  For many wave propagation problems with random sources it has been
demonstrated that cross correlations of wave fields are proportional to the
imaginary part of the Green function of the underlying wave equation. This
leads to the inverse problem to recover coefficients of a wave equation from
the imaginary part of the Green function on some measurement manifold. In this
paper we prove, in particular, local uniqueness results for the Schr{\"o}dinger
equation with one frequency and for the acoustic wave equation with unknown
density and sound speed and two frequencies. As the main tool of our analysis,
we establish new algebraic identities between the real and the imaginary part
of Green's function, which in contrast to the well-known Kramers-Kronig
relations involve only one frequency.
