  Full-waveform inversion (FWI) with extended sources 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. 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 recorded scattered data fitting problem. As a result, they are
computed by propagating backward in time the deblurred FWI data residuals,
where the deblurring operator is the inverse of the damped data-domain Hessian
of the scattering-source estimation problem. Estimating the deblurred data
residuals is the main computational bottleneck of time-domain extended-source
FWI (ES-FWI). To mitigate this issue, we first estimate them when the inverse
of the data-domain Hessians is approximated by matching filters in Fourier and
short-time Fourier domains. Second, we refine them with conjugate-gradient
iterations when necessary. Computing the matching filters and performing one
conjugate-gradient iteration each require two simulations per source.
Therefore, it is critical to design some workflows that minimize this
computational burden. We implement time-domain ES-FWI with the augmented
Lagrangian method. Moreover, we further extend its linear regime with a
multiscale frequency continuation approach, which is combined with grid
coarsening to mitigate the computational burden and regularize the inversion.
Finally, we use total-variation regularization to deal with large-contrast
reconstruction. We present synthetic cases where different inversion workflows
carried out with data-domain Hessians of variable accuracy were assessed with
the aim at converging toward accurate solutions while minimizing computational
cost.
