  While many geological and geophysical processes such as the melting of
icecaps, the magnetic expression of bodies emplaced in the Earth's crust, or
the surface displacement remaining after large earthquakes are spatially
localized, many of these naturally admit spectral representations, or they may
need to be extracted from data collected globally, e.g. by satellites that
circumnavigate the Earth. Wavelets are often used to study such nonstationary
processes. On the sphere, however, many of the known constructions are somewhat
limited. And in particular, the notion of `dilation' is hard to reconcile with
the concept of a geological region with fixed boundaries being responsible for
generating the signals to be analyzed. Here, we build on our previous work on
localized spherical analysis using an approach that is firmly rooted in
spherical harmonics. We construct, by quadratic optimization, a set of
bandlimited functions that have the majority of their energy concentrated in an
arbitrary subdomain of the unit sphere. The `spherical Slepian basis' that
results provides a convenient way for the analysis and representation of
geophysical signals, as we show by example. We highlight the connections to
sparsity by showing that many geophysical processes are sparse in the Slepian
basis.
