The search space of full-waveform inversion (FWI) can be extended via a relaxation of the wave equation to increase the linear regime of the inversion. This wave-equation relaxation is implemented by solving jointly (in a least-squares sense) the wave equation weighted by a penalty parameter and the observation equation such that the reconstructed wavefields closely match the data, hence preventing cycle skipping at receivers. Then, the subsurface parameters are updated by minimizing the temporal and spatial source extension generated by the wave-equation relaxation to push back the data-assimilated wavefields toward the physics. This extended formulation of FWI has been efficiently implemented in the frequency domain with the augmented Lagrangian method in which the overdetermined systems of the data-assimilated wavefields can be solved separately for each frequency with linear-algebra methods and the sensitivity of the optimization to the penalty parameter is mitigated through the action of the Lagrange multipliers. Applying this method in the time domain is, however, hampered by two main issues: the computation of data-assimilated wavefields with explicit time-stepping schemes and the storage of the Lagrange multipliers capturing the history of the source residuals in the state space. These two issues are solved by recognizing that the source residuals on the right side of the extended wave equation, when formulated in a form suitable for explicit time stepping, are related to the extended data residuals through an adjoint equation. This relationship first allows us to relate the extended data residuals to the reduced data residuals through a normal equation in the data space. Once the extended data residuals have been estimated by solving (exactly or approximately) this normal equation, the data-assimilated wavefields are computed with explicit time-stepping schemes by cascading an adjoint and a forward simulation. Second, the relationship between the source and the data residuals allows us to project the Lagrange multipliers tied to the wave-equation constraint in the data space at the expense of additional adjoint simulations. This leads to a formulation of the extended FWI that is very similar to that of the classic reduced-space FWI. The computational cost of the extended FWI is two times that of the classic reduced-space FWI, and the storage requirement is the same. Applications to different synthetic benchmarks involving large contrasts confirm the resilience of the method to cycle skipping even when the quadratic problem for the extended data residuals is solved approximately with one inner iteration of a steepest-descent method.

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