Snell traces

the radial-trace coordinate system can be used no matter what the velocity of the earth. But the coordinate system has a special advantage when the velocity is constant, because then it gathers all the energy of a fixed propagation angle. The logical generalization to stratified media is to gather all the energy with a fixed Snell parameter. A Snell trace is defined (Ottolini) as a trajectory on the (x,t)-plane where the stepout p=dt/dx would be constant if the velocity were v(z). Where the velocity increases with depth, the Snell traces bend upward. The Snell trace trajectory is readily found by integrating the ray equations:  

$$x \eq \int_0^z \ \tan \, \theta \ dz$$ (5)

 

$$t \eq \int_0^z \ {dz \over v \ \cos \, \theta }$$ (6)

To do moveout correction on the Snell traces, introduce the vertical travel-time depth t such that $dz=vd \tau$.The radial-trace moveout-correction equations become

$$\begin{eqnarray} x(p, \tau ) \ \ \ &=&\ \ \ \int_0^{\tau} \ {p\,v^2 \over \sqrt... ... &=&\ \ \ \int_0^{\tau} \ {1 \over \sqrt{1-p^2 v^2 } } \ \ d \tau\end{eqnarray}$$ (7) (8)

Where the earth velocity is stratified, Snell traces have a theoretical advantage over radial traces. However they have the disadvantage that the curves could become multibranched, so that the transformation would not be one-to-one. So in practice you might use a simplified velocity model instead of your best estimate of the true velocity.

More philosophically, the transition from constant-offset traces to radial traces is a big one, whereas the transition from radial traces to Snell traces is not so large. Since the use of radial traces is not widespread, we can speculate that the practical usefulness of Snell traces may be further limited.