Scalar and Vector Slepian Functions, Spherical Signal Estimation and Spectral Analysis

Abstract

It is a well-known fact that mathematical functions that are timelimited (or spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the finite precision of measurement and computation unavoidably bandlimits our observation and modeling scientific data, and we often only have access to, or are only interested in, a study area that is temporally or spatially bounded. In the geosciences we may be interested in spectrally modeling a time series defined only on a certain interval, or we may want to characterize a specific geographical area observed using an effectively bandlimited measurement device. It is clear that analyzing and representing scientific data of this kind will be facilitated if a basis of functions can be found that are "spatiospectrally" concentrated, i.e. "localized" in both domains at the same time. Here, we give a theoretical overview of one particular approach to this "concentration" problem, as originally proposed for time series by Slepian and coworkers, in the 1960s. We show how this framework leads to practical algorithms and statistically performant methods for the analysis of signals and their power spectra in one and two dimensions, and, particularly for applications in the geosciences, for scalar and vectorial signals defined on the surface of a unit sphere.

A combination of downward continuation and local approximation for harmonic potentials

“…This is essentially the expression that is minimized for the construction of Slepian functions (see, e.g., [38,39,42,43]). However, using (4.26) as a penalty term in the functional  from (3.7) would make it significantly harder to find its minimizers.…”

“…In order to deal with local/regional data sets, various types of localizing spherical basis functions have been developed during the last years and decades. Among them are spherical splines (e.g., [10], [41]), spherical cap harmonics (e.g., [23], [45]), and Slepian functions (e.g., [38], [39], [42], [43]). Spherical multiscale methods go a bit further and allow a scale-dependent adaptation of scaling and wavelet kernels (see, e.g., [6], [17], [25], and [40] for the early development).…”

Double-difference adjoint seismic tomography

“…The linear time-invariant model (27) is far from perfect (for a time-and-frequency dependent viewpoint, see, e.g., Kulesh et al 2005;Holschneider et al 2005): the transfer functions must be 'estimated'. Following Laske & Masters (1996), we begin by multiplying each of the individual time-domain records si(t) and di(t) by one of a small set of K orthogonal data windows denoted h T Ω k (t) (here, the prolate spheroidal functions of Slepian 1978), designed for a particular record length T and for a desired resolution angular-frequency half-bandwidth Ω (Mullis & Scharf 1991;Simons & Plattner 2015), and Fourier-transforming the result to yield the sets…”

Internal and external potential-field estimation from regional vector data at varying satellite altitude

“…Among the constructively 'spatio-spectrally localized' spherical functions (e.g. Lesur, 2006) features the general class of 'Slepian functions' Plattner and Simons, 2014;Simons and Plattner, 2015) upon which we build our present work.…”