Finite difference applied to the Eikonal and transport equations

The main feature of applying finite difference to the Eikonal equation, as opposed to the wave equation, is the efficiency gained in calculating traveltimes and computing synthetic sections. A major reason for the cost difference between using the wave equation and Eikonal equation is the difference in the dimensionality of the problem. Using the Eikonal equation we solve for $\tau(x,y,z)$, whereas by using the wave equation we solve for the time-varying wavefield, W(x,y,z,t). Add to that, the fine grid necessary to solve the wave equation to avoid dispersion. However the eikonal equation is a high frequency asymptotic approximation that would only provide traveltime information for the fastest arrival Vidale (1990). In complex media, the fastest arrival is not necessary the most energetic, and as a result, traveltime solutions obtained by applying the finite difference scheme on the eikonal equation might result in less sufficient solutions.