# Frequency-space implicit schemes

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Series Investigations in Geophysics Öz Yilmaz http://dx.doi.org/10.1190/1.9781560801580 ISBN 978-1-56080-094-1 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

 ${\displaystyle k_{z}={\frac {2\omega }{v}}\left[1-{\frac {v^{2}k_{x}^{2}}{8\omega ^{2}}}{\frac {1}{1-{\frac {v^{2}k_{x}^{2}}{16\omega ^{2}}}}}\right].}$ (18)

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):

 ${\displaystyle i{\frac {v}{4\omega }}{\frac {\partial ^{3}Q}{\partial z\partial x^{2}}}-{\frac {\partial ^{2}Q}{\partial x^{2}}}+i{\frac {4\omega }{v}}{\frac {\partial Q}{\partial z}}=0,}$ (19a)

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):

 ${\displaystyle i{\frac {1}{2\omega }}{\frac {\partial ^{3}Q}{\partial t\partial x^{2}}}-{\frac {\partial ^{2}Q}{\partial x^{2}}}+i{\frac {8\omega }{v^{2}}}{\frac {\partial Q}{\partial \tau }}=0,}$ (19b)

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.

Figure 4.1-23  An algorithmic description of frequency-space 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:

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

## Equations

 ${\displaystyle k_{z}={\frac {2\omega }{v}}{\sqrt {1-\left({\frac {vk_{x}}{2\omega }}\right)^{2}}},}$ (13b)

 ${\displaystyle k_{z}={\frac {2\omega }{v}}\left[1-{\frac {1}{2}}\left({\frac {vk_{x}}{2\omega }}\right)^{2}\right].}$ (14a)

 ${\displaystyle Q=P\exp(-i\omega \tau ),}$ (15a)

 ${\displaystyle \tau =\int _{0}^{z}{\frac {dz}{{\bar {v}}(z)}},}$ (15b)

 ${\displaystyle {\frac {\partial ^{2}Q}{\partial z\partial t}}={\frac {v}{4}}{\frac {\partial ^{2}Q}{\partial x^{2}}},}$ (16a)

 ${\displaystyle {\frac {\partial Q}{\partial z}}=2\left[{\frac {1}{{\bar {v}}(z)}}-{\frac {1}{v(x,z)}}\right]{\frac {\partial Q}{\partial t}},}$ (16b)

 ${\displaystyle {\frac {\partial ^{2}Q}{\partial \tau \partial t}}={\frac {v^{2}}{8}}{\frac {\partial ^{2}Q}{\partial x^{2}}}.}$ (17)

## References

1. Kjartansson, 1979, Kjartansson, E., 1979, Modeling and migration by the monochromatic 45-degree equation: Stanford Exploration Project Report No. 15, Stanford University.