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 Schrödinger 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.


  1. inverse scattering problems
  2. uniqueness for inverse problems
  3. passive imaging
  4. correlation data
  5. imaginary part of Green's function

MSC codes

  1. 35R30
  2. 35J08
  3. 35Q86
  4. 78A46

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Information & Authors


Published In

cover image SIAM Journal on Applied Mathematics
SIAM Journal on Applied Mathematics
Pages: 2865 - 2890
ISSN (online): 1095-712X


Submitted: 20 April 2018
Accepted: 29 August 2018
Published online: 23 October 2018


  1. inverse scattering problems
  2. uniqueness for inverse problems
  3. passive imaging
  4. correlation data
  5. imaginary part of Green's function

MSC codes

  1. 35R30
  2. 35J08
  3. 35Q86
  4. 78A46



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