The theory of general kriging, with applications to the determination of a local geoid

Two techniques are commonly used to predict values of a random field u(t) from a vector of observations Y; one, mainly used in geodesy, is collocation, the other, mainly used in hydrology and geology, is kriging. Both techniques are based on the same optimization principle, that is minimizing the mean square prediction error, and use the same class of predictors, namely linear in Y. However, in collocation theory u(t) is assumed to have zero average and the main statistic to be used is covariance; once covariance is estimated, it is easy to switch from one functional to another, through covariance propagation. In kriging theory, on the contrary, the average of the field can be an arbitrary constant (or even a function in a suitable finite space) and the main statistic to be estimated is the variogram. Due to its shape, there is no simple rule for variogram propagation from one functional to another; for this reason kriging has mainly dealt with simple evaluations of u(t), or possibly with block averages. This paper treats the unification of the two techniques under a method called by the authors general kriging. After an introduction discussing some typical problems where such theories can be usefully applied, in Section 2 we formalize our proposal in general mathematical terms. Basically one could say that a proper functional propagation based on variograms is established. A first 1-D simulated example follows. In Section 3, our proposal is contrasted on a conceptual basis with cokriging, showing that this method solves a different problem, and step-wise collocation with parameters, recalling some conceptual drawbacks of this approach. In Section 4, numerical examples taken from the problem of estimating a local gravimetric geoid are presented. In such a problem, in fact, both hypotheses, non-zero average and change of functionals, occur and general kriging has proved to be able to deal with them. Finally a short discussion of the results and on-going problems closes the paper.