Figure 4 shows a family of hyperbolas.

hyptrunc
Hyperbolas truncated at a particular time.
Figure 4 |

Notice that these hyperbolas do not extend to infinite time
but they truncate at a cut-off time *t*_{c}.
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 *t*_{c},
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 *t _{0}* to be truncated at some time

# Modeling with time truncation att_{c}ModelFT[model(x,z)] For all and allk_{x}. For ..., 0 { For all { For all { if () { sine if( sine ) aperture. else aperture. } else aperture. apertureModel(k_{x},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.

10/31/1997