A general approach to regularizing inverse problems with regional data using Slepian wavelets

Abstract

Slepian functions are orthogonal function systems that live on subdomains (for example, geographical regions on the Earth's surface, or bandlimited portions of the entire spectrum). They have been firmly established as a useful tool for the synthesis and analysis of localized (concentrated or confined) signals, and for the modeling and inversion of noise-contaminated data that are only regionally available or only of regional interest. In this paper, we consider a general abstract setup for inverse problems represented by a linear and compact operator between Hilbert spaces with a known singular-value decomposition (svd). In practice, such an svd is often only given for the case of a global expansion of the data (e.g. on the whole sphere) but not for regional data distributions. We show that, in either case, Slepian functions (associated to an arbitrarily prescribed region and the given compact operator) can be determined and applied to construct a regularization for the ill-posed regional inverse problem. Moreover, we describe an algorithm for constructing the Slepian basis via an algebraic eigenvalue problem. The obtained Slepian functions can be used to derive an svd for the combination of the regionalizing projection and the compact operator. As a result, standard regularization techniques relying on a known svd become applicable also to those inverse problems where the data are regionally given only. In particular, wavelet-based multiscale techniques can be used. An example for the latter case is elaborated theoretically and tested on two synthetic numerical examples.

Multi-physics adjoint modeling of Earth structure: combining gravimetric, seismic, and geodynamic inversions

“…Geophysical observables that are measured at Earth's surface ultimately derive from partial differential equations (PDEs), which lend themselves to theoretical analysis (Mead and Ford 2020;Michel and Simons 2017) and numerical simulation. One can synthetically create measurements based on an initial model-a starting guess for the subsurface structure-typically, but not exclusively, as we shall see, a "low-wavenumber" smooth model upon which "high-wavenumber" sharp contrasts are sought to be superposed (Bunks et al 1995; Yuan and Simons 2014).…”

“…We limit our scope to gradient-based inversions, including possible regularization, discussing their use in resolving the geometry and amplitude of subsurface anomalies. We do not consider operator inversions (Michel and Simons 2017;Mead and Ford 2020), subspace methods (Geng et al 2020) or statistical frameworks (Fichtner and Simutė 2018) that invert for anomalous subsurface structure by exploring the costfunction space by a combination of statistical and deterministic methods. Deterministic gradient-based methods such as those presented here can only find the minimum of a function that is closest to the initial guess, which is not necessarily the global minimum.…”

Slepian Scale-Discretised Wavelets on Manifolds

“…Other times, the Slepian functions on the sphere are used directly [61]. A variety of other approaches include: established regularisation techniques based on a known singular value decomposition [62], and wavelet-like representations without explicit inverse transforms [15]. Extending wavelets and wavelet techniques to manifold and graph domains has been extensively reviewed (e.g.…”

Slepian Scale-Discretised Wavelets on the Sphere

“…Slepian frames on the sphere are constructed in [25], which provide a wavelet-like representation but do not constitute a tight frame with an explicit inverse transform. An approach to solve inverse problems with regional data on the sphere is presented in [46], which results in Slepian functions that can be used to derive a singular value decomposition (SVD) approach. Standard regularisation techniques based on a known SVD can then be applied to inverse problems where data are only defined on a region.…”