Introduction

The least-squares migration method (Duquet et al., 2000; Lailly, 1984; Schuster, 1993; Cole and Karrenbach, 1992; Nemeth et al., 1999) has been shown to sometimes produce migration images with better quality than those computed by conventional migration. Its original implementation was with Kirchhoff migration (Duquet et al., 2000; Nemeth et al., 1999), but was later developed for phase shift migration algorithms (Huang and Schuster, 2012a; Kaplan et al., 2010). When least-squares migration is implemented with the reverse time migration method (Dai et al., 2010; Wong et al., 2011; Dai et al., 2012; Tang and Biondi, 2009; Dai and Schuster, 2010a), 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); Dai et al. (2011); Schuster et al. (2011); Dai et al. (2012); Tang and Biondi (2009); Krebs et al. (2009) 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 provides 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.

However, the random encoding functions used by Romero et al. (2000); Schuster et al. (2011); Krebs et al. (2009) and Dai et al. (2012), cannot be applied to a seismic survey with a marine streamer geometry (Huang and Schuster, 2012a; Routh et al., 2011) because, although the calculated synthetic data are of fixed spread geometry, the observed data are recorded with a marine streamer geometry. As a remedy, Routh et al. (2011) proposed a cross-correlation based misfit functional to mitigate the effect of a recording pattern mismatch. Alternatively, Huang and Schuster (2012a) proposed a frequency-selection encoding strategy for least-squares phase shift migration, which is applicable to marine data.

The frequency-selection encoding strategy can also be used with least-squares reverse time migration, where the time-domain simulation are performed with a single frequency harmonic source instead of the conventional broadband source. Nihei and Li (2006) proposed to use a time-domain finite-difference method to obtain the single frequency seismic response of a velocity model. Compared to the conventional frequency domain method, their method has significantly lower arithmetic complexity and storage requirements in the 3D case.

In this chapter, the frequency-selection encoding method is applied with least-squares reverse time migration and tested on the Marmousi2 model to show that LSRTM can produce better images than conventional RTM with comparable cost for marine datasets.