Spherical elastic ray invariant ($p=r\sin i/\alpha (r)$)

For an isotropic radial wave speed $\alpha (r)$, the elastic ray equations imply conservation of $\mathbf{x}\times \mathbf{y}$, where $\mathbf{y}$ is the slowness vector. Indeed, its derivative is $\alpha \mathbf{y}\times \mathbf{y}+\mathbf{x}\times \nabla (1/\alpha )=0$. Thus each nonradial elastic ray lies in a fixed plane through the centre and $p=|\mathbf{x}\times \mathbf{y}|$ is constant. The angle $i$ is measured from the radial direction; tangential slowness continuity preserves $p$ across spherical interfaces.

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.

In a constant-speed spherical shell with outer radius $R$ and inner radius $a$, a symmetric single reflection from the inner boundary has the displayed travel time. The boundary lies on the angular bisector of source and receiver. The branch exists only for $0\le \Delta \le 2\arccos (a/R)$, since the incident line must remain in the shell. At the endpoint it joins the mantle-only chord branch with the same slope $a/c$. Continuing the formula beyond that endpoint would make the line intersect the inner boundary before its supposed reflection point.