Slant stacking

To take advantage of the information that non-zero offsets provide, we can perform a slant stack () along the offset axis at each depth step before imaging. In this way we obtain an image as a function of the offset ray parameters p_hx and p_hy:

23#23 (10) (11) (12)

With this method we can extract angle-domain common image gathers (CIGs) from the new wavefield 24#24at fixed midpoint locations (). The right panel in Figure  shows the angle-domain CIG corresponding to the downward-continued offset gather shown in the left panel. By downward continuing the wavefields with the DSR equation and slant stacking at every depth step, this method can handle lateral velocity variations, unlike the slant stacking method proposed by ().

Strictly speaking, the CIGs obtained by this procedure are function of the offset ray parameters 25#25and not of the opening angle 5#5.However, in 2-D p_hx is linked to 5#5 by the following simple trigonometric relationship (a similar relationship can be found for the 3-D problem):  

 26#26 (13)

where 6#6 is the geological dip along the in-line direction and 27#27 is the velocity function. A rigorous treatment of CIGs in 3-D can be found in ().

 

hcig.pcig
Figure 5 Left: Offset panel after downward continuation. Right: Angle-domain CIG

() demonstrated that it is possible to perform downward-continuation migration in such a way as to get CIGs that have an axis of reflection angle rather than offset ray parameter. This method of obtaining CIGs is based on the relationship between the reflection angle 5#5 and the local subsurface offset as

28#28 (14)

where k_h is the local subsurface offset wavenumber. Since the offset involved in their methodology is related to the subsurface rather than the surface, the CIGs produced will be free of many of the artifacts seen in CIGs from surface related techniques.