A Gauss-Newton full-waveform inversion for material profile reconstruction in 1D PML-truncated solid media

This paper discusses a Gauss-Newton full-waveform inversion procedure for material profile reconstruction in semi-infinite solid media. Given surficial measurements of the solid's response to interrogating waves, the procedure seeks to find an unknown wave velocity profile within a computational domain truncated by Perfectly-Matched-Layer (PML) wave-absorbing boundaries. To this end, the inversion procedure minimizes a Lagrangian functional composed of a cost functional augmented by PML-endowed wave equations via Lagrange multipliers. Enforcing the stationarity of the Lagrangian leads to KKT (Karush-Kuhn-Tucker) conditions comprising time-dependent state, adjoint, and time-invariant control problems. The material parameter is updated by iteratively solving the KKT conditions in the reduced space of the control variable. The update of the control variable is determined by a Gauss-Newton-Krylov optimization algorithms. Super-linear convergence behavior of the Gauss-Newton inversion has been observed in one-dimensional implementations. Regularization and frequency-continuation schemes were used to relieve the ill-posedness of the inverse problem.