# Internal and external potential-field estimation from regional vector data at varying satellite altitude

## TL;DR

This paper introduces a localized method for estimating planetary potential fields from regional satellite vector data at varying altitudes, effectively mitigating noise and artifacts in the inversion process.

## Contribution

It develops a theory for inverting potential fields using spatiospectrally localized bases, enabling high-resolution regional modeling from incomplete, noisy satellite data.

## Key findings

- The method reduces artifacts from external fields in regional inversions.
- It constructs optimal localized bases with manageable computational demands.
- Numerical examples demonstrate improved accuracy in potential field estimation.

## Abstract

When modeling global satellite data to recover a planetary magnetic or gravitational potential field and evaluate it elsewhere, the method of choice remains their analysis in terms of spherical harmonics. When only regional data are available, or when data quality varies strongly with geographic location, the inversion problem becomes severely ill-posed. In those cases, adopting explicitly local methods is to be preferred over adapting global ones (e.g., by regularization). Here, we develop the theory behind a procedure to invert for planetary potential fields from vector observations collected within a spatially bounded region at varying satellite altitude. Our method relies on the construction of spatiospectrally localized bases of functions that mitigate the noise amplification caused by downward continuation (from the satellite altitude to the planetary surface) while balancing the conflicting demands for spatial concentration and spectral limitation. Solving simultaneously for internal and external fields in the same setting of regional data availability reduces internal-field artifacts introduced by downward-continuing unmodeled external fields, as we show with numerical examples. The AC-GVSF are optimal linear combinations of vector spherical harmonics. Their construction is not altogether very computationally demanding when the concentration domains (the regions of spatial concentration) have circular symmetry, e.g., on spherical caps or rings - even when the spherical-harmonic bandwidth is large. Data inversion proceeds by solving for the expansion coefficients of truncated function sequences, by least-squares analysis in a reduced-dimensional space. Hence, our method brings high-resolution regional potential-field modeling from incomplete and noisy vector-valued satellite data within reach of contemporary desktop machines.