A Comparison of Random Field Models Beyond Bivariate Distributions
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A Comparison of Random Field Models Beyond Bivariate Distributions Xavier Emery · Julián M. Ortiz
Received: 27 June 2007 / Accepted: 25 May 2010 / Published online: 19 October 2010 © International Association for Mathematical Geosciences 2010
Abstract In order to determine to what extent a spatial random field can be characterized by its low-order distributions, we consider four models (specifically, random spatial tessellations) with exactly the same univariate and bivariate distributions and we compare the statistics associated with various multiple-point configurations and the responses to specific transfer functions. The three- and four-point statistics are found to be the same or experimentally hardly distinguishable because of ergodic fluctuations, whereas change of support and flow simulation produce very different outcomes. This example indicates that low-order distributions may not discriminate between contending random field models, that simulation algorithms based on such distributions may not reproduce the spatial properties of a given model or training image, and that the inference of high-order distribution may require very large training images. Keywords Multivariate distributions · Poisson hyperplane tessellation · Stable iterated tessellation · Dead leaves tessellation · Random mosaic · Multiple-point statistics 1 Introduction Geostatistical simulation is widely used in the analysis of regionalized data in order to quantify spatial uncertainty and to perform spatial prediction. It relies on the definition of a random field model over a Euclidean space (Rd , d ≥ 1) and the conX. Emery () · J.M. Ortiz Department of Mining Engineering, University of Chile, Santiago, Chile e-mail: [email protected] J.M. Ortiz e-mail: [email protected] X. Emery · J.M. Ortiz ALGES Laboratory, Advanced Mining Technology Center, University of Chile, Santiago, Chile
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Math Geosci (2011) 43: 183–202
struction of multiple realizations of this random field, conditioned to the available data. In applications, the Gaussian random field model is very popular because it is fully characterized by a mean value and a covariance function that measures bivariate dependencies. To add flexibility and to describe a wider class of regionalized phenomena, other simulation approaches have been proposed, based on the modeling of multiple-point statistics, that is, statistics that depend on more than two points and therefore measure multivariate dependencies (Guardiano and Srivastava 1993; Strebelle 2002; Ortiz and Deutsch 2004; Daly 2005; Chugunova and Hu 2008). Multiple-point statistics are often inferred from a training image deemed representative of the random field to be reproduced. In practice, however, such statistics only give an insight into the low-order distributions of this random field and high-order distributions remain unspecified. Indeed, a training image provides a realization of a random vector of finite size n and, therefore, it cannot provide information on distributions of orders greater than n. Furthermore, i
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