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 skeletonizing the migration operator $\mathcal G({\bf {r}},{\bf {s}},{\bf {x}},t)=g({\bf {s}},t\vert{\bf {x}},0)*g({\bf {r}},t\vert{\bf {x}},0)$ into a skeletonized one $\hat{\mathcal G}({\bf {r}},{\bf {s}},{\bf {x}},t)=\hat{g}({\bf {s}},t\vert{\bf {x}},0)*\hat{g}({\bf {r}},t\vert{\bf {x}},0)$ with only a few nonzero samples; each sample is at the arrival time of an important early-arrival event (e.g., a primary reflection or a multiple arrival reflection). Thus the onerous storage costs (Zhou and Schuster, 2002; Cao, 2007) of a migration kernel trace with 1001 samples is reduced to a sparse trace with just 10 or so samples. The sparsity of this migration kernel can also reduce migration artifacts by eliminating unnecessary events for high-quality migration images.

Once the skeletonized 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) or waveform inversion. If it is combined with phase-encoded multisource technology (Dai and Schuster, 2009; Krebs et al., 2009), the cost savings can be even greater.

This chapter is divided into three parts: theory, numerical results, and conclusions. I briefly introduce the theory first, followed by the synthetic tests on the 2D SEG/EAGE salt model that demonstrate the effectiveness of this method. At the end, I draw some conclusions.