A Combination of Downward Continuation and Local Approximation for Harmonic Potentials

Abstract

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.

Cited by in corpus (5)

• Internal and external potential-field estimation from regional vector data at varying satellite altitude
• A General Approach to Regularizing Inverse Problems with Regional Data using Slepian Wavelets
• Approximation properties of the double Fourier sphere method
• A double Fourier sphere method for $d$-dimensional manifolds
• A Parameter Choice Strategy for the Inversion of Multiple Observations