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# The general theory

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

 l = vg t, (3)

where vg is the half of the group velocity, best expressed in terms of its components, as follows:

Here vgx denotes the horizontal component of group velocity, vv is the vertical P-wave velocity, and is the vv-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 px is the horizontal component of slowness, and is the normalized (again by the vertical P-wave velocity vv) 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 px, respectively, we need to eliminate px 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