Conventional migration (Claerbout, 1971) computes the reflectivity image by applying the adjoint operator just once to the data (Plessix, 2006) rather than its iterative application as required by waveform inversion (Tarantola, 1984). Migration can also be thought as the first iteration of iterative full wave inversion, but the Hessian of the misfit functional is not computed, because it is too large to store and invert. Therefore, a variety of approximations have been made to improve the migration images by approximating the inverse Hessian matrix with a fully populated inverse matrix (Yu et al., 2006; Wang and Yang, 2010; Aoki and Schuster, 2009; Guitton, 2004; Plessix and Mulder, 2004; Hu et al., 2001). The simplest approximation is the reciprocal of the diagonal matrix (Beydoun and Mendes, 1989; Rickett, 2003) that is applied to the image to compensate for uneven illumination. This is computationally inexpensive, but it only compensates for amplitude distortation but does not correct for aliasing artifacts or strong footprint noise.
To indirectly account for the inverse Hessian matrix, an iterative conjugate gradient method can be used to solve either the linear (Duquet et al., 2000; Lailly, 1984; Nemeth et al., 1999) or non-linear (Mora, 1987; Tarantola, 1984) optimization problem. In this way, the effects of the source wavelet, limited recording aperture, geometric spreading, etc, are taken into account to produce images with reduced acquisition footprints, balanced amplitudes and improved resolution. Following the work of Lailly (1984) and Beydoun and Mendes (1989), Nemeth et al. (1999) proposed an iterative linear inversion method they denoted as least-squares migration (LSM). They used an operator which is the adjoint to the Kirchhoff migration operator as the forward modeling operator and tested their algorithm with both synthetic data and field data. Duquet et al. (2000) tested a Kirchhoff LSM method but they also used a regularization term which penalized the difference in images from those estimated in different offset gathers. Their regularization was effective even with migration velocity errors up to 20%.
The inverse problem can also be formulated in the model space instead of the data space to avoid large I/O demands associated with 3D seismic data. To reduce computational cost, Hu et al. (2001) and Yu et al. (2006) computed the inverse Hessian in the wavenumber domain by assuming a locally layered medium to deconvolve the migration Green's function. Mulder and Plessix (2004a) derived both linear and quasi-linear approaches for LSM with the two-way wave equation, but their implementation was in the frequency domain. Another means for reducing the cost of computing the Hessian inverse was proposed by Tang (2009) who used phase-encoding technique to efficiently calculate the Hessian for a targeted area and performed the inversion in model space. The images he obtained showed high resolution and balanced amplitudes.
In this chapter, I propose an efficient multisource least-squares reverse time migration algorithm for a VTI medium and test it on synthetic traces generated for the 2D HESS VTI and 3D SEG/EAGE salt models. The acquisition geometries are for land data, and the application to marine data geometries requires a different phase encoding scheme (Huang and Schuster, 2012b). In order to increase computational efficiency, the multisource phase-encoding technique (Romero et al., 2000; Dai and Schuster, 2009,2010b; Krebs et al., 2009) employs encoding functions with random time shifts and random source polarities. Our synthetic results demonstrate that LSRTM can mitigate both migration artifacts and crosstalk noise introduced by phase encoding, balance the amplitudes of reflectors, and improve the spatial resolution of the image. Moreover, the efficiency of multisource LSRTM can be significantly higher than conventional RTM depending on the number of shots encoded in one supergather, the number of migration operations at every iteration, and the number of iterations needed for an image of acceptable quality. However, these results are sensitive to large errors in the migration velocity model.
This chapter is organized into four parts. The first is this introduction, which is followed by theory and the description of the numerical scheme for implementing LSRTM. The third part presents the numerical results for both the 2D HESS and the 3D SEG/EAGE salt models. In the end, a short summary is provided.