Discussion

For both acoustic RTM and $Q$-RTM in the Marmousi example, with a time step size of $2\;ms$, we used the low-rank approximation with a rank of $4$, corresponding to $5$ complex-to-complex FFTs per time step (with one additional forward FFT). Therefore, both methods had the same computational cost. A pseudo-spectral method would require $4$ real-to-complex FFTs per time step to calculate the two fractional Laplacians in the second-order wave equation (equation 6). On the other hand, the SLS model with $L$ relaxation mechanisms would require to solve $3+L$ equations in the $2D$ case or $4+L$ equations in the $3D$ case (Zhu et al., 2013), and has an effective cost of $4$ real-to-complex FFTs per time step when implemented using a pseudo-spectral method. However, a pseudo-spectral implementation poses a strict limit on time step size due to its finite-difference approximation of time derivatives, and thus may require a larger number of time steps to propagate the same length of time compared with the proposed method (Sun and Fomel, 2013).