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The elastic modulus matrix of an HTI medium with a symmetry axis along
the x direction (which I will denote as
) has the
following form (in compressed notation):
| ![\begin{displaymath}
C_{
\alpha
\beta }=
\left( {
\matrix{c_{11}&c_{13}&c_{13}&0...
..._{44}&0&0\cr 0&0&0&0&c_{55}&0\cr 0&0&0&0&0&c_{55}\cr }}
\right)\end{displaymath}](img19.gif) |
(3) |
This representation can easily be obtained by a
rotation of the VTI's elastic matrix along the y axis. Incorporating
this expression into equation (1) and identifying
as the velocity
of the plane wave with propagation
direction (kx,ky,kz) gives us
| ![\begin{eqnarray}
\left(
\begin{array}
{ccc}
c_{11} k_x^2 + c_{55} (k_y^2 +k_z^2)...
...
\begin{array}
{c}
v_x \\ v_y \\ v_z\end{array}\right) \nonumber\end{eqnarray}](img21.gif) |
|
| (4) |
| |
From here on, to simplify notation, I use the variables
and
, introduced by Dellinger 1991. Thus
we rewrite equation (4) as
| ![\begin{eqnarray}
\left(
\begin{array}
{ccc}
c_{11} k_x^2 + c_{55} (k_y^2 +k_z^2)...
...
\begin{array}
{c}
v_x \\ v_y \\ v_z\end{array}\right) \nonumber\end{eqnarray}](img25.gif) |
|
| (5) |
| |
HTIprop
Figure 1 Definition of propagation
angles. The azimuthal angle is measured with respect to the
symmetry axis, and the incidence angle with respect to the
vertical axis.
|
| ![HTIprop](../Gif/HTIprop.gif) |
Next: Plane wave modes
Up: TRANSVERSE ISOTROPY
Previous: TRANSVERSE ISOTROPY
Stanford Exploration Project
11/12/1997