Spherical Slepian functions and the polar gap in geodesy
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
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
- Figure 01
Sketch illustrating the
geometry of the spherical concentration problem to
the single and double polar cap.
- Figure 02
Spatial
eigensolutions to the problem of
concentrating a bandlimited spherical harmonic
expansion to the latitudinal belt.
- Figure 03
Spatial
eigensolutions to the problem of
concentrating a bandlimited spherical harmonic
expansion to the double polar cap.
- Figure 04
First four of
Grünbaum's eigenfunctions on the belt with L=18
and a double polar cap of radius 30°.
- Figure 05
Last four of
Grünbaum's eigenfunctions on the belt with L=18
and a double polar cap of radius 30°.
- Figure 06
Cumulative energy of
the latitudinal belt tapers in the space-domain
- Figure 07
Cumulative energy of
the double polar cap tapers in the space-domain
- Figure 08
Ranked eigenvalues of the
concentration to a latitudinal belt for a fixed bandwidth.
- Figure 09
Ranked Grünbaum
eigenvalues for L=18 and different sizes of the double
polar cap.
- Figure 10
Average error-to-signal
ratio for TH=10 and L=45 for different signal-to-noise
ratios and spherical harmonic damping levels.
- Figure 11
Error-to-signal ratio
for TH=10 and L=45 for different signal-to-noise ratios and
spherical harmonic damping levels.
- Figure 12
Average error-to-signal
ratio for TH=10 and L=45 for different signal-to-noise
ratios and Slepian truncation levels.
- Figure 13
Error-to-signal ratio
for TH=10 and L=45 for different signal-to-noise ratios and
Slepian truncation levels.
- 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.
- Figure X1
Spectral eigensolutions
to the problem of concentrating a spacelimited function
within a spectral band. (Not included in paper.)
- 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.)
- Figure X3
Up- and downward continuation
of spherical Slepian functions. (Not included in paper.)
Frederik Simons
Last modified: Wed Apr 12 23:06:25 EDT 2023
