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# 3D dispersion relation in tilted Ti media

In 3D VTI media, the phase velocity of P- and SV-waves in Thomsen's notation can be expressed as follows Tsvankin (1996):
 (1)
where is the phase angle of the propagating wave, and f=1-(VS0/VP0)2. VP0 and VS0 are the P- and SV- wave velocities in the vertical direction, respectively. The anisotropy parameters and are defined by Thomsen (1986):

where Cij are elastic moduli. In equation (1), is the P-wave phase-velocity when the sign in front of the square root is positive, and the SV-wave phase velocity for a negative sign. Let , and be the wavenumbers for VTI media in Cartesian coordinates. For plane-wave 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 kx, ky, and kz 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 re-organize the the dispersion relation and obtain the equation for the wavenumber kz as follows:

 a4kz4+a3kz3+a2kz2+a1kz+a0=0, (6)

where
 (7)
Equation (6) is a quartic equation in kz. Given kx, kz, the velocity vp0, 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 down-going P-wave, and the other two are related to the up- and down-going SV-wave, 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 kz by rotating coordinates (kx,ky) as follows:
 (8)
Figure shows kz as a function of kx and ky 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
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5/3/2005