Biased Kriging on The Sphere?

Abstract

It is well known among geodesists that homogeneous-isotropic processes on the sphere which are both, Gaussian and ergodic, do not exist. Thus we have to give up one of the two properties as soon as we consider global phenomena such as the (residual) topography, the disturbing gravity field, etc. In case of a non-Gaussian process we are facing problems in deriving the proper distribution functions for our test statistics, beside the fact that such a process is not completely defined by its first two moments. On the other hand, a non-ergodic process does not allow us to equivalently express “expectation”, and “covariance”, as spatial integral over any of its realizations. Hence our usual methods to derive the covariance function (or the semi-variogram) fail to produce consistent - or even unbiased - estimates.

This may turn out to be not as critical as it seems if we only could control the mean square error of the predicted phenomenon. For this purpose we introduce, in a second step, homBLIP (Best homogeneously LInear Prediction) as a biased alternative to Ordinary Kriging with a slightly reduced mean square error, thereby using new types of homeograms. By a similar approach of searching for biased predictors with smaller mean square error risk, we may also attempt to improve Universal Kriging in a later stage.