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Substituting the theoretical interval velocity
from equation (40)
into the definition of
RMS velocity
(equation (25))
yields:
| ![\begin{eqnarray}
\tau \ V^2(\tau) &=& \int_{0}^{\tau} v^2(\tau') \ d \tau'
\\ &=& v_0^2 \ \frac {e^{\alpha \tau} - 1} {\alpha} .\end{eqnarray}](img73.gif) |
(47) |
| (48) |
Thus the desired expression for RMS velocity
as a function of traveltime depth is:
| ![\begin{displaymath}
V(\tau) \eq v_0 \
\sqrt{
\frac{e^{\alpha \tau} - 1 }{\alpha \tau}
}\end{displaymath}](img74.gif) |
(49) |
For small values of
,this can be approximated as
| ![\begin{displaymath}
V(\tau) \quad\approx \quad v_0\ \sqrt{1 + \alpha \tau / 2} .\end{displaymath}](img76.gif) |
(50) |
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Stanford Exploration Project
12/26/2000