Early-reverberation imaging functions for bounded elastic domains

For the ultrasonic inspection of bounded elastic structures, finite-duration imaging functions are derived in the Fourier–Laplace domain, where efficient computational tools exist for solving problems in linear elasticity. The signals involved are exponentially windowed, so that early reflections are taken into account more strongly than later ones in the imaging methodology. Applying classical approaches to the general case of anisotropic elasticity, we express the Fréchet derivatives of the relevant data-misfit functional with respect to arbitrary perturbations of the mass density and stiffnesses in terms of forward and adjoint solutions. Their definitions incorporate the exponentially decaying weighting. The proposed finite-duration imaging functions are then defined on that basis. As some areas of the structure are less insonified than others, it is necessary to define normalized imaging functions to compensate for these variations. Our approach in particular aims to overcome the difficulty of dealing with bounded domains containing defects not located in direct line of sight from the transducers and measured signals of long duration. In this preliminary and methodological work, we demonstate the potential of the proposed approach on a two-dimensional test case featuring the imaging of mass and elastic stiffness variations in a region of a bounded isotropic medium that is not directly visible from the transducers. The results show that the early-reverberation imaging (ERI) method allows for mapping anomalies in masked regions of a structure with reasonable computational efforts.