An augmented time reversal method for source and scatterer identification

We propose a computational method which improves the identification of sources and scatterers in a two-dimensional elastic medium using the Time Reversal (TR) technique. The identification procedure makes use of partial numerical data available at a small number of sensors. While in realistic applications these data are obtained by lab or field measurements, in the present work they are synthesized by solving the corresponding forward problem. We introduce an “augmentation” procedure that is utilized in addition to the TR technique and investigate its benefits. The augmentation procedure involves the solution of a suitable elliptic problem at selected time steps. In practice, one needs to solve, as a pre-process, a small number of “fundamental” elliptic problems which is equal to the number of measured quantities. A version of the augmentation procedure, which is easier to implement and involves almost no computational cost, is also presented. The local elastic energy in the domain is employed to decide whether the TR solution has properly refocused at the source. Cost functionals, whose minima are sought, are constructed for the source and scatterer identification problems. We conclude that augmentation improves the identification performance, in particular in the presence of noisy measurements. In some cases augmented TR succeeds in the identification whereas non-augmented TR fails.