Modeling with traditional parameters

In this case 626#626 m/s, 627#627 and 628#628 Hz. Applying Equation  we get 629#629 m. The maximum depth of the target is 2000 m, so assuming an off-end cable (receivers to one side of the shot only) this will give about 67 receivers. This number rounds up nicely to 72, which means an actual maximum offset of 2160 m (assuming that the first receiver is at an offset of 30 m). The shot interval is chosen as 630#630 m so that the fold of coverage (number of receivers per CMP) is 36. The trace length was chosen to be 4 s. With these parameters I simulated acquisition using an analytical ray-tracing program. Obviously, we cannot expect to image the dips of the semicircular reflector up to 90 degrees because according to Equation  that would imply infinite aperture. The best we can do is image the maximum dip for which the reflection time is less than or equal to the trace length. In this case, given the simple geometry of the reflector, a quick computation shows that for the zero offset trace this corresponds to shot positions 631#631 m. The corresponding maximum dip is 632#632 degrees.

The acquisition proceeds from left to right in Figure . When the shot is to the left of the semicircle the longer offsets have shorter arrival times (from the semicircle) which means that we can actually achieve full fold at that point by extending the acquisition to the left by half cable-length (1080 m). The first shot is therefore at -7780 m. On the other hand, when the shot is to the right of the semicircle, longer offsets have longer arrival times and so we cannot expect to have full fold at 6700 m. Any shot past that point will only contribute reflections longer than the trace length. In summary, with the standard approach (using the off-end cable described above) we need shots between -7780 and 6700 m in order to image the maximum dip. At 30 m shot interval this implies 483 shots.

Figure  shows some of the modeled shot records. At both sides of the semicircle we see two reflections coming from the flat and the semicircular reflector, whereas above the semicircle only the reflection from the semicircular reflector is seen. Figure  shows some CMP gathers. Since the design is completely regular, the CMP's are also regular. This is further illustrated in Figure  which shows the fold diagram. Note that we have full fold at -6700 m but not at 6700 m.

Figure  shows the stacked section. The noise at the intersection between the flat and the dipping reflections reflects the inherent difficult in picking a stacking velocity appropriate to both (no DMO was applied). Finally, Figure  shows the post-stack Stolt-migrated section. As expected, dips in the semicircular reflector higher than 73 degrees were not recovered.

 

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Figure 2 Synthetic shot records modeled with the standard geometry.

 

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Figure 3 CMPs modeled with the standard geometry

 

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Figure 4 Fold diagram for the standard geometry

 

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Figure 5 Stacked section of modeled data generated with the standard design

 

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Figure 6 Post-stack migrated section of modeled data generated with the standard design