The factorization method for the acoustic transmission problem

Abstract

In the present work, the shape reconstruction problem of acoustically penetrable bodies from far-field data corresponding to time-harmonic plane wave incidence is investigated within the framework of the factorization method. Although the latter technique has received considerable attention in inverse scattering problems dealing with impenetrable scatterers, it has not been elaborated for inverse transmission problems with the only exception being a work by the first two authors and co-workers. Aimed at bridging this gap in the field of acoustic scattering, the paper at hand focuses on establishing rigorously the necessary theoretical framework for the application of the factorization method to the inverse acoustic transmission problem. The main outcome of the undertaken investigation is the derivation of an explicit formula for the scatterer's characteristic function, which depends solely on the far-field data feeding the inverse scattering scheme. Extended numerical examples in three dimensions are also presented, where a variety of different surfaces are successfully reconstructed by the factorization method, thus complementing the method's validation from the computational point of view.

The interior transmission eigenvalue problem for an inhomogeneous media with a conductive boundary

“…The derivation of the far-field pattern for a sphere of radius R is an easy task. Using the same ansatz as in [1,Section 4.2], we obtain…”

Two direct factorization methods for inverse scattering problems

“…subject to the Dirichlet boundary condition u = 0, on ∂D (2) and the Sommerfeld radiation condition (with r = |x|)…”

Remarks on the factorization and monotonicity method for inverse acoustic scatterings

“…The factorization method has first been introduced by Kirsch ( [16]) for the inverse acoustic obstacle scattering. It has been studied so far by many authors (see e.g., [1,3,4,7,11,17,18,19,20,22,24]), and it is well known as one of classical qualitative methods, which includes the linear sampling method of Colton and Kirsch ([5]), the singular sources method of Potthast ([23]), the probe method of Ikehata ([15]), etc. The monotonicity method, on the other hand, has been recently introduced by Harrach in [14] for the electrical impedance tomography.…”