Linear Inverse Problems, Quadratic Spectral Estimators & Spatiospectral Concentration to study Earth, Planets, and Space

Many modern techniques to analyze global planetary signals, from satellite gravity and the geoid, to magnetic fields, to the making of seismological models, heavily rely on a global function basis - the spherical harmonics. When the available data are incomplete and/or noisy, or when the desired models or the features to be extracted are spatially localized geographically, these have easily demonstrable shortcomings. Enter Slepian functions: firmly rooted in theory (for scalar, vectorial and tensorial applications), they are designed to be practical by combining the best of both the spatial and spectral viewpoints, achieving optimal spatio-spectral localization. A Slepian function basis for geophyiscal signals is often sparse, and when used as windows to taper data for spectral analysis, they allow for the robust extraction of localized power. Among the applications are the analysis of the time-variable gravity field for the recovery of the signal of melting ice caps in Greenland and Antarctica or of coseismic gravity perturbations, the localization of geologically relevant and coherent areas on the surface of Venus, the analysis of the Martian magnetic field for the signature of its crustal source depth and intensities, of the terrestrial magnetic field for the separation of the signature of oceanic and continental magnetic anomalies, the recovery of the power spectral density of the cosmic microwave background radiation in the presence of a sky cut, the analysis of local solar magnetism, and more.