Least-squares collocation: a spherical harmonic representer theorem

Abstract

SUMMARY The functional analysis of the least-squares collocation (LSC) for gravity potential modelling using m measurements is revisited starting from an explicit spherical harmonic expansion. A spherical harmonic representer theorem (SHRT) is given: the model of the potential is a linear combination of m kernels or covariances. This theorem is independent of the specific forms of the data-fitting loss and the regularizer, showing that it is a stronger result than the LSC theory. The corresponding reproducing kernel Hilbert space is explicitly specified. When the least-squares data-fitting loss and the quadratic regularizer are employed, the SHRT gives exactly the LSC method for variable prediction. The nominal prediction precision assessment of the SHRT and that of the LSC are also explicitly compared; this contributes to the unification of the deterministic and stochastic analyses of the LSC theory.