Derivatives of traveltimes

For a general variable-slowness model, traveltimes can be calculated on a regular grid by using a finite difference scheme. Two such algorithms are commonly used, Vidale's method and Van Trier's method. Vidale (1988) approximates the eikonal equation by finite differences and solves directly for traveltime using a planar or circular wavefront extrapolation. In Van Trier's method (Van Trier and Symes, 1990), the eikonal equation is transformed into a hyperbolic conservation law that relates the gradient components of traveltimes. A first-order upwind finite-difference scheme is used to solve this transformed equation and obtain the partial derivatives of the traveltimes. The traveltimes are then calculated by integrating these partial derivatives along any chosen axis.

Comparing his method with Vidale's method, Van Trier pointed out that his algorithm is vectorizable and thus more efficient. For the present purposes, there is also another reason to choose Van Trier's method. The partial differential equation (4) contains the partial derivatives of traveltimes, but not the traveltimes themselves. If we use Vidale's method, the partial derivatives of traveltimes have to be calculated by the finite difference approximations

$${\partial \tau \over \partial x} = {\Delta \tau \over \Delta... ...\partial \tau \over \partial z} = {\Delta \tau \over \Delta z}.$$

Since the use of finite difference operators increases numerical noise, it it should be avoided whenever possible. With Van Trier's method, we do not have this problem because the partial derivatives of traveltimes are intermediate results of the algorithm. For reasons of stability, we need the partial derivatives to be smooth. Van Trier's algorithm sometimes gives rough derivative functions because the finite difference scheme depends on the signs of the partial derivatives that are discontinuous functions. Therefore I take some special measures to maintain smooth solutions.