Immersed boundary parametrizations for full waveform inversion

Abstract

Full Waveform Inversion (FWI) is a successful and well-established inverse method for reconstructing material models from measured wave signals. In the field of seismic exploration, FWI has proven particularly successful in the reconstruction of smoothly varying material deviations. In contrast, non-destructive testing (NDT) often requires the detection and specification of sharp defects in a specimen. If the contrast between materials is low, FWI can be successfully applied to these problems as well. However, so far the method is not fully suitable to image defects such as voids, which are characterized by a high contrast in the material parameters. In this paper, we introduce a dimensionless scaling function γ to model voids in the forward and inverse scalar wave equation problem. Depending on which material parameters this function γ scales, different modeling approaches are presented, leading to three formulations of mono-parameter FWI and one formulation of two-parameter FWI. The resulting problems are solved by first-order optimization, where the gradient is computed by an ajdoint state method. The corresponding Fréchet kernels are derived for each approach and the associated minimization is performed using an L-BFGS algorithm. A comparison between the different approaches shows that scaling the density with γ is most promising for parameterizing voids in the forward and inverse problem. Finally, in order to consider arbitrary complex geometries known a priori, this approach is combined with an immersed boundary method, the finite cell method (FCM).

On the Use of Neural Networks for Inverse Problems

“…To this end, we consider the three key ingredients (a) the forward solver, (b) the Ansatz space of the optimization variable, and (c) the sensitivity computation. The empirical investigation is performed for full waveform inversion, where the unknown is a scaling function of the density field to locate internal voids [2]. PINN-based approaches, as presented in [3], represent both the solution and the scaling function with separate neural networks and perform a nested minimization of the emerging residuals.…”

“…Lastly, the sensitivity computation with automatic differentiation is partially substituted with the continuous adjoint method (c). These aspects are studied using two-dimensional benchmark problems [2] and complex three-dimensional cases based on CT scans of rare drill cores [4].…”