Orthogonal equation

Let us suppose (x,z) are the coordinates before the rotation and $(x^\prime,z^\prime)$ after the rotation. One can find the relations between these two coordinates:  

$$\left\{ \begin{array} {lll} x & = & x^\prime \cos \theta_0 -... ...prime \cos \theta_0 + x^\prime \sin \theta_0,\end{array}\right.$$ (7)

where $\theta_0$ is the incident angle of the plane waves. It is easy to show that both the eikonal equation and the orthogonal equation are invariant under the transformation defined in equation (7). Therefore, we can use equations (1) and (2) after replacing (x,z) by $(x^\prime,z^\prime)$. The new definition of the function p is as follows:  

$$p(x,z) = {\partial \tau \over \partial \theta} = -s_0 x_0,$$ (8)

where s₀ is the slowness at the surface, and x₀ is the surface location of a incident ray. I do not know the meaning of p defined here. I guess it is a horizontal traveltime. Along each ray, p maintains to be constant. Therefore, we can trace a ray by following the trajectory of a contour line of p.