CONCLUSIONS

The focusing eikonal equation exactly models the traveltimes in a heterogeneous medium parametrized by the vertical traveltime $\tau$ in place of depth z. The solutions of the focusing eikonal can be efficiently computed by solving the associated ray tracing equations.

A potential application of the focusing eikonal that we intend to investigate further is reflection tomography. Our preliminary analysis of the linearization errors for $\left(z,x\right)$ domain tomography and $\left(\tau,\xi\right)$ domain tomography shows that, when reflectors' movements interfere with velocity perturbations, $\left(\tau,\xi\right)$ tomography behaves more linearly than $\left(z,x\right)$ tomography. However, we feel that our analysis is only preliminary. A natural next step is to analyze the back-projection operators, and compare their properties. Further on we should perform full non-linear tomographic tests.

The first time that the first author (BB) heard about ``tracing rays in time'' was from Francis Muir few years ago. He heard about $\tau$-rays again this summer while visiting Elf's Geophysical Research Center in London.