Assessing the Variability of the Variogram Estimator

Abstract

Statistical estimation and modeling of the variogram are essential steps in the analysis of spatial variability in geostatistics. In spite of the frequent use of the method-of-moment variogram estimator, a precise quantification of the uncertainty of this estimator is rarely provided. In many cases, variogram selection and modeling are made on a somewhat subjective basis, without taking into account the distributional properties of these estimators. The variability of the variogram can be very high when few data are involved in its computations. Most of the approaches used to characterize the variogram variability are based on simulations, which are lenghty to obtain and provide approximate results. Exact theoretical results can however be obtained under some hypotheses. The first part of this paper provides the theoretical developments for the distribution of the method-of-moment variogram estimator and the least-squares variogram estimator. It is shown that, using an analytic approach based on the properties of characteristic functions in the frequency domain, the complete probability density functions of the method-of-moment variogram estimator can be obtained for each class of distance, as well as the covariance matrix of these estimators for different classes of distance. Using a Taylor expansion truncated to the first degree, the covariance matrix of the parameter estimators for the variogram model can also be obtained.

In the second part, the usefulness of these developments is illustrated with a practical case study. It is shown how the theoretical results can be used for obtaining confidence intervals and hypothesis testing. A detailed methodology for the selection of a variogram model is proposed and applied to concentration mesurements in the springs of the Dyle watershed, Belgium.