  In the last few decades a series of increasingly sophisticated satellite
missions has brought us gravity and magnetometry data of ever improving
quality. To make optimal use of this rich source of information on the
structure of Earth and other celestial bodies, our computational algorithms
should be well matched to the specific properties of the data. In particular,
inversion methods require specialized adaptation if the data are only locally
available, their quality varies spatially, or if we are interested in model
recovery only for a specific spatial region. Here, we present two approaches to
estimate potential fields on a spherical Earth, from gradient data collected at
satellite altitude. Our context is that of the estimation of the gravitational
or magnetic potential from vector-valued measurements. Both of our approaches
utilize spherical Slepian functions to produce an approximation of local data
at satellite altitude, which is subsequently transformed to the Earth's
spherical reference surface. The first approach is designed for
radial-component data only, and uses scalar Slepian functions. The second
approach uses all three components of the gradient data and incorporates a new
type of vectorial spherical Slepian functions which we introduce in this
chapter.
