Derivatives of spatial variances of growing windows and the variogram

Geostatistical analysis of spatial random functions frequently uses sample variograms computed from increments of samples of a regionalized random var iable. This paper addresses the theory of computing variograms not from inc rements but from spatial variances. The objective is to extract information about the pont support space from the average or larger support data. The variance is understood as a parametric and second moment average feature of a population. However, it is well known that when the population is for a stationary random function, spatial variance within a region is a function of the size and geometry of the region and not a function of location. Spat ial variance is conceptualized as an estimation variance between two physic al regions or a region and itself. If such a spatial variance could be meas ured within several sizes of windows, such variances allow the computation of the sample variogram. The approach is extended to covariances between at tributes that lead to the cross-variogram. The case of nonpoint sample supp ort of the blocks or elements composing each window is also included. A num erical example illustrates the application of this conceptualization.