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In a VTI medium, the phase velocity of qP and qSVwaves in Thomsen's notation
can be expressed as Tsvankin (1996):
 
(1) 
where is the phase angle of the propagating wave, and f=1(V_{S0}/V_{P0})^{2}. V_{P0} and V_{S0} are the qP and qSV wave velocities in the vertical direction, respectively.
and are anisotropy parameters defined by Thomsen (1986):
where C_{ij} are elastic moduli.
In equation (1), is qPwave phasevelocity when the sign in front of the square root is positive,
and the qSVwave phase velocity for a negative sign.
Rotating the symmetry axis from vertical to a tilted angle , we obtain the phase velocity of
a tilted TI medium whose symmetry axis forms an angle with the vertical direction:
 
(2) 
Here, in contrast to equation (1), and are now defined in a direction tilted by the angle
from the vertical direction. V_{P0} is the qPwave velocity in the direction parallel to the symmetry axis.
The phase angle is related to the wavenumbers k_{x} and k_{z} by:
 
(3) 
where is the temporal frequency. Let , and .We can obtain a dispersion relation equation from equations (2) and (3):

d_{4}S_{z}^{4}+d_{3}S_{z}^{3}+d_{2}S_{z}^{2}+d_{1}S_{z}+d_{0}=0,

(4) 
where the coefficients d_{0},d_{1},d_{2},d_{3}, and d_{4} are as follows:
Equation (4) is a quartic equation and there is no explicit expression for its solution.
Generally, the Padé approximation suggests that
if the function , then S_{z}(S_{r}) can be approximated by a rational function
R_{n,m}(S_{r}):
 
(5) 
where
and
are polynomials of degree n and m, respectively. The coefficients a_{i} and b_{i} can be obtained
either analytically by Taylorseries analysis or numerically by leastsquares fitting.
For an isotropic or VTI medium, S_{z} is an even function of S_{x}. We can approximate the dispersion
relation with even rational functions, such as S_{x}^{2}, S_{x}^{4}.
For TTI media, S_{z} is not an symmetric function of S_{x}.
Therefore, in addition to even rational functions, we need odd rational functions to approximate the
dispersion relation, such as S_{x}, S_{x}^{3}.
The fourth order approximation for the dispersion relation of TTI media is as follows:
 
(6) 
where S_{z0}=S_{z}(0) and the coefficients c_{1},b_{1},a_{1},c_{2},b_{2},a_{2} can be estimated by leastsquares methods.
They are functions of
the anisotropy parameters , and the tilting angle . When these parameters vary
laterally, the coefficients c_{1},b_{1},a_{1},c_{2},b_{2},a_{2} also vary laterally. It is too expensive to
run leastsquares estimation for each grid point during the wavefield extrapolation.
They can be calculated and stored in a table before the wavefield extrapolation.
During the wavefield extrapolation, given the anisotropy parameters , , and the tilting angle ,
we search for these coefficients from the table and put them into the finitedifference algorithm.
Given the coefficients found from the table,
the finite difference algorithm in TTI media is similar to the isotropic media.
Next: Finitedifference scheme
Up: Shan: Implicit migration for
Previous: Introduction
Stanford Exploration Project
1/16/2007