Ray parameter

Remarks

Equivalently, $p$ is the horizontal component of the slowness vector $\nabla T$, where $T$ is the travel time; in case 2, $p$ is the component along the great circle times $r$. If $T(X)$ is the travel time from a surface source to a surface receiver at distance $X$, then $dT/dX = p$ for the ray that arrives at $X$. Example: in a half-space with $v(z) = v_0 + g z$, $g > 0$, a ray leaving the surface at angle $i_0$ has $p = \sin i_0 / v_0$. It turns where $\sin i = 1$, that is at the depth $z$ with $v(z) = 1/p$, so $z = (1/p - v_0)/g$. Non-example: if $v$ also depends on the horizontal position, $\sin i / v$ changes along a ray, so case 1 does not define a single number for the ray.