Generalized covariance functions associated with the Laplace equation and their use in interpolation and inverse problems

This work examines which generalized covariance function when used in the s tochastic approach produces the flattest possible estimate of an unknown fu nction that is consistent with the data. Such an estimate is the plainest p ossible continuous function, thus in a sense eliminating details that are i rrelevant or unsupported by data. The answer is found from the solution of the following variational problem: Determine the function that reproduces t he data, has the smallest gradient (in the square norm sense), and has a gr adient that vanishes at large distances from the observations. The generali zed covariance functions are shown to be the Green's functions for the free -space Laplace equation: the linear distance, in one dimension; the logarit hmic distance in two dimensions; and the inverse distance in three dimensio ns. It is demonstrated that they are appropriate covariance functions for i ntrinsic random fields, a modification is proposed to facilitate numerical implementation, and a couple of examples are presented to illustrate the ap plicability of the methodology.