Spectral estimation on a sphere in geophysics and cosmology

F. A. Dahlen and Frederik J Simons

Geosciences Department
Princeton University
Princeton NJ 08544, USA

Abstract

We address the problem of estimating the spherical-harmonic power spectrum Sl of a statistically isotropic scalar signal s(r) from noise-contaminated data d(r) = s(r) + n(r) on a region R of the unit sphere. Three different methods of spectral estimation are considered: (i) the spherical analogue of the 1-D periodogram, (ii) the maximum likelihood method, and (iii) a spherical analogue of the 1-D multitaper method. The periodogram exhibits strong spectral leakage, especially for small regions of area A << 4π, and is generally unsuitable for spherical spectral analysis applications, just as it is in 1-D. The maximum likelihood method is particularly useful in the case of nearly whole-sphere coverage, A=4π, and has been widely used in cosmology to estimate the spectrum of the cosmic microwave background radiation from spacecraft observations. The spherical multitaper method affords easy control over the fundamental tradeoff between spectral resolution and variance, and is easily implemented, requiring neither non-linear iteration nor large-scale matrix inversion. As a result, the method is ideally suited for routine applications in geophysics, geodesy or planetary science, where the objective is to obtain a spatially localized estimate of the spectrum of a signal s(r) from data d(r) = s(r)+n(r) within a pre-selected and typically small region R.

Figures

1. Figure 01 Geometry of the problem
2. Figure 02 Power spectra of the single and the double polar boxcar caps for a varity of cap sizes
3. Figure 03 Power spectra of the single and the double polar boxcar cap with superimposed the asymptotic equivalent cap size wavelengths
4. Figure 04 Boxcar coupling kernels for various single and double cap sizes: perspective views
5. Figure 05 Boxcar coupling kernels for various single and double cap sizes [Log version]
6. Figure 06 Eigenvalue-weighted multitaper coupling kernels for various bandwidths: perspective views
7. Figure 07 Eigenvalue-weighted multitaper coupling kernels for various bandwidths [Log version]
8. Figure 08 Ratio of boxcar periodogram estimation variance over whole-sphere estimation variance
9. Figure 09 Ratio of maximum likelihood cut-sphere estimation variance over whole-sphere estimation variance
10. Figure 10 Ratio of eigenvalue-weighted multitaper estimation variance over whole-sphere estimation variance
11. Figure 11 Evolution of the eigenvalue-weighted multitaper variance ratio at large degrees
12. Figure 12 Cosmic Microwave Background radiation: spectra and uncertainties