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## Stopping phase-shift migration wraparound (S. Levin)

Figure 4 shows a family of hyperbolas.

 hyptrunc Figure 4 Hyperbolas truncated at a particular time.

Notice that these hyperbolas do not extend to infinite time but they truncate at a cut-off time tc. A Fourier method will be described to create such time-truncated data. The method leads to a phase-shift migration program without wraparound artifacts.

When the fast Fourier transformation algorithm first came into use people noticed that it could be used for filtering. Transient filtering could be done exactly in the periodic Fourier domain if signals and filters were surrounded by enough zero padding. The same concept applies with migration. If field data and migration hyperbolas are surrounded by enough zeroes in the time- and space-domain then migration can be done in the Fourier domain with no wraparound. The trick is to see how the truncated hyperbolas in Figure 4 can be constructed in the Fourier domain.

To have truncations at time tc, special point sources must be used. The deeper the source, the narrower must be its angular aperture. Take a hyperbola with first arrival at time t0 to be truncated at some time tc. The propagation angle of energy at the cutoff is given by .So exploding reflectors have their kx-spectrum truncated at .A 90 aperture implies echoes with an infinite time delay. Here is a sketch of the program.


# Modeling with time truncation at tc
ModelFT[model(x,z)]
For all  and all kx
.
For  ..., 0 {
For all  {
For all  {
if () {
sine
if( sine  )
aperture.		 		 		 		 else
aperture.		 		 		 		 }
else
aperture.		 		 		apertureModel(kx,z)
}
}
}
FT2D

The above modeling program may be converted to a migration program by running the depth z loop down instead of up and by multiplying the downward continued data by the aperture function. The modifications to the program not only improve the quality of the migration, but the calculation is faster.

Next: Controlling Stolt migration wraparound Up: TUNING UP FOURIER MIGRATIONS Previous: Dips greater than 90
Stanford Exploration Project
10/31/1997