Spectral estimation on a sphere in geophysics and cosmology

TL;DR

This paper compares three spectral estimation methods for analyzing scalar signals on a sphere, highlighting their advantages and limitations in geophysics and cosmology applications, especially for small regions.

Contribution

It introduces and evaluates spherical analogues of periodogram, maximum likelihood, and multitaper methods for spectral estimation on the sphere, emphasizing the multitaper method's practicality.

Findings

The periodogram suffers from spectral leakage in small regions.

Maximum likelihood is effective for near-whole-sphere data.

Multitaper method offers a good balance of resolution and variance control.

Abstract

We address the problem of estimating the spherical-harmonic power spectrum of a statistically isotropic scalar signal from noise-contaminated data on a region of the unit sphere. Three different methods of spectral estimation are considered: (i) the spherical analogue of the one-dimensional (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 $A\ll 4\pi$, 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, $A\approx 4\pi$, 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 trade-off between spectral resolution and variance, and is easily implemented regardless of the region size, requiring neither non-linear iteration nor large-scale matrix inversion. As a result, the method is ideally suited for most applications in geophysics, geodesy or planetary science, where the objective is to obtain a spatially localized estimate of the spectrum of a signal from noisy data within a pre-selected and typically small region.