Problem setup

Let us assume that reflecting interface i is given through two points belonging to it, A and B. Since traveltimes are computed independently for each CMP, we use a coordinate system with the origin in the midpoint between source and receiver (both located at the surface). In this CMP-centric coordinate system, interface i can be expressed as  

$$z = x \tan \theta_i+\frac{d_i}{\cos\theta_i},$$ (1)

where $\theta_i$ is the dip of the interface:  

$$\tan \theta_i = \frac{z_b-z_a}{x_b-x_a}$$ (2)

and d_i is the distance from the CMP point to the interface:  

$$d_i = z_a\cos\theta-x_a\sin\theta.$$ (3)

Figure shows a three-interface example.

 

peglegmodel Figure 1 Three interface reflectivity model for illustrating the meaning of the notations di and $\theta_i$. Notice the sign convention for angles.

Similar to Levin and Shah (1977), we use the method of images to compute traveltimes. Let us denote with $Q\left(x_q,z_q\right)$ the image of point $P\left(x_p,z_p\right)$ through reflector i (see Figure ). Through simple analytical geometry we find that  

$$\underbrace {\left[ {\begin{array} {*{20}c} {x_q } \\ {z_q... ...a_i } \\ \end{array}} \right]}_{{\bf b}\left(\theta_i\right) }.$$ (4)

 

imex Figure 2 Image point concept illustration.