Time-domain extended-source full-waveform inversion: algorithm and practical workflow

Extended-source full-waveform inversion (ES-FWI) first computes wavefields with data-driven source extensions such that the simulated data in inaccurate velocity models match the observed counterpart well enough to prevent cycle skipping. Then, the source extensions are minimized to update the model parameters toward the true medium. This two-step workflow is iterated until both data and sources are matched. It was recently shown that the source extensions are the least-squares solutions of the scattered data fitting problem. As a result, the source extensions are computed by propagating backward in time the deconvolved data residuals by the damped data-domain Hessian of the scattered data fitting problem. Estimating these weighted data residuals is the main computational bottleneck of time-domain ES-FWI. To mitigate this burden, we approximate the inverse data-domain Hessian by mono- and multi-dimensional matching filters with two simulations per source. We implement time-domain ES-FWI with the alternating-direction method of multiplier and total-variation regularization. Moreover, we apply ES-FWI with a multiscale approach involving frequency continuation and layer-stripping, with the latter being implemented with an offset-time dependent weighting operator. In this framework, we further regularize the inversions while mitigating their computational burden by matching the grid interval to the frequency bandwidth. Finally, the overall workflow combines ES-FWI and classical FWI during the early and late stages of the multiscale approach, respectively. We illustrate that the sensitivity of ES-FWI to the accuracy of the approximated inverse data-domain Hessian depends on the complexity of the targeted model, the data anatomy, and the accuracy of the starting model. In the case of the 2004 BP salt model, we demonstrate that the layer stripping is necessary when the inverse data-domain Hessian is approximated by a 2D Gabor matching filter and the starting model is crude, while this feature is not necessary with the Marmousi II model.