Small defects reconstruction in waveguides from multifrequency one-side scattering data

TL;DR

This paper introduces a multi-frequency inversion method for reconstructing small defects in 2D waveguides using one-sided scattering data, employing a Born approximation and Fourier inversion techniques.

Contribution

The work presents a novel multi-frequency inversion approach for small defect reconstruction in waveguides from partial data, incorporating a mode-by-mode Fourier inversion and stability analysis.

Findings

Stable reconstruction of various defect types achieved

Effective Fourier inversion method developed

Method's limitations and applications discussed

Abstract

Localization and reconstruction of small defects in acoustic or electromagnetic waveguides is of crucial interest in nondestructive evaluation of structures. The aim of this work is to present a new multi-frequency inversion method to reconstruct small defects in a 2D waveguide. Given one-side multi-frequency wave field measurements of propagating modes, we use a Born approximation to provide a L2-stable reconstruction of three types of defects: a local perturbation inside the waveguide, a bending of the waveguide, and a localized defect in the geometry of the waveguide. This method is based on a mode-by-mode spacial Fourier inversion from the available partial data in the Fourier domain. Indeed, in the available data, some high and low spatial frequency information on the defect are missing. We overcome this issue using both a compact support hypothesis and a minimal smoothness hypothesis on the defects. We also provide a suitable numerical method for efficient reconstruction of such defects and we discuss its applications and limits.