Tangential cone condition and Lipschitz stability for the full waveform forward operator in the acoustic regime

Abstract

Time-domain full waveform inversion (FWI) in the acoustic regime comprises a parameter identification problem for the acoustic wave equation: Pressure waves are initiated by sources, get scattered by the earth's inner structure, and their reflected parts are picked up by receivers located on the surface. From these reflected wave fields the two parameters, density and sound speed, have to be reconstructed. Mathematically, FWI reduces to the solution of a nonlinear and ill-posed operator equation involving the parameter-to-solution map of the wave equation. Newton-like iterative regularization schemes are well suited and well analyzed to tackle this inverse problem. Their convergence results are often based on an assumption about the nonlinear map known as tangential cone condition. In this paper we verify this assumption for a semi-discrete version of FWI where the searched-for parameters are restricted to a finite dimensional space. As a byproduct we establish that the semi-discrete seismic inverse problem is locally Lipschitz stable, in particular, it is conditionally well-posed.

Bi-level iterative regularization for inverse problems in nonlinear PDEs

“…For the upper-level problem, verification of the tangential condition (in θ) is another technical question. We refer to [30] for a study of such condition in several linear and nonlinear parabolic benchmark problems, to [21] for abstract linear elliptic problems, to [23] for quantitative elastography, to [24] for magnetic resonance elastography, to [36] for the electrical impedance tomography, to [11] for full wave from inversion, and recently, to [1,46] in the context of machine learning.…”