Residual-moveout equation

To perform residual-velocity analysis, we need to find RMO equations that describe how the image of a reflector moves as a function of the offset between the surface locations of the reflector and the shot. By substituting relation (10) and equation (7) into (8), we have  

$$\begin{array} {lll} x & = & x_t-(\gamma^2-1){\displaystyle{\... ...+2\theta)} \over \gamma \cos (\alpha+2\theta)}}z_t.\end{array}$$ (11)

where

$$\alpha = \arctan {x_t-x_s \over z_t}.$$

This pair of parametric equations defines the RMO equation for a common depth point (x_t, z_t) where the dipping angle of the reflector is $\theta$.As we mentioned before, if the migration velocities are not equal to the actual velocities of the media, the image of the reflector moves away both horizontally and vertically from the actual position of the reflector. Therefore the residual-moveout curve for the CDP is three-dimensional. Figure 4 shows the residual moveout curves for a given reflector location (x_t, z_t), with dipping angle $\theta$ and velocity ratio $\gamma$.

 

rmo3d
Figure 4 The residual moveout of a common depth point is a 3-D curve in the coordinates of surface location x, shot location x_s, and depth z. This figure shows the residual moveout of a reflector of dipping angle $\theta=10^\circ$ with $\gamma$ ranging from 0.9 to 1.1. The thick curve corresponds to a velocity ratio $\gamma = 1$.

 

rmo2d
Figure 5 The residual moveout of a common surface location is a 2-D curve in the coordinates of the offset x-x_s and depth z. This figure shows the residual moveout of a reflector of dipping angle ranging from $-25^\circ$ to $+25^\circ$: (a) $\gamma = 0.9$; (b) $\gamma = 1.1$. The thick curves correspond to horizontal reflectors.

The application of equation (11) requires the random access of whole 3-D dataset. This process is usually expensive and perhaps unnecessary for mildly varying structures. Let us assume that the structures are so smooth that, within the range of horizontal movement of the image (due to velocity errors), the lateral variation of the dipping angle of the reflector can be neglected. Under this assumption, we can consider the RMO in common-surface-location gathers. In Appendix B, we show that for a given surface location x, depth z_t, dipping angle $\theta$ and velocity ratio $\gamma$, the RMO equation for a CSL is  

$$\begin{array} {cll} x_s & = & x-{\displaystyle{\sin (\phi-\t... ...\phi+\theta)} \over \gamma \cos (\phi+\theta)}} z_t.\end{array}$$ (12)

We can compute the RMO depth z as a function of the offset x-x_s with open angle $\phi$ as a parametric variable. Figure 5 shows the CSL residual-moveout curves for several dipping angles.