Next: Extrapolation operator in laterally
Up: Shan and Biondi: Anisotropic
Previous: Introduction
In a VTI media, 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), the is qPwave phasevelocity when the sign in front of the square root is positive,
and the qSVwave 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. V_{P0} is the qPwave velocity in the direction parallel to the symmetry axis.
For planewave propagation, the phase angle is related to the wavenumbers k_{x} and k_{z} by:
 
(3) 
where is the temporal frequency. Squaring equation (2) and substituting (3) into (2), we
can obtain a dispersion relation equation:

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

(4) 
where the coefficients d_{0},d_{1},d_{2},d_{3}, and d_{4} 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 upgoing and downgoing
qP and qSV waves, respectively. For a medium without lateral change in the velocity V_{P0} and anisotropy parameters
and , the wavefield can be extrapolated by the phaseshift method Gazdag (1978):
 
(5) 
where k_{z}^{a} 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