Introduction

The least-squares migration method (Lailly, 1984; Cole and Karrenbach, 1992; Schuster, 1993; Nemeth et al., 1999; Duquet et al., 2000) has been shown to sometimes produce migration images with better quality than those computed by conventional migration. Its original implementation 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.

For iterative phase-encoded multisource migration, many shot gathers are encoded with random encoding functions and blended together to form a supergather. One supergather can be modeled and migrated with one finite-difference solution to the wave equation for multiple sources and so provide a high computational efficiency compared to standard LSM. With increasing iteration number, the crosstalk between different shots will be increasingly suppressed. Consequently, the computational cost of LSRTM is reduced to a level comparable to conventional reverse time migration or even lower, depending on the acquisition geometry.

There are two significant problems with LSRTM. The problems and my proposed solutions are now presented.

1. The standard multisource LSRTM combined with the random encoding method 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 . The encoded source represents a physically realizable planar or line source on the surface, given that the sampling of the shot location is dense, regular, and continuous (Liu et al., 2006). Hence, the blending process with linear phase encoding is identical to a tau-p transformation that is used to transform shot-domain data to plane waves for plane-wave migration (Zhang et al., 2005). Liu et al. (2006) described the relationship between linear time-shift encoding and a plane-wave transformation. They also reported the existence of crosstalk when the sampling of shots is too coarse, and proposed to stack over many different encoding functions (different surface shooting angles) to reduce the crosstalk, which is similar to the crosstalk reduction procedure for random phase encoding. In this way, Vigh and Starr (2008) implemented full waveform inversion in the plane-wave domain and achieved significant computational savings.

2. 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 C).

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.

This chapter is organized into four sections. The first one is this introduction, which is followed by the theory of LSRTM. The synthetic and field data examples are then presented in the numerical results section, and this is followed by the summary.