Herglotz–Wiechert inversion ($r(\mu )=a\exp [-{\pi }^{-1}{\int }_{\mu }^{{\eta }_a}\Delta (p)(p^2-{\mu }^2{)}^{-1/2}dp]$)

For a spherically symmetric speed with strictly increasing $\eta (r)=r/v(r)$, a turning ray of parameter $p$ has $\Delta (p)=2p{\int }_p^{{\eta }_a}(d\log r/d\eta )({\eta }^2-p^2{)}^{-1/2}d\eta$. Integrate this against $(p^2-{\mu }^2{)}^{-1/2}$ and interchange the triangular integration region. The inner kernel integral is $\pi /2$, giving the displayed radius and then $v(r(\mu ))=r(\mu )/\mu$. Complete turning-ray branch data are required.

Where $\eta =r/v$ decreases outwards, it increases as a surface ray travels inward, so no surface ray turns inside that interval. Penetrating rays may traverse it and turn deeper. The radius is then not a single-valued function of $\eta$, and angular ray data constrain combined branch contributions rather than the single function inverted by Herglotz–Wiechert inversion. Additional structural or observational information is needed to locate the hidden profile.