  Extended full-waveform inversion (FWI) has shown promising results for
accurate estimation of subsurface parameters when the initial models are not
sufficiently accurate. Frequency-domain applications have shown that the
augmented Lagrangian (AL) method solves the inverse problem accurately with a
minimal effect of the penalty parameter choice. Applying this method in the
time domain, however, is limited by two main factors: (1) The challenge of
data-assimilated wavefield reconstruction due to the lack of an explicit
time-stepping and (2) The need to store the Lagrange multipliers, which is not
feasible for the field-scale problems. We show that these wavefields are
efficiently determined from the associated data (projection of the wavefields
onto the receivers space) by using explicit time stepping. Accordingly, based
on the augmented Lagrangian, a new algorithm is proposed which performs in
"data space" (a lower dimensional subspace of the full space) in which the
wavefield reconstruction step is replaced by reconstruction of the associated
data, thus requiring optimization in a lower dimensional space (convenient for
handling the Lagrange multipliers). We show that this new algorithm can be
implemented efficiently in the time domain with existing solvers for the FWI
and at a cost comparable to that of the FWI while benefiting from the
robustness of the extended FWI formulation. The results obtained by numerical
examples show high-performance of the proposed method for large scale
time-domain FWI.
