UCL Discovery

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.