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Let the surface radius be $a$, let $\Delta$ denote epicentral angle in radians, and put $\eta (r)=r/v(r)$. The spherical elastic ray invariant gives $\sin i=p/\eta$. Along either radial leg, if $l$ is arclength and $\theta$ polar angle in the ray plane,For a turning ray with minimum radius $r_p$ defined by $\eta (r_p)=p$, the outgoing and incoming legs therefore giveThe square-root turning singularity is integrable for a regular turning point.

Introduce the reduced travel timeIts integrand vanishes at the moving lower endpoint, so differentiation gives ${\tau }^{\prime }(p)=-\Delta (p)$. Consequentlyon every regular travel-time branch. This is also the endpoint variation of travel time: the tangential endpoint slowness multiplied by the surface radius is $p$. Angles expressed in degrees would introduce the corresponding conversion factor; the displayed equality uses radians.

For Herglotz–Wiechert inversion, assume $\eta$ is strictly increasing so that $r=r(\eta )$ is single-valued, and set ${\eta }_a=a/v(a)$ and $h(\eta )=d\log r/d\eta$. The angular integral becomes an Abel integral equation:For any $0<\mu <{\eta }_a$, multiply by $(p^2-{\mu }^2{)}^{-1/2}$ and integrate. Interchanging the triangular region $\mu <p<\eta <{\eta }_a$ givesThe inner integral equals $\pi /2$: the substitution $p^2={\mu }^2+({\eta }^2-{\mu }^2){\sin }^2\theta$ reduces it to ${\int }_0^{\pi /2}d\theta$. Thus the full constructive inversion isThe surface radius and surface slowness fix the normalization. Varying $\mu$ reconstructs radius and speed parametrically; the centre is recovered by a limit when the data extend that far. The calculation assumes smooth regular turning rays and complete branch data, not merely the first arrival at each epicentral angle.

For a check, a constant speed $v_0$ gives ${\eta }_a=a/v_0$, $\Delta (p)=2\arccos (p/{\eta }_a)$ and $T(p)=2\sqrt{a^2-p^2{v}_0^2}/v_0$. These satisfy $dT/d\Delta =p$, and the inversion returns $r(\mu )=v_0\mu$.

If $\eta$ decreases outwards on an interval, then it increases as a surface ray travels inward through that interval. Such a ray cannot turn there, since turning requires the decreasing inward value of $\eta$ to reach $p$. Rays that penetrate this interval can traverse it and turn deeper, but the map $\eta \mapsto r$ now has several branches. The angular data constrain combinations of those branch contributions, rather than the single function $h(\eta )$ used above. This is a hidden turning-ray zone from nonmonotone spherical slowness. The stated single-valued inversion fails and does not uniquely locate the speed profile in that interval without additional information. Missing turning rays and possible multiple travel-time branches are the obstruction, not a failure of the algebraic Abel-kernel identity.

Because $s$ is arc length, $|d\mathbf{x}/ds|=1$. Thus $\mathbf{y}=\mathbf{n}/\alpha$, where $\mathbf{n}$ is the unit tangent to the elastic ray. In an isotropic medium this is also the wavefront normal. Consequently $\mathbf{y}$ is the vector slowness, of magnitude $1/\alpha$; locally $\mathbf{y}=\nabla T=\mathbf{k}/\omega$, with $T$ the travel time and $\mathbf{k}$ the wavevector.

For depth-dependent wave speed $\alpha (z)$, the ray equation gives $d{y}_x/ds=d{y}_y/ds=0$. The horizontal component of $d\mathbf{x}/ds$ therefore has a fixed direction, and the elastic ray lies in the vertical plane determined by that direction and its initial point. A vertical elastic ray is the limiting case. If $i$ is the angle to the vertical, its conserved horizontal slowness givesAt a horizontal interface, matching the tangential wavevector at fixed frequency gives the same invariant on both sides: this is Snell's law.

For radial wave speed $\alpha (r)$, take the cross product of position with slowness. The ray equations givesince the second vector is radial. A nonzero constant $\mathbf{L}=\mathbf{x}\times \mathbf{y}$ is perpendicular to every position on the elastic ray, so the ray lies in a fixed plane through the centre. Radial rays obey the same conclusion with a nonunique plane. The spherical elastic ray invariant is thereforewhere $i$ is measured from the radial direction. Across a spherical interface the radius is common and tangential slowness is continuous, so this form of Snell's law survives an interface discontinuity.

In a homogeneous sphere a P wave follows the straight chord between the two surface points. If their principal angular separation is $0\le \Delta \le \pi$, its length is $2R\sin (\Delta /2)$. For constant P wave speed $c_P$,Now put $a=R^{\prime }$ and denote the mantle and core P wave speeds by $c_m>c_c$. Equal incidence and reflection angles put the core-boundary reflection point on the angular bisector. Each mantle leg has length $L$ given by $L^2=R^2+a^2-2Ra\cos (\Delta /2)$. Hence the core-reflected travel time isThe stated domain is essential. At the reflection point the incoming line approaches from the mantle only if $R\cos (\Delta /2)\ge a$. Otherwise it has already crossed the core boundary. The mantle-only direct chord also exists only for $\Delta \le {\Delta }_g$, with $T_D=2R\sin (\Delta /2)/c_m$. In this interval,so the reflected arrival is later. Its intercept is $2(R-a)/c_m$, whereas the direct intercept is zero. At grazing incidence the branches meet at $T=2\sqrt{R^2-a^2}/c_m$, and both have slope $dT/d\Delta =a/c_m$. A continuation of the homogeneous-sphere direct formula beyond this endpoint is not a mantle-only arrival in the modified model.

The core-transmitted elastic rays consist of three straight legs. Their impact distances in mantle and core are $b_m=p{c}_m$ and $b_c=p{c}_c$, respectively. At core entry,Since $c_c<c_m$, the elastic ray bends towards the normal on entry and away from it on exit. The mantle leg subtends $i_m-i_R$ at the centre, and the core chord subtends $\pi -2i_c$. Thus the total angular sweep along the transmitted path is $2(i_m-i_R)+\pi -2i_c$. This sweep need not equal the principal separation if it exceeds $\pi$; the diagram uses a path with sweep below $\pi$.