Two Ordinary Kriging Approaches to Predicting Block Grade Distributions
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Two Ordinary Kriging Approaches to Predicting Block Grade Distributions1 Xavier Emery2 Multigaussian kriging aims at estimating the local distributions of regionalized variables and functions of these variables (transfer or recovery functions) at unsampled locations. In this paper, we focus on the evaluation of the recoverable reserves in an ore deposit accounting for a change of support and information effect caused by ore/waste misclassifications. Two approaches are proposed: the multigaussian model with Monte Carlo integration and the discrete Gaussian model. The latter is simpler to use but requires stronger hypotheses than the former. In each model, ordinary multigaussian kriging gives unbiased estimates of the recoverable reserves that do not utilize the mean value of the normal score data. The concepts are illustrated through a case study on a copper deposit which shows that local estimates of the metal content based on ordinary multigaussian kriging are close to the optimal conditional expectation when the data are abundant and are not dominated by the global mean when the data are scarce. The two proposed approaches (Monte Carlo integration and discrete Gaussian model) lead to similar results when compared to two other geostatistical methods: service variables and ordinary indicator kriging, which show strong deviations from conditional expectation. KEY WORDS: change of support, information effect, multigaussian kriging, discrete Gaussian model, Monte Carlo integration, conditional expectation
INTRODUCTION Geostatistical studies often aim at quantifying the uncertainty in estimation of unsampled values of regionalized variables, which plays an important role in risk assessment and decision-making. In many fields of application, e.g. ore reserve evaluation, reservoir characterization, hydrogeology, soil and environmental sciences, the quantities of interest (grade, permeability, conductivity, nutrient or contaminant concentration, etc.) refer to bigger supports than that of the available samples. To account for such a change of support in geostatistical estimations, several approaches have been developed, including indicator kriging with an affine or a lognormal correction, disjunctive kriging and conditional expectation (Journel, 1Received
12 January 2005; accepted 26 January 2006; Published online: 26 January 2007. of Mining Engineering, University of Chile, Avenida Tupper 2069, Santiago, Chile e-mail: [email protected]
2Department
801 C 2006 International Association for Mathematical Geology 0882-8121/06/1100-0801/1
802
Emery
1984; Rivoirard, 1994; Chil`es and Delfiner, 1999, p. 435). The first two methods only make use of the two-point distributions of the available data. In contrast, the latter takes advantage of their multivariate distribution. Both disjunctive kriging and conditional expectation rely on the mean value of (a transform of) the variable of interest, which may be considered demanding and undesirable. To avoid using this mean value, alternative techniques have been proposed,
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