Gazdag's v(x) method

The phase-shift method of migration is attractive because it allows for arbitrary depth variation in velocity and arbitrary angles of propagation up to 90$^\circ$.Unfortunately, lateral variation in velocity is not permitted because of the Fourier transformation over the x-axis. To alleviate this difficulty, Gazdag and Sguazzero [1984] proposed an interpolation method. Recall from chapter that the phase-shift method 2-D Fourier transforms the data p(x,t) to $P ( k_x , \omega )$.Then $P ( k_x , \omega )$ is downward continued in steps of depth by multiplication with $\exp [ i k_z ( \omega , k_x )\,\Delta z ]$.Gazdag proposed several reference velocities, say, v₁, v₂, v₃, and v₄. He downward continued one depth step with each of the velocities, obtaining several reference copies of the downward-continued data, say, P₁, P₂, P₃, and P₄. Then he inverse Fourier transformed each of the P_j over k_x to $p_j ( x , \omega )$.At each x, he interpolated the reference waves of nearest velocity to get a final value, say, $p(x, \omega )$ which he retransformed to $P ( k_x , \omega )$ ready for another step. This appears to be an inefficient method since it duplicates the usual migration computation for each velocity. Surprisingly, the method seems to be successful, perhaps because of the peculiar nature of computation using an array processor.

EXERCISES:

1. To obtain a sharp cutoff in time t_c requires a broad bandwidth in the spectral domain. Given that Figure 4 is expressed on a 1000$\,\times\,$1000 mesh, deduce the uncertainty in the cutoff t_c.
2. The phase-shift method tends to produce a migration that is periodic with z because of the periodicity of the Fourier transform over t. Ordinarily, this is not troublesome because we do not look at large z. The upcoming wave at great depth should be zero $\ \it\hbox{before}$ t=0. Kjartansson pointed out that periodicity in z could be avoided if the wave at t=0 is subtracted from the wavefield before the computation descends further. Thus, information could never get to negative time and ``wrap around.'' Indicate how the program should be changed.