  Full waveform inversion (FWI) is a highly nonlinear and ill-posed problem. On
one hand, it can be easily trapped in a local minimum. On the other hand, the
inversion results may exhibit strong artifacts and reduced resolution because
of inadequate constraint from data. Proper regularizations are necessary to
reduce such artifacts and steer the inversion towards a good direction. In this
study, we propose a novel adaptive sparsity-promoting regularization for FWI in
the model-derivative domain which exploits nonlocal similarity in the model.
This regularization can be viewed as a generalization of total variation (TV)
with multi-class learning-based dictionaries. The dictionaries incorporate the
prior information of nonlocal similarity into the inversion, exploiting the
fact that geological patterns at different places are similar to some others up
to affine transformations (translation, rotation and scaling). Such nonlocal
similarity priors effectively reduce the degrees of freedom in model
parameters, and may also mitigate the problem of local minima. The formulated
optimization problem is solved by the Alternating Direction Method of
Multipliers (ADMM). By interpreting the iterative scheme, we find our method
closely connected with image processing techniques and convolutional neural
networks (CNN). We test our proposed method on a modified BP 2004 velocity
model and a smoothed Marmousi model. Compared with traditional FWI, our
technique can better reconstruct sharp edges such as salt body boundaries, and
is also able to effectively reduce artifacts. Compared with TV, our result is
less blocky and more geologically realistic. Quantitatively, our result has the
highest structural similarity index (SSIM) and also the lowest model mean
square error.
