Positive Hessian of a layered travel-time functional
= Positive Hessian of a layered travel-time functional
For three unit-height layers with positive speeds $u,v,w$, the straight-segment travel time is $t=\sqrt{1+x_1^2}/u+\sqrt{1+(x_2-x_1)^2}/v+\sqrt{1+(X-x_2)^2}/w$. Its <Hessian matrix> has form $\begin{pmatrix}p+q&-q\\-q&q+r\end{pmatrix}$ with positive $p,q,r$. Its leading minor is positive and its determinant is $pq+pr+qr>0$. Hence this time function is strictly convex, and its stationary path is the unique minimum.