Spatiospectral concentration and spectral analysis of potential fields on the sphere

Physical properties, such as the elastic strength or the magnetization depth of a planetary lithosphere can be estimated from the cross-spectral properties of potential fields. Such data are most commonly available as bandlimited spherical harmonic coefficients, measured by artificial satellites or spacecraft. In many if not most applications, planetary curvature prohibits the use of locally flat approximations. Thus, the determination of spatially localized estimates of planetary properties requires spatiospectral localization methods that go beyond those available in the plane. Single spherical windows or tapers have been developed and applied in a number of recent studies; however, these are neither optimally concentrated, nor as reliable as an orthogonal family of multitapers in the extraction of robust localized statistical information from bandlimited spherical data. Here, we pose and solve the analogue of Slepian's time-frequency concentration problem on the surface of the unit sphere to determine an orthogonal family of strictly bandlimited functions that are optimally concentrated within a closed region of the sphere, or, alternatively, of strictly spacelimited functions that are optimally concentrated within the spherical harmonic domain. Such a basis of simultaneously spatially and spectrally concentrated functions should be a useful data analysis and representation tool in a variety of geophysical and planetary applications, as well as in medical imaging, computer science, cosmology and numerical analysis. The spherical Slepian functions can be found either by solving an algebraic eigenvalue problem in the spectral domain or by solving a Fredholm integral equation in the spatial domain. The associated eigenvalues are a measure of the spatiospectral concentration. When the concentration region is an axisymmetric polar cap the spatiospectral projection operator commutes with a Sturm-Liouville operator; this enables the eigenfunctions to be computed extremely accurately and efficiently, even when their area-bandwidth product, or Shannon number, is large. In the asymptotic limit of a small concentration region and a large spherical harmonic bandwidth the spherical concentration problem approaches its planar equivalent, which exhibits self-similarity when the Shannon number is kept invariant. Our examples show families of bandlimited spherical harmonic expansions that are localized to Earth's continents. In a related presentation, we investigate the ability of our orthogonal data tapers to obtain spectral estimates by analyzing the bias and variance properties of the multitaper estimator constructed using our windows.