# Difference between revisions of "Frequency-space implicit schemes"

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==Migration principles== | ==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 ({{EquationNote|13b}}) is achieved by continued fractions expansion (Section D.4) as | + | 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 ({{EquationNote|13b}}) is achieved by continued fractions expansion ([[Mathematical foundation of migration#D.4 Frequency-space implicit schemes|Section D.4]]) as |

{{NumBlk|:|<math>k_z=\frac{2\omega}{v}\left[1-\frac{v^2k_x^2}{8\omega^2}\frac{1}{1-\frac{v^2k_x^2}{16\omega^2}}\right].</math>|{{EquationRef|18}}}} | {{NumBlk|:|<math>k_z=\frac{2\omega}{v}\left[1-\frac{v^2k_x^2}{8\omega^2}\frac{1}{1-\frac{v^2k_x^2}{16\omega^2}}\right].</math>|{{EquationRef|18}}}} | ||

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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 <ref name=ch04r19>Kjartansson, 1979, Kjartansson, E., 1979, Modeling and [[migration]] by the monochromatic 45-degree equation: Stanford Exploration Project Report No. 15, Stanford University.</ref>. | 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 <ref name=ch04r19>Kjartansson, 1979, Kjartansson, E., 1979, Modeling and [[migration]] by the monochromatic 45-degree equation: Stanford Exploration Project Report No. 15, Stanford University.</ref>. | ||

− | Refer to the steps described earlier and replace the Taylor expansion given by equation ({{EquationNote|14a}}) with the continued fractions expansion given by equation ({{EquationNote|18}}). Follow the subsequent steps to derive the corresponding differential equation associated with the 45-degree diffraction term (Section D.4): | + | Refer to the steps described earlier and replace the Taylor expansion given by equation ({{EquationNote|14a}}) with the continued fractions expansion given by equation ({{EquationNote|18}}). Follow the subsequent steps to derive the corresponding differential equation associated with the 45-degree diffraction term ([[Mathematical foundation of migration#D.4 Frequency-space implicit schemes|Section D.4]]): |

{{NumBlk|:|<math>i\frac{v}{4\omega}\frac{\partial^3Q}{\partial z\partial x^2}-\frac{\partial^2Q}{\partial x^2}+i\frac{4\omega}{v}\frac{\partial Q}{\partial z}=0,</math>|{{EquationRef|19a}}}} | {{NumBlk|:|<math>i\frac{v}{4\omega}\frac{\partial^3Q}{\partial z\partial x^2}-\frac{\partial^2Q}{\partial x^2}+i\frac{4\omega}{v}\frac{\partial Q}{\partial z}=0,</math>|{{EquationRef|19a}}}} | ||

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

− | When recast for time [[migration]], equation ({{EquationNote|19a}}) becomes (Section D.4): | + | When recast for time [[migration]], equation ({{EquationNote|19a}}) becomes ([[Mathematical foundation of migration#D.4 Frequency-space implicit schemes|Section D.4]]): |

{{NumBlk|:|<math>i\frac{1}{2\omega}\frac{\partial^3Q}{\partial t\partial x^2}-\frac{\partial^2Q}{\partial x^2}+i\frac{8\omega}{v^2}\frac{\partial Q}{\partial \tau}=0,</math>|{{EquationRef|19b}}}} | {{NumBlk|:|<math>i\frac{1}{2\omega}\frac{\partial^3Q}{\partial t\partial x^2}-\frac{\partial^2Q}{\partial x^2}+i\frac{8\omega}{v^2}\frac{\partial Q}{\partial \tau}=0,</math>|{{EquationRef|19b}}}} | ||

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[[file:ch04_fig1-23.png|thumb|{{figure number|4.1-23}} An algorithmic description of frequency-space [[migration]].]] | [[file:ch04_fig1-23.png|thumb|{{figure number|4.1-23}} An algorithmic description of frequency-space [[migration]].]] | ||

− | The 45-degree approximation given by equation ({{EquationNote|19b}}) actually is fairly accurate in practice up to 60 degrees. As described in Section D.4, the basic 45-degree equation ({{EquationNote|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]]). | + | The 45-degree approximation given by equation ({{EquationNote|19b}}) actually is fairly accurate in practice up to 60 degrees. As described in [[Mathematical foundation of migration#D.4 Frequency-space implicit schemes|Section D.4]], the basic 45-degree equation ({{EquationNote|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: | 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: |

## Latest revision as of 17:14, 7 October 2014

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

**(**)

**(**)

**(**)

**(**)

**(**)

**(**)

**(**)

## 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