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# Anisotropic phase-shift in tilted TI media

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

where Cij are elastic moduli. In equation (1), the is qP-wave phase-velocity when the sign in front of the square root is positive, and the qSV-wave phase velocity for a negative sign.

If we rotate 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. VP0 is the qP-wave velocity in the direction parallel to the symmetry axis.

For plane-wave propagation, the phase angle is related to the wavenumbers kx and kz by:
 (3)
where is the temporal frequency. Squaring equation (2) and substituting (3) into (2), we can obtain a dispersion relation equation:

 d4kz4+d3kz3+d2kz2+d1kz+d0=0, (4)

where the coefficients d0,d1,d2,d3, and d4 are as follows:

The dispersion relation equation (4) is a quartic equation. It can be solved analytically Abramowitz and Stegun (1972) or numerically by Newton's Method Stoer and Bulirsch (1992). Equation (4) has four roots, which are related to up-going and down-going qP- and qSV- waves, respectively. For a medium without lateral change in the velocity VP0 and anisotropy parameters and , the wavefield can be extrapolated by the phase-shift method Gazdag (1978):
 (5)
where kza is one of the roots of equation (4).

Next: Extrapolation operator in laterally Up: Shan and Biondi: Anisotropic Previous: Introduction
Stanford Exploration Project
10/23/2004