Orthogonal equation

Let us assume the location of the point source is x₀ on the surface. With the coordinate transformation:

$$\left\{ \begin{array} {lll} x & = & x_0+r\sin \theta \\ z & = & r\cos \theta,\end{array}\right.$$ (3)

we can derive the orthogonal equation in the polar coordinates:  

$$\tau_rp_r+{1 \over r^2}\tau_\theta p_\theta=0,$$ (4)

where $\tau=\tau(r,\theta,x_0)$ and $p=p(r,\theta,x_0)$. The function p has a physical meaning of surface horizontal slowness:  

$$p = {\partial \tau \over \partial x_0} = -s_0\sin \theta_0,$$ (5)

where s₀ is the slowness at the source location, $\theta_0$ is the angle between incident ray at the surface and the vertical. Along each ray, the surface horizontal slowness p maintains to be constant. Therefore, we can trace a ray by following the trajectory of a contour line of p.