Spherical Slepian functions and the polar gap in geodesy

Frederik J Simons1 and F. A. Dahlen2

1 Earth Sciences Department
University College London
London, WC1E 6BT, UK

2 Geosciences Department
Princeton University
Princeton NJ 08544, USA

Geoph. J. Int., 2006, 166 (3), 1039-1061, doi:10.1111/j.1365-246X.2006.03065.x
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Abstract

The estimation of potential fields such as the gravitational or magnetic potential at the surface of a spherical planet from noisy observations taken at an altitude over an incomplete portion of the globe is a classic example of an ill-posed inverse problem. We show that this potential-field estimation problem has deep-seated connections to Slepian's spatiospectral localization problem which seeks bandlimited spherical functions whose energy is optimally concentrated in some closed portion of the unit sphere. This allows us to formulate an alternative solution to the traditional damped least-squares spherical harmonic approach in geodesy, whereby the source field is now expanded in a truncated Slepian function basis set. We discuss the relative performance of both methods with regard to standard statistical measures such as bias, variance and mean-square error, and pay special attention to the algorithmic efficiency of computing the Slepian functions on the region complementary to the axisymmetric polar gap characteristic of satellite surveys. The ease, speed, and accuracy of our method make the use of spherical Slepian functions in earth and planetary geodesy practical.

Figures

  1. Figure 01 Sketch illustrating the geometry of the spherical concentration problem to the single and double polar cap.
  2. Figure 02 Spatial eigensolutions to the problem of concentrating a bandlimited spherical harmonic expansion to the latitudinal belt.
  3. Figure 03 Spatial eigensolutions to the problem of concentrating a bandlimited spherical harmonic expansion to the double polar cap.
  4. Figure 04 First four of Grünbaum's eigenfunctions on the belt with L=18 and a double polar cap of radius 30°.
  5. Figure 05 Last four of Grünbaum's eigenfunctions on the belt with L=18 and a double polar cap of radius 30°.
  6. Figure 06 Cumulative energy of the latitudinal belt tapers in the space-domain
  7. Figure 07 Cumulative energy of the double polar cap tapers in the space-domain
  8. Figure 08 Ranked eigenvalues of the concentration to a latitudinal belt for a fixed bandwidth.
  9. Figure 09 Ranked Grünbaum eigenvalues for L=18 and different sizes of the double polar cap.
  10. Figure 10 Average error-to-signal ratio for TH=10 and L=45 for different signal-to-noise ratios and spherical harmonic damping levels.
  11. Figure 11 Error-to-signal ratio for TH=10 and L=45 for different signal-to-noise ratios and spherical harmonic damping levels.
  12. Figure 12 Average error-to-signal ratio for TH=10 and L=45 for different signal-to-noise ratios and Slepian truncation levels.
  13. Figure 13 Error-to-signal ratio for TH=10 and L=45 for different signal-to-noise ratios and Slepian truncation levels.
  14. Figure 14 Breakdown of mse, var, and bias, as a function of damping/truncation level and, for the optimum values, as a function of colatitude.
  15. Figure X1 Spectral eigensolutions to the problem of concentrating a spacelimited function within a spectral band. (Not included in paper.)
  16. Figure X2 Eigenfunctions of the double circular polar cap of radius 40° for a fixed bandwidth of L= 18 in decreasing order of concentration. (Not included in paper.)
  17. Figure X3 Up- and downward continuation of spherical Slepian functions. (Not included in paper.)
Frederik Simons
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