Seminar in Numerical Analysis: Florian Faucher (Université de Pau)
We study the inverse problem associated with the propagation of time-harmonic waves. In the seismic context, the available measurements correspond with partial reflection data, obtained from one side illumination (only from the Earth surface). The inverse problem aims at recovering the subsurface Earth medium parameters and we employ the Full Waveform Inversion (FWI) method, which relies on an iterative minimization algorithm of the difference between the measurement and simulation.
We investigate the deployment of new devices developed in the acoustic setting: the dual-sensors, which are able to capture both the pressure field and the vertical velocity of the waves. For solving the inverse problem, we define a new cost function, adapted to these two types of data and based upon the reciprocity. We first note that the stability of the problem can be shown to be Lipschitz, assuming piecewise linear parameters. In addition, reciprocity waveform inversion allows a separation between the observational and numerical acquisitions. In fact, the numerical sources do not have to coincide with the observational ones, offering new possibilities to create adapted computational acquisitions, consequently reducing the numerical cost. We illustrate our approach with three-dimensional medium reconstructions, where we start with minimal information on the target models. We also extend the methodology for elasticity.
Eventually, if time allows, we shall explore the model representation in numerical seismic inversion, where the adaptive eigenspace method appears as a promising approach to have a compromise between number of unknowns and resolution.
References
[1] G. Alessandrini, M. V. de Hoop, F. Faucher, R. Gaburro and E. Sincich, Inverse problem for the Helmholtz equation with Cauchy data: reconstruction with conditional well-posedness driven iterative regularization, ESAIM: M2AN (2019).
[2] E. Beretta, M. V. De Hoop, F. Faucher, and O. Scherzer, Inverse boundary value problem for the Helmholtz equation: quantitative conditional Lipschitz stability estimates. SIAM Journal on Mathematical Analysis, 48(6), pp.3962-3983 (2016).
[3] M. J. Grote, M. Kray, and U. Nahum, Adaptive eigenspace method for inverse scattering problems in the frequency domain. Inverse Problems, 33(2), 025006 (2017).
[4] H. Barucq, F. Faucher, and O. Scherzer, Eigenvector Model Descriptors for Solving an Inverse Problem of Helmholtz Equation. arXiv preprint arXiv:1903.08991 (2019).
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