Common-reflection-point versus common-reflection-surface stacking
Conventional stacking is based on the notion of a common midpoint (CMP) and migration is based on the notion of a common reflection point (CRP). In both cases, we assume that reflections are represented by hyperbolas and reflectors are represented by points. Consider a seismic line from the Alberta Plains of Western Canada. The subsurface is just as flat as the surface over which you have recorded the data. When stacking the data, you can almost picture summing of the amplitudes in a CMP gather over all offsets along a hyperbolic trajectory associated with a zero-offset time and the resulting stacked amplitude being placed at a point reflector where the CMP raypaths converge conveniently. When migrating the stacked data, again, you sum the amplitudes along a hyperbolic trajectory and place the result at the apex of the hyperbola. You conveniently associate the apex of the latter hyperbola with a point diffractor situated on the reflector. Whether it is a reflection hyperbola associated with a point reflector or a diffraction hyperbola associated with a point diffractor, the process of stacking and migrating the data involves summation of amplitudes along hyperbolas and placing the resulting sum to a point in the subsurface.
Now consider a seismic line from the Alberta Foothills where the cascaded flanks of the Rocky Mountains rise steeply. The subsurface is just as steep as the surface over which you have recorded the data. A diffractor still is a point whether it is in the Plains or in the Foothills, and you can still think of diffractions as hyperbolas so long as they are situated below a simple overburden. Fortunately, you can also think of reflections as hyperbolas. No longer, however, can you associate a reflection hyperbola on your CMP gather with a single point reflector; instead, you have reflection points dispersed along the reflector (Section E.1). This is when you have to introduce DMO correction to account for the reflection-point dispersal (principles of dip-moveout correction). Once the reflection-point dispersal is removed, the resulting stack can be considered equivalent to a zero-offset section which you can migrate, again, using the hyperbolic summation rule. Thus you have been able to overcome the Foothills problem of steeply dipping events.
Finally, consider a seismic line from the Canadian Thrust Belt west of the Foothills. The subsurface is just as complex as it appears on the surface over which you have recorded the data. A diffractor situated below the complex overburden structure caused by the overthrust tectonics stays as a point; no longer, however, can you associate it with a hyperbolic traveltime. Instead, when migrating the data, you have to deal with a complex, distorted traveltime trajectory. And neither can you associate the reflections with hyperbolic traveltimes. Instead, when stacking the data, you have to deal with a nonhyperbolic moveout trajectory associated with many reflection points scattered around in the subsurface .
So the simple hyperbolic and point rules of the Plains or the Foothills are no longer applicable in the Thrust Belt. To overcome the first problem — migration of data in the presence of strong lateral velocity variations associated with complex overburden structures in the Thrust Belt, you may decide to do the imaging in depth (introduction to earth imaging in depth) instead of imaging in time (introduction to migration). Earth imaging in depth requires earth modeling in depth (introduction to earth modeling in depth) — a challenge much higher than the imaging itself. To overcome the second problem — stacking of data with nonhyperbolic moveout, you may combine it with the first problem and pursue a rigorous solution by doing the imaging not just in depth but also before stack.
Note that, insofar as stacking and imaging in time or in depth, we choose to map events to common-reflection points. By way of DMO correction, we map events to common-reflection points in their unmigrated positions. Similarly, by way of prestack time migration, we map events to common-reflection points in their migrated positions. De Bazelaire  and Gelchinsky  pointed out that, rather than associating the recorded data with common-reflection points, we should associate the recorded data with common-reflection surfaces. As such, when stacking the data, rather than constricting the summation of amplitudes to within a single CMP gather, data may be focused to a common-reflection surface (CRS) using multiple shots and receivers  and .
The theory and practice of the CRS stacking method        suggest that it is indeed related to prestack time migration when the medium velocities are judged to be within the acceptable bounds for the latter. The CRP gathers from prestack time migration and the CRS gathers both are created by including in the summation aperture multiple numbers of CMP gathers. The difference is that the CRP gathers are associated with events in their migrated positions, whereas the CRS gathers are associated with events in their unmigrated positions. So, migration of the CRS stack should resemble the section from prestack time migration.
Figure 5.3-40 Migration of the common-reflection-surface (CRS) stack shown in Figure 5.3-39.
Figure 5.3-41 A common-reflection-point (CRP) stack of the data as in Figure 5.3-40 derived from prestack time migration.
Figure 5.3-39 shows a CRS stack from an area with low-relief structures. The field data were recorded with 240-fold coverage. For CRS processing, the negative offsets were discarded and the near one-half of the positive offsets were kept; thus the reduced data used to create the CRS stack had 60-fold coverage. For prestack time migration, the original 240-fold data were first reduced to 60-fold by partial stacking. The migrated CRS stack shown in Figure 5.3-40 is comparable to the CRP stack derived from prestack time migration shown in Figure 5.3-41. In the present example, five CMP gathers were used to create one CRS gather with 300 traces as shown in Figure 5.3-42. The 60-fold CRP gather at the same location as for the CRS gather of Figure 5.3-42 is shown in Figure 5.3-43.
Practical refinements to various approaches to CRS stacking and imaging, including extensions to 3-D, are under current investigation. As to what extent the technique will be applied routinely to seismic data remains to be seen.
- de Bazelaire, 1988, de Bazelaire, E., 1988, Normal-moveout correction revisited: Inhomogeneous media and curved interfaces: Geophysics, 53, 143–157.
- Gelchinsky (1988), Gelchinsky, B., 1988, The common-reflecting-element (CRE) method: ASEG/SEG Internat. Geophys. Conf., Extended Abstracts, 71–75.
- Höcht, 1998, Höcht, G., 1998, Common-reflection-surface stack: Master’s thesis, Karlsruhe University.
- Tygel et al., 1997, Tygel, M., Müller, T., Hubral, P., and Schleicher, J., 1997, Eigenwave-based multiparameter traveltime expansions: 67th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1770–1773.
- Gelchinsky et al., 1999a, Gelchinsky, B., Berkovitch, A., and Keydar, S., 1999a, Multifocusing homoemorphic imaging: Part I: Basic concepts and formulas: J. Appl. Geophys., 42, 229–242.
- Gelchinsky et al., 1999a, Gelchinsky, B., Berkovitch, A., and Keydar, S., 1999b, Multifocusing homoemorphic imaging: Part II: Multifold data set and multifocusing: J. Appl. Geophys., 42, 243–260.
- Berkovitch et al., 1998, Berkovitch, A., Keydar, S., Landa, E., and Trachtman, P., 1998, Multifocusing in practice: 68th Ann. Internat. Mtg., Soc. Expl. Geophys., Expanded Abstracts, 1748–1751.
- Landa et al., 1999, Landa, E., Gurevich, B., Keydar, S., and Trachtman, P., 1999, Application of multifocusing method for subsurface imaging: J. Appl. Geophys., 42, 283–300.