APPENDIX B

We want to find the moveout equation of a CSL at surface location x. Suppose a reflector of dipping angle $\theta$ is located at (x_t,z_t); then equation (11) gives the coordinates, (x,z), to which the reflector is mapped after migration. According to our assumption, the lateral change of the dipping angle around the surface location x can be neglected. Let $\hat{z}$ be the true depth of the reflector at surface location x, then we have following relations:

$$\begin{array} {l} z_t-x_t \tan \theta = \hat{z}-x \tan \theta, \\ x_t-x_s =z_t \tan \theta. \end{array}\eqno(B.1)$$

Solving x_t and z_t from this pair of equations,

$$\begin{array} {l} x_t = x_s+{\displaystyle{\sin \alpha \over... ...\hat{z}\cos \theta +(x_s-x)\sin \theta]. \end{array}\eqno(B.2)$$

Replacing x_t and z_t in equation (11) with equation (B.2) and reorganizing the expressions in (11), we obtain equation (12), in which the variable $\alpha$ is replaced by $\phi$, as defined in equation (10):

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