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Levin and Shah's Traveltime Equations

Levin and Shah (1977) show that for a flat subsea reflector and dipping seabed, the moveout equation of the ``S102G'' (source-seabed-surface-reflector-receiver) pegleg multiple is:  
 \begin{displaymath}
(tV)^2 = \left[\tau^* \cos{\theta} + \tau \cos{\phi}\right]^...
 ...eta)}}{V} - \tau^* V \sin{\theta} - \tau V \sin{\phi}\right]^2,\end{displaymath} (5)
where $\phi$ and $\theta$ are the dip angle (in radians) of the seabed and subsea reflector, respectively. For small dip angles (i.e., less than 5 degrees), we can make the small angle approximation for angles $\phi$, $\theta$, and $\phi+\theta$to update equation (5) accordingly:  
 \begin{displaymath}
(tV)^2 = \left[\tau^* + \tau \right]^2 
 + \left[x - \tau^* V \theta - \tau V \phi\right]^2.\end{displaymath} (6)
Multiplying out the squares in equation (6) and collecting terms gives:  
 \begin{displaymath}
t^2 = \left[\tau^* + \tau \right]^2 + \frac{x^2}{V^2} 
 - 2\...
 ... - 2\frac{\phi \tau x}{V}
 + (\tau^* \theta)^2 + (\tau \phi)^2.\end{displaymath} (7)
The $\theta^2$ and $\phi^2$ terms are negligible for small angles, so we can ignore these terms and further simplify equation (7):  
 \begin{displaymath}
t^2 = \left[\tau^* + \tau \right]^2 + \frac{x^2}{V^2} 
 - 2\frac{ (\theta \tau^* + \phi \tau) x }{V}.\end{displaymath} (8)
Notice that equation (8) is equivalent to the previously derived equation (1) for a first-order pegleg, with the addition of an offset-dependent correction term for the dipping layers.

Although explicit seabed and subsea reflector dip angles, $\phi$ and $\theta$,are contained in equation (14), they were introduced only to show equivalence to equation (8). Locally-planar reflectors are not required to implement equation (3).


next up previous print clean
Next: Approximation to Levin and Up: Acknowledgment Previous: Acknowledgment
Stanford Exploration Project
7/8/2003