Universal kriging

This chapter discusses universal kriging, the kriging of a random function Z(x) which is not intrinsic and exhibits an expectation E [Z(x)] = m(x) variable over the space. For the function m(x), called the drift of Z(x), a model has to be chosen, usually polynomial of degree 1, 2, or 3. Then, the best polynomial coefficients are provided by a Lagrange approach similar to that used for simple kriging. However, drift estimation is not required for the true kriging, that is, for the best estimate of Z itself from a given sampling. The associated system of equations is still linear and can be formally calculated in dimension 1 for the usual drifts and variograms. Universal kriging is compared with least squares methods, and a few variants are studied, including universal kriging in the case of an implicit drift, and universal kriging of a vector random function, or cokriging.

Introduction

Geological data, notably geochemical data, often take the form of a regionalized composition. The concept of regionalized composition combines the concepts of composition and coregionalization. A composition, also known in the literature as a closed array (Chayes 1962), is a random vector whose components add up to a constant. A coregionalization is a set of two or more regionalized variables defined over the same spatial domain, which is modeled as a realization of a vector random function. Here the term regionalized composition is used both for the vector random function used to model a composition and for the realization that we can observe. A regionalized composition can be, for example, a heavy-mineral suite along a river valley. The minerals are quantitatively determined through frequency counts and represented as percent-proportions of the entire heavy-mineral occurrence. Another example is the set of grades in a lead-copper-zinc deposit. In this instance, all components of each specimen are not quantitatively recorded and the grades are also not expressed as proportions of the whole of the measured components: only a small fraction of the composition in ppm is accounted for in each specimen. The problem with the statistical analysis of compositions has been stated historically in terms of correlations: the covariances are subject to essential nonstochastic controls, i.e., distortions which are due to the constant-sum constraint. These numerically induced covariances and correlations arise also with regionalized compositions and are called spurious spatial correlations. They falsify the picture of the spatial covariance structure and can lead to misinterpretations. This problem arises not only when the whole regionalized composition is analyzed, but also when interest lies only in a subvector. A second problem, singularity of the covariance matrix of a composition, has generally been considered only from a numerical point of view. Singularity is a direct consequence of the constant-sum constraint and, as in other multivariate methods, it rules out the use of estimation techniques such as cokriging of all components. Numerically the problem can be tackled either by taking generalized inverses or, equivalently, leaving one component out to avoid singularity of the matrices of coefficients.