- 1.
- Input an image, migrated with velocity
*v*._{0} - 2.
- Transform the time axis
*t*to the squared time coordinate: . - 3.
- Apply a fast Fourier transform (FFT) on both the squared time
and the midpoint axis. The squared time transforms to the
frequency , and the midpoint coordinate
*x*transforms to the wavenumber*k*. We can safely assume that in the post-migration domain seismic images are uniformly sampled in*x*, which allows us to use the FFT technique. In the case of 3-D data, FFT should be applied in both midpoint coordinates. - 4.
- Apply a phase-shift operator to transform to different velocities
*v*:(1) - 5.
- Apply an inverse FFT to transform from and
*k*to and*x*. - 6.
- Apply an inverse time stretch to transform from to
*t*.

To generalize algorithm (1) to the prestack case, we first need to include the residual NMO term Fomel (1996). Residual normal moveout can be formulated with the help of the differential equation:

(2) |

(3) |

(4) |

(5) |

- 1.
- Input a set of common-offset images, migrated with velocity
*v*._{0} - 2.
- Transform the time axis
*t*to the squared time coordinate: . - 3.
- Apply a fast Fourier transform (FFT) on both the squared time
and the midpoint axis. The squared time transforms to the
frequency , and the midpoint coordinate
*x*transforms to the wavenumber*k*. - 4.
- Apply a phase-shift operator to transform to different velocities
*v*:(6) - 5.
- Apply an inverse FFT to transform from and
*k*to and*x*. - 6.
- Apply an inverse time stretch to transform from to
*t*.

The complete theory of prestack velocity continuation also requires a residual DMO operator Etgen (1990); Fomel (1996, 1997). However, the difficulty of implementing this operator is not fully compensated by its contribution to the full velocity continuation. For simplicity, I decided not to include residual DMO in the current implementation.

Figure shows impulse responses of prestack velocity continuation. The input for producing this figure was a time-migrated constant-offset section, corresponding to an offset of 1 km and a constant migration velocity of 1 km/s. In full accordance with the theory Fomel (1996), three spikes in the input section transformed into shifted ellipsoids after continuation to a higher velocity and into shifted hyperbolas after continuation to a smaller velocity.

Figure 1

Figure compares the result of a constant-velocity
prestack migration with the velocity of 1.8 km/s, applied to the
infamous Gulf of Mexico dataset from *Basic Earth Imaging*
Claerbout (1995) and the result of velocity continuation to the
same velocity from a migration with a smaller velocity of 1.3 km/s.
The differences in the top part of the images are explained by
differences in muting. In the first case, muting was applied after
migration, and in the second case, muting was applied prior to
velocity continuation. The other parts of the sections look very
similar, as expected from the theory.

Figure 2

Velocity continuation creates a time-midpoint-velocity cube (four-dimensional for 3-D data), which we can use for picking RMS velocities in the same way as we would use the result of common-midpoint or common-reflection-point velocity analysis. The important difference is that velocity continuation provides an optimal focusing of the reflection energy by properly taking into account both vertical and lateral movements of reflector images with changing migration velocity. Figure compares velocity spectra (semblance panels) at a CRP location of about 11.5 km after residual NMO and after prestack velocity continuation. Although the overall difference between the two panels is small, the velocity continuation panel shows a noticeably better focusing, especially in the region of conflicting dips between 1 and 2 seconds. The next section discusses the velocity picking step in more details.

Figure 3

4/20/1999