Inverse Acoustic Obstacle Scattering

Abstract

With the analysis of the preceding chapters, we now are well prepared for studying inverse acoustic obstacle scattering problems. We recall that the direct scattering problem is, given information on the boundary of the scatterer and the nature of the boundary condition, to find the scattered wave and in particular its behavior at large distances from the scatterer, i.e., its far field. The inverse problem starts from this answer to the direct problem, i.e., a knowledge of the far field pattern, and asks for the nature of the scatterer. Of course, there is a large variety of possible inverse problems, for example, if the boundary condition is known, find the shape of the scatterer, or, if the shape is known, find the boundary condition, or, if the shape and the type of the boundary condition are known for a penetrable scatterer, find the space dependent coefficients in the transmission or resistive boundary condition, etc. Here, following the main guideline of our book, we will concentrate on one model problem for which we will develop ideas which in general can also be used to study a wider class of related problems. The inverse problem we consider is, given the far field pattern for one or several incident plane waves and knowing that the scatterer is sound-soft, to determine the shape of the scatterer. We want to discuss this inverse problem for frequencies in the resonance region, that is, for scatterers D and wave numbers k such that the wavelengths 2π∕k is less than or of a comparable size to the diameter of the scatterer. This inverse problem turns out to be nonlinear and improperly posed. Although both of these properties make the inverse problem hard to solve, it is the latter which presents the more challenging difficulties. The inverse obstacle problem is improperly posed since, as we already know, the determination of the scattered wave us from a given far field pattern u∞ is improperly posed. It is nonlinear since, given the incident wave ui and the scattered wave us, the problem of finding the boundary of the scatterer as the location of the zeros of the total wave ui + us is nonlinear.