Investigations into Green's function as inversion-free solution of the Kriging equation, with Geodetic applications

Statistical interpolation has been proven to be a legitimate and efficient approach for data processing in the field of geodetic and geophysical sciences. Pursuing the minimization of the mean squared prediction error, the technique, known as Kriging or least-squares collocation, is able to densify, respectively filter a spatially and/or temporally referenced dataset, provided that its associated covariance model is given or estimated in advance. The involvement of the covariance matrix which to some extent reflects the physical behavior of the underlying process may, however, potentially lead to an ill-conditioned situation when the data are observed at a relatively high sampling rate. A new perspective, interpreting the Kriging equation in the continuous sense, is therefore proposed in this research so that, instead of matrix terms, a convolution equation is set up for the Green’s function where the covariance function is preserved in its analytic form. Two methods to approximate the solution of such a convolution equation are employed: One transforms the unknown Green’s function into a series consisting of a linear combination of (partial) derivatives of the covariance function so that the approximation of the Green’s function can be determined through a term-by-term approach; the other one manipulates the convolution equation in the spectral domain where the inversion can be treated within the space of real number. The proposed approach has been applied to various covariance models, especially several more recently established spatial-temporal models which have attracted increasing interests for geophysical applications. Examples from geodetic science include the cases of data fusion and terrain profile monitoring; although based on simulated data, the demonstration of this innovative approach shows great potential.