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Approximation to Levin and Shah's Traveltime Equations for mild reflector dip

In a constant-velocity medium, the expression for xp derived earlier, equation (4), simplifies to:  
 \begin{displaymath}
x_p = \frac{\tau}{\tau+\tau^*}x\end{displaymath} (9)
Then x-xp, which will be needed later, simplifies to:  
 \begin{displaymath}
x-x_p = \frac{\tau^*}{\tau+\tau^*}x\end{displaymath} (10)
Using equations (9) and (10), we can directly write the (two-way) zero offset traveltime of the seabed and subsea reflection as a function of midpoint location, y0:
      \begin{eqnarray}
\tau^*(y_0-x_p/2) &=& \tau^*(y_0) - \frac{x_p \sin{\phi}}{V} \n...
 ...(y_0) - \frac{\theta \ \tau^*(y_0) \ x}{V(\tau(y_0)+\tau^*(y_0))},\end{eqnarray}
(11)
(12)
where the small angle approximation was employed as before. Utilizing equations (9)-(12), we can now manipulate equation (3) to make it resemble Levin and Shah's moveout equation (8):
      \begin{eqnarray}
t^2 &=& \left[\tau(y_0) + \tau^*(y_0) 
 - \frac{ (\phi \ \tau(y...
 ...(\phi \ \tau(y_0) + \theta \ \tau^*(y_0))x }{V}
 + \frac{x^2}{V^2}\end{eqnarray} (13)
(14)
Equation (14) is equivalent to equation (8). Therefore, we have proven the equivalence of the moveout equations of the true and approximate raypaths shown in Figure [*], subject to the small dip angle approximation. As before, $\phi^2$ and $\theta^2$ terms were dropped in going from equation (13) to equation (14).


next up previous print clean
Next: REFERENCES Up: Acknowledgment Previous: Levin and Shah's Traveltime
Stanford Exploration Project
7/8/2003