# Reverse time migration

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 |

## Contents

## Migration principles

Another migration method, known as *reverse time migration* ^{[1]}, extrapolates an initially zero *x − z* plane *backward in time*, bringing in the seismic data *P*(*x, z* = 0, *t*) as a boundary condition *z* = 0 at each time step to compute snapshots of the *x − z* plane at different times. At time *t* = 0, this *x − z* plane contains the migration result *P*(*x, z, t* = 0) (Figure 4.1-21).

The algorithmic structure for the reverse time migration is illustrated schematically in Figure 4.1-22. Start with the *x − t* section at the surface, *z* = 0. Also, consider an *x − z* frame at *t _{max}*. This frame is blank except for the first row which is equal to the bottom row of the

*x − t*section at

*t*. Extrapolate this snapshot at

_{max}*t*=

*t*to

_{max}*t*=

*t*− Δ

_{max}*t*by using the phase-shift operator exp(

*iω*Δ

*t*). This yields a new snapshot of the

*x − z*frame at

*t*=

*t*− Δ

_{max}*t*. The first row of numbers in this frame is identical to the row in the

*x − t*plane — the original unmigrated section, at

*t*=

*t*− Δ

_{max}*t*. Hence, replace the first row in the snapshot at

*t*=

*t*− Δ

_{max}*t*with the row of the

*x − t*section at

*t*=

*t*− Δ

_{max}*t*and continue the extrapolation back in time. The last snapshot is at

*t*= 0 that represents the final migrated section.

**Figure 4.1-21**Reverse time migration: Start with an all-zero*x − z*plane at the bottom of the data cube and extrapolate backward in time toward*t*= 0 to compute snapshots of the*x − z*plane at different times. These snapshots of the subsurface are indicated by the horizontal planes; the direction of extrapolation — reverse in time, is indicated by the thick arrow. At each time level, include the boundary value (*x*-slice at*z*= 0, indicated by the dotted lines) into the*x − z*plane from the seismic section. The migrated section is the*x − z*plane at*t*= 0 (the top horizontal plane).

## Finite-difference migration in practice

In migration principles, a migration algorithm based on extrapolation back in time while using the stacked section to be the boundary condition at *z* = 0 was discussed. The impulse response of this algorithm, which is known as reverse time migration, is shown in Figure 4.3-21. Note that the algorithm can handle dips up to 90 degrees with the accuracy of phase-shift migration. The important consideration is that the extrapolation step Δ*t* in reverse time migration must be taken quite small, usually a fraction of the input temporal sampling interval. This then makes the algorithm computationally intensive.

Figure 4.3-22 shows a portion of a CMP-stacked section and its reverse time migration. The steep flanks of the salt diapirs have been imaged accurately, enabling delineation of the geometry of the top-salt boundary with confidence. Reverse time migration, albeit its simple and elegant implementation (migration principles), has not been used widely in practice. Again, this is primarily because it requires very small extrapolation step in time, which increases the computational cost of the algorithm.

## References

- ↑ Baysal et al., 1983, Baysal, E., Kosloff, D., and Sherwood, J.W.C., 1983, Reverse time migration: Geophysics, 48, 1514–1524.

## See also

- Kirchhoff migration
- Diffraction summation
- Amplitude and phase factors
- Kirchhoff summation
- Finite-difference migration
- Downward continuation
- Differencing schemes
- Rational approximations for implicit schemes
- Frequency-space implicit schemes
- Frequency-space explicit schemes
- Frequency-wavenumber migration
- Phase-shift migration
- Stolt migration
- Summary of domains of migration algorithms
- Depth step size
- Velocity errors
- Cascaded migration