Derivation of Snell Resampling Operator  

In the following appendix, I derive the Snell resampling operation, equation (). 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, or spatial derivative, of the traveltime curves of the two events is the same at x and x_p. First we compute the stepout of the primary event, starting from the standard NMO equation:

$$\begin{eqnarray} t^2_p &=& \tau + \frac{x_p^2}{V^2} \ \frac{d}{dx_p}\left(t^... ...= \frac{2x_p}{V^2} \ \frac{d t_p}{dx_p} &=& \frac{x_p}{t_p V^2}.\end{eqnarray}$$ (49) (50) (51)

Using equations () and (), we can similarly compute the stepout of the corresponding j^th-order pegleg multiple:  

$$\frac{d t_m}{dx} = \frac{x}{t_m V_{eff}^2}.$$ (52)

Finally, we compute x_p as a function of x by squaring equations () and (), setting them equal, and substituting traveltime equations () and () for t_m and t_p, respectively:

$$\begin{eqnarray} \frac{x_p^2}{t_p^2 V^4} &=& \frac{x^2}{t_m^2 V_{eff}^4}. \ x^... ...{x^2 \tau^2 V^4}{(\tau+j\tau^*)^2 V_{eff}^4 + x^2(V_{eff}^2-V^2)}.\end{eqnarray}$$ (53) (54) (55)