A spatiospectral localization approach for analyzing and representing vector-valued functions on spherical surfaces

ABSTRACT We review the construction of three different Slepian bases on the sphere, and illustrate their theoretical behavior and practical use for solving ill-posed satellite inverse problems. The first basis is scalar, the second vectorial, and the third suitable for the vector representation of the harmonic potential fields on which we focus our analysis. When data are noisy and incompletely observed over contiguous domains covering parts of the sphere at satellite altitude, expanding the unknown solution in terms of a Slepian basis and seeking truncated expansions to achieve least-squares data fit has advantages over conventional approaches that include the ease with which the solutions can be computed, and a clear statistical understanding of the competing effects of solution bias and variance in modulating the mean squared error, as we illustrate with several new examples.