Least-squares reverse time migration with sparse regularization in the 2D wavelet domain

The inadequate sampling of seismic data in the spatial dimension results in migration artifacts. Conventional least-squares reverse time migration (LSRTM) could improve the image quality. However, even LSRTM will not work in some inadequately sampling situations. To mitigate the impact of migration artifacts, we have developed a new LSRTM method with a sparse regularization, which takes advantage of the effective sparse representation of the subsurface reflectivity model in the 2D undecimated wavelet transform (UWT) domain. Different from other sparse regularizations, a sparseness constraint in the 2D UWT domain is applied on the angle slices of the image. To efficiently solve the least-squares inversion problem, we employ an inversion scheme using the conjugate gradient method that uses a soft threshold method to achieve sparse constraint in updating the conjugate gradient direction. Compared with the sparse constraint based on the discrete wavelet transform (DWT), the threshold in this method is angle-dependent and is determined according to the energy distribution of each angle slice. To avoid overregularization that can lead to instability and increase the number of iterations, we also apply an exponential threshold strategy. Numerical tests on synthetic datasets demonstrate that our method is capable of improving the image quality by enhancing the resolution and suppressing migration artifacts caused by inadequately sampled seismic data. The method can converge more rapidly than conventional LSRTM. Because this method performs sparse regularization on several slopes, it achieves better performance on enhancing complex structures with discontinuities such as the steeply dipping faults compared to DWT-based regularization.