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When modeling global satellite data to recover a planetary magnetic orgravitational potential field and evaluate it elsewhere, the method of choiceremains their analysis in terms of spherical harmonics. When only regional dataare available, or when data quality varies strongly with geographic location,the inversion problem becomes severely ill-posed. In those cases, adoptingexplicitly local methods is to be preferred over adapting global ones (e.g., byregularization). Here, we develop the theory behind a procedure to invert forplanetary potential fields from vector observations collected within aspatially bounded region at varying satellite altitude. Our method relies onthe construction of spatiospectrally localized bases of functions that mitigatethe noise amplification caused by downward continuation (from the satellitealtitude to the planetary surface) while balancing the conflicting demands forspatial concentration and spectral limitation. Solving simultaneously forinternal and external fields in the same setting of regional data availabilityreduces internal-field artifacts introduced by downward-continuing unmodeledexternal fields, as we show with numerical examples. The AC-GVSF are optimallinear combinations of vector spherical harmonics. Their construction is notaltogether very computationally demanding when the concentration domains (theregions of spatial concentration) have circular symmetry, e.g., on sphericalcaps or rings - even when the spherical-harmonic bandwidth is large. Datainversion proceeds by solving for the expansion coefficients of truncatedfunction sequences, by least-squares analysis in a reduced-dimensional space.Hence, our method brings high-resolution regional potential-field modeling fromincomplete and noisy vector-valued satellite data within reach of contemporarydesktop machines.

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