NONSTATIONARITY OF THE MEAN AND UNBIASED VARIOGRAM ESTIMATION - EXTENSION OF THE WEIGHTED LEAST-SQUARES METHOD

When concerned with spatial data, it is not unusual to observe a nonst ationarity of the mean. This nonstationarity may be modeled through li near models and the fitting of variograms or covariance functions perf ormed on residuals. Although it usually is accepted by authors that a bias is present if residuals are used, its importance is rarely assess ed In this paper, an expression of the variogram and the covariance fu nction is developed to determine the expected bias. It is shown that t he magnitude of the bias depends on the sampling configuration, the im portance of the dependence between observations, the number of paramet ers used to model the mean, and the number of data. The applications o f the expression are twofold The first one is to evaluate a priori the importance of the bias which is expected when a residuals-based vario gram model is used for a given configuration and a hypothetical data d ependence. The second one is to extend the weighted least-squares meth od to fit the variogram and to obtain an unbiased estimate of the vari ogram. Two case studies show that the bias can be negligible or larger than 20%. The residual-based sample variogram underestimates the tota l variance of the process bur rite nugget variance may be overestimate d.