  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. Here we show that the geodetic estimation problem
has deep-seated connections to Slepian's spatiospectral localization problem on
the sphere, which amounts to finding 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 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 this new method makes the use of
spherical Slepian functions in earth and planetary geodesy practical.
