Kohn–Vogelius criterion applied to inversion problems in acoustic wave propagation in time domain

Abstract

This article deals with the problem of identifying inclusions in a homogeneous acoustic medium using the full-waveform inversion (FWI) approach. The aim is to evaluate an alternative to the standard way of calculating the objective function, introducing the Kohn–Vogelius criterion for this purpose. While in the standard approach the objective function is calculated as a measure of the difference between the simulated acoustic pressure and the corresponding value experimentally acquired at a finite number of receivers, in the Kohn–Vogelius procedure the experimental signal is used as an inhomogeneous boundary condition for an auxiliary problem. Thus, the objective function is calculated as a measure of the difference between the original and auxiliary propagation problems. It is then expected that both being solved by the same discretized model, improvements in identification can be achieved when compared to the standard procedure. The sensitivity is obtained through conventional operations of the adjoint method. The spatial discretization was performed using finite elements and the temporal discretization by the Newmark method. The acoustic propagation velocity distribution is controlled through a level set approach that allows for a sharp interface between two media with previously known propagation velocities. Numerical results showed that both, standard and Kohn–Vogelius-based objective functions provided comparable results when the inverse crime was included. However, in a series of examples where the inverse crime was avoided, the objective function based on the Kohn–Vogelius criterion introduced relevant improvements in the quality of the inversion. Despite these encouraging improvements, the Kohn–Vogelius procedure still presents approximately twice the computational cost compared to the Standard case. Therefore, further investigations will be necessary to consider this procedure as an efficient alternative to the problem under study.