Theory overview

The residual moveout of primaries in ADCIGs, under the approximation of stationarity of the rays (local constant velocity) is given by Biondi and Symes (2004):  

$$\Delta \mathbf{n}_{\mbox{RMO}}=\frac{\rho-1}{\rho}\frac{\sin^2\gamma}{(\cos\alpha-\sin^2\gamma)\cos\alpha}\bar{z}\mathbf{n},$$ (1)

where $\Delta \mathbf{n}_{\mbox{RMO}}$ is the residual moveout function with respect to the aperture angle $\gamma$, $\rho$ is the ration between the migration and the true slowness, $\alpha$ is the reflector dip, $\bar{z}$ is the true (unknown) depth of the reflector and $\mathbf{n}$ is the unit normal vector to the reflector in the direction of decreasing depth. For a flat reflector ($\alpha=0$) equation 1 reduces to  

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

For primaries, we can estimate the true depth $\bar{z}$ using the migration depth at normal incidence z₀ Biondi and Symes (2004) as  

$$z_0=\frac{\bar{z}}{\rho}$$ (3)

which leads to the simple result  

$$\Delta \mathbf{n}_{\mbox{RMO}}=(\rho-1)\tan^2\gamma z_0 \mathbf{n}.$$ (4)

For specularly-reflected multiples, Alvarez 2005 showed that, for a flat reflector, the functional dependence between the image depth and the aperture angle is given by  

$$z_{\xi_\gamma}=\frac{z_{\xi_\gamma}(0)}{1+\rho)}\left[1+\fra... ...^2-(1-\rho^2)\tan^2\gamma)}{\sqrt{\rho^2-\sin^2\gamma}}\right],$$ (5)

where $z_{\xi_\gamma}(0)$ is the normal-incidence migrated-depth, (i.e. z₀) in the previous equations. There is an important and unfortunate difference in notation, however, because $\rho$ in equations 1 through 4 is the ratio of the migration to the true slowness whereas $\rho$ in equation 5 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 5 replacing $\rho$ by $1/\rho$ and $z_{\xi_\gamma}(0)$ with z₀ to get:  

$$z_{\xi_\gamma}=\left[1+\frac{\cos\gamma(\rho^2-(\rho^2-1)\tan^2\gamma)}{\sqrt{1-\rho^2\sin^2\gamma}}\right]\frac{z_0}{1+\rho}.$$ (6)

Finally, since $\Delta \mathbf{n}_{\mbox{RMO}}=z_0-z_{\xi_\gamma}$ we get:  

$$\Delta \mathbf{n}_{\mbox{RMO}}=\left[1-\frac{\cos\gamma(1-(\... ...sqrt{1-\rho^2\sin^2\gamma}}\right]\frac{z_0}{1+\rho}\mathbf{n},$$ (7)

which, for small $\gamma$, reduces to  

$$\Delta \mathbf{n}_{\mbox{RMO}}=(\rho^2-1)\tan^2\gamma\frac{z_0}{1+\rho}=(\rho-1)\tan^2\gamma z_0\mathbf{n}.$$ (8)

This is the same as equation 4. This result is intuitively appealing because it shows that the approximation of neglecting the ray bending at the reflecting interface deteriorates as the aperture angle increases which is when the ray bending is larger.

Figure  shows a comparison of the residual moveout curves for an ADCIG computed with equations 7 (ray-bendinga pproximation) and 8 (straight-ray approximation). The residual moveouts correspond to a water-bottom multiple from the flat interface of a two layer model where the top layer is water and the second layer is a half space. The velocity of the water layer is 1500 m/s and its thickness is 500 m. The velocity of the half space is 2500 m/s. The migration was done with the true velocity model. Therefore, there is significant ray bending of the multiple at the reflecting interface. Figure  shows the actual ADCIG with the depth moveout as a function of angle superimposed for both approximations. For large aperture angles the departure of the straight ray approximation can be significant.

 

rmos2 Figure 1 Residual moveout curves for an ADCIG from a two flat-layer model. The curves correspond to straight ray and the ray-bending approximations to a water-bottom multiple.

 

adcig1 Figure 2 ADCIG for a water-bottom multiple from a two flat-layer model. The dotted curve corresponds to the straight ray approximation whereas the solid curve corresponds to the ray-bending approximation.