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 $\Omega_R$ of radius $R$ (e.g., a satellite's orbit) with locally available data on a sphere $\Omega_r$ 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 $\Phi_N$ deals with the downward continuation from $\Omega_R$ to $\Omega_r$, while in a second step, the result is locally refined by a convolution on $\Omega_r$ with a wavelet kernel $\tilde{\Psi}_N$. Different from earlier multiscale approaches, it is not the primary goal to obtain an adaptive spatial localization but to simultaneously optimize the related kernels $\Phi_N$, $\tilde{\Psi}_N$ in such a way that the former behaves well for the downward continuation while the latter shows a good localization on $\Omega_r$ in the region where data is available. The concept is indicated for scalar as well as vector potentials.