When a stacked section is migrated, we use the migration theory applicable to data recorded with a coincident source and receiver (zero-offset). To develop a conceptual framework for discussing migration of zero-offset data, we now examine two types of recording schemes.
A zero-offset section is recorded by moving a single source and a single receiver along the line with no separation between them (Figure 4.0-6). The recorded energy follows raypaths that are normal incidence to reflecting interfaces. This recording geometry obviously is not realizable in practice.
Now consider an alternative geometry (Figure 4.0-6) that will produce the same seismic section. Imagine exploding sources that are located along the reflecting interfaces . Also, consider one receiver located on the surface at each CMP location along the line. The sources explode in unison and send out waves that propagate upward. The waves are recorded by the receivers at the surface. The earth model described by this experiment is referred to as the exploding reflectors model.
The seismic section that results from the exploding reflectors model is largely equivalent to the zero-offset section, with one important distinction. The zero-offset section is recorded as two-way traveltime (from source to reflection point to receiver), while the exploding reflectors model is recorded as one-way traveltime (from the reflection point at which the source is located to the receiver). To make the sections compatible, we can imagine that the velocity of propagation is half the true medium velocity for the exploding reflectors model.
These concepts now are applied to the velocity-depth model in Figure 4.0-7. Consider source-receiver pairs placed along the earth’s surface at every tenth midpoint. In this case, a zero-offset section is modeled. At midpoint 130, five different arrivals are associated with rays that are normal incidence to the first interface. Alternatively, imagine receivers placed along the earth’s surface at every tenth midpoint and sources placed along the interface where the rays emerge at the right angle to the interface (equivalent to the normal-incidence rays of the zero-offset section). In the latter case, the velocities indicated in Figure 4.0-7 must be halved to match the time axis with that associated with the zero-offset section.
The interface can be sampled more densely by placing receivers and sources at closer spacing (Figure 4.0-8a). The next deeper interface can be modeled; that is, the traveltime trajectory can be computed by placing sources along this interface and leaving the receivers where they were on the surface (Figure 4.0-8b). Finally, the same experiment can be repeated for the third interface (Figure 4.0-8c). To derive the composite response from the velocity-depth model in Figure 4.0-8d (the left-hand column), individual responses shown in Figures 4.0-8a, 4.0-8b and 4.0-8c from each interface are superimposed. The result is shown in Figure 4.0-8d (the right-hand column). We can imagine that sources were placed at all three interfaces and turned on simultaneously. Such an experiment would cause the rays emerging from the three interfaces to be recorded at receivers placed on the surface, along the line.
Figure 4.0-6 Geometry of zero-offset recording (left), and hypothetical simulation of the zero-offset experiment using exploding reflectors (right) .
Actually, Figure 4.0-8d (the right-hand column) represents the modeled zero-offset traveltime section. Seismic wavefields, however, are represented not only by wave traveltimes but also by wave amplitudes. Figure 4.0-9a shows the modeled zero-offset wavefield section based on the same velocity-depth model in Figure 4.0-8d (the left-hand column). The shallow complex interface (horizon 1 in Figure 4.0-8a) caused the complicated response of the two simple interfaces (horizons 2 and 3) in this zero-offset traveltime section.
How valid is the assumption that a stacked section is equivalent to a zero-offset section? The conventional CMP recording geometry provides the wavefield at nonzero offsets. During processing, we collapse the offset axis by stacking the data onto the midpoint-time plane at zero offset. For CMP stacking, we normally assume hyperbolic moveout. Figure 4.0-10 shows selected CMP gathers modeled from the velocity-depth model in Figure 4.0-8d (the left-hand column). Because of the presence of strong lateral velocity variations, the hyperbolic assumption may not be appropriate for some reflections on some CMP gathers (Figure 4.0-10a); however, it may be valid for others (Figure 4.0-10b). We obtain a stacked section (Figure 4.0-9b) that resembles the zero-offset section (Figure 4.0-9a) to the extent that the hyperbolic moveout assumption is valid. The assumption that a conventional stacked section is equivalent to a zero-offset section also is violated to varying degrees in the presence of strong multiples and conflicting dips with different stacking velocities (dip-moveout correction and prestack migration). While migration of unstacked data is discussed in dip-moveout correction and prestack migration, our main focus in this chapter is on migration after stack.
Figure 4.0-8 Exploding-reflector modeling of zero-offset traveltimes associated with (a) a water bottom, (b) a flat, and (c) a dipping reflector. (d) The superposition of the normal-incidence traveltime responses in (a), (b), and (c). Shown on the velocity-depth models in the left-hand column are the normal-incidence rays used to compute the traveltime trajectories. The time sections shown on the right-hand column are equivalent to a zero-offset traveltime section with the vertical axis in two-way time.
Figure 4.0-9 (a) The zero-offset wavefield section equivalent to the zero-offset traveltime section in Figure 4.0-8d (the right-hand column); (b) the CMP stack generated from the CMP gathers as in Figure 4.0-10. Modeling by .
Figure 4.0-10 Selected CMP gathers modeled from the velocity-depth model in Figure 4.0-8d (the left-hand column). (a) Gathers from the complex part of the velocity-depth model, (b) gathers from the simpler part of the velocity-depth model. CMP locations are indicated in Figure 4.0-8d. The CMP stack is shown in Figure 4.0-9b. Modeling by .
- Migration strategies
- Migration algorithms
- Migration parameters
- Aspects of input data
- Migration velocities
- Loewenthal et al., 1976, Loewenthal, D., Lu, L., Roberson, R., and Sherwood, J.W.C., 1976, The wave equation applied to migration: Geophys. Prosp., 24, 380–399.
- Kjartansson and Rocca, 1979, Kjartansson, E. and Rocca, F., 1979, The exploding reflector model and laterally variable media: Stanford Exploration Project Report No. 16, Stanford University.
- Claerbout, 1985, Claerbout, J.F., 1985, Imaging the earth’s interior: Blackwell Scientific Publications.
- Deregowski and Barley, 1981, Deregowski, S. and Rocca, F., 1981, Geometrical optics and wave theory for constant-offset sections in layered media: Geophys. Prosp., 29, 374–387.