The simplest type of aliasing related to imaging
operator is
**image-space aliasing** .
It occurs when
the spatial sampling of the image is too coarse
to represent adequately the steeply dipping reflectors that
the imaging operator attempts to build during the imaging process.
In other words, image-space aliasing is caused by
too coarse sampling of the image space,
and consequent aliasing of the migration ellipsoid.
A simple way of avoiding image aliasing is to make the
spatial sampling of the image denser.
But for a given spatial sampling of the image,
to avoid image aliasing we need to limit,
during the migration process,
the frequency content of the image as a function of reflectors' dips.
This goal can be accomplished by performing a dip-dependent
temporal lowpass filtering of the input data
during the summation process.
An efficient method to lowpass time-domain data with
variable frequency is the triangle-filters method
described in
*Basic Earth Imaging* Claerbout (1995).
An alternative method is to precompute lowpassed versions
of the input traces,
and select the appropriate input data during summation
Gray (1992).
This second method is potentially more accurate
than the triangle filtering,
and it is more computationally efficient
when each input trace is summed into many output traces,
as happens for 3-D migrations
Abma (1998).
However, it may require storing many versions
of the input data.
To reduce the storage requirements,
without affecting the accuracy,
I linearly interpolate the lowpassed traces
along the frequency axis during the summation.

The anti-aliasing constraints to avoid image aliasing can be easily derived from basic sampling theory. For the case of time migration, when the coordinates of the image space are ,the pseudo-depth frequency must fulfill the following inequalities:

(1) |

As discussed above, it is convenient to apply
an anti-aliasing filter as a band-pass filter of the input traces,
before summing
their contributions to the image
into the output.
Therefore, we need to recast the constraints
on the pseudo-depth frequency
of equations (1) into constraints
on the input data frequency .This distinction is important because
the frequency content of the seismic wavelet changes during
the imaging process because of stretching, or compression,
of the time axis.
In particular, during migration the wavelet is always
stretched; neglecting this stretch would lead
to anti-aliasing constraints that are too stringent.
Notice, that the seismic wavelet may get compressed,
instead of stretched, by other imaging operators, such as
inverse DMO and AMO Biondi et al. (1998).
The pseudo-depth frequency of the image and the temporal frequency
of the data are thus linked by the **wavelet-stretch factor**
, as
.Taking into account the wavelet-stretch factor,
we can write the constraints on the data frequency
that control image aliasing, as a function of the
image sampling rates and ,the image dips and ,and the wavelet-stretch factor ,

(2) |

7/5/1998