Huygens’ secondary source signature is a semicircle in the x − z plane and a hyperbola in the x − t plane. This characterization of point sources in the subsurface leads to two practical migration schemes. Figure 4.1-14a shows a zero-offset section that consists of a single arrival at a single trace. This event migrates to a semicircle (Figure 4.1-14b). From Figure 4.1-14, note that the zero-offset section recorded over a constant-velocity earth model consisting of a semicircular reflecting interface contains a single blip of energy at a single trace as in Figure 4.1-14a. Since this recorded section consists of an impulse, the migrated section in Figure 4.1-14b can be called the migration impulse response. An alternate scheme for migration results from the observation that a zero-offset section consisting of a single diffraction hyperbola migrates to a single point (Figure 4.1-15b).
The first method of migration is based on the superposition of semicircles, while the second method is based on the summation of amplitudes along hyperbolic paths. The first method was used before the age of digital computers. The second method, which is known as the diffraction summation method, was the first computer implementation of migration.
The migration scheme based on the semicircle superposition consists of mapping the amplitude at a sample in the input x − t plane of the unmigrated time section onto a semicircle in the output x − z plane. The migrated section is formed as a result of the superposition of the many semicircles.
The migration scheme based on diffraction summation consists of searching the input data in the x − t plane for energy that would have resulted if a diffracting source (Huygens’ secondary source) were located at a particular point in the output x − z plane. This search is carried out by summing the amplitudes in the x − t plane along the diffraction curve that corresponds to Huygens’ secondary source at each point in the x − z plane. The result of this summation then is mapped onto the corresponding point in the x − z plane. As noted early in this section, within the context of time migration, however, the summation result actually is mapped onto the x − τ plane, where τ is the event time in the migrated position.
The curvature of the hyperbolic trajectory for amplitude summation is governed by the velocity function. The equation for this trajectory can be derived from the geometry of Figure 4.1-15. A formal derivation also is provided in Section D.2. Assuming a horizontally layered velocity-depth model, the velocity function used to compute the traveltime trajectory is the rms velocity at the apex of the hyperbola at time τ (normal moveout). From the triangle COA in Figure 4.1-15a, we note that
Having computed the input time t, the amplitude at input location B is placed on the output section at location A, corresponding to the output time τ at the apex of the hyperbola.
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).
From normal moveout, reflection traveltimes in a layered earth approximate small-spread hyperbolas. This may seem to impose a serious restriction on the aperture width — the lateral extent of the diffraction hyperbola, in the summation process. However, the small-spread approximation is valid even at large distances from the apex, and the errors associated with it are insignificant at late times. In practice, this approximation is not usually an issue.
- Kirchhoff migration
- Amplitude and phase factors
- Kirchhoff summation
- Finite-difference migration
- 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