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

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