Narrow Results

In this review paper, the use of the Time Reversal (TR) method as a computational tool for solving some classes of inverse problems is surveyed. The basics of computational TR are explained, using the scalar wave equation as a simple model. The application of TR to various problems in acoustics and elastodynamics is reviewed, in a selective and biased way as it leans on the author's personal view, referring to representative articles published on the subject.

Time reversal is a powerful procedure in application fields involving wave propagation. It is based on the invariance of the wave equations, in the absence of dissipation, in the time direction. This allows going backward in time to recover past events. We use time reversal to recover the location of a source applied at the initial time based on measurements at a later time. We generalize the procedure previously developed for the scalar wave equation¹ to elastodynamics. We show that the technique is quite robust, sometimes even in the presence of very high noise levels. Also it is not very sensitive to the medium characterizations, when a sufficient amount of measurement data is available. We extend previous work to get good refocusing for multiple sources. We introduce a new score to assess the quality of the numerical solution for the refocusing problem which produces good results. Furthermore, we use the refocusing technique as a basis for scatterer location recovery. By adding noise in a controlled manner we improve the scheme of finding the location of the scatterer.

The convective wave equation deals with wave propagation in a moving media. We focus on the underwater acoustic wave equation where the convective element is the flow of water inside a river, along its length. The main thrust of this paper is the ill-posed “refocusing” problem. The initial condition simulates an explosion in a small compact region and the response is recorded over time at several microphones. Having only partial and noisy information we expect that small perturbations will destroy the ability to recover the complete initial data. We use the time reversal (TR) technique to determine the location of the original explosion, given limited spatial observations. We test the effectiveness of this scheme under conditions including dissipation, dispersion, etc. We use finite differences and implement absorbing boundary conditions to simulate an unbounded region.