Five-planes synthetic data set migration results

We can gain a intuitive understanding of the effects of applying the coplanarity condition, and of the trade-off when setting the parameters for narrow-azimuth migration, by analyzing the results of two full source-receiver migrations of the synthetic data set with five dipping planes, which we introduced in the previous section. I used 8 cross-line offsets for both migrations; for the first migration the cross-line sampling was 100 meters whereas for the second one the cross-line sampling was 50 meters. There is a trade-off between the coarser and finer cross-line offset sampling. With the finer sampling we expect stronger artifacts caused by the circular boundary condition because the offset range is narrower (only 400 meters vs. 800 meters). On the other hand, with the finer offset sampling the cross-line dip range is wider than with the coarse offset sampling and thus we expect better imaging of the events reflected with wide-aperture angles from the steeply dipping planes.

These ``theoretical'' predictions are confirmed by the zero offset images (panels a) and the ADCIGs (panels b) displayed in Figure  ($\Delta y_h=100$ meters) and Figure  ($\Delta y_h=50$ 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.

 

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Figure 5 (a) Stack (i.e. image at zero offset) and (b) CIG produced by full prestack migration with N_yh=8 and $\Delta y_h=100$ meters.

 

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Figure 6 (a) Stack (i.e. image at zero offset) and (b) CIG produced by full prestack migration with N_yh=8 and $\Delta y_h=50$ meters.

Figure  shows the same section and ADCIG as in Figure  but after applying the coplanarity condition for zero azimuth (i.e. $\beta$=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.

 

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Figure 7 (a) Stack (i.e. image at zero offset) and (b) CIG for reflections with 0 degrees azimuth at the image point. This image cube was produced starting from the image cube obtained by full prestack migration with N_yh=8 and $\Delta y_h=100$ meters.

 

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Figure 8 (a) Stack (i.e. image at zero offset) and (b) CIG produced by common-azimuth migration.

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 ($\Delta y_h=100$ 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 ($\Delta y_h=50$ 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.

 

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Figure 9 (a) Stack (i.e. image at zero offset) and (b) CIG for reflections with 18 degrees azimuth at the image point. This image cube was produced starting from the image cube obtained by full prestack migration with N_yh=8 and $\Delta y_h=100$ meters.

 

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Figure 10 (a) Stack (i.e. image at zero offset) and (b) CIG for reflections with 18 degrees azimuth at the image point. This image cube was produced starting from the image cube obtained by full prestack migration with N_yh=8 and $\Delta y_h=50$ meters.

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.

 

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Figure 11 (a) Stack (i.e. image at zero offset) and (b) CIG for reflections with azimuth within the 0-30 degrees range. This image cube was produced starting from the image cube obtained by full prestack migration with N_yh=8 and $\Delta y_h=100$ meters.

 

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Figure 12 (a) Stack (i.e. image at zero offset) and (b) CIG for reflections with azimuth within the 0-30 degrees range. This image cube was produced starting from the image cube obtained by full prestack migration with N_yh=8 and $\Delta y_h=50$ meters.