Optimal acoustic measurements

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 undetectabl e because their scattered fields are below the precision of the measuring i nstrument. 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-splitti ng in the accessible half-space: What downgoing wave will result in an upgo ing wave of greatest energy? A closely related question arises in the case when we have a guess about th e configuration of the inaccessible half-space. What measurements should we make to determine whether our guess is accurate? In this case we compare t he scattered field to the field computed from the guessed configuration. Ag ain we look for the incident field that results in the greatest energy diff erence. We show that the optimal incident field can be found by an iterative proces s involving time reversal mirrors. For band-limited incident fields and com pactly supported scatterers, in the generic case this iterative process con verges to a single time-harmonic field. In particular, the process automati cally tunes to the best frequency. This analysis provides a theoretical fou ndation for the frequency-shifting and pulse-broadening observed in certain computations [E. Cherkaeva and A. C. Tripp, SEG97 Expanded Abstracts, 67th Annual Meeting of Society of Exploration Geophysicists, SEG Publications, Tulsa, OK, 1997, pp. 438-441] and time-reversal experiments [ C. Prada and M. Fink, Wave Motion, 20 (1994), pp. 151-163], [ C. Prada, J.-L. Thomas, an d M. Fink, J. Acoust. Soc. Amer., 97 (1995), pp. 62-71].