The inverse medium problem in 1D PML-truncated heterogeneous semi-infinite domains

Abstract

We discuss a total wavefield-based inversion approach for the reconstruction of the material profile of heterogeneous semi-infinite domains, directly in the time domain, based on surficial measurements of the domain's response to prescribed wave illumination. The ability to recover the in-depth profile of moduli/wave velocities typically associated with geotechnical site characterization applications is of particular interest. We address four key issues associated with the wavefield-based inversion: (a) to limit the semi-infinite extent of the heterogeneous physical domain, a perfectly-matched-layer (PML) is introduced at the truncation interface; (b) to account for the introduction of the PML, we use a mixed unsplit-field PML formulation for the coupled PML-regular-domain problem; (c) to tackle the inversion, we adopt a partial-differential-equation (PDE)-constrained optimization framework that formally leads to a classic Karush–Kuhn–Tucker (KKT) system comprising the initial-value state, final-value adjoint and time-independent control problems; and (d) to narrow the feasibility space and alleviate the inherent solution multiplicity, we discuss Tikhonov and Total Variation regularization schemes, endowed with a regularization factor continuation algorithm. We also limit the total observation time to optimally account for the domain's heterogeneity during inversion iterations. We report on the theory and results that lead efficiently to the reconstruction of both sharp and smooth profiles in one dimension.