Verifying the high-order consistency of training images with data for multiple-point geostatistics

Abstract

Parameter inference is a key aspect of spatial modeling. A major appeal of variograms is that they allow inferring the spatial structure solely based on conditioning data. This is very convenient when the modeler does not have a ready-made geological interpretation. To date, such an easy and automated interpretation is not available in the context of most multiple-point geostatistics applications. Because training images are generally conceptual models, their preparation is often based on subjective criteria of the modeling expert. As a consequence, selection of an appropriate training image is one of the main issues one must face when using multiple-point simulation. This paper addresses the development of a geostatistical tool that addresses two separate problems. It allows (1) ranking training images according to their relative compatibility to the data, and (2) obtaining an absolute measure quantifying the consistency between training image and data in terms of spatial structure. For both, two alternative implementations are developed. The first one computes the frequency of each pattern in each training image. This method is statistically sound but computationally demanding. The second implementation obtains similar results at a lesser computational cost using a direct sampling approach. The applicability of the methodologies is successfully evaluated in two synthetic 2D examples and one real 3D mining example at the Escondida Norte deposit.