According to equation (9), as the geological dip increases the horizontal-offset axis is stretched. At the limit, when is equal to 90 degrees, the relation between the horizontal-offset and the geological-dip offset becomes singular. Similarly, VOCIGs have problems when the geological dip is close to flat ( degrees) and equation (10) becomes singular. This dip-dependent offset-stretching of the offset-domain CIGs causes artifacts in the corresponding ADCIGs.

The fact that relationships (9) and (10) diverge only for isolated dips (0, 90, 180, and 270 degrees) may falsely suggest that problems are limited to rare cases. However, in practice there are two factors that contribute to make the computation of ADCIGs in presence of geological dips prone to artifacts:

- To limit the computational cost, we would like to compute the offset-domain gathers over a range of offsets as narrow as possible. This is particularly true for shot-profile migrations, where the computation of the imaging conditions by equation (2) can add substantially to the computational cost when it is carried over a wide range of subsurface offsets.
- The attractive properties of the ADCIGs that we demonstrated above, including the elimination of the image-point dispersal, depend on the assumption of locally constant velocity. In particular, velocity is assumed to be constant along the ray segments , , , and drawn in Figure . The longer those segments are, the more likely it is that the constant velocity assumption will be violated sufficiently to cause substantial errors.

These considerations suggest that, in presence of complex structures, high-quality ADCIGs ought to be computed using the information present in both HOCIGs and VOCIGs. There are two alternative strategies for obtaining a single set of ADCIGs from the information present in HOCIGs and VOCIGs. The first method merges HOCIGs with VOCIGs after they have been transformed to GOCIGs by the application of the offset stretching expressed in equation (16). The merged GOCIGs are then transformed to ADCIGs by applying the radial-trace transformation expressed in equation (18). The second method merges HOCIGs with VOCIGs directly in the angle domain, after both have been transformed to ADCIGs by the radial-trace transforms expressed in equations (4) and (7).

The two methods are equivalent if the offset range is infinitely wide, but they may have different artifacts when the offset range is limited. Since the first method merges the images in the offset domain, it can take into account the offset-range limitation more directly, and thus it has the potential to produce more accurate ADCIGs. However, the second method is more direct and simpler to implement. In both methods, an effective, though approximate, way for taking into account the limited offset ranges is to weight the CIGs as a function of the apparent dips in the image. A simple weighting scheme is:

(20) |

7/8/2003