We construct spherical vector bases that are bandlimited and spatially concentrated, or alternatively, spacelimited and spectrally concentrated, suitable for the analysis and representation of real-valued vector fields on the surface of the unit sphere, as arises in the natural and biomedical sciences, and engineering. Building on the original approach of Slepian, Landau, and Pollak we concentrate the energy of our function basis into arbitrarily shaped regions of interest on the sphere and within a certain bandlimit in the vector spherical-harmonic domain. As with the concentration problem for scalar functions on the sphere, which has been treated in detail elsewhere, the vector basis can be constructed by solving a finite-dimensional algebraic eigenvalue problem. The eigenvalue problem decouples into separate problems for the radial, and tangential components. For regions with advanced symmetry such as latitudinal polar caps, the spectral concentration kernel matrix is very easily calculated and block-diagonal, which lends itself to efficient diagonalization. The number of spatiospectrally well-concentrated vector fields is well estimated by a Shannon number that only depends on the area of the target region and the maximal spherical harmonic degree or bandwidth. The Slepian spherical vector basis is doubly orthogonal, both over the entire sphere and over the geographic target regions. Like its scalar counterparts it should be a powerful tool in the inversion, approximation and extension of bandlimited fields on the sphere: vector fields such as gravity and magnetism in the earth and planetary sciences, or electromagnetic fields in optics, antenna theory and medical imaging.

We present two high-resolution local models for the crustal magnetic field of the Martian south polar region. Models SP130 and SP130M were derived from three-component measurements made by Mars Global Surveyor at nighttime and at low altitude (<200 km). The availability area for these data covers the annulus between latitudes −76 ∘ and −87 ∘ and contains a strongly magnetized region (southern parts of Terra Sirenum) adjacent to weakly magnetized terrains (such as Prometheus Planum). Our localized field inversions take into account the region of data availability, a finite spectral bandlimit (spherical harmonic degree L = 130), and the varying satellite altitude at each observation point. We downward continue the local field solutions to a sphere of Martian polar radius 3376 km. While weakly magnetized areas in model SP130 contain inversion artifacts caused by strongly magnetized crust nearby, these artifacts are largely avoided in model SP130M, a mosaic of inversion results obtained by independently solving for the fields over individual subregions. Robust features of both models are magnetic stripes of alternating polarity in southern Terra Sirenum that end abruptly at the rim of Prometheus Planum, an impact crater with a weak or undetectable magnetic field. From a prominent and isolated dipole-like magnetic feature close to Australe Montes, we estimate a paleopole with a best fit location at longitude 207 ∘ and latitude 48 ∘ . From the abruptly ending magnetic field stripes, we estimate average magnetization values of up to 15 A/m.

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. The 'altitude-cognizant' gradient vector Slepian functions (AC-GVSF) were first employed in a preceding paper. They enjoy a noise tolerance under downward continuation that is much improved relative to the 'classical' gradient vector Slepian functions (CL-GVSF), which do not factor satellite altitude into their construction. Furthermore, venturing beyond the realm of their first application, in the present article we extend the theory to being able to handle both internal and external potential-field estimation. 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. We explain our solution strategies on the basis of analytic expressions for the behavior of the estimation bias and variance of models for which signal and noise are uncorrelated, (essentially) spaceand bandlimited, and spectrally (almost) white. 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.

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. The 'altitude-cognizant' gradient vector Slepian functions (AC-GVSF) were first employed in a preceding paper. They enjoy a noise tolerance under downward continuation that is much improved relative to the 'classical' gradient vector Slepian functions (CL-GVSF), which do not factor satellite altitude into their construction. Furthermore, venturing beyond the realm of their first application, in the present article we extend the theory to being able to handle both internal and external potential-field estimation. 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. We explain our solution strategies on the basis of analytic expressions for the behavior of the estimation bias and variance of models for which signal and noise are uncorrelated, (essentially) space-and bandlimited, and spectrally (almost) white. 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.