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The same approach used in homogeneous media is followed here. Starting with the
phase of exponential for VTI v(z) media which is given by
 

 (26) 
Using Taylor series, I expand equation (26) around p_{h}=0 and p_{x}=0, which corresponds to small offsets
and dips. In the Taylor series expansion of equation (26),
I drop terms beyond the quartic power in p_{h} and p_{x}. Thus,
 

 (27) 
As mentioned earlier, the analytical homogeneousmedium equations can be used
to calculate traveltimes in vertically inhomogeneous media, granted that the medium
parameters are replaced by their equivalent averages in v(z) media. Thus, equation (20) becomes
 
(28) 
The Taylor series expansion of equation (28) around p_{h}=0 and p_{x}=0, with terms
beyond the quartic power in p_{h} and p_{x} dropped, yields
 

 (29) 
Matching coefficients of terms with the same power in p_{x} and p_{h}
in equations (27) and (29) provides
us with two key relations:
 
(30) 
and
 
(31) 
The stationary points (p_{h} and p_{x}), in vertically inhomogeneous media, satisfy
 

 (32) 
with solutions best solved numerically. Again by expanding this equation in powers of p_{x} and p_{h} and
matching its coefficients with the coefficients of an equivalent expansion of the effective equation, we obtain
equations (30) and (31) again.
In summary, equations (30) and (31) provide us with the equivalent relations
necessary to use the offsetmidpoint traveltime equation for homogeneous VTI media in v(z) media.
C
Next: Stationary phase approximation
Up: Alkhalifah: Analytical traveltimes in
Previous: Stationary point solutions
Stanford Exploration Project
7/5/1998