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In the case of zero-offset reflection, the ray travel distance,
*l*, from the source to the reflection point is related to the
two-way zero-offset time, *t*, by the simple equation

where *v*_{g} is the half of the group velocity, best expressed in
terms of its components, as follows:
Here *v*_{gx} denotes the horizontal component of group velocity,
*v*_{v} is the vertical *P*-wave velocity, and is the
*v*_{v}-normalized vertical component of the group velocity. Under the
assumption of zero shear-wave velocity in VTI media, these components
have the following analytic expressions:
| |
(4) |

and
| |
(5) |

where *p*_{x} is the horizontal component of slowness, and is the normalized (again by the vertical *P*-wave velocity *v*_{v})
vertical component of slowness. The two components of the slowness
vector are related by the following eikonal-type equation
Alkhalifah (1997):
| |
(6) |

Equation (6) corresponds to a normalized version of the
dispersion relation in VTI media.
If we consider *v* and as imaging parameters (migration
velocity and migration anisotropy coefficient), the ray length *l* can
be taken as an imaging invariant. This implies that the partial
derivatives of *l* with respect to the imaging parameters are zero.
Therefore,

| |
(7) |

and
| |
(8) |

Applying the simple chain rule to equations (7) and
(8), we obtain
| |
(9) |

where , and the two-way vertical traveltime is
given by
Combining equations (7-9) eliminates the two-way
zero-offset time *t*, which leads to the equations
| |
(10) |

and
| |
(11) |

After some tedious algebraic manipulation, we can transform
equations (5) and (6) to the general form

| |
(12) |

and
| |
(13) |

Since the residual migration is applied to migrated data, with the
time axis given by and the reflection slope given by
, instead of *t* and *p*_{x},
respectively, we need to eliminate *p*_{x} from
equations (12) and (13). This task can be
achieved with the help of the following explicit relation, derived
in Appendix A,

| |
(14) |

where =, and
Inserting equation (14) into equations (12)
and (13) yields exact, yet complicated equations,
describing the continuation process for *v* and . In
summary, these equations have the form

| |
(15) |

and
| |
(16) |

Equations of the form (15) and (16) contain all
the necessary information about the kinematic laws of anisotropy
continuation in the domain of zero-offset migration.

** Next:** Linearization
** Up:** Alkhalifah and Fomel: Anisotropy
** Previous:** Introduction
Stanford Exploration Project

11/11/1997