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 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 .Strong aliasing artifacts are visible in both the time slices and the vertical section.
Figure 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 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 .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 , 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 shows the effects of image-space aliasing on the migrated results from the salt-dome data set shown in Figure . Figure a shows the non-antialiased migration results with ;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 (Figure b). It is useful to notice that the traces in Figure a are exactly the same as the odd traces in Figure b. 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 (), further improves the image quality, in particular for the shallower events (Figure c). Although the image in Figure c is less noisy than the images in Figure a and Figure b, 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.