Introduction

Wave equation migration methods can be very expensive compared to diffraction-stack migration methods. The conventional RTM approach requires a numerical solution to the wave equation for every source position. Hence, much research effort has been spent in reducing the costs of RTM.

I propose to reduce the costs of both standard RTM and least-squares RTM by calculating the blended migration operator $\tilde{\Gamma}({\bf {r}},{\bf {x}}) = \sum_s N({\bf {s}}) G({\bf {x}}\vert{\bf {r}})^* G({\bf {x}}\vert{\bf {s}})^*$ , and then save it on disk. When the blended migration kernel is ready, the migration image can be obtained by a dot product of $\mathcal{F}^{-1}(\tilde{\Gamma})$ and the recorded shot gathers in the $x-t$ domain; here, $\mathcal{F}^{-1}$ denotes the inverse Fourier transform. Because the migration kernels for the whole model space is too large to be saved, I phase-encode the migration kernels as well as phase-encode the recoded shot gathers before doing the dot product. This reduces the memory cost by at least an order-of-magnitude.

Once the blended migration kernel is saved, it does not need to be recomputed at each LSM iteration so this can result in almost two orders-of-magnitude reduction in cost for iterative least-squares migration (Nemeth et al., 1999; Aoki and Schuster, 2009). A further decrease in cost can be achieved by phase-encoding and multisource migration.

In this chapter, I briefly introduce the least-squares phase-encoded GDM theory first, followed by numerical tests on the 2D SEG/EAGE salt model that demonstrate the effectiveness of this method. Conclusions are drawn at the end.