It is often advantageous to investigate the relationship between two
geophysical data sets in the spectral domain by calculating
admittance and coherence functions. While there exist powerful
Cartesian windowing techniques to estimate spatially localized
(cross-)spectral properties, the inherent sphericity of planetary
bodies sometimes necessitates an approach based in spherical
coordinates. Direct localized spectral estimates on the sphere can
be obtained by tapering, or multiplying the data by a suitable
windowing function, and expanding the resultant field in spherical
harmonics. The localization, or concentration, of a window in space
and its spectral bandlimitation jointly determine the quality of the
spatiospectral estimation. We construct two kinds of axisymmetric
windows: bandlimited functions that maximize their spatial energy
within a cap of angular radius, and spacelimited functions
maximizing their spectral power within a spherical harmonic
bandwidth L. Both concentration criteria yield an eigenvalue
problem solved by an orthogonal family of data tapers. The
properties of our new windows depend almost entirely upon the
space-bandwidth product N0=(L+1)θ0/π, with the first N0-1
windows nearly perfectly concentrated. The concentration of the best
window approaches a lower bound imposed by a quantum-mechanical
uncertainty principle. In order to make robust localized estimates
of the admittance and coherence between two fields on the sphere, we
propose a method that uses the optimally concentrated data tapers
calculated with extreme computational efficiency. We show that the
expectation of localized (cross-)power spectra calculated using our
data tapers is nearly unbiased when the input spectrum is white and
averages are made over all possible realizations of the random
variables. In physical situations, only one realization of such a
process will be available, but in this case, a weighted average of the
spectra obtained using multiple data tapers will approach the expected
spectrum; the approximation improves with the number of tapers
used. While developed primarily to solve problems in planetary
science, our method has applications in all areas of science that
investigate spatiospectral relationships between data fields defined
on a sphere.
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