Full Reciprocity-Gap Waveform Inversion in the frequency domain, enabling sparse-source acquisition

Abstract

We perform quantitative sub-surface Earth imaging by setting up the Full Reciprocity-gap Waveform Inversion (FRgWI ) method. The reconstruction relies on iterative minimization of a misfit functional which is specifically designed for data obtained from dual-sensor devices. In addition to the pressure field, the dual-sensor devices are able to measure the normal velocity of the waves, and have been deployed in geophysical exploration. The use of reciprocity-based misfit functional provides additional features compared to the more traditional least-squares approaches with, in particular, that the observational and computational acquisitions can be different. Therefore, the source positions and wavelets that generate the measurements are not needed for the reconstruction procedure and, in fact, the numerical acquisition (for the simulations) can be arbitrarily chosen. We illustrate our method with three-dimensional experiments, where we first show that the reciprocity-gap inversion behaves better than the Full Waveform Inversion (FWI) in the same context. Next, we investigate arbitrary numerical acquisitions in two ways: firstly, when few measurements are given, we use a dense numerical acquisition (compared to the observational one). On the other hand, with a dense observational acquisition, we employ a sparse computational acquisition, with multiple-point sources, to reduce the numerical cost. We highlight that the reciprocity-gap functional is efficient in both situations and is more robust with respect to cross-talk than shot-stacking.

Adjoint-state method for Hybridizable Discontinuous Galerkin discretization, application to the inverse acoustic wave problem

“…where the x k are a discrete set of positions, that is, the forward problem gives measurements obtained from n rcv receivers. Here we have considered measurements of the pressure fields (commonly employed in seismic applications), but we can proceed similarly with the velocity, or with both, see, e.g., [2,32].…”

“…Therefore, one would need more advanced techniques to obtain both fields with similar accuracy, while this is natural with the HDG discretization. It has motivated its used in application of inverse problem where both the velocity and the pressure field are employed, e.g., [32].…”

“…Our misfit criterion in (1) is the L 2 difference and least-squares problems are further analyzed in [17,8]. Several alternatives for the misfit have been investigated, we refer to, e.g., [53,68,36,14,55,31,74,2,32]. Furthermore, one can incorporate a regularization term in (1) to reduce the ill-posedness, by the means of additional constraints, see, e.g., [64,27,48,45,44,34].…”

Eigenvector models for solving the seismic inverse problem for the Helmholtz equation

“…The L1 norm is studied by Brossier et al (2010) while approaches based upon optimal transport are considered by Métivier et al (2016); Yang et al (2018). In the context where different fields are measured, Alessandrini et al (2019); Faucher et al (2019) advocate for a reciprocity-based functional, which further connects to the correlation-based formulas (Faucher et al, 2019). In the case of accurate knowledge of the background velocity, the inverse problem is close to linear or quasi-linear as the Born approximation holds and then, alternative methods of linear inverse problem can be applied, such as the Backus-Gilbert method (Backus & Gilbert, 1967, 1968.…”