Traveltime computation with the linearized eikonal equation

ALGORITHM

Linearization of the eikonal equation suggests the following algorithm of traveltime computation:

1. Start with an initial traveltime field $\tau_0$. The initial traveltime may be the result of a previous computation or (for simple models) the result of an approximate analytic evaluation.
2. Compute the finite-difference gradient $\nabla \tau_0$ and the corresponding slowness model $n_0$ with equation (2).
3. Compute the slowness perturbation $n_1$ as the difference between the true slowness model $n$ and $n_0$. Exit the computation if the perturbation is smaller than the desired accuracy.
4. Solve numerically equation (5) for the traveltime perturbation $\tau_1$.
5. Update the traveltime field $\tau_0$ by adding $\tau_1$ to it.
6. Repeat the loop.

Equation (5) can be solved numerically with a simple explicit upwind finite-difference method.