  Full waveform inversion (FWI) is an iterative nonlinear waveform matching
procedure subject to wave-equation constraint. FWI is highly nonlinear when the
wave-equation constraint is enforced at each iteration. To mitigate
nonlinearity, wavefield-reconstruction inversion (WRI) expands the search space
by relaxing the wave-equation constraint with a penalty method. The pitfall of
this approach resides in the tuning of the penalty parameter because increasing
values should be used to foster data fitting during early iterations while
progressively enforcing the wave-equation constraint during late iterations.
However, large values of penalty parameter lead to ill-conditioned problems.
Here, this tuning issue is solved by replacing the penalty method by an
augmented Lagrangian method equipped with operator splitting (IR-WRI as
iteratively-refined WRI). It is shown that IR-WRI is similar to a penalty
method in which data and sources are updated at each iteration by the running
sum of the data and source residuals of previous iterations. Moreover, the
alternating direction strategy exploits the bilinearity of the wave equation
constraint to linearize the subsurface model estimation around the
reconstructed wavefield. Accordingly, the original nonlinear FWI is decomposed
into a sequence of two linear subproblems, the optimization variable of one
subproblem being passed as a passive variable for the next subproblem. The
convergence of WRI and IR-WRI are first compared with a simple transmission
experiment, which lies in the linear regime of FWI. Under the same conditions,
IR-WRI converges to a more accurate minimizer with a smaller number of
iterations than WRI. More realistic case studies performed with the Marmousi II
and the BP salt models show the resilience of IR-WRI to cycle skipping and
noise.
