Shape and property identification of an elastic inclusion via a full waveform inversion and adjoint method

The problem of identifying the precise shape and material properties of a soft inclusion in a two-dimensional elastic medium is considered. The identification is performed based on time-dependent displacement response to a given wave source, measured by sensors at discrete locations. "Inclusion" here means an unknown local region whose material properties differ significantly from those of the background medium, which are assumed to be known. Applications include identification of unknown scatterers in solid earth geophysics and non-destructive testing of structures. The identification method is based on Full Waveform Inversion (FWI) and an adjoint scheme. In a previous publication, a precise shape identification method was developed for a cavity of an arbitrary shape in an acoustic medium (or a hole in a membrane). The present paper extends this publication in three ways. First, the medium is elastic rather than acoustic. Second, the scatterer is an inclusion rather then a cavity (the latter can be regarded as the limiting case of an inclusion). Third, an attempt is made to identify both the precise shape and the material properties of the inclusion, simultaneously. This attempt is only partly successful, and the reasons for this are analyzed. Numerical examples are used to demonstrate the proposed methodology.