Residual Moveout in ADCIGs

In this appendix I show that, for a flat reflector, the residual moveout of the multiples in ADCIGs reduces to the tangent-squared expression derived by Biondi and Symes (2004) for the residual moveout of under-migrated primaries:

$$\Delta \mathbf{n}_{\mbox{RMO}}=(\rho-1)\tan^2\gamma z_0 \mathbf{n}.$$ (7)

Start with Equation 

$$z_{\xi_\gamma}=\frac{z_{\xi_\gamma}(0)}{1+\rho}\left[1+\frac... ...2-(1-\rho^2)\tan^2\gamma)}{\sqrt{\rho^2-\sin^2\gamma}}\right],$$ (8)

where $z_{\xi_\gamma}(0)$ is the normal-incidence migrated-depth, (i.e. $z_0$) in the previous equations.

There is an important and unfortunate difference in notation here, however, because $\rho$ in equation A-1 is the ratio of the migration to the true slowness whereas $\rho$ in equation A-2 is the ratio of the migration to the true velocity. Therefore, in order to get a better idea of how the approximation for the RMO of the multiples (accounting for ray bending at the reflector interface) relates to that of the primaries (neglecting ray bending), I rewrite equation A-2 replacing $\rho$ by $1/\rho$ and $z_{\xi_\gamma}(0)$ with $z_0$ to get:

$$z_{\xi_\gamma}=\left[\rho+\frac{\cos\gamma(1-(\rho^2-1)\tan^2\gamma)}{\sqrt{1-\rho^2\sin^2\gamma}}\right]\frac{z_0}{1+\rho}.$$ (9)

Since $\Delta n_{\mbox{RMO}}=z_0-z_{\xi_\gamma}$ we get:

$$\Delta n_{\mbox{RMO}}=\left[1-\frac{\cos\gamma(1-(\rho^2-1)\... ...\gamma}{\sqrt{1-\rho^2\sin^2\gamma}}\right]\frac{z_0}{1+\rho}.$$ (10)

For small $\gamma$, $\sin\gamma\approx 0$ and $\cos\gamma\approx 1$, therefore

$$\Delta n_{\mbox{RMO}}=(\rho^2-1)\tan^2\gamma\frac{z_0}{1+\rho}=(\rho-1)\tan^2\gamma z_0.$$ (11)

This is the same as equation A-1 save for the unit vector $\mathbf{n}$. This result is intuitively appealing because it shows that the approximation of neglecting ray bending at the reflecting interface deteriorates as the aperture angle increases which is when the ray bending is larger.