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.
We describe a new filter that simultaneously achieves spectral filtering and image replication to yield a two-dimensional, snapshot spectral imager. Filtering is achieved by spectral demultiplexing; that is without rejection of light; so optical throughput efficiency is, in principle, unity. The principle of operation can be considered as a generalisation of the Lyot filter to achieve multiple bandpasses. We report on the design and experimental implementation of an eight-band system for use in the visible and the design of an eight-band long-wave infrared system.