Next: From elastic constants to
Up: FORWARD MAPPING
Previous: From elastic constants to
Expanding equation (1a) around
and neglecting
terms in
, results in:
| ![\begin{eqnarray}
2 W_{P,SV} (\theta) & = & (W_{33} + W_{44}) \cos^2 \theta + (W_...
...} + W_{44})^2}{W_{33} - W_{44}} \sin^2 \theta
\right).
\nonumber\end{eqnarray}](img7.gif) |
(2) |
| |
Choosing the positive root yields the
P-wave phase velocity near the vertical axis, as follows:
| ![\begin{displaymath}
W_{P} (\theta) \ =\ W_{P,z} \ c^2 + W_{P,xnmo} \ s^2 ,\end{displaymath}](img8.gif) |
(3) |
where
,
,
| ![\begin{displaymath}
W_{P,z} \ =\ W_{33},\end{displaymath}](img11.gif) |
(4) |
and
| ![\begin{displaymath}
W_{P,xnmo} \ =\ W_{44} + \frac{(W_{13} + W_{44})^2}{W_{33} - W_{44}}.\end{displaymath}](img12.gif) |
(5) |
WP,z is the vertical P-wave phase velocity squared and WP,xnmo
is the horizontal normal moveout (NMO) phase velocity squared.
Choosing the negative root in equation
(2) yields SV-wave phase velocities near the vertical axis,
as follows:
| ![\begin{displaymath}
W_{SV} (\theta) \ =\ W_{SV,z} \ c^2 + W_{SV,xnmo} \ s^2,\end{displaymath}](img13.gif) |
(6) |
where
| ![\begin{displaymath}
W_{SV,z} \ =\ W_{44},\end{displaymath}](img14.gif) |
(7) |
and
| ![\begin{displaymath}
W_{SV,xnmo} \ =\ W_{11} - \frac{(W_{13} + W_{44})^2}{W_{33}- W_{44}}.\end{displaymath}](img15.gif) |
(8) |
The previous expressions for the NMO
velocities agree with the results
of Thomsen (1986) and Vernik and Nur (1992).
The expression for SH-wave phase velocities near the
vertical axis is
| ![\begin{displaymath}
W_{SH} (\theta) \ =\ W_{SH,z} \ c^2 + W_{SH,xnmo} \ s^2,\end{displaymath}](img16.gif) |
(9) |
where
| ![\begin{displaymath}
W_{SH,z} \ =\ W_{44},\end{displaymath}](img17.gif) |
(10) |
and
| ![\begin{displaymath}
W_{SH,xnmo} \ =\ W_{SH,x} \ =\ W_{66}.\end{displaymath}](img18.gif) |
(11) |
Next: From elastic constants to
Up: FORWARD MAPPING
Previous: From elastic constants to
Stanford Exploration Project
11/17/1997