  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.
