Image anti-aliasing for 3-D prestack time migration

For 3-D prestack time migration, the reflectors dips $p^{{\rm \xi}}_x$ and $p^{{\rm \xi}}_y$, and the wavelet-stretch factor ${dt_{D}}/{d\tau_\xi}$,can be analytically derived as functions of the input and output trace geometry and the input time. Appendix A presents the derivation of the analytical relationships that I apply for the following examples.

 

Imp-noantialias Figure 5 Image obtained by applying Kirchhoff migration without anti-aliasing.

 

Imp-antialias-equal Figure 6 Image obtained by applying Kirchhoff migration with image-space anti-aliasing.

 

Imp-nostretch Figure 7 Image obtained by applying too strong of anti-aliasing by ignoring the effects of the wavelet stretch on the frequency content of the image.

To gain intuition about the effects of incorporating an anti-aliasing filter in migration operator, it is instructive to analyze the images generated by migrating one single input trace into a cube. Figure 5 shows the result of migrating one trace without the application of any anti-aliasing filter. The input trace was recorded at an offset of 2.4 km. The image sampling was 20 m in each direction $\left(\Delta x_\xi= \Delta y_\xi= 20 {\rm m}\right)$.Strong aliasing artifacts are visible in both the time slices and the vertical section.

Figure 6 shows the result of migrating the same data trace with an appropriate anti-aliased operator. Notice that the aliasing artifacts disappear as the frequency content of the imaged reflectors progressively decreases as the dips increase. Figure 7 shows the result of migrating the same data trace when the effects of the wavelet stretch are not taken into account; that is, by setting ${dt_{D}}/{d\tau_\xi}=1$.In this case the anti-aliasing filter over-compensates for the image dip and valuable resolution is lost at steep dips. Examining the time slice shown on the top of Figure 7, we notice that the loss of resolution is larger for regions of the migration ellipses with a steep dip along the cross-line direction.

Figure 8 shows the effects of image-space aliasing on the migrated results from the salt-dome data set shown in Figure 1. Figure 8a shows the non-antialiased migration results with $\Delta x_\xi=36~{\rm m}$;that is the original sampling of the zero-offset data. Shallow dipping reflectors and the steep high-frequency event at about 2.2 seconds are badly aliased. The quality of the image improves by halving the image spatial sampling to $\Delta x_\xi=18~{\rm m}$(Figure 8b). It is useful to notice that the traces in Figure 8a are exactly the same as the odd traces in Figure 8b. Therefore, image aliasing does not add noise to the image, it just makes the image more difficult and ambiguous to interpret. The application of the image-space anti-aliasing constraints, expressed in equation (2), further improves the image quality, in particular for the shallower events (Figure 8c). Although the image in Figure 8c is less noisy than the images in Figure 8a and Figure 8b, it is still contaminated by aliasing artifacts. These artifacts are caused by operator aliasing , or data-space aliasing. Next section analyzes the causes of operator aliasing, and presents anti-aliasing constraints to eliminate it.

 

Comp-noanti-mod
Figure 8 Migrated section from the Gulf of Mexico salt dome: (a) $\Delta x_\xi=36~{\rm m}$ and no anti-aliasing, (b) $\Delta x_\xi=18~{\rm m}$ and no anti-aliasing, (c) $\Delta x_\xi=18~{\rm m}$ and image-space anti-aliasing.