The Semivariogram

Three good reasons may be cited to explain why the semivariogram is important in geostatistics: 1. The semivariogram is a statistic that assesses the average decrease in similarity between two random variables as the distance between the variables increases, leading to some applications in exploratory data analysis. 2. It has been demonstrated by the foregoing algorithms and exercises that kriging is not possible without knowledge of the semivariogram or the covariance. In the formulation of our exercises, the covariance has been a known analytical expression—which, incidentally, is what the rigorous application of the algorithms demands. Yet, in practice, neither the covariance nor the semivariogram is known. The way in which geostatistics sidesteps this impasse is by use of an estimate of the semivariogram or the covariance instead of the true moments of the random function model, an approximation for which the derivation of the normal equations does not account. 3. In the previous chapter we have also seen that the practice is to solve the kriging system of equations in terms of covariances. This is primarily for convenience in the handling of the square matrices, despite the slight loss in generality. Yet in terms of determining the spatial correlation, the practice continues to be to estimate the semivariogram and then, provided that the covariance exists, to use the following Corollary 5.2 for converting semivariograms into covariances.