Migration Method

The migration method is considered next. I choose prestack split-step migration based on the following two reasons: first, the fact that sources are subject to phase and/or frequency encoding demands that prestack migration is the method of choice. Second, aside from computational efficiency and the absence of operator aliasing, the fact that phase shift migration is a spectral technique makes it particularly convenient to perform frequency encoding. To handle smooth lateral variations in the velocity field, I opt for the split-step migration (Stoffa et al., 1990), as did Kuehl and Sacchi (1999). It is a straightforward procedure to adapt this to RTM, with the finite-difference method replacing the spectral method.

The use of LSM requires both the forward modeling and the migration operations. The use of prestack migration requires both a source field and downward continued data field. The details of this migration method are relegated to Appendix B, which is included because of the usefulness in assessing the computational complexities of the algorithms studied in this chapter.

To demonstrate the effectiveness of LSM, its performance will be compared to that of is. In contrast to the iterative refinement of LSM, IS of encoded migration images (Schuster et al., 2011) at the $k^{\textrm{th}}$ iteration produces a sum of $k$ realizations of migration images. For IS, dynamic encoding is used, so that at each iteration the input supergather, specifically the source wavefield at surface $P(x,z=0,\omega)$ in Figure B.1(a), is formed using a new frequency assignment.

It is of interest to analyze how the proposed method would fare compared to the standard migration in terms of saving computational cost. This analysis is provided in Appendix C. In addition, the results in Appendix C allow us to compare the convergence performances of LSM and IS on the basis of the same computational cost.