Spatiospectral concentration of vector fields on a sphere

Abstract

We construct spherical vector bases that are bandlimited and spatially concentrated, or alternatively, spacelimited and spectrally concentrated, suitable for the analysis and representation of real-valued vector fields on the surface of the unit sphere, as arises in the natural and biomedical sciences, and engineering. Building on the original approach of Slepian, Landau, and Pollak we concentrate the energy of our function basis into arbitrarily shaped regions of interest on the sphere and within a certain bandlimit in the vector spherical-harmonic domain. As with the concentration problem for scalar functions on the sphere, which has been treated in detail elsewhere, the vector basis can be constructed by solving a finite-dimensional algebraic eigenvalue problem. The eigenvalue problem decouples into separate problems for the radial, and tangential components. For regions with advanced symmetry such as latitudinal polar caps, the spectral concentration kernel matrix is very easily calculated and block-diagonal, which lends itself to efficient diagonalization. The number of spatiospectrally well-concentrated vector fields is well estimated by a Shannon number that only depends on the area of the target region and the maximal spherical harmonic degree or bandwidth. The Slepian spherical vector basis is doubly orthogonal, both over the entire sphere and over the geographic target regions. Like its scalar counterparts it should be a powerful tool in the inversion, approximation and extension of bandlimited fields on the sphere: vector fields such as gravity and magnetism in the earth and planetary sciences, or electromagnetic fields in optics, antenna theory and medical imaging.

High‐resolution local magnetic field models for the Martian South Pole from Mars Global Surveyor data

“…The "gradient vector Slepian" eigenfunctions of equation (11) were first introduced by Plattner and Simons [2014a]. They are different from the "vector Slepian" functions developed by Plattner and Simons [2014b], which are applicable to vector fields in general, not only gradients of inner-source scalar potentials as stipulated by our definitions in equations (4) and (7).…”

“…Their approach led to a method called "Revised Spherical Cap Harmonic Analysis. " Holschneider et al [2003], Previously, Plattner and Simons [2014a] discussed the use of scalar and vector [Jahn and Bokor, 2012;Plattner et al, 2012;Plattner and Simons, 2014b] Slepian functions for magnetic field inversions. These "canonical" or "classical" Slepian functions, optimized exclusively to maximize spatial concentration within a certain region for a given bandwidth, are of great utility for inverting relatively low-bandwidth data collected at relatively low altitudes.…”

A Further Look at Time-and-Band Limiting for Matrix Orthogonal Polynomials

“…For more details on computational issues see [2,18,22]. For applications involving (sometimes) vector-valued quantities on the sphere, see [17,26,29,30].…”

Internal and external potential-field estimation from regional vector data at varying satellite altitude

“…Our method traces its history to the one-dimensional theory of 'prolate spheroidal wave functions' by Slepian and Pollak (1961), its applications in signal processing (Slepian, 1983), and especially its extensions to scalar spherical fields by and , to spherical vector fields by Plattner and Simons (2014), and to gradient vector spherical functions (curl-free potential fields) by Plattner and Simons (2015b). In the above cited works, satellite altitude, though explicitly considered within the context of the inverse problem, was never a factor in the optimization construction of the Slepian functions, and so we will term them 'canonical' or 'classical'.…”

“…We test both methods on a simulated data set in Section 8 and investigate the effect of neglecting to account for an external field. In Sections 9 and 10 we provide a more in-depth analysis of the relationship of our new Slepian functions to the generic vector spherical Slepian functions presented by Plattner and Simons (2014) and showcase their mathematical and statistical properties. We summarize our findings in Section 11 and explain methods to significantly decrease the computational costs of high spherical-harmonic degree calculations in the Appendix.…”