Elastic Wave Diffraction Tomography

Abstract

Following the guidelines of a recently developed unified theory of scalar inverse scattering within the linearizing Born or Kirchhoff approximations — culminating in a 3D imaging system for ultrasonic NDT applications — we extend its basic ideas to elastic wave inverse scattering, especially for the case of a scatterer with stress-free boundaries being illuminated by a plane pressure or shear wave. Two inversion schemes, both yielding the same result, i.e. the “visible” part of the singular function of the scattering surface, are proposed: far-field inversion, provided the data are collected in the remote region of the defect, or, far-field inversion combined with a near-field far-field transformation of the measured data for the case of defects closed to a measurement surface. The latter is computationally very effective in terms of Fourier transforms provided the mesurement surface is planar or circular cylindrical. In addition, we point out why the powerful idea of backpropagation resulting in the scalar Porter-Bojarski integral equation is not very appropriate for the elastodynamic case. Our algorithms are checked against simulations and discussed as elastodynamic extensions of the scalar SAFT-algorithm in terms of a mode-matched SAFT; their use in advanced imaging systems is presently only limited due to the requirement of tangential as well as normal components of the displacement vector to be measured on the material surface.