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Acknowledgment

WesternGeco acquired and released the Mississippi Canyon dataset in 1997. A  Derivation of Snell Resampling Operator In the following appendix, I derive the Snell resampling operation, equation (1). The graphical basis for the derivation is Figure [*]. Since the pegleg multiple and primary in the figure have the same emergence angle, $\theta$, the stepout of the two events is the same at x and xp. First we compute the stepout of the primary event (standard NMO equation):
         \begin{eqnarray}
t^2_p &=& \tau + \frac{x_p^2}{V^2}
\  
 \frac{d}{dx_p}\left(t^...
 ...= \frac{2x_p}{V^2}
\  \frac{d t_p}{dx_p} &=& \frac{x_p}{t_p V^2}.\end{eqnarray} (10)
(11)
(12)
Brown (2002) derived an extension to the conventional NMO equation which flattens peglegs to the zero-offset traveltime of the reflector of interest.  
 \begin{displaymath}
t_m = \sqrt{ (\tau+j\tau^{*})^2 + \frac{x^2}{V_{eff}^2} }\end{displaymath} (13)
Using equations (13) and (2), we can similarly compute the stepout of the corresponding jth-order pegleg multiple:  
 \begin{displaymath}
\frac{d t_m}{dx} = \frac{x}{t_m V_{eff}^2}.\end{displaymath} (14)
Finally, we compute xp as a function of x. We square equations (12) and (14), set them equal, then substitute equation (13) for tm and tp.
         \begin{eqnarray}
\frac{x_p^2}{t_p^2 V^4} &=& \frac{x^2}{t_m^2 V_{eff}^4}.
\  x^...
 ...c{x^2 \tau^2 V^4}{(\tau+j\tau^*)^2 V_{eff}^4 + x^2(V_{eff}^2-V^2)}\end{eqnarray} (15)
(16)
(17)


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Next: REFERENCES Up: Brown: Pegleg amplitudes Previous: Conclusions
Stanford Exploration Project
7/8/2003