Estimation of correlation structure for a homogeneous isotropic random field: A simulation study

In this paper we investigate the effect of sampling density on the estimation of the covariance and semivanogram for homogeneous, isotropic, random fields. Two methods based on the least-squares principle, and a third method known as the Minimum Interpolation Error method are studied when the analytic form of the covariance or semivariogram model is known a priori. The analysis is accomplished through a single realization simulation experiment which is felt to represent the type of conditions usually encountered in real world environmental and geophysical field problems. The Turning Bands method is used to generate the field at randomly distributed sampling points in a fixed field for three types of correlation structure: exponential, Bessel, and Gaussian models. The performance of the three estimation methods is evaluated for varying sampling densities and correlation distances. The main results are: the least-squares methods work best for preserving the pattern of correlation in most situations examined; for a domain of fixed size, the ratio of the correlation distance to the length scale of the field is a measure of the "information" contained in the field, and when this ratio exceeded ≈0.2 the statistics of the process became inaccurate. On the other hand, when this ratio is {precedes above almost equal to}0.2 reasonable estimates for the mean and variance were determined even for small sampling densities (≈ 25-50). The implications for practical problems are discussed.