Asymptotic behavior of the indicator function in the inverse problem of the wave equation for media with multiple types of cavities

This paper discusses an inverse problem of recovering information about cavities by measuring the solutions of a wave equation. When the medium has multiple types of cavities and the distances from the observation site to the ‘positive cavity’ and to the ‘negative cavity’ coincide, the sign of the indicator function of the enclosure method cannot be determined. Thus, sign cancellation may occur, and information may be lost. In this study, by using an asymptotic solution and examining the top term of the indicator function in detail, it is shown that the shortest distance to the cavity can be obtained, even in the above case.