Semivariogram Models Based on Geometric Offsets
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Semivariogram Models Based on Geometric Offsets1 Michael J. Pyrcz2 and Clayton V. Deutsch3 Kriging-based geostatistical models require a semivariogram model. Next to the initial decision of stationarity, the choice of an appropriate semivariogram model is the most important decision in a geostatistical study. Common practice consists of fitting experimental semivariograms with a nested combination of proven models such as the spherical, exponential, and Gaussian models. These models work well in most cases; however, there are some shapes found in practice that are difficult to fit. We introduce a family of semivariogram models that are based on geometric shapes, analogous to the spherical semivariogram, that are known to be conditional negative definite and provide additional flexibility to fit semivariograms encountered in practice. A methodology to calculate the associated geometric shapes to match semivariograms defined in any number of directions is presented. Greater flexibility is available through the application of these geometric semivariogram models. KEY WORDS: nested structures, kriging, stochastic simulation, geostatistics.
INTRODUCTION Kriging-based geostatistics is routinely used for estimation and simulation of continuous and categorical geologic properties. The random function paradigm of geostatistics involves three main steps: (1) definition of the variable and the stationary domain for the variable {Z(u), u∈A}, which involves the definition of rock types/facies and large scale trends, (2) establish a semivariogram model for the variable, γ (h), that is valid for all distances and directions found in the domain A, and (3) make inferences with kriging and Monte Carlo simulation. The reasonableness of the inferences depends on the first two steps (Pyrcz and others, 2006). The expert site-specific decision of a stationary domain is arguably the most important; however, the calculation and fitting of a semivariogram model is also very important. The inference step is largely automatic once the first two steps are 1Received
22 March 2005; accepted 15 September 2005; Published online: 7 November 2006. Stratigraphy, Earth Science R&D, Chevron Energy Technology Company, Houston, TX 77002; e-mail: [email protected] 3Department of Civil and Environmental Engineering, University of Alberta, Edmonton, AB, T6G 2W2 Canada; e-mail: [email protected] 2Quantitative
475 C 2006 International Association for Mathematical Geology 0882-8121/06/0500-0475/1
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taken. This paper is aimed at the second step of establishing a valid semivariogram model. The conventional method of modeling semivariograms by nested structures is reviewed. A suite of geometric semivariograms and a method for constructing new geometries that match custom continuity styles are presented. These geometric semivariogram models allow for greater flexibility in the generation of permissible semivariogram models. CONVENTIONAL SEMIVARIOGRAM MODELING The semivariogram characterizes spatial variability of the variable under co
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