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Estimating a Green’s function from “field-field” correlations in a random medium. (English) Zbl 1172.62030

Summary: Traditional imaging methods use coherent signals as data. Here, we discuss recent developments in imaging that aim at exploiting as data incoherent noisy signals that are not associated with well-defined arrival times. Indeed, signal constituents that in a classical setting may be regarded as noise may contain important information about the medium to be imaged. We show how it is possible to use the statistics of such noisy signals, specifically, the second-order statistics, for imaging.
We consider two particular situations: first, the estimation of an (“empirical”) Green’s function from noisy signals which can subsequently be used in imaging; second, the localization of a cluster of random sources from noisy signals (passive imaging). The analysis presented here is based on assuming a remote sensing scaling and the paraxial approximation, and it uses in part the results set forth by G. Papanicolaou, L. Ryzhik and K. Solna [SIAM J. Appl. Math. 64, No. 4, 1133–1155 (2004; Zbl 1065.35058)] that relate to time-reversal, statistical stability, and superresolution. Robustness with respect to modeling assumptions is illustrated by considering other scaling regimes also. We demonstrate how the estimation problem and its robustness can be considered as a dual to that of time-reversal and stable superresolution. We obtain a novel analysis and foundation for the use of ambient seismic noise in body-wave (tomographic) imaging, motivated by the recent successes of surface-wave tomography using ambient seismic noise.

MSC:

62M40 Random fields; image analysis
60H30 Applications of stochastic analysis (to PDEs, etc.)
60H20 Stochastic integral equations

Citations:

Zbl 1065.35058
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