Seismic rays in a linear velocity gradient
= Seismic rays in a linear velocity gradient
{title2=$T(X)=2\operatorname{arsinh}(\beta X/(2v_s))/\beta$}
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 $T=2\operatorname{arcosh}(v_t/v_s)/\beta$. Differentiation gives $dT/dX=p$, the <seismic ray parameter>. In the limit $\beta\to0$ at fixed range, $T\to X/v_s$.