We develop a method to estimate the power spectrum of a stochastic
process on the sphere from data of limited geographical coverage. Our
approach can be interpreted either as estimating the global power
spectrum of a stationary process when only a portion of the data are
available for analysis, or estimating the power spectrum from local
data under the assumption that the data are locally stationary in a
specified region. Restricting a global function to a spatial subdomain
- whether by necessity or by design - is a windowing operation,
and an equation like a convolution in the spectral domain relates the
expected value of the windowed power spectrum to the underlying global
power spectrum and the known power spectrum of the localization
window. The best windows for the purpose of localized spectral
analysis have their energy concentrated in the region of interest
while possessing the smallest effective bandwidth as possible. Solving
an optimization problem in the sense of Slepian (1960) yields a family
of orthogonal windows of diminishing spatiospectral localization, the
best concentrated of which we propose to use to form a weighted
multitaper spectrum estimate in the sense of Thomson (1982). Such an
estimate is both more representative of the target region and reduces
the estimation variance when compared to estimates formed by any
single bandlimited window. We describe how the weights applied to the
individual spectral estimates in forming the multitaper estimate can
be chosen such that the variance of the estimate is minimized.
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