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.Comment: Under revision for Geophys. J. Int. Supported by NASA grant
NNX14AM29