To describe the physical basis of finite-difference migration, recall the harbor example of Figure 4.1-9. Instead of taking the section recorded along the beach, which contains the diffraction hyperbola, then collapsing it to get the migrated section in Figure 4.1-15, consider the following alternative procedure. Again, start with the wavefield recorded along the beach (Figure 4.1-16a). Assume that the barrier is 1250 m from the beach. Now move the recording cable into the water, 250 m from the beach. Start recording at the instant the plane wave hits the barrier. The recorded section is shown in Figure 4.1-16b. Move the cable 500 m from the beach and record the section in Figure 4.1-16c, followed by a recording 750 m from the beach to obtain the section in Figure 4.1-16d. Finally, 1000 m from the beach, record the section shown in Figure 4.1-16e.
Figure 4.1-9 The gap in the barrier acts as Huygens’ secondary source, causing the circular wavefronts that approach the beach line. Adapted from Claerbout.
Figure 4.1-15 Principles of migration based on diffraction summation. (a) Zero-offset section (trace interval, 25 m; constant velocity, 2500 m/s), (b) migration. The amplitude at input trace location B along the flank of the traveltime hyperbola is mapped onto output trace location A at the apex of the hyperbola by equation (4).
Note that each recording yields a hyperbola in which the apex moves closer to zero time. The actual extent of the recording cable is denoted by the solid line on top of each frame. Had we recorded at the barrier (1250 m from the beach), the apex of the hyperbola would be positioned at t = 0.
In Kirchhoff migration, the diffraction hyperbola is collapsed by summing the amplitudes, then placing them at the apex. The alternative approach implied by the result of the experiment shown in Figure 4.1-16 is to use the hyperbola recorded a distance away from the beach to construct the hyperbola that would be recorded at another distance closer to the source of the diffraction hyperbola. The process is stopped when the hyperbola collapses to its apex. In the harbor experiment, this collapse occurs when the receiver cable coincides with the barrier, or, equivalently, when t = 0. As stated in the introductory section, this is called the imaging principle.
- Claerbout, 1985, Claerbout, J.F., 1985, Imaging the earth’s interior: Blackwell Scientific Publications.
- Kirchhoff migration
- Diffraction summation
- Amplitude and phase factors
- Kirchhoff summation
- Downward continuation
- Differencing schemes
- Rational approximations for implicit schemes
- Reverse time migration
- Frequency-space implicit schemes
- Frequency-space explicit schemes
- Frequency-wavenumber migration
- Phase-shift migration
- Stolt migration
- Summary of domains of migration algorithms