  In the previous paper "Stabilizing Inverse Problems by Internal Data", the
authors introduced a simple procedure that allows one to detect whether and
explain why internal information arising in several novel coupled physics
(hybrid) imaging modalities could turn extremely unstable techniques, such as
optical tomography or electrical impedance tomography, into stable,
good-resolution procedures. It was shown that in all cases of interest, the
Frechet derivative of the forward mapping is a pseudo-differential operator
with an explicitly computable principal symbol. If one can set up the imaging
procedure in such a way that the symbol is elliptic, this would indicate that
the problem was stabilized. In the cases when the symbol is not elliptic, the
technique suggests how to change the procedure (e.g., by adding extra
measurements) to achieve ellipticity.
  In this article, we consider the situation arising in acousto-optical
tomography (also called ultrasound modulated optical tomography), where the
internal data available involves the Green's function, and thus depends
globally on the unknown parameter(s) of the equation and its solution. It is
shown that the technique of "Stabilizing Inverse Problems by Internal Data" can
be successfully adopted to this situation as well. We also obtain results on
generic uniqueness for the linearized problem in a variety of situations,
including those arising in acousto-electric and quantitative photoacoustic
tomography.
