The next step is to use non-linear inversion to find the optimum source and receiver positions from the different targets. In this case, since there is only one target, the inversion reduces to a simple binning to honor the constraint that the receivers should be at equal distance along the line profile. I chose for the receiver interval the same value obtained in the standard design so that the results can be easily compared. The shots were also binned at the same receiver interval to further regularize the design (and guarantee equal distance between CMPs and between the stacked traces). Only shots that contribute at least half the number of traces of the standard geometry shot (that is 36) were considered. A total of 402 shots met this criterion (which is rather arbitrary). The number of traces per shot, and hence the maximum offset, was allowed to change from shot to shot. In this example this is the only degree of freedom that I used to adapt the acquisition effort to variations in the subsurface dip. In a real case, where the reservoir location is known or suspected, we could locally vary the receiver group interval or more likely the shot interval. In 3-D there are extra degrees of freedom associated with the azimuth and the choice of geometry template.
The next step is to simulate the acquisition of the data using the computed shot and receiver positions. Again, this was done with an analytic ray tracer. Figure shows some of the shots. Note that they have different number of traces. Also, they look irregular because the plotting program places the traces together at the same distance irrespective of their offset. Figure shows some CMPs along the line profile. As with the shots, the number of traces changes from CMP to CMP. Also note that there are ``holes'' in the CMP's illustrating the difference between uniform offsets and uniform illumination. Figure shows the fold diagram. In this case there are differences in the fold coverage from CMP to CMP. As long as the minimum CMP fold is maintained, this shouldn't be a problem. More importantly, note the large offsets at both sides of the semicircle and the smaller offsets above the semicircle (compare with Figure ). Figure shows the stacked section. Comparison with Figure does not reveal any striking difference because the stack smooths out the effect of the irregular offsets. The important difference between the two figures is the lateral extent of the semicircular reflection. Figure shows the migrated section and Figure shows a comparison with the migrated section obtained with the standard acquisition. Not surprisingly, the two images are almost the same, since they were computed with the same aperture. The proposed design, however, required about 80 fewer shots.
This example is rather artificial in that the savings in the number of shots comes simply from a realization that not all shots contribute the same number of traces to the subsurface image. In the real case a more important consideration would be to what part of the image every shot contributes. Those shots that contribute to the reservoir location (or any other critical part of the image) will be kept even if they contribute only a small number of traces. This flexibility is important when faced with obstacles which force us to displace shots or receivers. The effort that we put into it may depend on the relative contribution of those shots and receivers to the critical parts of the image as opposed to the standard approach in which all shots and receivers are considered equally important.
In order to see the importance of the fewer shots in the quality of the image, I modeled the data again with the standard approach but using only 402 shots (the same that I used in the proposed approach). The first shot will now be at -5350 m which translates to a maximum dip angle of 69 degrees with one fold and 60 degrees with full fold. Figure shows a comparison with the proposed approach. The difference in the high dips of the images on the left-hand-side of the semicircle is clearly visible.
An obvious improvement to the above methodology consists in acquiring, for every shot, not only those receiver positions obtained form the inversion, but also those in between. After all, if the intermediate receiver stations are available, why not use them? Figure shows the fold diagram in this case. The number of shots is the same as in the previous case, and the increase in fold is due entirely to the intermediate receivers.