Precise solution for a finite set of spherical coefficients from equiangular gridded data

摘要

An important goal of geodesy is to determine the anomalous potential and its derivatives outside of the earth. Representing the surface anomalies by a series of spherical harmonics is useful since it is then possible to do a term by term solution of Laplace's equation and upward continuation. The problem of finding such a spherical harmonic series for anomaly values given on an equiangular surface grid is addressed. (This is a first step toward the more complicated problem of finding a function such that locally averaged values fit a grid of mean anomalies.) Three approaches to this fitting problem are discussed and compared: the discrete Fourier technique, the discrete integral technique, and a new approach. The peculiar nature of the equiangular grid, with its increasing density of (noisy) data toward the poles, causes each method to exhibit a different type of difficulty. The new method is shown to be practical as well as precise since the numerical conditioning problems which appear can be successfully handled by such well-known techniques as a (simple) Kalman filter.