We review the construction of three different Slepian bases on the sphere, and illustrate their theoretical behavior and practical use for solving ill-posed satellite inverse problems. The first basis is scalar, the second vectorial, and the third suitable for the vector representation of the harmonic potential fields on which we focus our analysis. When data are noisy and incompletely observed over contiguous domains covering parts of the sphere at satellite altitude, expanding the unknown solution in terms of a Slepian basis and seeking truncated expansions to achieve least-squares data fit has advantages over conventional approaches that include the ease with which the solutions can be computed, and a clear statistical understanding of the competing effects of solution bias and variance in modulating the mean squared error, as we illustrate with several new examples.
Many flexible parameterizations exist to represent data on the sphere. In addition to the venerable spherical
harmonics, we have the Slepian basis, harmonic splines, wavelets and wavelet-like Slepian frames. In this paper
we focus on the latter two: spherical wavelets developed for geophysical applications on the cubed sphere,
and the Slepian "tree", a new construction that combines a quadratic concentration measure with wavelet-like
multiresolution. We discuss the basic features of these mathematical tools, and illustrate their applicability in
parameterizing large-scale global geophysical (inverse) problems.
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.
Satellites mapping the spatial variations of the gravitational or magnetic fields of the Earth or other planets
ideally fly on polar orbits, uniformly covering the entire globe. Thus, potential fields on the sphere are usually
expressed in spherical harmonics, basis functions with global support. For various reasons, however, inclined
orbits are favorable. These leave a "polar gap": an antipodal pair of axisymmetric polar caps without any data
coverage, typically smaller than 10° in diameter for terrestrial gravitational problems, but 20° or more in some
planetary magnetic configurations. The estimation of spherical harmonic field coefficients from an incompletely
sampled sphere is prone to error, since the spherical harmonics are not orthogonal over the partial domain of
the cut sphere. Although approaches based on wavelets have gained in popularity in the last decade, we present
a method for localized spherical analysis that is firmly rooted in spherical harmonics. We construct a basis of
bandlimited spherical functions that have the majority of their energy concentrated in a subdomain of the unit
sphere by solving Slepian's (1960) concentration problem in spherical geometry, and use them for the geodetic problem at hand. Most of this work has been published by us elsewhere. Here, we highlight the connection of the "spherical Slepian basis" to wavelets by showing their asymptotic self-similarity, and focus on the computational considerations of calculating concentrated basis functions on irregularly shaped domains.
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