Accurate and efficient data-assimilated wavefield reconstruction in the time domain

Abstract

Wavefield reconstruction inversion (WRI) mitigates cycle skipping in full-waveform inversion by computing wavefields that do not exactly satisfy the wave equation to match data with inaccurate velocity models. We refer to these wavefields as data assimilated wavefields because they are obtained by combining the physics of wave propagation and the observations. Then, the velocity model is updated by minimizing the wave-equation errors, namely, the source residuals. Computing these data-assimilated wavefields in the time domain with explicit time stepping is challenging. This is because the right-hand side of the wave equation to be solved depends on the back-propagated residuals between the data and the unknown wavefields. To bypass this issue, a previously proposed approximation replaces these residuals by those between the data and the exact solution of the wave equation. This approximation is questionable during the early WRI iterations when the wavefields computed with and without data assimilation differ significantly. We have developed a simple backward-forward time-stepping recursion to refine the accuracy of the data-assimilated wavefields. Each iteration requires us to solve one backward and one forward problem, the former being used to update the right side of the latter. An application to the BP salt model indicates that a few iterations are enough to reconstruct data-assimilated wavefields accurately with a crude velocity model. Although this backward-forward recursion leads to increased computational overheads during one WRI iteration, it preserves its capability to extend the search space.

Multipliers waveform inversion

“…Even though the alternating direction strategy (alternating direction method of multipliers (ADMM), Boyd, Parikh, Chu, Peleato, & Eckstein, 2010) allowed breaking the primal optimization over the fullspace into smaller subproblems over each parameter class separately, the algorithm face two main challenges from the computation and memory point of views: 1) reconstruction of the data-assimilated wavefield, as required at each iteration of the optimization algorithm, is time consuming and 2) storage of the Lagrange multipliers associated to the wave-equation constraint is memory demanding because it is of the size of the wavefield. These challenges prevent the algorithm to be fully applied in the time domain applications (Aghamiry et al, 2020b) or even in the frequencydomain for large scale applications where direct methods can not be used to solve the data-assimilated system. Several attempts have been made to improve these limitations.…”

“…van Leeuwen & Herrmann (2013) do this by relaxing the wave-equation constraint by using a penalty formulation of the unreduced FWI. The extra parameter introduced is the so called data-assimilated wavefield (Aghamiry et al, 2020b) and the try is to approach this wavefield to the original reduced wavefield by forcing the wave-equation constraint. Huang et al (2018) introduced nonphysical sources and tried to collapse them into the physical source through iterations.…”

“…The main drawback is a need to compute the data-assimilated wavefields at each iteration and also storage of the multipliers vector associated with the wave-equation constraint. These may limit somewhat its applicability for time-domain formulations (Aghamiry et al, 2020b;Gholami et al, 2020;Rizzuti et al, 2019;Wang et al, 2016).…”

Solution of an Acoustic Transmission Inverse Problem by Extended Inversion

“…Model extension is an old idea in seismic data processing (Symes, 2008). In recent years, many extended inversion algorithms have been proposed as cures for cycle-skipping (Symes & Carazzone, 1991;Plessix et al, 2000;Symes, 2009;Luo & Sava, 2011;Biondi & Almomin, 2012;van Leeuwen & Herrmann, 2013;Liu et al, 2014;Warner & Guasch, 2014;Lameloise et al, 2015;Warner & Guasch, 2016;Chauris & Cocher, 2017;Symes & Hou, 2018;Aghmiry et al, 2020;Métivier & Brossier, 2020). The approach studied in this paper uses source extension: the energy source component of the experimental model is permitted to have more (or less constrained) parameters than the experimental design suggests.…”

Large-scale highly-accurate extended full waveform inversion using convergent Born series

“…In the (IR-)WRI framework, the wavefields are the least-squares solutions of an overdetermined system gathering the observation equation and the wave equation, the latter being weighted by the penalty parameter. These wavefields were referred to as data-assimilated (DA) wavefields by Aghamiry et al (2020a) since they are computed with a feedback term to the observables. The DA wavefields are the solution of a normal-equation system, which is denser, has a wider numerical bandwidth, and a poorer condition number than the original wave-equation system involved in classical FWI, where the wave-equation constraint is strictly satisfied at each iteration.…”