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In 3D angle gathers, the multiples and the primaries behave differently not only in terms of aperture angles but also in terms of reflection azimuth. For computational convenience, I first windowed the SODCIGs in depth and computed the ADCIGs for the water-bottom primary only. Panel (a) of Figure 18 shows the result for the ADCIG at CMP-X=8500 m and CMP-Y=837.5 m. It is flat for a given azimuth although the range of aperture angles as a function of the azimuth is limited. For zero aperture angle, by definition there is no azimuth resolution (the incident and reflected rays coincide) and therefore all azimuths are seen as equally likely to contribute to the image. Similarly, I windowed the water-bottom multiple and computed the ADCIG as shown in panel (b) of Figure 18. The depth slice shows different azimuth dependency compared with the primary. While the range of azimuths decrease with increasing aperture angle for the primary, it does not for the multiples (compare the two depth slices). As a function of aperture angle for a given azimuth, the multiple shows the expected over-migrated residual moveout. At zero aperture angle there is no azimuth resolution.

Figure 18.
3D ADCIG at CMP-X=8500 m and CMP-Y=837.5 m for the water-bottom primary (a) and the water-bottom multiple (b). The depth slice is taken at the depth of the primary (1220 m) in panel (a) and at the depth of the multiple (3840 m) in panel (b). The primary was correctly migrated and therefore its moveout is flat. The multiple was over-migrated and so exhibits residual moveout as a function of the aperture angle.

It is interesting to analyze the angle gathers as a function of azimuth for different aperture angles. The water-bottom primary and the water-bottom multiple behave very differently as shown in Figures 19 and 20. For the primary, as the aperture angle increases, the angle gather as a function of azimuth becomes narrower as a consequence of the increase in azimuth resolution. For large enough aperture angles, the incident and reflected rays are no longer any where near co-linear and so nicely define a reflection plane whose azimuth is well defined (see panels (d) an (e)), except for an artifact due to insufficient padding when computing the ADCIG. For the multiple, as the aperture angle increases, the gather as a function of azimuth curves up because it is over-migrated. There is no similar increase in azimuth resolution because for the multiple there is not a clear notion of a reflection plane. Recall that, because of the crossline dip, the ray from the source to the reflector to the surface bounce point is contained in a plane normal to the reflector, whereas the ray from the bounce point to the reflector to the receiver is contained in a plane normal to the surface, that is, a vertical plane. The full multiple trajectory, from the source to the receiver, is therefore not contained on a single plane.

az-gath1
Figure 19.
3D ADCIG for the primary water-bottom reflection as a function of azimuth. The different panels correspond to different aperture angles: (a) 0, (b) 5, (c) 10, (d) 15 and (e) 20 degrees. As the aperture angle increases, the azimuth resolution increases as well.

az-gath2
Figure 20.
3D ADCIG for the water-bottom multiple reflection as a function of azimuth. The different panels correspond to different aperture angles: (a) 0, (b) 5, (c) 10, (d) 15 and (e) 20 degrees. As with the primaries, there is no azimuth resolution at zero aperture angle. In contrast to the primary, however, the azimuth resolution does not increase with increasing aperture angle.

Another interesting piece of information that 3D angle gathers can give us is the range of aperture angles that are illuminated at a particular reflection azimuth angle. Figure 21 shows the angle gathers of the water-bottom primary as a function of aperture angle for five reflection azimuth angles: (a)-40, (b)-20, (c)0, (d)20 and (e)40 degrees. The moveout of the primary is flat for those aperture angles that are actually illuminated at the corresponding reflection azimuth. Given that both the water-bottom and the deep reflector dip in the same crossline direction, the reflection azimuth coverage is good in that direction but is poor in the opposite direction (positive azimuths with my sign convention). Notice that the angle gather may even curve down as if it were over-migrated for those reflection azimuths not actually illuminating the reflector. At first glance this is counter-intuitive, since the primary was migrated with the exact primary velocity (, ). The curvature is due to poor illumination and not to velocity errors.

ap-gath1
Figure 21.
3D ADCIG for the primary water-bottom reflection as a function of aperture angle. The different panels correspond to different reflection azimuth angles: (a)-40, (b)-20, (c)0, (d)20 and (e)40 degrees.

ap-gath2
Figure 22.
3D ADCIG for the first-order water-bottom multiple reflection as a function of aperture angle. The different panels correspond to different reflection azimuth angles: (a)-40, (b)-20, (c)0, (d)20 and (e)40 degrees.

Similarly, Figure 22 shows the corresponding gathers for the water-bottom multiple reflection. Unlike the primary, the multiple shows residual curvature at all azimuths. Not only that, but because of the crossline dip, the over-migrated multiple illuminates the reflector at positive and negative azimuths.

Next: Azimuth Illumination Up: Image space mapping of Previous: Source-Receiver Migration

2007-10-24