A study on finding a buried obstacle in a layered medium having the influence of the total reflection phenomena via the time domain enclosure method

TL;DR

This paper extends the time domain enclosure method to detect a penetrable obstacle in a two-layered medium with total reflection phenomena, showing that reflections do not affect the main indicator function.

Contribution

It introduces a new analysis accounting for total reflection effects in layered media, demonstrating the robustness of the enclosure method in more complex wave propagation scenarios.

Findings

Total reflection phenomena do not influence the leading profile of the indicator function.

The method successfully detects obstacles despite complex reflection effects.

The analysis extends the applicability of the enclosure method to layered media with different wave speeds.

Abstract

An inverse obstacle problem for the wave governed by the wave equation in a two layered medium is considered under the framework of the time domain enclosure method. The wave is generated by an initial data supported on a closed ball in the upper half-space, and observed on the same ball over a finite time interval. The unknown obstacle is penetrable and embedded in the lower half-space. It is assumed that the propagation speed of the wave in the upper half-space is greater than that of the wave in the lower half-space, which is excluded in the previous study: Ikehata and Kawashita (2018) to appear, Inverse Problems and Imaging. In the present case, when the reflected waves from the obstacle enter the upper layer, the total reflection phenomena occur, which give singularities to the integral representation of the fundamental solution for the reduced transmission problem in the background medium. This fact makes the problem more complicated. However, it is shown that these waves do not have any influence on the leading profile of the indicator function of the time domain enclosure method.