Imaging Extended Reflectors in a Terminating Waveguide

Abstract

We consider the problem of imaging extended reflectors in terminating waveguides. We form the image by back-propagating the array response matrix projected on the waveguide's non evanescent modes. The projection is adequately defined for any array aperture size covering fully or partially the waveguide's vertical cross-section. We perform a resolution analysis of the imaging method and show that the resolution is determined by the central frequency while the image's signalto-noise ratio improves as the bandwidth increases. The robustness of the imaging method is assessed with fully non-linear scattering data in terminating waveguides with complex geometries.

Imaging junctions of waveguides

“…The interpretation of these results is quite clear. For h " 1, the identification results are better ifh " 0.5 than ifh " 0.75 (figure 10), because the amount of reflected waves at the interface between the two half-waveguides is larger (at the limit whenh tends to 0, our waveguide tends to a terminating waveguide like in [22], which can be seen as an ideal situation). On the contrary, forh " 1, the Figure 7.…”

“…Note that in [3], our method was successfully tested in the presence of real data coming from an ultrasonic NDT experiment on a steel plate. We here mention several other works based on sampling-type methods in acoustic waveguides [15,20,21,22,5,6] and electromagnetic waveguides [7,19].…”

“…Let us now assume that there is a Dirichlet obstacle O to retrieve in the junction B. For all y P W , we introduce the total field up¨, yq which satisfies (22) and the scattered field u s p¨, yq which satisfies (23) with f "´Gp¨, yq| BO . For j " 0, .…”

“…We assume there exists an obstacle O within the waveguide W , more precisely O is a smooth and bounded open (not necessarily connected) domain such that O Ă W , with Ω " W zO a connected open domain. For some y P Ω, let us consider the problem: find up¨, yq P L2 loc pΩq such that WW Obstacles within the waveguide(22)…”

A direct approach to imaging in a waveguide with perturbed geometry

“…The wave equation in such empty waveguides can be solved with separation of variables and the wave field is a superposition of propagating, evanescent and possibly radiating modes that do not interact with each other. A sample of the existing mathematical literature is [17,12,23,24,10,31,32] and examples of imaging with experimental validation are in [25,26].…”