  Seismic data quality is vital to geophysical applications, so methods of data
recovery, including denoising and interpolation, are common initial steps in
the seismic data processing flow. We present a method to perform simultaneous
interpolation and denoising, which is based on double-sparsity dictionary
learning. This extends previous work that was for denoising only. The original
double sparsity dictionary learning algorithm is modified to track the traces
with missing data by defining a masking operator that is integrated into the
sparse representation of the dictionary. A weighted low-rank approximation
algorithm is adopted to handle the dictionary updating as a sparse recovery
optimization problem constrained by the masking operator. Compared to
traditional sparse transforms with fixed dictionaries that lack the ability to
adapt to complex data structures, the double-sparsity dictionary learning
method learns the signal adaptively from selected patches of the corrupted
seismic data while preserving compact forward and inverse transform operators.
Numerical experiments on synthetic seismic data indicate that this new method
preserves more subtle features in the dataset without introducing pseudo-Gibbs
artifacts when compared to other directional multiscale transform methods such
as curvelets.
