Time reversal with partial information for wave refocusing and scatterer identification

Time reversal is a well-known procedure in application fields involving wave propagation. Among other uses, it can be applied as a computational tool for solving certain inverse problems. The procedure is based on advancing the solution of the relevant wave problem " backward in time" One important use of numerical time-reversal is that of refocusing, where a reverse run is performed to recover the location of a source applied at an initial time based on measurements at a later time. Usually, only partial, noisy, information is available, at certain measurement locations, on the field values that serve as data for the reverse run. In this paper, the question concerning the amount and characterization of the available data needed for a successful refocusing is studied for the scalar wave equation. In particular, a simple procedure is proposed which exploits multiple measurement times, and is shown to be very beneficial for refocusing. A tradeoff between availability of spatial and temporal information is discussed. The effect of measurement noise is studied, and the technique is shown to be quite robust, sometimes even in the presence of very high noise levels. The use of the technique as a basis for scatterer identification is also discussed. A numerical study of these effects is presented, employing finite elements in space and a standard explicit marching scheme in time. In contrast to some previous studies, the propagation medium is taken to be homogeneous.