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

TL;DR

This paper develops an enclosure method using bistatic data from dynamical acoustic scattering to determine the shape and boundary deviations of a sound-soft obstacle over a finite time interval.

Contribution

It introduces two analytical formulas based on bistatic data that enable shape reconstruction and boundary deviation analysis of unknown obstacles, including a proof of uniqueness for spherical obstacles.

Findings

The first formula extracts the maximum spheroid enclosing the obstacle.

The second formula indicates boundary deviations under certain assumptions.

A constructive proof of uniqueness for spherical obstacles is provided.

Abstract

This paper is concerned with an inverse obstacle problem which employs the dynamical scattering data of acoustic wave over a finite time interval. The unknown obstacle is assumed to be 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 center 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 a uniqueness of a spherical obstacle using the bistatic data is given.