Spatiospectral concentration on a sphere

Frederik J Simons and F. A. Dahlen

Geosciences Department
Princeton University
Princeton NJ 08544, USA

Mark A. Wieczorek

Département de Géophysique Spatiale et Planétaire
Institut de Physique du Globe de Paris
94701 St. Maur-des-Fossés, France

Abstract

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 spatial region and a large spherical harmonic bandwidth, the spherical concentration problem reduces to its planar equivalent, which exhibits self-similarity when the Shannon number is kept invariant.

Figures

1. Figure 01 Sketch illustrating the geometry of the spherical concentration problem.
   
2. Figure 02 Spectral eigensolutions to the problem of concentrating a spacelimited function within a spectral band.
   
3. Figure 03 Spatial eigensolutions to the problem of concentrating a bandlimited spherical harmonic expansion to a circular polar cap.
   
4. Figure 04 Eigenvalue spectra of the concentration problem to a circular polar cap, with partial Shannon numbers. (not included in paper)
   
5. Figure 05 Ranked eigenvalues of the concentration to circular polar caps for a fixed bandwidth.
   
6. Figure 06 Cumulative space-domain energy of the tapers for the spherical cap problem.
   
7. Figure 07 Ranked Grünbaum eigenvalues for L=18 and different concentration colatitudes. (not included in paper)
   
8. Figure 08 Grünbaum eigenfunctions for high-degree spherical caps. (not included in paper)
   
9. Figure 09 First and last four of Grünbaum eigenfunctions of the concentration with L=18 and radius 40°.
   
10. Figure 10 Eigenfunctions of the concentration to a circular polar cap of radius 40° for a fixed bandwidth of L= 18 in decreasing order of concentration.
    
11. Figure 11 Eigenvalue structure for the problems of localizing to different continents, for different values of the bandwidth L.
    
12. Figure 12 Concentrating Australia with bandlimited spherical harmonics (L=18).
    
13. Figure 13 Concentrating North America with bandlimited spherical harmonics (L=18) (not included in paper).
    
14. Figure 14 Concentrating Africa with bandlimited spherical harmonics (L=18).
    
15. Figure 15 Progressive buildup of energy by adding the square of the continental tapers weighted by their eigenvalue.
    
16. Figure 16 Comparison of several exact scaled kernels (grey) with their asymptotic approximation (black). (not included in paper)
    
17. Figure 17 Approximate number of significant eigenvalues given by the exact expression versus the asymptotic results (not included in paper).
    
18. Figure 18 Comparison of scaled spatial eigenfunctions (not included in paper).
    
19. Figure 19 Comparison of scaled spectral eigenfunctions (not included in paper).