Fast Spherical Collocation: A General Implementation

Abstract

Spherical harmonic analysis of gridded (and noisy) data on a sphere (with uniform error for a fixed latitude) gives rise to simple systems of equations. This has for the method of least-squares collocation (using an isotropic covariance function or reproducing kernel) been implemented with as much generality as the theory allows. The data only needs to be at the same altitude and of the same kind for each latitude. This permits for example the combination of gravity data at the surface of the Earth and data at satellite altitude.

Suppose that data are associated with the points of a grid with N values in latitude and M values in longitude. The latitudes do not need to be spaced uniformly. Also suppose we want to determine the spherical harmonic coefficients to a maximal degree and order K. Then the method will require that we solve K systems of equations each having a symmetric positive definite matrix of size N * N, only.

Results of three simulation studies using the method are described.