Signal-to-Noise Ratio analysis for time-reversal based imaging techniques in bounded domains

We consider the problem of localizing small material defects in rectangular bounded domains. The scalar acoustic equation is used to model wave propagation in this context. Our data is the scattered field collected at one or more receivers and due to impulsive excitations at one or more source positions. To localize the defect we use an imaging method that consists in back-propagating the recorded field in the domain of interest. The back-propagation is performed numerically using a model for the Green's function in the bounded medium. For the source localization problem this imaging technique is equivalent to computational Time Reversal (TR). We study in this paper the quality of imaging in terms of the Signal to Noise Ratio (SNR) both for the source and the defect localization problems. SNR here is defined as the value of the image at the true source (defect) location, divided by the maximal value of the image outside a small region around the true source (defect) location. Our theoretical analysis carried out for the simpler one-dimensional case allows us to correctly predict the performance of the method. Our results indicate that for the source localization problem the SNR increases linearly with the number of receivers while for the defect localization its maximal value is 2 and can only be attained by decreasing the time of the experiment so as to minimize the boundary effects.