University College London

London, WC1E 6BT, UK

Princeton University

Princeton NJ 08544, USA

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## 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.

- 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