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Generalized covariance functions associated with the Laplace Equation and Their use in interpolation and inverse problems
Author(s) -
Kitanidis Peter K.
Publication year - 1999
Publication title -
water resources research
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.863
H-Index - 217
eISSN - 1944-7973
pISSN - 0043-1397
DOI - 10.1029/1999wr900026
Subject(s) - covariance , covariance function , mathematics , matérn covariance function , law of total covariance , rational quadratic covariance function , covariance mapping , interpolation (computer graphics) , laplace transform , mathematical analysis , function (biology) , inverse problem , dimension (graph theory) , inverse , covariance intersection , statistics , computer science , pure mathematics , geometry , animation , computer graphics (images) , evolutionary biology , biology
This work examines which generalized covariance function when used in the stochastic approach produces the flattest possible estimate of an unknown function that is consistent with the data. Such an estimate is the plainest possible continuous function, thus in a sense eliminating details that are irrelevant or unsupported by data. The answer is found from the solution of the following variational problem: Determine the function that reproduces the data, has the smallest gradient (in the square norm sense), and has a gradient that vanishes at large distances from the observations. The generalized covariance functions are shown to be the Green's functions for the free‐space Laplace equation: the linear distance, in one dimension; the logarithmic distance in two dimensions; and the inverse distance in three dimensions. It is demonstrated that they are appropriate covariance functions for intrinsic random fields, a modification is proposed to facilitate numerical implementation, and a couple of examples are presented to illustrate the applicability of the methodology.