Detecting a hidden obstacle via the time domain enclosure method. A scalar wave case

Abstract

The characterization problem of the existence of an unknown obstacle behind a known obstacle is considered by using a singe observed wave at a place where the wave is generated. The unknown obstacle is invisible from the place by using a visible ray. A mathematical formulation of the problem using the classical wave equation is given. The main result consists of two parts: (a) one can make a decision whether the unknown obstacle exists or not behind a known impenetrable obstacle by using a single wave over a finite time interval under some a‐priori information on the position of the unknown obstacle; (b) one can obtain a lower bound on the Euclidean distance of the unknown obstacle to the center point of the support of the initial data of the wave. The proof is based on the idea of the time domain enclosure method and employs some previous results on the Gaussian lower/upper estimates for the heat kernels and domination of semigroups.

Extracting discontinuity using the probe and enclosure methods

“…As a closely related problem, it would be interested to apply the enclosure method with the monostatic data to the problem of finding information about the geometry of an unknown impenetrable obstacle behind a known one. In [118] an asymptotic relation between an indicator function and some information about the geometry of the unknown obstacle has been established.…”