HEMNO Equivalence with Levin and Shah's Equations  

In this appendix, I prove that the HEMNO equation is equivalent to Levin and Shah's traveltime equation Levin and Shah (1977) in the limit of small dip angle. They show that in a constant velocity medium with dipping target reflector and multiple generator, the moveout equation of the ``S102G'' pegleg multiple (see Figure ) is:  

$$t^2 = \left[\tau^* \cos{\theta} + \tau \cos{\phi}\right]^2 ... ...+\theta)}}{V} - \tau^* \sin{\theta} - \tau \sin{\phi}\right]^2,$$ (39)

where $\phi$ and $\theta$ are the dip angle (in radians) of the multiple generator and target reflector, respectively. $\tau^*$ and $\tau$ are the zero-offset traveltimes to the two reflectors, x is offset, and V is the medium velocity. 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 () accordingly:  

$$t^2 = \left[\tau^* + \tau \right]^2 + \left[\frac{x}{V} - \tau^* \theta - \tau \phi\right]^2.$$ (40)

Multiplying out the squares in equation () and collecting terms gives:  

$$t^2 = \left[\tau^* + \tau \right]^2 + \frac{x^2}{V^2} - 2\... ... - 2\frac{\phi \tau x}{V} + (\tau^* \theta)^2 + (\tau \phi)^2.$$ (41)

The $\theta^2$ and $\phi^2$ terms are negligible for small angles, so we can ignore these terms and further simplify equation ():  

$$t^2 = \left[\tau^* + \tau \right]^2 + \frac{x^2}{V^2} - 2\frac{ (\theta \tau^* + \phi \tau) x }{V}.$$ (42)

I will now show that the HEMNO equation () is equivalent to the Levin/Shah equation () under the constant velocity and small dip angle assumptions. First I make some preliminary definitions. In a constant-velocity medium, the expression for x_p, equation (), simplifies to:  

$$x_p = \frac{\tau}{\tau+\tau^*}x.$$ (43)

Then x-x_p, which will be needed later, simplifies to:  

$$x-x_p = \frac{\tau^*}{\tau+\tau^*}x$$ (44)

Since the reflectors in this derivation are assumed planar and the velocity is assumed constant, using equations () and (), we can directly write the (two-way) zero offset traveltime to the seabed and subsea reflection at any midpoint as a function of the corresponding zero-offset traveltimes at the midpoint location, y₀:

$$\begin{eqnarray} \tau^*(y_0-x_p/2) &=& \tau^*(y_0) - \frac{x_p \sin{\phi}}{V} \n... ...frac{\theta \ \tau^*(y_0) \ x}{( \ \tau(y_0)+\tau^*(y_0) \ ) \ V},\end{eqnarray}$$ (45) (46)

where the small angle approximation was employed as before. Substituting the zero-offset traveltimes () and () into the HEMNO equation () yields:

$$\begin{eqnarray} t^2 &=& \left[\tau(y_0) + \tau^*(y_0) - \frac{ \left( \ \phi ... ...y_0) + \theta \ \tau^*(y_0) \ \right) \ x }{V} + \frac{x^2}{V^2}.\end{eqnarray}$$ (47) (48)

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