Estimating and Modeling Space-Time Variograms

Abstract As with a spatial variogram or spatial covariance, a principal purpose of estimating and modeling a space-time variogram is to quantify the spatial temporal dependence reflected in a data set. The resulting model might then be used for spatial interpolation and/or temporal prediction which might take several forms, e.g. kriging and Radial Basis functions. There are significant problems to overcome in both the estimation and the modeling stages for space-time problems unlike the purely spatial application where estimation is the more difficult step. The key point is that a spatial-temporal variogram as a function must be conditionally negative definite (not just semi-definite) which can be a difficult condition to verify in specific cases. In the purely spatial context one relies on a known list of isotropic valid models, e.g., the Matern class as well as the exponential and gaussian models, as well as on positive linear combinations of known valid models. Bochner’s Theorem (or the extension given for generalized covariances by Matheron) characterizes nonnegative definite functions but does not easily distinguish the strictly positive definite functions.