Impulse responses

Figure 1 shows the impulse responses of the optimized implicit finite-difference method. The medium is homogeneous, in which the vertical velocity of the medium is 2000 m/s, the anisotropy parameter $\varepsilon$ is 0.4, the anisotropy parameter $\delta$ is 0.2 and the tilting angle is $30^\circ$. The travel time of the impulse responses are 0.4s, 0.6s and 0.8s, respectively. The impulse location is at x=4000m. For comparison, I also present the impulse responses of phase-shift (Figure 2) and the impulse responses of plane-wave migration in tilted coordinates (Figure 3). Notice in Figure 2, although we use the phase-shift method, we can not achieve $90^\circ$in the right section. The reason is that the waves on the right side overturn when the propagation direction is close to the horizontal and the phase-shift method is still a one-way equation based method. Figure 3 shows the impulse responses generated by plane-wave migration in tilted coordinates Shan and Biondi (2004), which extrapolate the wavefield accurately even when the waves overturn. Comparing the left section of the impulse responses of Figure 1-3, I find that the impulse responses of optimized implicit finite-difference is very close to the other two. In the right section in Figure 1, optimized implicit finite-difference loses accuracy when the waves propagate almost horizontally or they overturn, since it is still a one-way based method. The impulse responses in Figure 1 have heart-shaped noise, which are typical evanescent energy in implicit finite-difference.