Communications in Mathematical Sciences

Volume 13 (2015)

Number 3

Special Issue in Honor of George Papanicolaou’s 70th Birthday

Guest Editors: Liliana Borcea, Jean-Pierre Fouque, Shi Jin, Lenya Ryzhik, and Jack Xin

A quantitative study of source imaging in random waveguides

Pages: 749 – 776



Liliana Borcea (Department of Mathematics, University of Michigan, Ann Arbor, Mich., U.S.A.)

Josselin Garnier (Laboratoire de Probabilités et Modèles Aléatoires, Laboratoire Jacques-Louis Lions, Université Paris Diderot, Paris, France)

Chrysoula Tsogka (Department of Applied Mathematics, University of Crete, Heraklion, Greece)


We present a quantitative study of coherent array imaging of remote sources in randomly perturbed waveguides with bounded cross-section. We study how long range cumulative scattering by perturbations of the boundary and the medium impedes the imaging process. We show that boundary scattering effects can be mitigated with filters that enhance the coherent part of the data. The filters are obtained by optimizing a measure of quality of the image. The point is that there is an optimal trade-off between the robustness and resolution of images in such waveguides, which can be found adaptively, as the data are processed to form the image. Long range scattering by perturbations of the medium is harder to mitigate than scattering by randomly perturbed boundaries. Coherent imaging methods do not work and more complex incoherent methods, based on transport models of energy, should be used instead. Such methods are neither useful, nor needed in waveguides with perturbed boundaries. We explain all these facts using rigorous asymptotic stochastic analysis of the wave field in randomly perturbed waveguides. We also analyze the adaptive coherent imaging method and obtain a quantitative agreement with the results of numerical simulations.


waveguides, source imaging, random media

2010 Mathematics Subject Classification

35Q61, 35R60

Published 3 March 2015