# Frequency-space migration in practice

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 |

The basis of the steep-dip implicit algorithms is the dispersion relation of equation (**18**). Finite-difference schemes with steep-dip accuracy are implemented conveniently in the frequency-space domain. An important advantage of the implicit method is its exceptional ability to handle velocity variations, whether vertical or lateral. Its accuracy for the lateral velocity problem results from the fact that the time shift associated with the thin-lens term (equation **16b**) can be implemented exactly in the frequency domain. For these reasons, the algorithm is most appropriate for depth migration to image targets beneath complex structures (earth imaging in depth).

**Figure 4.3-17**(a) A zero-offset section that contains a diffraction hyperbola with 2500-m/s velocity, (b) desired migration using the phase-shift method, (c) 15-degree finite-difference migration using a depth step of 20 ms; cascaded application of the 15-degree migration using (d) 4 cascades with 80-ms depth step, (e) 10 cascades with 200-ms depth step, and (f) 20 cascades with 400-ms depth step.**Figure 4.3-18**(a) A zero-offset section that contains dipping events with 3500-m/s velocity, (b) desired migration using the phase-shift method, (c) 15-degree finite-difference migration using a depth step of 20 ms; cascaded application of the 15-degree migration using (d) 4 cascades with 80-ms depth step, (e) 10 cascades with 200-ms depth step, and (f) 20 cascades with 400-ms depth step. For comparison, event*AB*with the steepest dip is labeled on the desired migration (b) and the cascaded migration (f).**Figure 4.3-19**(a) A zero-offset section that contains three diffraction hyperbolas with a vertically varying velocity, (b) 15-degree finite-difference migration using a depth step of 20 ms; the output from the last stage of cascaded application of the 15-degree migration using (c) 4 cascades with 80-ms depth step, (d) 10 cascades with 200-ms depth step, (e) 20 cascades with 400-ms depth step, and (f) desired migration using the phase-shift method.**Figure 4.3-20**(a) A zero-offset section that contains three diffraction hyperbolas with a vertically varying velocity; four-stage cascaded migration using the phase shift-method with 80-ms depth step: (b) first-stage, (c) second stage, (d) third stage, (e) fourth stage, and (f) desired migration using the phase-shift method only once with 20-ms depth step. Compare the fourth stage (e) with the output of the four-stage cascaded migration using a dip-limited algorithm as in Figure 4.3-19c.**Figure 4.3-21**Impulse response of a reverse time migration algorithm.

The frequency-space, sometimes referred to as *ω − x* or *f − x*, migration also has the important operational advantage that each frequency can be processed separately. This property can reduce computer memory requirements significantly and, thus, decrease input-output operations for large data sets. Also, in frequency-space migration, some accuracy features can be conveniently implemented. For example, wave extrapolation can be limited to a specified signal bandwidth. Each frequency component can, in principle, be downward continued using an optimum depth step size that yields a minimum acceptable phase error, leading to a minimum amount of dispersive noise on the migrated section.

## See also

- Introduction to migration
- Migration principles
- Kirchhoff migration in practice
- Finite-difference migration in practice
- Frequency-wavenumber migration in practice
- Further aspects of migration in practice
- Exercises
- Mathematical foundation of migration