Princeton University

Princeton NJ 08544, USA

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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.

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

Frederik Simons | Back |