A Comparison of Spatial Semivariogram Estimators and Corresponding Ordinary Kriging Predictors

Predicting values of a spatially distributed variable, such as the concentration of a mineral throughout an ore body or the level of contamination around a toxic-waste dump, can be accomplished by a regression procedure known as kriging. Kriging and other types of statistical inference for spatially distributed variables are based on models of stochastic processes Y t: t ∊ D called random-field models. A commonly used class of random-field models are the intrinsic models, for which the mean is constant, and half of the variance of Yt, – Ys. is a function, called the semivariogram, of the difference t – s. The type of kriging corresponding to an intrinsic model is called ordinary kriging. The semivariogram, which typically is taken to depend on one or more unknown parameters, must be estimated prior to ordinary kriging. Various estimators of the semivariogram's parameters have been proposed. For two Gaussian intrinsic random-field models, we compare, by a Monte Carlo simulation study, the performance of seven estimators—ordinary least squares (OLS), Cressie's weighted least squares (WLS-1 and WLS-2), Delfiner's weighted least squares (WLS-3). maximum likelihood (ML), restricted maximum likelihood (REML), and generalized minimum variance quadratic unbiased (GMIVQU) estimators. In addition, we compare the performance of the standard 95% confidence interval for the ordinary kriging predictor corresponding to each of the estimators. Our results indicate that the relatively easy-to-compute OLS, WLS-1, and WLS- 2 estimators and their corresponding prediction intervals perform as well or nearly as well as the more computationally demanding ML, REML, and GMIVQU estimators and their corresponding prediction intervals, but that the WLS3 estimator and its corresponding prediction interval perform poorly.