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## 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.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 .Then is downward continued in steps of depth by multiplication with .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 Pj over kx to .At each x, he interpolated the reference waves of nearest velocity to get a final value, say, which he retransformed to 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 tc requires a broad bandwidth in the spectral domain. Given that Figure 4 is expressed on a 10001000 mesh, deduce the uncertainty in the cutoff tc.
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 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.

Next: IMPEDANCE Up: TUNING UP FOURIER MIGRATIONS Previous: Stolt stretch
Stanford Exploration Project
10/31/1997