The estimation of potential fields such as the gravitational or
magnetic potential at the surface of a spherical planet from noisy
observations taken at an altitude over an incomplete portion of the
globe is a classic example of an ill-posed inverse problem. We show
that this potential-field estimation problem has deep-seated
connections to Slepian's spatiospectral localization problem which
seeks bandlimited spherical functions whose energy is optimally
concentrated in some closed portion of the unit sphere. This allows us
to formulate an alternative solution to the traditional damped
least-squares spherical harmonic approach in geodesy, whereby the
source field is now expanded in a truncated Slepian function basis
set. We discuss the relative performance of both methods with regard
to standard statistical measures such as bias, variance and
mean-square error, and pay special attention to the algorithmic
efficiency of computing the Slepian functions on the region
complementary to the axisymmetric polar gap characteristic of
satellite surveys. The ease, speed, and accuracy of our method make
the use of spherical Slepian functions in earth and planetary geodesy
practical.
|