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Extended full-waveform inversion (FWI) has shown promising results foraccurate estimation of subsurface parameters when the initial models are notsufficiently accurate. Frequency-domain applications have shown that theaugmented Lagrangian (AL) method solves the inverse problem accurately with aminimal effect of the penalty parameter choice. Applying this method in thetime domain, however, is limited by two main factors: (1) The challenge ofdata-assimilated wavefield reconstruction due to the lack of an explicittime-stepping and (2) The need to store the Lagrange multipliers, which is notfeasible for the field-scale problems. We show that these wavefields areefficiently determined from the associated data (projection of the wavefieldsonto the receivers space) by using explicit time stepping. Accordingly, basedon the augmented Lagrangian, a new algorithm is proposed which performs in"data space" (a lower dimensional subspace of the full space) in which thewavefield reconstruction step is replaced by reconstruction of the associateddata, thus requiring optimization in a lower dimensional space (convenient forhandling the Lagrange multipliers). We show that this new algorithm can beimplemented efficiently in the time domain with existing solvers for the FWIand at a cost comparable to that of the FWI while benefiting from therobustness of the extended FWI formulation. The results obtained by numericalexamples show high-performance of the proposed method for large scaletime-domain FWI.

Extended full waveform inversion in the time domain by the augmented Lagrangian method