Approximations of High-Order Spatial Statistics Through Decomposition

Abstract

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.