Nonparametric cross-covariance modeling as exemplified by soil heavy metal concentrations from the Swiss Jura

Geostatistics is used in soil science to map spatial distributions of soil properties from limited samples. Covariance models provide the basic measure of spatial continuity which is used to weight the information available at different sample locations, as in kriging. Traditionally, a closed-form analytical model is fitted to allow for interpolation of sample covariance values while ensuring the positive definiteness condition. For cokriging where several different properties are cross correlated with each other, the simultaneous modeling of several (cross) covariances is made more difficult because of the restrictions imposed by the linear coregionalization model. An algorithm for automatic joint modeling of multiple (cross) covariance tables is proposed, building on an extension of Bochner's positive definiteness theorem and eigenvalue correction. The objective of this paper is to present the new methodology and demonstrate its application to modeling large cross-covariance matrix. Such task is not easily done or, more bluntly, seldom done in the conventional way when the number of coregionalized variables becomes large. The data set used for the case study relates to heavy metal pollution in soil. (C) 1999 Elsevier Science B.V. All rights reserved.