Cokriging, kernels, and the SVD: Toward better geostatistical analysis.

Abstract

Three forms of multivariate analysis, one very classical and the other two relatively new and little-known, are showcased and enhanced: the first is the Singular Value Decomposition (SVD), which is at the heart of many statistical, and now geostatistical, techniques; the second is the method of Variogram Analysis, which is one way of investigating spatial correlation in one or several variables; and the third is the process of interpolation known as cokriging, a method for optimizing the estimation of multivariate data based on the information provided through variogram analysis. The SVD is described in detail, and it is shown that the SVD can be generalized from its familiar matrix (two-dimensional) case to three, and possibly n, dimensions. This generalization we call the "Tensor SVD" (or TSVD), and we demonstrate useful applications in the field of geostatistics (and indicate ways in which it will be useful in other areas). Applications of the SVD to the tools of geostatistics are described: in particular, applications dependent on the TSVD, including variogram modelling in coregionalization. Variogram analysis in general is explored, and we propose broader use of an old tool (which we call the "corhogram ", based on the variogram) which proves useful in helping one choose variables for multivariate interpolation. The reasoning behind kriging and cokriging is discussed, and a better algorithm for solving the cokriging equations is developed, which results in simultaneous kriging estimates for comparison with those obtained from cokriging. Links from kriging systems to kernel systems are made; discovering kerneIs equivalent to kriging systems will be useful in the case where data are plentiful. Finally, some results of the application of geostatistical techniques to a data set concerning nitrate pollution in the West Salt River Valley of Arizona are described.