Preamble: the ray parameter in three dimensions

In a two-dimensional earth model, the ray parameter is given at any point of the ray by the relationship  

$$p = \frac{\sin\theta }{v} ,$$ (1)

where $\theta$ is the inclination of the ray path with respect to the vertical axis and v is the local velocity. We can see that the ray parameter is simply the projection length of the ray path vector $\bf r$($\Vert{\bf r}\Vert = r = 1/v$) on the earth surface. In a 3-D v(z) model of the earth, the rays travel in a vertical plane. In this case, the ray parameter, being the projection of the ray path vector, is a two-dimensional vector that can be expressed in either cartesian or polar coordinates, as follows:  

$${\bf p} = \left( \begin{array} {c} p_x \\ p_y \end{array}... ...\begin{array} {c} \cos\phi \\ \sin\phi \end{array} \right) ,$$ (2)

where $\phi$ is the strike of the vertical plane containing the ray and $p = \sqrt{ p_x^2 + p_y^2 }$.