Estimating the theoretical semivariogram from finite numbers of measurements

We investigate from a theoretical basis the impacts of the number, location, and correlation among measurement points on the quality of an estimate of the semivariogram. The unbiased nature of the semivariogram estimator ŷ(r) is first established for a general random process Z(x). The variance of ŷ_Z(r) is then derived as a function of the sampling parameters (the number of measurements and their locations). In applying this function to the case of estimating the semivariograms of the transmissivity and the hydraulic head field, it is shown that the estimation error depends on the number of the data pairs, the correlation among the data pairs (which, in turn, are determined by the form of the underlying semivariogram γ(r)), the relative locations of the data pairs, and the separation distance at which the semivariogram is to be estimated. Thus design of an optimal sampling program for semivariogram estimation should include consideration of each of these factors. Further, the function derived for the variance of ŷ_Z(r) is useful in determining the reliability of a semivariogram developed from a previously established sampling design.