Seismic ray parameter in horizontal layering ($p=\sin i/v=dT/dx$)

In horizontally layered isotropic media, horizontal slowness is conserved along a ray. The incidence angle $i$ is measured from vertical. A smooth turning depth satisfies $pv=1$. The range and travel time are $x=2\int pvdz/\sqrt{1-p^2{v}^2}$ and $T=2\int dz/[v\sqrt{1-p^2{v}^2}]$ for a symmetric surface-to-surface turning ray.

A ray's intercept time subtracts its tangent travel-time contribution at the receiver range. For a smooth branch, $dT/dx=p$ implies $d\tau /dp=-x$. Thus an intercept-time curve encodes range as its negative slope. Separate turning branches around a low-velocity zone must be retained rather than joined by fictitious rays.