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P- and SV-wave phase velocities
near the horizontal axis can be obtained
by interchanging W11 and
W33 and c2 and s2 wherever they occur in equations
(3) through (8). Thus, for P-waves the result is
| ![\begin{displaymath}
W_P (\theta) \ =\ W_{P,x} \ s^2 + W_{P,znmo} \ c^2,\end{displaymath}](img19.gif) |
(12) |
where
| ![\begin{displaymath}
W_{P,x} \ =\ W_{11},\end{displaymath}](img20.gif) |
(13) |
and
| ![\begin{displaymath}
W_{P,znmo} \ =\ W_{44} + \frac{(W_{13} + W_{44})^2}{W_{11} - W_{44}}.\end{displaymath}](img21.gif) |
(14) |
For SV-waves, the expression for the phase velocity near the
horizontal
axis is
| ![\begin{displaymath}
W_{SV} (\theta) \ =\ W_{SV,x} \ s^2 + W_{SV,znmo} \ c^2,\end{displaymath}](img22.gif) |
(15) |
where
| ![\begin{displaymath}
W_{SV,x} \ =\ W_{44},\end{displaymath}](img23.gif) |
(16) |
and
| ![\begin{displaymath}
W_{SV,znmo} \ =\ W_{33} - \frac{(W_{13} + W_{44})^2}{W_{11}- W_{44}}.\end{displaymath}](img24.gif) |
(17) |
Near the horizontal axis the SH-wave phase velocities are
| ![\begin{displaymath}
W_{SH} (\theta) \ =\ W_{SH,x} \ s^2 + W_{SH,znmo} \ c^2,\end{displaymath}](img25.gif) |
(18) |
where
| ![\begin{displaymath}
W_{SH,x} \ =\ W_{66},\end{displaymath}](img26.gif) |
(19) |
and
| ![\begin{displaymath}
W_{SH,znmo} \ =\ W_{SH,z} \ =\ W_{44}.\end{displaymath}](img27.gif) |
(20) |
In the rest of the paper I refer to the elliptical parameters
WP,x, WP,z, WP,xnmo,
WP,znmo, WSV,x, WSV,z, WSV,xnmo, WSV,znmo,
WSH,x, WSH,z, WSH,xnmo, and WSH,znmo as
W*, direct or NMO phase velocity squared for P-, SV-, and
SH-waves.
Next: INVERSE MAPPING
Up: FORWARD MAPPING
Previous: From elastic constants to
Stanford Exploration Project
11/17/1997