Focusing Waves in Unknown Media by Modified Time Reversal Iteration

Abstract

We study the wave equation on a bounded domain M in $\mathbb{R}^m$ or on a compact Riemannian manifold with boundary. Assume that we do not know the coefficients of the wave equation but are given only the hyperbolic Robin-to-Dirichlet map that corresponds to physical measurements on a part of the boundary. In this paper we show that at a fixed time $t_0$ a wave can be cut off outside a suitable set. That is, if $N\subset M$ is a union of balls in the travel time metric having centers at the boundary, then we can modify a given Robin boundary value of a wave such that at time $t_0$ the modified wave is arbitrarily close to the original wave inside N and arbitrarily small outside N. Also, at time $t_0$ the time derivative of the modified wave is arbitrarily small in all of M. We apply this result to construct a sequence of Robin boundary values so that at a time $t_0$ the corresponding waves converge to a delta distribution $\delta_{\widehat{x}}$ while the time derivative of the waves converge to zero. Such boundary values are generated by an iterative sequence of measurements. In each iteration step we apply time reversal and other simple operators to measured data and compute boundary values for the next iteration step. A key feature of this result is that it does not require knowledge of the coefficients in the wave equation, that is, of the material parameters inside the media. However, we assume that the point $\widehat{x}$ where the wave focuses is known in travel time coordinates, and $\widehat{x}$ satisfies a certain geometrical condition.