** Next:** Application
** Up:** Vaillant & Fomel: On
** Previous:** Introduction

Stolt time migration can be summarized as the following sequence of
transformations:

| |
(1) |

where
| |
(2) |

The function is the dispersion relation and has
the following expression in the constant velocity case:
| |
(3) |

The approximation suggested by Stolt 1978 for
extending the method to *v*(*z*) media involves a change of the time
variable (Stolt-stretch):

| |
(4) |

where *s*(*t*) is the stretched time variable, *v*_{0} is an arbitrarily chosen constant velocity, and *v*_{rms}(*t*)
is the root mean square velocity along the vertical ray,
defined by
| |
(5) |

This change of variable yields a transformed wave-equation for the
wavefield extrapolation, in which Stolt replaces a slowly
varying complicated function of several parameters (denoted by *W*)
by its average value. Making this
approximation yields a new dispersion relation in the transformed
coordinate system:
| |
(6) |

This factor *W* contains all the information about the heterogeneities
of the medium. However, it has to be determined a priori, that is,
before migration.
This empirical choice for *W* was one of the drawbacks of the
Stolt-stretch method. Fomel 1995 derived an
analytical formulation of the Stolt-stretch parameter, based on
Malovichko's formula for approximating traveltimes in vertically
inhomogeneous media Malovichko (1978):

| |
(7) |

where the function *S*(*t*) defines the so-called parameter of heterogeneity:
| |
(8) |

Fomel proved that, for a given depth (or vertical traveltime), the
optimal value of *W* is

| |
(9) |

where *v*_{rms}(*t*) is the root mean square velocity along the
vertical ray, and the vertical
traveltime. The value of *W* used during Stolt migration is the
average along the vertical profile of these *W*(*t*).
In the case of an homogeneous constant-velocity model, *W* is equal to
1.0, whereas it has to be less than 1.0 if the velocity increases
monotonically with depth.
We can sum up the application of Stolt-stretch algorithm with the
optimal parameter *W* by the following sequence of steps:

**1.** Stretch the time axis and determine |

the value of *W* along the vertical profile |

**2.** Interpolate stretched time to a regular grid |

**3.** 2-D FFT |

**4.** Apply Stolt migration with the dispersion relation (6) |

**5.** 2-D inverse FFT |

**6.** Unstretch the time axis |

** Next:** Application
** Up:** Vaillant & Fomel: On
** Previous:** Introduction
Stanford Exploration Project

10/25/1999