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For many wave propagation problems with random sources it has beendemonstrated that cross correlations of wave fields are proportional to theimaginary part of the Green function of the underlying wave equation. Thisleads to the inverse problem to recover coefficients of a wave equation fromthe imaginary part of the Green function on some measurement manifold. In thispaper we prove, in particular, local uniqueness results for the Schr{\"o}dingerequation with one frequency and for the acoustic wave equation with unknowndensity and sound speed and two frequencies. As the main tool of our analysis,we establish new algebraic identities between the real and the imaginary partof Green's function, which in contrast to the well-known Kramers-Kronigrelations involve only one frequency.

The imaginary part of the scattering Green function: monochromatic relations to the real part and uniqueness results for inverse problems