The enclosure method for inverse obstacle scattering problems with dynamical data over a finite time interval: III. Sound-soft obstacle and bistatic data

Abstract

This paper is concerned with an inverse obstacle problem which employs the dynamical scattering data of an acoustic wave over a finite time interval. The unknown obstacle is assumed to be a sound-soft one. The governing equation of the wave is given by the classical wave equation. The wave is generated by the initial data localized outside the obstacle and observed over a finite time interval at a place which is not necessary the same as the support of the initial data. The observed data are the so-called bistatic data. In this paper, an enclosure method which employs the bistatic data and is based on two main analytical formulae is developed. The first one enables us to extract the maximum spheroid with focal points at the centre of the support of the initial data and that of the observation points whose exterior encloses the unknown obstacle of general shape. The second one, under some technical assumption for the obstacle including convexity as an example, indicates the deviation of the geometry of the boundary of the obstacle and the maximum spheroid at the contact points. Several implications of those two formulae are also given. In particular, a constructive proof of the uniqueness of a spherical obstacle using the bistatic data is given.