Monochromatic Identities for the Green Function and Uniqueness Results for Passive Imaging
Abstract
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.
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Information & Authors
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Published In
SIAM Journal on Applied Mathematics
Pages: 2865 - 2890
DOI: 10.1137/18M1182218
ISSN (online): 1095-712X
Copyright
© 2018, Society for Industrial and Applied Mathematics.
History
Submitted: 20 April 2018
Accepted: 29 August 2018
Published online: 23 October 2018
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