Residual moveout in ADCIGs

The inconsistencies between the migrated images at different aperture angles are the primary source of information for velocity updating during Migration Velocity Analysis (MVA). An effective and robust method for measuring inconsistencies between images is to compute first semblance scans as a function of one ``residual moveout'' (RMO) parameter, and then pick the maxima of the semblance scan. This procedure is most effective when the residual moveout function used for computing the semblance scans closely approximates the true moveouts in the images.

Appendix A presents the derivation of a RMO functions for scanning 3-D ADCIGs. The starting point of our derivation is the 2-D RMO function presented by Biondi and Symes (2003). Its generalization is based on the 3-D geometrical understanding developed in the previous sections, and it leads to the following expression:  

$${\bf \Delta n}_{\rm RMO}= \frac{\rho-1}{\cos \alpha} \frac{\... ...2 \left( \eta- \phi\right)- \sin^2\gamma\right)} z_0\; {\bf n},$$ (20)

where $\rho$ is the ratio between the true velocity and the migration velocity, z₀ is the migrated depth of the normal incidence event, $\alpha$ is the geological dip, and $\eta$ is the azimuth angle of the normal to the geological dip (see Figure 13).

Equation (20) is the general expression of the RMO function in 3-D. For flat reflectors (i.e. $\alpha= 0$)equation (20) becomes the much simpler RMO function Biondi and Symes (2003) that follows:  

$${\bf \Delta n}_{\rm RMO}= \left(\rho-1\right)\tan^2\gamma z_0\; {\bf n}.$$ (21)

It is also easy to verify that when the reflection azimuth is aligned with the geological dip azimuth (i.e. ``dip reflection'' with $\eta= \phi$), equation (20) simplifies to the following 2-D RMO function:  

$${\bf \Delta n}_{\rm 2D-RMO}= \frac{\rho-1} {\cos\alpha_x{'}}... ...a}{\left(\cos^2\alpha_x{'}- \sin^2\gamma\right)} z_0\; {\bf n},$$ (22)

with $\alpha_x{'}=\alpha$.

Similarly, when the reflection azimuth is orthogonal to the geological dip (i.e. ``strike reflection'' with $\eta- \phi=\pm 90^\circ$), equation (20) simplifies into the following expression:  

$${\bf \Delta n}_{\rm RMO}= \frac{\rho-1}{\cos\alpha}\tan^2\gamma z_0\; {\bf n}.$$ (23)

Equation (23) is the RMO function for flat events, [equation (21)], except for the scaling factor of $1/\cos \alpha$ (with $\alpha= \alpha_y{'}$) that takes into account the inclination of the tilted plane in Figure 13.

The special cases of dip-reflections and strike-reflections depend on the orientation of the geological dip relative to the reflection azimuth and not on the data azimuth. They are thus subsurface (wavefields at depth) properties and not surface (recorded data) properties; that is, they depend on the velocity model in the overburden as well as on the relative orientation of the acquisition direction with respect to the geological dips.