Inverse acoustic scattering using high-order small-inclusion expansion of misfit function

Abstract

This article concerns an extension of the topological derivative concept for 3D inverse acoustic scattering problems involving the identification of penetrable obstacles, whereby the featured data-misfit cost function \begin{document} $\mathbb{J}$ \end{document} is expanded in powers of the characteristic radius \begin{document} $a$ \end{document} of a single small inhomogeneity. The \begin{document} $O(a^6)$ \end{document} approximation \begin{document} $\mathbb{J}_6$ \end{document} of \begin{document} $\mathbb{J}$ \end{document} is derived and justified for a single obstacle of given location, shape and material properties embedded in a 3D acoustic medium of arbitrary shape. The generalization of \begin{document} $\mathbb{J}_6$ \end{document} to multiple small obstacles is outlined. Simpler and more explicit expressions of \begin{document} $\mathbb{J}_6$ \end{document} are obtained when the scatterer is centrally-symmetric or spherical. An approximate and computationally light global search procedure, where the location and size of the unknown object are estimated by minimizing \begin{document} $\mathbb{J}_6$ \end{document} over a search grid, is proposed and demonstrated on numerical experiments, where the identification from known acoustic pressure on the surface of a penetrable scatterer embedded in a acoustic semi-infinite medium, and whose shape may differ from that of the trial obstacle assumed in the expansion of \begin{document} $\mathbb{J}$ \end{document} , is considered.