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In this section, I will summarize the derivation of
the VTI eikonal equation for the (x)domain, derived originally by Alkhalifah et al. (1997).
Obviously, the twoway vertical traveltime is related to depth,
 
(1) 
where v_{v} is the vertical Pwave velocity, which can vary vertically as well as laterally.
As a result, the stretch applied to the depth axis is laterally variant.
Alkhalifah (1997a) derived a simple form of the eikonal equation for VTI media, based on setting
the shear wave velocity to zero. For 2D media, it is
 

 (2) 
This equation, based on the acoustic medium assumption in VTI media,
though not physically possible, yields extremely
accurate traveltime solutions that are close to what we find for
typical elastic media.
Clearly, equation 2 includes firstorder derivatives of traveltime with respect
to position. In order to transform this eikonal equation from
depth to time, we need to replace x with , as well.
Using the chain rule, in the eikonal equation 2
is given by
 
(3) 
where , derived from equation (1),
 
(4) 
Likewise, the partial derivative in z of the eikonal equation is
 
(5) 
Therefore, the
transformation from (x, z) to (, ) is governed
by the following Jacobian matrix in 2D media:
 
(6) 
Substituting equations (3) and (5)
into the eikonal equation (2) yields an eikonal equation in the ()domain given by
 

 (7) 
which is also indirectly independent on the vertical velocity. However,
according to equation (4),
still depends on the vertical Pwave velocity.
Alkhalifah et al. (1997) demonstrated that
if the medium was factorized laterally (), then
 
(8) 
which is independent of the vertical velocity.
The departure of the medium from this special
condition of laterally factorized media
will cause errors in traveltime
calculation; these errors, however, are generally small.
Using the method of characteristics, Alkhalifah et al. (1997) derived
a system of ordinary differential equations that define the
ray trajectories in the ()domain.
Numerical solutions of the raytracing equations,
as opposed to the eikonal equation, provide multiarrival traveltimes and amplitudes,
a feature that is key to properly image the Marmousi model Geoltrain and Brac (1993). Thus, in the Kirchhoff migration examples,
I use ()domain traveltime maps extracted from ray tracing.
Next: The Marmousi Model
Up: Alkhalifah: Prestack time migration
Previous: Introduction
Stanford Exploration Project
7/5/1998