Abstract
We consider the problem of obtaining information about an inaccessible half-space from acoustic measurements made in the accessible half-space. If the measurements are of limited precision, some scatterers will be undetectable because their scattered fields are below the precision of the measuring instrument. How can we make measurements that are optimal for detecting the presence of an object? In other words, what incident fields should we apply that will result in the biggest measurements? There are many ways to formulate this question, depending on the measuring instruments. In this paper we consider a formulation involving wave-splitting in the accessible half-space: what downgoing wave will result in an upgoing wave of greatest energy? A closely related question arises in the case when we have a guess about the configuration of the inaccessible half-space. What measurements should we make to determine whether our guess is accurate? In this case we compare the scattered field to the field computed from the guessed configuration. Again we look for the incident field that results in the greatest energy difference. We show that the optimal incident field can be found by an iterative process involving time reversal ``mirrors''. For band-limited incident fields and compactly supported scatterers, in the generic case this iterative process converges to a single time-harmonic field. In particular, the process automatically tunes to the best frequency. This analysis provides a theoretical foundation for the frequency-shifting and pulse-broadening observed in certain computations and time-reversal experiments.
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