A combination of downward continuation and local approximation for harmonic potentials

This paper presents a method for the approximation of harmonic potentials that combines downward continuation of globally available data on a sphere ΩR of radius R (e.g., a satellite's orbit) with locally available data in a sub-region Γr of the sphere Ωr of radius r<R (e.g., the spherical Earth's surface). The approximation is based on a two-step algorithm motivated by spherical multiscale expansions: first, a convolution with a scaling kernel ΦN deals with the downward continuation from ΩR to Ωr, while in a second step, the result is locally refined by a convolution on Ωr with a wavelet kernel Ψ-tilde N. The kernels ΦN and Ψ-tilde N are optimized in such a way that the former behaves well for the downward continuation while the latter shows a good localization in Γr. The concept is indicated for scalar as well as vector potentials. (paper)