Statistical Foundations of Collocation,

Abstract
The paper deals with mathematical models suitable as a basis for the statistical treatment of collection. As a preparation, stochastic processes on the circle are discussed first; such processes are simple to understand and exhibit already essential features of the problem. Then the paper treats stochastic processes on the sphere, which may be suitable as statistical models for collocation. Lauritzen's theorem on the nonexistence of ergodic Gaussian stochastic process models for collocation is seen to be essentially dependent on the Gaussian character. Two non-Gaussian ergodic models are given, one of a genuinely probabilistic character similar to Lauritzen's model, and another based on a formal probability theory in rotation group space. This second model gives a statistical foundation of the usual homogeneous and isotropic covariance analysis of the anomalous gravity field; it also provides a basis for the study of the statistical distribution of quantities related to this field. This model allows a formal statistical treatment of the anomalous gravitational field which is independent of an interpretation of this field as some genuinely physical stochastic process and seems, therefore, to be preferable. (Author)

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