Mathematical Geosciences

, Volume 42, Issue 5, pp 457–485

High-order Stochastic Simulation of Complex Spatially Distributed Natural Phenomena

Open Access
Special Issue

DOI: 10.1007/s11004-010-9291-8

Cite this article as:
Mustapha, H. & Dimitrakopoulos, R. Math Geosci (2010) 42: 457. doi:10.1007/s11004-010-9291-8


Spatially distributed and varying natural phenomena encountered in geoscience and engineering problem solving are typically incompatible with Gaussian models, exhibiting nonlinear spatial patterns and complex, multiple-point connectivity of extreme values. Stochastic simulation of such phenomena is historically founded on second-order spatial statistical approaches, which are limited in their capacity to model complex spatial uncertainty. The newer multiple-point (MP) simulation framework addresses past limits by establishing the concept of a training image, and, arguably, has its own drawbacks. An alternative to current MP approaches is founded upon new high-order measures of spatial complexity, termed “high-order spatial cumulants.” These are combinations of moments of statistical parameters that characterize non-Gaussian random fields and can describe complex spatial information. Stochastic simulation of complex spatial processes is developed based on high-order spatial cumulants in the high-dimensional space of Legendre polynomials. Starting with discrete Legendre polynomials, a set of discrete orthogonal cumulants is introduced as a tool to characterize spatial shapes. Weighted orthonormal Legendre polynomials define the so-called Legendre cumulants that are high-order conditional spatial cumulants inferred from training images and are combined with available sparse data sets. Advantages of the high-order sequential simulation approach developed herein include the absence of any distribution-related assumptions and pre- or post-processing steps. The method is shown to generate realizations of complex spatial patterns, reproduce bimodal data distributions, data variograms, and high-order spatial cumulants of the data. In addition, it is shown that the available hard data dominate the simulation process and have a definitive effect on the simulated realizations, whereas the training images are only used to fill in high-order relations that cannot be inferred from data. Compared to the MP framework, the proposed approach is data-driven and consistently reconstructs the lower-order spatial complexity in the data used, in addition to high order.


Conditional sequential simulation High-order spatial cumulants Legendre polynomials 
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© The Author(s) 2010

Authors and Affiliations

  1. 1.COSMO – Stochastic Mine Planning Laboratory, Department of Mining and Materials EngineeringMcGill UniversityMontrealCanada

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