Travel-time curve

Remarks

Where the curve is a smooth function $T(X)$ near a ray, its slope is the ray parameter of the ray: $dT/dX = p$. Writing $T = pX + \tau(p)$ defines the intercept time $\tau(p)$, and $d\tau/dp = -X$. Example: in a layer of thickness $h$ and speed $v_1$ over a half-space of speed $v_2 > v_1$, with the source at the surface, the direct wave has $T = X/v_1$. The head wave refracted along the interface has $T = X / v_2 + 2h\sqrt{1/v_1^2 - 1/v_2^2}$, a line of slope $1/v_2$. Its slope $1/v_2 = \sin i_c / v_1$ is the ray parameter of the ray at the critical angle $i_c$, as expected. The two lines cross at $X = 2h \sqrt{(v_2 + v_1)/(v_2 - v_1)}$, beyond which the head wave arrives first. Non-example: a single seismogram, ground motion against time at one station, is not a travel-time curve; it gives only one value of $X$. The curve can be multivalued, with triplications, when the speed increases fast with depth.