Vector-valued crustal magnetic field estimation using vector Slepian functions

To solve for the terrestrial or a planetary magnetic field from vector-valued measurements made by a satellite, an inversion needs to be performed that correctly maps the noisily and incompletely observed data down to the source level. For the case of the scalar potential, powerful localization techniques have aided in regularizing the ill-posed inverse problem of making global inference from local data, or vice versa. One can use splines, wavelets, cap harmonics, and also Slepian functions, which are optimally concentrated spatio-spectrally with respect to a function-energy norm. The Slepian functions, in particular, have been very useful in gravity geodesy, but also for the study of planetary magnetic fields available as spherical-harmonic potential expansions. It is clear that the benefits of localization are not available to vector data by simply focusing on the vector components individually. Rather, for the most common problems where a potential-description remains adequate, a dedicated function basis needs to be sought that is harmonic, vectorial in nature, bandlimited, and localized to target regions on the surface of the sphere. We have recently developed such a "vectorial spherical Slepian basis", and are thus ready to tackle the study of the satellite geomagnetic inverse problems of the future. In this presentation we apply vector-Slepian functions to the estimation of the vector-valued crustal magnetic field from vector-valued data at satellite altitude. The downward continuation of vector functions that satisfy the source-free assumption can be performed similarly to the scalar downward continuation if the data are described in a suitably transformed vector-spherical harmonic basis. We test our approach with artificial data using different data and noise power spectra and for different target regions. The vector Slepian approach should be beneficial in cases where we either have only local data at satellite altitude but want to obtain as much crustal field information as possible or if we have global data at satellite altitude but want to perform a purely local analysis of the crustal field, as we show by example.