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:
where tk 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:
assuming the recording surface is horizontal at .
The spatial derivative of (1) is needed to determine the migration operator aliasing criterion:
To remain unaliased, the temporal period T of any local waveform on a seismic trace at time tk must be greater than :
I evaluate the effective rms spatial sampling interval for the prestack migration operator as
where I define and in the prestack case as
Here, and are the true inline and crossline sampling intervals of the seismic trace data. For the poststack case, (3) reduces to
where is the surface distance from the midpoint to the image point surface projection, and (5) reduces to
where dxm and dym are still validly defined by (6) and (7). Evidently,
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 vint(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 tk. This local anti-alias lowpass filter is a function of trace time tk 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 ts and tr, 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:
For poststack geometries, this result reduces to: