Vector-Valued Localizing Basis Functions on the Sphere for Satellite Geomagnetic Data Analysis

For the analysis and inversion of scalar (potential) fields on the sphere, spatiospectrally localized ``Slepian'' functions have proven to be a viable tool, enjoying a growing popularity, especially in geodesy but also in seismology and geodynamics. These functions, linear combinations of spherical harmonics, are typically spectrally bandlimited while spatially concentrated, by quadratic optimization, to a target region on the sphere. The family of functions that is obtained in this manner remains a globally orthornomal basis but is also orthogonal over the area of interest, which makes them suitable for the representation and analysis of global as well as regional data sets, even those showing features that warrant a multiresolution approach. Localized spherical analysis of vector fields such as would be demanded by geomagnetic satellite missions has not been attempted in this consistent framework. In this presentation we construct bandlimited vector-spherical-harmonics based Slepian basis sets for the study of such data, on regular symmetric and irregularly shaped domains on the surface of the sphere. These will be ideally suited for the separation of lithospheric and deeper sources of the magnetic field from processed solutions of the terrestrial magnetic field, in new inversions for crustal magnetization from primary data, in the analysis of solar flares, and so on. We pay special attention to algorithmic and numerical efficiency in the construction of the vectorial Slepian basis even to high degrees, and present several alternative approaches by which they can be computed depending on the application of interest and the acquisition geometry of the data. A first example of their use in an inversion scheme will use a Slepian basis set that, by judicious truncation, provides a natural regularization and hence an alternative solution to the traditional damped least-squares vector-spherical-harmonic approaches that are computationally more demanding and numerically more poorly conditioned than what we show can be achieved using Slepian functions.