# Frequency-space implicit schemes

Series | Investigations in Geophysics |
---|---|

Author | Öz Yilmaz |

DOI | http://dx.doi.org/10.1190/1.9781560801580 |

ISBN | ISBN 978-1-56080-094-1 |

Store | SEG Online Store |

## Migration principles

As discussed in finite-difference migration in practice, in practice the 15-degree finite-difference migration can handle dips up to 35 degrees with sufficient accuracy. A steep-dip approximation to equation (**13b**) is achieved by continued fractions expansion (Section D.4) as

**(**)

This dispersion equation is known as the 45-degree approximation and is the basis of the most common implementation of steep-dip implicit finite-difference schemes ^{[1]}.

Refer to the steps described earlier and replace the Taylor expansion given by equation (**14a**) with the continued fractions expansion given by equation (**18**). Follow the subsequent steps to derive the corresponding differential equation associated with the 45-degree diffraction term (Section D.4):

**(**)

where *Q*(*x, z, ω*) is the retarded wavefield in the frequency-space domain.

When recast for time migration, equation (**19a**) becomes (Section D.4):

**(**)

where *τ* is the time variable associated with the migrated data.

Note that dropping the first term in equation (**19a**) and inverse Fourier transforming in time yields the 15-degree diffraction equation (**16a**). Similarly, dropping the first term in equation (**19b**) yileds the 15-degree equation (**17**) for time migration.

As for the 15-degree equation, the thin-lens equation (**16b**) also applies for the 45-degree equation. When implemented in the frequency-space domain, the thin-lens term is represented by the phase-shift operator of equation (**15a**). Again, the final step in the procedure is to write down the difference forms of the differential operators in implicit form to be used in finite-difference solution of the 45-degree equation (**19**) for migration. Kjartansson ^{[1]} provides an implicit scheme in which the extrapolation is in *z*. Nevertheless, as for the 15-degree equation (**17**), it is trivial to adapt his scheme for time migration with the extrapolation in *τ* of equation (**15b**). The phase-shift operator of equation (**15a**) is velocity-dependent when implemented for depth migration, and it is velocity-independent when implemented for time migration.

The 45-degree approximation given by equation (**19b**) actually is fairly accurate in practice up to 60 degrees. As described in Section D.4, the basic 45-degree equation (**19b**) also can be adapted to obtain extrapolation schemes for imaging steeper dips up to 90 degrees. Nevertheless, a penalty is paid for steep-dip accuracy in terms of dispersive noise incurred by implicit schemes (finite-difference migration in practice).

Steep-dip finite-difference algorithms may be more conveniently implemented in the frequency-space domain than in the time-space domain. A general framework for implementing such algorithms involve a loop over the depth step *z*, and a loop over the frequency *ω* (Figure 4.1-23). For each depth step:

- Apply the shift term (equation
**15a**). - Apply the diffraction term (equation
**19**) by performing implicit extrapolation of each of the frequency components of the wavefield. - Sum over the frequencies to invoke the imaging principle which is equivalent to setting
*t*= 0. - Repeat the computation for all the depth steps to complete the imaging.

## Equations

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## References

- ↑
^{1.0}^{1.1}Kjartansson, 1979, Kjartansson, E., 1979, Modeling and migration by the monochromatic 45-degree equation: Stanford Exploration Project Report No. 15, Stanford University.

## See also

- Kirchhoff migration
- Diffraction summation
- Amplitude and phase factors
- Kirchhoff summation
- Finite-difference migration
- Downward continuation
- Differencing schemes
- Rational approximations for implicit schemes
- Reverse time migration
- Frequency-space explicit schemes
- Frequency-wavenumber migration
- Phase-shift migration
- Stolt migration
- Summary of domains of migration algorithms