. Mirowski PW, Trtzlaff DM, Davies RC, McCormick DS, Williams N, Signer C (2008) Stationary scores on training images for multipoint geostatistics. Math Geosci 41:447–474 CrossRefGoogle Scholar
  26. Mustapha H, Dimitrakopoulos R (2010) High-order stochastic simulations for complex non-Gaussian and non-linear geological patterns. Math Geosci 42(5) Google Scholar
  27. Nikias CL, Petropulu AP (1993) Higher–order spectra analysis: A nonlinear signal processing framework. PTR Prentice Hall, Upper Saddle River Google Scholar
  28. Nowicki T, Crawford B, Dyck D, Carlson J, McElroy R, Oshust P, Helmstaedt H (2004) The geology of the kimberlite pipes of the Ekati property, Northwest Territories, Canada. Lithos 76:1–27 CrossRefGoogle Scholar
  29. Remy N, Boucher A, Wu J (2009) Applied geostatistics with SGeMS: A user’s guide. Cambridge University Press, Cambridge Google Scholar
  30. Rendu JM, Ready L (1982) Geology and the semivariogram—A critical relationship. In: APCOM’82, pp 771–783 Google Scholar
  31. Rosenblatt M (1985) Stationary sequences and random fields. Birkhäuser, Boston Google Scholar
  32. Scheidt C, Caers J (2009) Representing spatial uncertainty using distances and kernels. Math Geosci 41:397–419 CrossRefGoogle Scholar
  33. Shiryaev AN (1960) Some problems in the spectral theory of higher order moments I. Theory Probab Its Appl 5(3):265–284 CrossRefGoogle Scholar
  34. Strebelle S (2002) Conditional simulation of complex geological structures using multiple point statistics. Math Geol 34:1–22 CrossRefGoogle Scholar
  35. Tjelmeland H (1998) Markov random fields with higher order interactions. Scand J Statist 25:415–433 CrossRefGoogle Scholar
  36. Tjelmeland H, Eidsvik J (2004) Directional Metropolis: Hastings updates for posteriors with nonlinear likelihoods. In: Geostatistics Banff 2004. Springer, Berlin, pp 95–104 Google Scholar
  37. Webster R, Oliver MA (2007) Geostatistics for environmental scientists. Wiley, New York CrossRefGoogle Scholar
  38. Wu J, Boucher A, Zhang T (2008) SGeMS code for pattern simulation of continuous and categorical variables: FILTERSIM. Comput Geosci 34:1863–1876 CrossRefGoogle Scholar
  39. Zhang F (2005) A high order cumulants based multivariate nonlinear blind source separation method source. Mach Learn 61(1–3):105–127 CrossRefGoogle Scholar
  40. Zhang T, Switzer P, Journel AG (2006) Filter-based classification of training image patterns for spatial simulation. Math Geol 38(1):63–80 CrossRefGoogle Scholar

Copyright information

© International Association for Mathematical Geosciences 2009

Authors and Affiliations

  • Roussos Dimitrakopoulos
    • 1
  • Hussein Mustapha
    • 1
  • Erwan Gloaguen
    • 1
  1. 1.COSMO–Stochastic Mine Planning Laboratory, Department of Mining and Materials EngineeringMcGill UniversityMontrealCanada

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