  We analyze the refocusing properties of time reversed waves that propagate in
two different media during the forward and backward stages of a time-reversal
experiment. We consider two regimes of wave propagation modeled by the paraxial
wave equation with a smooth random refraction coefficient and the
It\^o-Schr\"odinger equation, respectively. In both regimes, we rigorously
characterize the refocused signal in the high frequency limit and show that it
is statistically stable, that is, independent of the realizations of the two
media. The analysis is based on a characterization of the high frequency limit
of the Wigner transform of two fields propagating in different media. The
refocusing quality of the back-propagated signal is determined by the cross
correlation of the two media. When the two media decorrelate, two distinct
de-focusing effects are observed. The first one is a purely absorbing effect
due to the loss of coherence at a fixed frequency. The second one is a phase
modulation effect of the refocused signal at each frequency. This causes
de-focusing of the back-propagated signal in the time domain.
