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

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Abstract

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.

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References

  • Arpat GB, Caers J (2007) Conditional simulation with patterns. Math Geosci 39(2):177–203

    Google Scholar 

  • 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 

  • Boucher A (2009) Considering complex training images with search tree partitioning. Comput Geosci 35:1151–1158

    Article  Google Scholar 

  • Caers J (2005) Petroleum geostatistics. SPE–Pennwell Books, Houston

    Google Scholar 

  • Chilès JP, Delfiner P (1999) Geostatistics: modeling spatial uncertainty. Wiley, New York

    Google Scholar 

  • Chugunova TL, Hu Ly (2008) Multiple-point simulations constrained by continuous auxiliary data. Math Geosci 40:133–146

    Article  Google Scholar 

  • Cox DR (1972) Regression models and life-tables. J R Stat Soc B 34:45–58

    Google Scholar 

  • Cressie NA (1993) Statistics for spatial data. Wiley, New York

    Google Scholar 

  • 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 

  • David M (1977) Geostatistical ore reserve estimation. Elsevier, Amsterdam

    Google Scholar 

  • David M (1988) Handbook of applied advanced geostatistical ore reserve estimation. Elsevier, Amsterdam

    Google Scholar 

  • Dreiss N, Johnson S (1989) Hydrostratigraphic interpretation using indicator geostatistics. Water Resour Res 25(12):2501–2510

    Article  Google Scholar 

  • Gaztanaga EP, Fosalba P, Elizalde E (2000) Gravitational evolution of the large-scale probability density distribution. Astrophys J 539:522–531

    Article  Google Scholar 

  • Gloaguen E, Dimitrakopoulos R (2009) Two-dimensional conditional simulations based on the wavelet decomposition of training images. Math Geosci 41(6):679–701

    Article  Google Scholar 

  • Goovaerts P (1998) Geostatistics for natural resources evaluation. Cambridge University Press, Cambridge

    Google Scholar 

  • 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 

  • Journel AG (1997) Deterministic geostatistics: A new visit. In: Baafy E, Shofield N (eds) Geostatistics Woolongong 1996. Kluwer, Dordrecht, pp 213–224

    Google Scholar 

  • Journel AG, Huijbregts ChJ (1978) Mining geostatistics. Academic Press, San Diego

    Google Scholar 

  • Kitanidis PK (1997) Introduction to geostatistics—Applications in hydrogeology. Cambridge Univ Press, Cambridge

    Google Scholar 

  • Krishnan S, Journel AG (2003) Spatial connectivity: From variograms to multiple-point measures. Math Geol 35:915–925

    Article  Google Scholar 

  • 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

  • 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

  • Matheron G (1971) The theory of regionalized variables and its applications. Cahier du Centre de Morphologie Mathematique, No 5, Fontainebleau

  • 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

    Article  Google Scholar 

  • 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

    Article  Google Scholar 

  • Mustapha H, Dimitrakopoulos R (2010) High-order stochastic simulations for complex non-Gaussian and non-linear geological patterns. Math Geosci 42(5)

  • Nikias CL, Petropulu AP (1993) Higher–order spectra analysis: A nonlinear signal processing framework. PTR Prentice Hall, Upper Saddle River

    Google Scholar 

  • 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

    Article  Google Scholar 

  • Remy N, Boucher A, Wu J (2009) Applied geostatistics with SGeMS: A user’s guide. Cambridge University Press, Cambridge

    Google Scholar 

  • Rendu JM, Ready L (1982) Geology and the semivariogram—A critical relationship. In: APCOM’82, pp 771–783

  • Rosenblatt M (1985) Stationary sequences and random fields. Birkhäuser, Boston

    Google Scholar 

  • Scheidt C, Caers J (2009) Representing spatial uncertainty using distances and kernels. Math Geosci 41:397–419

    Article  Google Scholar 

  • Shiryaev AN (1960) Some problems in the spectral theory of higher order moments I. Theory Probab Its Appl 5(3):265–284

    Article  Google Scholar 

  • Strebelle S (2002) Conditional simulation of complex geological structures using multiple point statistics. Math Geol 34:1–22

    Article  Google Scholar 

  • Tjelmeland H (1998) Markov random fields with higher order interactions. Scand J Statist 25:415–433

    Article  Google Scholar 

  • 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 

  • Webster R, Oliver MA (2007) Geostatistics for environmental scientists. Wiley, New York

    Book  Google Scholar 

  • Wu J, Boucher A, Zhang T (2008) SGeMS code for pattern simulation of continuous and categorical variables: FILTERSIM. Comput Geosci 34:1863–1876

    Article  Google Scholar 

  • Zhang F (2005) A high order cumulants based multivariate nonlinear blind source separation method source. Mach Learn 61(1–3):105–127

    Article  Google Scholar 

  • Zhang T, Switzer P, Journel AG (2006) Filter-based classification of training image patterns for spatial simulation. Math Geol 38(1):63–80

    Article  Google Scholar 

Download references

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Authors and Affiliations

  1. COSMO–Stochastic Mine Planning Laboratory, Department of Mining and Materials Engineering, McGill University, Montreal, Qc, Canada, H3A 2A7

    Roussos Dimitrakopoulos, Hussein Mustapha & Erwan Gloaguen

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  1. Roussos Dimitrakopoulos
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  2. Hussein Mustapha
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  3. Erwan Gloaguen
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Correspondence to Roussos Dimitrakopoulos.

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Dimitrakopoulos, R., Mustapha, H. & Gloaguen, E. High-order Statistics of Spatial Random Fields: Exploring Spatial Cumulants for Modeling Complex Non-Gaussian and Non-linear Phenomena. Math Geosci 42, 65–99 (2010). https://doi.org/10.1007/s11004-009-9258-9

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  • DOI: https://doi.org/10.1007/s11004-009-9258-9

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