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To evaluate the integrals in equation (7), using the stationary phase method, we need to calculate
the maximum of the phase function, which for VTI media is given by
 
(20) 
Since the relation between the sourcereceiver rayparameters (p_{s} and p_{g}) and the offsetmidpoint
rayparameters (p_{h} and p_{x}) is linear, we can evaluate the stationary points by solving for
p_{s} and p_{g} instead of p_{h} and p_{x},
 
(21) 
Setting the derivative of equation 21 in terms of p_{s} and p_{g} to
zero provides us with two independent equations that can be solved for p_{s} and p_{g},
separately. Physically, this implies that we are solving
for the sourcetoimagepoint traveltime and receivertoimagepoint traveltime, separately,
which makes complete sense.
The p_{s} and p_{g} stationary point solutions can be used later to evaluate p_{h} and p_{x}.
First, p_{s} is evaluated by solving
 
(22) 
where y_{s}=2(xx_{0}h_{0}). Similarly, p_{g} is evaluated by solving
 
(23) 
where y_{g}=2(xx_{0}+h_{0}). Equation 23 is similar to equation 22, with y_{s} replaced by y_{g}, and
p_{s} replaced by p_{g}. Therefore, solving for p_{s} will yield an equation that can be used to
solve for p_{g} as well.
To remove the square root in equation 22, I move y_{s} to the other side of the
equation and square both sides. This will allow us to right equation 21
in a polynomial form as follows,
 

 (24) 
This is a fourthorder polynomial in p_{s}^{2}, which can be solved exactly for the four roots in p_{s}^{2}.
However, these four roots are given by highly complicated equations which include square roots as
well as powers of the order . Such equations are not useful for practical use.
Thus, I elect to use Shanks transform to obtain approximations that are almost exact, yet more useful
for practical implementations. Also, since Shanks transform is based on perturbation theory, it will provide us
with the desired solution of p_{s}^{2} among the four possible solutions;
the solution based on perturbation from an isotropic model.
Using Shanks transform, described in Appendix D, we obtain
 
(25) 
Again, p_{g} is given by the same equation but with y_{s} replaced by y_{g}.
B
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Stanford Exploration Project
7/5/1998