  We propose an extended full-waveform inversion formulation that includes
general convex constraints on the model. Though the full problem is highly
nonconvex, the overarching optimization scheme arrives at geologically
plausible results by solving a sequence of relaxed and warm-started constrained
convex subproblems. The combination of box, total-variation, and successively
relaxed asymmetric total-variation constraints allows us to steer free from
parasitic local minima while keeping the estimated physical parameters
laterally continuous and in a physically realistic range. For accurate starting
models, numerical experiments carried out on the challenging 2004 BP velocity
benchmark demonstrate that bound and total-variation constraints improve the
inversion result significantly by removing inversion artifacts, related to
source encoding, and by clearly improved delineation of top, bottom, and flanks
of a high-velocity high-contrast salt inclusion. The experiments also show that
for poor starting models these two constraints by themselves are insufficient
to detect the bottom of high-velocity inclusions such as salt. Inclusion of the
one-sided asymmetric total-variation constraint overcomes this issue by
discouraging velocity lows to buildup during the early stages of the inversion.
To the author's knowledge the presented algorithm is the first to successfully
remove the imprint of local minima caused by poor starting models and
band-width limited finite aperture data.
