The one-way-in-depth scalar wave equation is the basis for common migration algorithms. These algorithms do not explicitly model multiple reflections, converted waves, surface waves, or noise. Any such energy present in data input to migration is treated as primary reflections. Migration algorithms can be classified under three main categories:
- those that are based on the integral solution to the scalar wave equation,
- those that are based on the finite-difference solutions, and
- those that are based on frequency-wavenumber implementations.
Whatever the algorithm, it should desirably:
- handle steep dips with sufficient accuracy,
- handle lateral and vertical velocity variations, and
- be implemented, efficiently.
Figure 4.0-11 is a migrated CMP stacked section with a major unconformity. The undermigration — incomplete imaging of the unconformity, is not because of erroneously too low velocities. Although we should always be aware of velocity errors when migrating seismic data, the undermigration in Figure 4.0-11 is the result of using a dip-limited algorithm. By using a steep-dip algorithm, we can achieve a more accurate imaging of the unconformity (Figure I-9).
The three principle migration techniques are discussed in this chapter in their historical order of development as outlined below. The first migration technique developed was the semicircle superposition method that was used before the age of computers. Then came the diffraction-summation technique, which is based on summing the seismic amplitudes along a diffraction hyperbola whose curvature is governed by the medium velocity. The Kirchhoff summation technique introduced later , but actually in use earlier, basically is the same as the diffraction summation technique with added amplitude and phase corrections applied to the data before summation. These corrections make the summation consistent with the wave equation in that they account for spherical spreading (gain applications), the obliquity factor (angle-dependency of amplitudes), and the phase shift inherent in Huygens’ secondary sources (migration principles).
Another migration technique  is based on the idea that a stacked section can be modeled as an upcoming zero-offset wavefield generated by exploding reflectors. Using the exploding reflectors model, migration can be conceptualized as consisting of wavefield extrapolation in the form of downward continuation followed by imaging. To understand imaging, consider the shape of a wavefield at observation time t = 0 generated by an exploding reflector. Since no time has elapsed and, thus, no propagation has occurred, the wavefront shape must be the same as the reflector shape that generated the wavefront. The fact that the wavefront shape at t = 0 corresponds to the reflector shape is called the imaging principle. To define the reflector geometry from a wavefield recorded at the surface, we only need to extrapolate the wave-field back in depth then monitor the energy arriving at t = 0. The reflector shape at any particular extrapolation depth directly corresponds to the wavefront shape at t = 0.
Downward continuation of wavefields can be implemented conveniently using finite-difference solutions to the scalar wave equation. Migration methods based on such implementations are called finite-difference migration. Many different differencing schemes applied to the differential operators in the scalar wave equation exist both in time-space and frequency-space domains. Claerbout  provides a comprehensive theoretical foundation of finite-difference migration and its practical aspects.
After the developments on Kirchhoff summation and finite-difference migrations,  introduced migration by Fourier transform. This method involves a coordinate transformation from frequency (the transform variable associated with the input time axis) to vertical wavenumber axis (the transform variable associated with the output depth axis), while keeping the horizontal wavenumber unchanged. The Stolt method is based on a constant-velocity assumption. However, Stolt modified his method by introducing stretching in the time direction to handle the types of velocity variations for which time migration is acceptable. Stolt and Benson  combine theory with practice in the field of migration with an emphasis on the frequency-wavenumber methods.
Another frequency-wavenumber migration is the phase-shift method . This method is based on the equivalence of downward continuation to a phase shift in the frequency-wavenumber domain. The imaging principle is invoked by summing over the frequency components of the extrapolated wavefield at each depth step to obtain the image at t = 0.
A reason for the wide range of migration algorithms used in the industry today is that none of the algorithms fully meets the important criteria of handling all dips and velocity variations while still being cost-effective.
Migration algorithms based on the integral solution to the scalar wave equation, commonly known as Kirchhoff migration, can handle all dips up to 90 degrees, but they can be cumbersome in handling lateral velocity variations.
Finite-difference algorithms can handle all types of velocity variations, but they have different degrees of dip approximations. Furthermore, differencing schemes, if carelessly designed, can severely degrade the intended dip approximation.
Finally, frequency-wavenumber algorithms have limited ability in handling velocity variations, particularly in the lateral direction. As a result of limitations of the three main categories of migration algorithms — integral, finite-difference, and frequency-wavenumber methods, migration software has expanded further to additional extensions and combinations of the basic algorithms. Residual migration — phase-shift or constant-velocity Stolt migration followed by the application of a dip-limited algorithm is one example.
- Exploding reflectors
- Migration strategies
- Migration parameters
- Aspects of input data
- Migration velocities
- Schneider, 1978, Schneider, W., 1978, Integral formulation for migration in two and three dimensions: Geophysics, 43, 49–76.
- Claerbout and Doherty, 1972, Claerbout, J.F. and Doherty, S.M., 1972, Downward continuation of moveout-corrected seismo-grams: Geophysics, 37, 741–768.
- Claerbout, 1985, Claerbout, J.F., 1985, Imaging the earth’s interior: Blackwell Scientific Publications.
- Stolt (1978), Stolt, R.H., 1978, Migration by Fourier transform: Geophysics, 43, 23–48.
- Stolt and Benson (1986), Stolt, R.H. and Benson, A.K., 1986, Seismic migration: theory and practice: Geophysical Press.
- Gazdag, 1978, Gazdag, J., 1978, Wave-equation migration by phase shift: Geophysics, 43, 1342–1351.