Open AccessTangential cone condition and Lipschitz stability for the full waveform forward operator in the acoustic regime
Author(s) -
Matthias Eller,
Andreas Rieder
Publication year - 2021
Publication title -
inverse problems
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.003
H-Index - 119
eISSN - 1361-6420
pISSN - 0266-5611
DOI - 10.1088/1361-6420/ac11c5
Subject(s) - lipschitz continuity , mathematics , acoustic wave equation , mathematical analysis , inverse problem , wave equation , nonlinear system , operator (biology) , regularization (linguistics) , waveform , perturbation (astronomy) , acoustic wave , acoustics , physics , computer science , radar , biochemistry , chemistry , repressor , quantum mechanics , transcription factor , gene , telecommunications , artificial intelligence
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.