In geophysics, potential field data (gravitational
anomalies and magnetic field variations) and topography are the primary
sources of information on the internal structure of the planets. As one
example, the elastic properties of the lithosphere (the rigid outer layer)
can be estimated by comparing a planet's gravity signature with its topography
by analyzing the cross-spectral properties of these fields. On the Earth,
such calculations are usually performed in the Fourier domain on locally
flat surfaces. On the Moon, Mars and Mercury, however, the effects of curvature
become too important to neglect. Moreover spherical harmonic representations
of the gravity field from satellite orbital determinations usually provide
the primary data.
Adequate methods of spatio-spectral localization are required
in order to extract spectral information from spatially localized regions.
The joint optimization between spectral and spatial resolution is well studied
on flat surfaces and has led to a variety of methods employing multiple data
tapers or wavelets. In one dimension, D. Slepian found that prolate spheroidal
wave functions on finite intervals are ideally concentrated in frequency.
In two dimensions, I. Daubechies showed how weighted Hermite polynomials achieve
a joint optimization in the time-frequency phase plane. We solve the analogous
problem of finding optimally space-concentrated band-limited data windows
on the sphere, using spherical harmonics expansions.
The solution of the associated eigenvalue problem leads to
a new class of data windows that are mutually orthogonal on the sphere, and
which can be calculated semi-analytically. We discusss their characteristics
and point to their usage. The most concentrated window of the orthogonal
family is shown to satisfy quantum-mechanical uncertainty principles of location
(via the center of mass) and total angular momentum (via the angular degree
l), and for the case where the angular order m=0, its properties are nearly
identical to those of squeezed coherent states on the circle.