Figure 7 illustrates the geometry of the offset-domain CIGs for a single event recorded at the surface for the source location S and receiver location R. The crucial assumption of our geometric construction is that the traveltime along the source ray summed with the traveltime along the receiver ray is the same for all the offset directions and equal to the recording time of the event .
In this sketch, the migration velocity is assumed to be lower than the true velocity, and thus the reflections are imaged too shallow and above the point where the source ray crosses the receiver ray (SR). The line passing through SR, and bisecting the angle formed by the source and receiver ray, is oriented at an angle with respect to the vertical direction. The angle is the apparent geological dip of the event after imaging. It would correspond to the true geological dip if the migration velocity were correct. Half of the angle formed between the source and receiver ray is the aperture angle .
When HOCIGs are computed, the end point of the source ray (Sxh) and the end point of the receiver ray (Rxh) are at the same depth. The imaging point Ixh is in the middle between Sxh and Rxh and the imaging offset is xh=Rxh-Sxh. Similarly, when VOCIGs are computed, the end point of the source ray (Szh) and the end point of the receiver ray (Rzh) are at the same horizontal location. The imaging point Izh is in the middle between Szh and Rzh and the imaging offset is zh=Rzh-Szh. When the offset direction is oriented along the apparent geological dip (what we called the optimally focusing offset direction), the end point of the source ray is S0 and the end point of the receiver ray is R0. The imaging point I0 is in the middle between S0 and R0 and the imaging offset is h0=R0-S0. It is easy to demonstrate that both Ixh and Izh lie on the line passing through .The demonstration is based on the assumption that and .
The offsets along the different directions are linked by the following simple relationship, which can be readily derived by trigonometry applied to Figure 7; that is,
Also the shift of the imaging points Ixh and Izh can be easily expressed in terms of the offset h0 and the angles and as:
The fact that all three imaging points are aligned along the apparent geological dip allows our transformation to remove the image-point dispersal, and it is crucial to the effectiveness of DDOCIGs. In other words, to transform one set of CIGs into another set we just need to transform the offset axis; the image is then automatically shifted along the apparent geological dip by the right amount. Appendix A demonstrates this fact.
The proposed CIG transformation is a simple dip-dependent non-uniform stretching of the the offset-axis according to the relationships in equations (1) and (2). The transformation is easily implemented in the wavenumber (kz,kx) domain, by taking advantage of the well known relationship .
After both the HOCIGs and the VOCIGs are transformed, they can be merged together. A simple scheme to merge them is a weighted average, where the weights wxh and wzh are set to