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.
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