A Combination of Downward Continuation and Local Approximation for Harmonic Potentials
This work addresses the problem of combining global and local potential field data for geoscience applications, offering an incremental improvement over existing multiscale approaches.
The paper proposes a method combining downward continuation of satellite data with local surface data to approximate harmonic potentials, using optimized scaling and wavelet kernels to improve accuracy and localization.
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.