  Full waveform inversion (FWI) delivers high-resolution images of the
subsurface by minimizing iteratively the misfit between the recorded and
calculated seismic data. It has been attacked successfully with the
Gauss-Newton method and sparsity promoting regularization based on fixed
multiscale transforms that permit significant subsampling of the seismic data
when the model perturbation at each FWI data-fitting iteration can be
represented with sparse coefficients. Rather than using analytical transforms
with predefined dictionaries to achieve sparse representation, we introduce an
adaptive transform called the Sparse Orthonormal Transform (SOT) whose
dictionary is learned from many small training patches taken from the model
perturbations in previous iterations. The patch-based dictionary is constrained
to be orthonormal and trained with an online approach to provide the best
sparse representation of the complex features and variations of the entire
model perturbation. The complexity of the training method is proportional to
the cube of the number of samples in one small patch. By incorporating both
compressive subsampling and the adaptive SOT-based representation into the
Gauss-Newton least-squares problem for each FWI iteration, the model
perturbation can be recovered after an l1-norm sparsity constraint is applied
on the SOT coefficients. Numerical experiments on synthetic models demonstrate
that the SOT-based sparsity promoting regularization can provide robust FWI
results with reduced computation.
