These ``theoretical'' predictions are confirmed by the zero offset images (panels a) and the ADCIGs (panels b) displayed in Figure ( meters) and Figure ( meters). The ADCIG shown in Figure b shows stronger artifacts than the ADCIG shown in Figure b. Even the ```stacked'' image (i.e. zero-offset image) shown in Figure a has strong artifacts, at least in the shallow part of the section. On the other hand, the finer offset sampling allows a slight better imaging of the wide-aperture reflection from the 60-degree plane, as the comparison of the deepest event in Figure b and Figure b demonstrates.
Figure shows the same section and ADCIG as in Figure but after applying the coplanarity condition for zero azimuth (i.e. =0). As expected, the events from the flattish reflectors are preserved since their azimuth at the reflection point is close to zero. In contrast, the reflections from the steeper reflectors are attenuated because their azimuth at the reflection point is larger than zero.
An interesting side-benefit of the capability of selecting reflections with a given azimuthal direction from full prestack migration, is the possibility to demonstrate the differences between the zero azimuth image shown in Figure , and the result obtained by common-azimuth migration shown in Figure . In constant velocity, the image produced by common-azimuth migration is equivalent to the zero-azimuth image. However, in variable velocity the two images are substantially different. The zero azimuth image (Figure ) contains only the events that were close to zero azimuth at the reflection point. In contrast, the common-azimuth migration image (Figure ) contains all the events. Common-azimuth downward continuation propagates all the events assuming that they are coplanar along the zero azimuth. In variable velocity this assumption is incorrect for some of the events, which are therefore slightly mispositioned in the image.
As the common-azimuth migration image illustrates, the challenge of this data set is to image properly the wide-aperture reflections from the 60-degree plane. Simple ray-tracing modeling indicates that those reflections occur along an azimuth oriented approximately at 18 degrees with respect to the acquisition geometry. Figure shows the image obtained by selecting the reflections with 18-degrees azimuth from the results of full prestack migration with the coarser offset sampling ( meters). Figure shows the image obtained by selecting the reflections with 18-degrees azimuth from the results of full prestack migration with the finer offset sampling ( meters). Both images show significantly weaker artifacts than the corresponding images with full azimuth (Figure and Figure ). Figure has weaker artifacts than Figure . The reflections for the 60-degree plane is flat as a function of the reflection angle for both ADCIGs (panels b), but Figure b has broader angular bandwidth (up to 40 degrees for as compared with up to 35 degrees) than Figure b.
The best-quality image can be obtained by stacking the images corresponding to a range of azimuths. This range can be fairly narrow because of the narrow-azimuth nature of streamer data. In this example, I stacked the image corresponding to azimuths within the 0-30 degrees range. Figure and Figure are the result of this averaging process. Notice the further attenuation of the artifacts as compared with both the full-azimuth images (Figure and Figure ) and the 18-degrees azimuth images (Figure and Figure ). As before, there is a trade-off between the better signal-to-noise in Figure , and the wider angular bandwidth in Figure .
The last two figures show that the stacking over azimuth decreases the amplitude of the reflections with wide reflection angles relatively to the narrow reflection angles. The intuitive explanation of this phenomenon is that the narrow reflection angles are enhanced by the stacking over azimuth because they are more stationary as a function of azimuth than the wide reflection angles. I believe that this effect can be compensated by applying an appropriate jacobian during the integration over azimuth, but I have not derived such a factor yet.