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This paper is concerned with an inverse obstacle problem which employs thedynamical scattering data of acoustic wave over a finite time interval. Theunknown obstacle is assumed to be sound-soft one. The governing equation of thewave is given by the classical wave equation. The wave is generated by theinitial data localized outside the obstacle and observed over a finite timeinterval at a place which is not necessary the same as the support of theinitial 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 mainanalytical formulae, is developed. The first one enables us to extract themaximum spheroid with focal points at the center of the support of the initialdata and that of the observation points whose exterior encloses the unknownobstacle of general shape. The second one, under some technical assumption forthe obstacle including convexity as an example, indicates the deviation of thegeometry of the boundary of the obstacle and the maximum spheroid at thecontact points. Several implications of those two formulae are also given. Inparticular, a constructive proof of a uniqueness of a spherical obstacle usingthe bistatic data is given.

inverse problems

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