Chapter 3: Plane-wave Least-squares Reverse Time Migration

The original implementation of least-squares migration was with Kirchhoff migration (Nemeth et al., 1999; Duquet et al., 2000), but was later developed for phase shift migration algorithms (Kaplan et al., 2010; Huang and Schuster, 2012). When least-squares migration is implemented with the reverse time migration method (Tang and Biondi, 2009; Dai and schuster, 2010; Dai et al., 2010; Wong et al., 2011; Dai et al., 2012), it can reduce not only the acquisition footprint but also the artifacts in the RTM image, while enhancing the image resolution. In addition, Romero et al. (2000); Krebs et al. (2009); Tang and Biondi (2009); Schuster et al. (2011); Dai et al. (2011, 2012) employed a phase-encoding multisource approach to increase the computational efficiency by more than an order-of-magnitude compared to conventional LSRTM.

One significant problem with random encoding LSRTM is that it requires all the encoded shot gathers to share the same receivers (fixed spread geometry). Therefore, it is not applicable to marine streamer data which are recorded by a towed receiver array (Routh et al., 2011; Huang and Schuster, 2012). To remedy this problem, I devise a plane-wave LSRTM method that can be applied to both land and marine datasets (An alternative remedy is to use frequency selection encoding, as proposed by Huang and Schuster (2012)).

Another drawback of multisource least-squares reverse time migration algorithm is that its convergence is sensitive to the accuracy of the velocity model. When the velocity model contains large bulk errors, the migration images from different shots are inconsistent with each other, so the stacking process become less effective in reducing crosstalk noise and the resolution of the final image is spoiled. In addition, when many shots are blended together, it is difficult to separate them to produce common image gathers as quality control tools.

This problem is now remedied by incorporating a regularization term into the LSRTM method that penalizes misfits between the images in the plane-wave domain. In this way the defocusing due to velocity errors is reduced. The formulation is similar to differential semblance optimization (Symes and Carazzone, 1991) which inverted for the velocity model, but in this chapter only the reflectivity image is produced. In contrast to a stacked image, the prestack image ensemble accommodates more unknowns to allow for better fitting of the observed data, and so the convergence of least-squares migration is improved (see Appendix A).

In summary, I present a plane-wave prestack least-squares migration method where the migration image of each shot is updated separately and an ensemble of prestack images is produced with common image gathers. The advantage over conventional LSRTM where all the shot gathers are explained by a single migration image is that it is relatively less sensitive to bulk errors in the migration velocity. The plane-wave encoding technique can significantly reduce the computational and input/output (I/O) cost. In contrast to conventional multisource least-squares migration with phase-encoded supergathers, it can be applied to marine data.