Seismic rays in a linear velocity gradient ($T(X)=2\mathrm{arsinh}(\beta X/(2v_s))/\beta$)

For $v(z)=v_s+\beta z$ with $\beta >0$, a surface-to-surface diving ray is circular. Its turning speed is $v_t=1/p$, and its range and time are $X=2\sqrt{v_t^2-v_s^2}/\beta$ and $T=2\mathrm{arcosh}(v_t/v_s)/\beta$. Differentiation gives $dT/dX=p$, the seismic ray parameter. In the limit $\beta \to 0$ at fixed range, $T\to X/v_s$.