Waveform-Based Geometrical Inversion of Obstacles

Full-waveform inversion (FWI) can produce previously unobtainable levels of accuracy and is revolutionizing the field of wave imaging. The basic principle is that a numerically produced data set is matched to the measured waveforms, enabling a high-resolution image to be produced since the model being inverted fully captures the physical behavior without approximation. This is achieved by gradually updating the numerical model using optimization algorithms. Currently, most FWI methods aim to recover material properties of a medium containing penetrable scatterers; however, there are many applications that, instead, require the boundary shapes of impenetrable objects to be reconstructed. Conventional velocity-style FWI will be trapped in local minima, with such problems being due to the extremely sharp contrast at the boundary. We propose a FWI procedure to directly recover the geometrical parameters of impenetrable obstacles via shape optimizations. The geometry is reconstructed by iteratively deforming the boundary of the target, following the negative direction of the geometrical boundary gradient. The boundary gradient is calculated from the shape derivatives of mass and stiffness matrices of a finite-element (FE) representation, when distorting the elements attached at the boundary. In addition, multiple-scattering events, which are more likely to occur between impenetrable obstacles, can be utilized automatically to provide substantial information for the inversion. Numerical and experimental results are shown to demonstrate the accuracy of the procedure for an example taken from the field of nondestructive evaluation, giving sizing within fractions of a wavelength for the tested cases; this step change in accuracy could be critical in sizing defects, enabling significantly more reliable decisions to be made about whether it is safe to continue using a component. Mathematical derivations and physical reasons for the success of our approach are illustrated.