Single-field identification of inclusions and cavities in an elastic medium

The inverse problem of identifying a small unknown inclusion or cavity in an elastic medium is considered. Important applications are structural damage identification, medical imaging and geophysical exploration; the latter is the focus of the present investigation. It is assumed that the (isotropic but possibly heterogeneous) material properties of the medium are known, except for the presence of a small inclusion of a different material or of a small cavity. The goal is to find the location, size and shape of the inclusion or cavity. Identification is performed using full waveform inversion (FWI) and the adjoint method for the efficient calculation of the gradient of the misfit function. The identification is accomplished by considering a single unknown material-property field. The inclusion manifests itself in the inverse solution as a local region where the unknown material property becomes significantly different than the known background property. The limiting case of cavity identification requires special treatment in the minimization process to avoid failure, as Bürchner et al. showed empirically for the scalar wave equation. Their (Formula presented.) -scaling approach, whose success is explained mathematically here for the first time, is extended to elastodynamics. The performance of the proposed method is demonstrated via numerical examples, involving two geophysical models: a homogeneous model and the heterogeneous Marmousi model, which is a standard testing model in geophysical research.