SYNTHETIC EXPERIMENT

To examine the amplitude behavior of the imaging operators in the sequence, we designed a synthetic reflectivity model which simulates a flat reflector with amplitude anomalies along its surface. The model is 10,000 by 10,000 feet wide, which is about the size of the real survey area. The anomaly zones take the form of circles of high reflectivity. Figure 6 shows a cross section of the synthetic model taken at the top of the flat reflector. The edges of the model are tapered to avoid numerical diffractions from the modeling algorithm. For the sake of simplicity, we consider a medium with a constant velocity of 10,000 ft/sec.

In a first experiment, we modeled the reflector using regular constant offset geometry. The survey simulates a zero-azimuth acquisition, and an offset of 8000 ft with regular mid-point spacing of 62.5 ft. Figure 7 shows the migration of the regular CA/CO cube. As expected, the migration produced a nice image and successfully inverted for the reflectivity function along the reflector. The edges of the circular anomalies have been slightly smeared along the zero-azimuth acquisition direction (in-line direction).

In a second experiment, we modeled the reflector using the Button-Patch geometry of the real 3-D survey. We extracted a total of 24,000 traces whose source-receiver azimuth is between -30^o and 30^o with an absolute offset range from 7000 to 9000 ft. The results of the migrated image are shown in figure 7. The reflectivity map has a very poor resolution and suffers amplitude distortions scattered along all the flat reflector. A direct correlation is noted between the areas of low coverage and the resolution of the reflectivity map. The two anomalies on the left of section suffered the most distortion since many of the missing traces correspond to that side of the map. Figure 9 shows the location of the anomalies overlaying the mid-point geometry of the input traces. The two left anomalies are poorly sampled, much beyond aliasing. We conclude that Kirchhoff migration on its own was unable to preserve the amplitude of the reflector and resolve for the anomalous locations in the presence of very sparse and irregular coverage.

In a final experiment, we applied the AMO transformation to regularize the geometry and reconstruct the data as a zero-azimuth cube with an 8000 ft effective absolute offset and constant mid-point spacing of 62.5 ft. The CA/CO cube is then imaged using the 3-D prestack depth migration algorithm. Figure 10 shows the results of the migration after applying AMO. The migrated image shows better resolution than the previous experiment. The AMO transformation before migration eliminated most of the amplitude distortions along the horizontal reflector and nicely imaged the location of the anomalies. The lower left zone is better defined, whereas the upper left anomaly is still not fully recovered. That anomaly corresponds to an extreme case of poor trace coverage. The AMO operator in its Kirchhoff implementation could not perfectly reconstruct the reflectivity at that location. We conclude that interpolating the data and regularizing its geometry before migration can help a great deal in preserving its amplitude. AMO lends itself as a convenient tool for organizing the data for common-azimuth processing while correcting for its sparse and irregular coverage.

 

model
Figure 6 Synthetic reflectivity model of a flat reflector. The white circles are zones of high reflection coefficient of 2.5 in a constant background of unit reflectivity.

 

mig-reg
Figure 7 Image of the reflectivity map obtained by migrating a regular cube from a zero-azimuth constant-offset experiment.

 

mig-irr
Figure 8 Reflectivity map obtained by direct Kirchhoff migration traces modeled using the real 3-D irregular geometry

 

overlay
Figure 9 Location of the amplitude anomalies overlaying the mid-point geometry of the recording traces.

 

amo
Figure 10 Reflectivity map obtained by migrating the data after applying the AMO transformation to an effective zero-azimuth and a constant offset of 8000 ft.