Considering the case of integral DMO, Hale
1991 points out that the steep parts of the
operator, while aliased in the space (midpoint) coordinate, are not
aliased with respect to the time coordinate. He suggests replacing the
conventional *t*(*x*) parameterization of the DMO impulse response by
*x*(*t*) parameterization. Conventionally, the integral operators are
implemented by shifting the input traces in space and transforming
them in time. According to Hale's method, the traces are shifted in
time and transformed along the *x*(*t*) trajectories in
space. Interpolation in time, required in the conventional approach,
is replaced by interpolation in space. The idea of Hale's method is
related to the idea of the ``pixel-precise velocity transform'',
introduced by Claerbout
.

The steep parts of the operator satisfy the criterion

(10) |

To eliminate spatial aliasing, simply never allow successive time shifts applied to the input trace to differ by more than one time sampling interval. Further restrict the difference between time shifts so that the spacing between the corresponding output trajectories never exceeds the CMP sampling interval

We illustrate the idea of Hale's method in Figure 6. Increasing the density of spatial sampling by small successive time shifts implies increasing the Nyquist boundaries of the spatial spectrum (wavenumber). Further interpolation is a low-pass spatial filter removing the parts of the spectrum beyond the Nyquist frequency of the output. If the dip of the operator does not vary between neighboring traces (the operator is a straight line as in the slant stack case), Hale's approach produce essentially the same result as low-pass filtering. Triangle filters in this case approximately correspond to linear interpolation in space between adjacent traces Nichols (1993). The difference between the two approaches occurs if the local dip varies in space (the case of a curved operator, such as DMO). In this case, Hale's approach provides a more accurate space interpolation of the operator and preserves the high-frequency part of its spectrum from distortion.

Figure 6

Hale's method has proven to preserve the amplitude of flat reflectors from aliasing distortions, which is the simplest antialiasing test on a DMO operator. We see the most valuable advantage of this method in the fact that the implied low-pass spatial filtering (interpolation) does not depend on the operator dip and is controlled by the Nyquist boundary of the spectrum only (compare Figures 4 and 6). This is especially important, when the local dip of the operator changes rapidly and therefore cannot be estimated precisely by finite-difference approximation at spatially separated traces. Such a situation is common in DMO and AMO integral operators, as well as in prestack Kirchhoff migration.

A weakness of the method is the necessity to switch from interpolation in space to two-dimensional interpolation in both the time and the space variables, when trying to construct the flat part of the operator. In the case of AMO, the 2-D spatial interpolation arises as a result of building the operator in the transformed coordinate system. However, we would prefer to avoid the expense of the additional time interpolation required by Hale's method of antialiasing.

11/12/1997