Next: Wavefield extrapolation operator
Up: Shan and Biondi: 3D
Previous: Introduction
In 3D VTI media, the phase velocity of P and SVwaves in Thomsen's notation can be expressed as follows 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 P and SV wave velocities in the vertical direction, respectively.
The anisotropy parameters and are defined by Thomsen (1986):
where C_{ij} are elastic moduli.
In equation (1), is the Pwave phasevelocity when the sign in front of the square root is positive,
and the SVwave phase velocity for a negative sign. Let , and be the
wavenumbers for VTI media in Cartesian coordinates.
For planewave propagation, the phase angle is related to
the wavenumbers , and by the following relations:
 
(2) 
where is the temporal frequency, and .From equations (1) and (2), we can derive the dispersion relation for 3D VTI media as follows:
 
(3) 
where
 
(4) 
For tilted TI media, the symmetry axis deviates from the vertical direction.
We need two angles to describe the tilting direction, the tilting angle
and the azimuth of the tilting direction . We first assume , that is the symmetry axis is in the plane y=0.
Then we generalize the dispersion relation to the case that by coordinate rotation.
For a tilted TI medium, if we rotate the coordinates so that the symmetry axis is the axis ,
it becomes a VTI medium in the new coordinates. Let k_{x}, k_{y}, and k_{z} be the wavenumbers for
a tilted TI medium in Cartesian coordinates. , and ,
which are the wavenumbers for VTI media in Cartesian coordinates, can also be considered as the wavenumbers
for tilted TI media in the rotated coordinates.
For the case that , the dispersion relation can be obtained from equation (3) by rotating the coordinates as follows:
 
(5) 
We can reorganize the the dispersion relation and obtain the equation for the wavenumber k_{z}
as follows:

a_{4}k_{z}^{4}+a_{3}k_{z}^{3}+a_{2}k_{z}^{2}+a_{1}k_{z}+a_{0}=0,

(6) 
where
 
(7) 
Equation (6) is a quartic equation in k_{z}. Given k_{x}, k_{z}, the velocity v_{p0}, the anisotropy parameters
and , and the tilting angle , we can calculate all the coefficients of equation (6),
and it can be solved analytically Abramowitz and Stegun (1972). Usually there are four solutions for equation (6). Two
of them are related to the up and downgoing Pwave, and the other two are related to the up and downgoing SVwave, respectively.
Let , and be the wavenumbers for tilted TI media with
a general in the original coordinate system.
For general , after solving equation (6), we can get the wavenumber k_{z} by rotating coordinates (k_{x},k_{y}) as follows:
 
(8) 
Figure shows k_{z} as a function of k_{x} and k_{y} in
a constant tilted TI medium. In this medium, the velocity is 2000 m/s,
is 0.4, is 0.2, is and is 0.
The frequency used in Figure 57 Hz.
dispersion
Figure 1 Dispersion relation of 3D tilted TI media.
Next: Wavefield extrapolation operator
Up: Shan and Biondi: 3D
Previous: Introduction
Stanford Exploration Project
5/3/2005