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.

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