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The dispersion relation of TTI media can be characterized by a quartic equation as follows:

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

(1) 
where the coefficients d_{0},d_{1},d_{2},d_{3}, and d_{4} are defined as follows:
where and are Thomsen anisotropy parameters Thomsen (1986) and is
the tilting angle of the media.
Theoretically, equation 1 can be solved analytically, but there is no explicit analytical expression for its solution.
The solid line in Figure shows how the dispersion relation
looks, given the anisotropy parameters , and the tilting angle .Note that S_{z} is not a symmetric function of S_{x}. And S_{z} has two branches when S_{x}>0.8. One of them represents the upgoing
waves and the other one represents the down going waves. Therefore, in a TTI medium waves may overturn even though it is homogeneous.
Conventional implicit finitedifference methods are designed by truncating the Taylor series of the dispersion relation.
The dispersion relation for TTI media is so complex that it is difficult to derive an analytical Taylor series
used for an implicit finitedifference scheme.
Generally, the Padé approximation suggests that
if the function , then S_{z}(S_{x}) can be approximated by a rational function
R_{n,m}(S_{x}):
 
(2) 
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.
S_{z} is an even function of S_{x} for isotropic and VTI media. In contrast,
S_{z} is not an symmetric function of S_{x} for TTI media.
It's well known that an general function can be decomposed into an even function and an odd function.
We approximate the even part of the dispersion with the even rational functions, such as
S_{x}^{2}, S_{x}^{4} and approximate the odd part with odd rational functions, such as S_{x},S_{x}^{3}.
dispersion
Figure 1 Comparison of the true and approximate dispersion relations for a TTI medium with , and : the solid line
is the true dispersion relation for TTI media; the dashed line is the approximate dispersion relation for finitedifference
scheme. The dispersion relation for the finitedifference scheme is very close to the true one for negative
S_{x}. When the phaseangle is close to or more than for the positive S_{x}, the dispersion for the finitedifference scheme
diverge from the true one.
Considering the stability of the finitedifference scheme, I approximate the dispersion relation of TTI media with
rational functions as follows:
 
(3) 
where S_{z0}=S_{z}(0) and the coefficients a_{1},b_{1},c_{1},a_{2},b_{2},c_{2} are estimated by leastsquares optimization.
They are functions of the anisotropy parameters , and the tilting angle .
Figure compares the true dispersion relation with the approximate dispersion relation.
The solid line is the true dispersion relation (equation 1) and the dashed line is the approximate dispersion relation for the finitedifference
scheme ( equation 3). The dispersion relation for the finitedifference scheme is very close to the true one for the negative
S_{x}. When the phaseangle is close to or more than for positive S_{x},
the dispersion for the finitedifference scheme diverges from the true one.
Figure shows the relative dispersion error defined as follows:
 
(4) 
where S_{z}^{true}(S_{x}) is the value of S_{z} calculated from equation 1 and S_{z}^{fd}(S_{x}) is
the value of S_{z} from equation 3 using the coefficients from the leastsquares estimation.
err
Figure 2 Relative dispersion relation error of finitedifference approximation for a TTI medium with , and .
For a laterally varying medium, the anisotropy parameters vary laterally. As a consequence, the coefficients for
the finitedifference scheme vary laterally.
It is too expensive to estimate these coefficients for each discrete grid during the wavefield extrapolation.
After estimating the minimum and maximum value of the anisotropy parameters and the tilting angle,
I compute the coefficients for the anisotropy parameters and the tilting angle in these ranges and store them in a table before the migration.
During the wavefield extrapolation, given the anisotropy parameters , , and the tilting angle ,
I search the coefficients in the table and put them into the finitedifference scheme.
Given the coefficients found from the table, the finite difference algorithm in TTI media is the same as the isotropic media.
The table of coefficients is small, and the computation cost for tablesearching is trivial compared to
that of solving the finitedifference equation. Therefore, the cost of the optimized implicit finitedifference for TTI media
is similar to that of the conventional finitedifference methods for isotropic media.
model
Figure 3 The velocity model and anisotropy parameters: (a) the tilting angle of the TTI medium; (b) the velocity paralleling the symmetry axis; (c) the anisotropy parameter ; (d) the anisotropy parameter .
Next: 2D synthetic data example
Up: Shan: Optimized finitedifference
Previous: Introduction
Stanford Exploration Project
5/6/2007