We propose sparse least-squares reverse time migration (LSRTM) using seislets as a basis for the reflectivity distribution. This basis is used along with a dip-constrained preconditioner that emphasizes image updates only along prominent dips during the iterations. These dips can be estimated from the standard migration image or from the gradient using plane-wave destruction filters or structural tensors. Numerical tests on synthetic datasets demonstrate the benefits of this method for mitigation of aliasing artifacts and crosstalk noise in multisource least-squares migration.
Wavelet transforms provide a compact basis for data decomposition which in turn is useful for formulating efficient signal processing and depth imaging algorithms. Such transforms usually exploit the directional properties of an image through the use of suitable basis functions. They provide a perfect reconstruction of the parameters after forward and inverse transforms, are efficient to compute, and use minimal redundancy. Thus, different wavelet-like transforms such as the digital wavelet transform (DWT), curvelets, or projection onto convex sets (POCS) algorithms are often used in geophysical applications like data compression, interpolation, data regularization and denoising (Foster et al., 1994; Dessing, 1997; Wapenaar et al., 2005; Abma and Kabir, 2006; Candes et al., 2006a,b; Herrmann et al., 2009).
Fomel and Liu (2010) introduced the theory of the seislet transform that is more suitable for representing seismic data. They use basis functions that are aligned along dominant seismic events or dips. In 2D or 3D, the basis functions from the seislet transform follow locally linear events obtained from the input data using local plane-wave destruction filters (Claerbout, 1992; Fomel, 2002). Through numerical tests, they demonstrated the superior compression, interpolation and denoising properties of the seislet transform over the digital wavelet transform.
The above listed properties of the seislet transform makes it an appealing tool for use in seismic imaging problems such as least-squares migration (LSM) or full waveform inversion (FWI). LSM has been shown to produce images with better balanced amplitudes, fewer artifacts and better resolution than standard migration (Lailly, 1984; Nemeth et al., 1999; Duquet et al., 2000; Plessix and Mulder, 2004; Dai and Schuster, 2009; Tang, 2009; Wong et al., 2011). However, the computational cost of least-squares migration (LSM) makes the application of this algorithm prohibitive for large-scale industrial 3D problems. Morton and Ober (1998) and Romero et al. (2000) proposed blended source migration where they blended several shotgathers into one supergather which is then migrated. This approach, although very effective in reducing the computational cost, suffers from crosstalk noise which severely degrades the quality of the migrated image. Later, Dai and Schuster (2009) and Schuster et al. (2011) extended the blended source migration technique to multisource least-squares migration and showed that the crosstalk noise can be mitigated by an iterative migration of supergathers.