Claerbout (1992) proposed a method to anti-alias Kirchhoff space-time operators by local lowpass triangle filtering. Here, I extend Claerbout's anti-aliasing method to the Kirchhoff prestack time migration operator, which is given by:

(1) |

where *t*_{k} is the two-way Kirchhoff migration reflection traveltime,
is the one-way vertical
traveltime to the reflection point (pseudodepth) as shown in the geometry
of Figure , *v* is the rms migration
velocity which may vary over in general, and () is
the distance measured along the planar recording surface between the source
(receiver) and the vertical projection of the image point to the recording
surface:

(2) |

assuming the recording surface is horizontal at .

Figure 1

The spatial derivative of (1) is needed to determine the migration operator aliasing criterion:

(3) |

To remain unaliased, the temporal period *T* of any *local* waveform
on a seismic trace at time *t*_{k} must be greater than :

(4) |

I evaluate the effective rms spatial sampling interval for the prestack migration operator as

(5) |

where I define and in the prestack case as

(6) |

(7) |

Here, and are the true inline and crossline sampling intervals of the seismic trace data. For the poststack case, (3) reduces to

(8) |

where is the surface distance from the midpoint to the image point surface projection, and (5) reduces to

(9) |

where *dx*_{m} and *dy*_{m} are still validly defined by (6) and
(7).
Evidently,

(10) |

as expected. As an aside, the anti-aliasing
criterion (4) may also be a good approximation for the case of
Kirchhoff *depth* migration with the use of the rms migration
velocity equivalent to the depth migration interval
velocity model *v*_{int}(*x*,*y*,*z*).

The Kirchhoff migration operator anti-aliasing criterion (4) suggests
an anti-aliasing method by local lowpass filtering of the input trace
waveform in the vicinity of *t*_{k}. This local anti-alias lowpass filter
is a function of trace time *t*_{k} and migration operator dip given by
(3), in general. In particular, the anti-aliasing filter is not
necessarily a simple function of offset or aperture. For example,
a constant offset trace will be subject to relatively stronger lowpass
filtering at early traveltimes *t*_{s} and *t*_{r}, compared to later arrival
times (as expected by the greater relative curvature of a shallow versus deep
migration impulse response).

I implement the local lowpass filters as triangular smoothing.
Claerbout (1992) showed that for an arbitrary *N*-point triangle filter,
the smoothing can be implemented efficiently by using only 3 filter
coefficients instead of *N*, if the input trace data are first subjected to
a sequential process of causal followed by acausal temporal integration.
This observation offers a large savings in computational effort when
applying the triangle filters.
I relate an *N*-point triangle to the aliasing period *T*
by the relation , where is the true temporal
trace sample interval. My chosen anti-aliasing criterion (4)
in terms of triangle filter length *N* becomes:

(11) |

For poststack geometries, this result reduces to:

(12) |

11/17/1997