Upwind finite-difference calculation of traveltimes

Jos van Trier and William W. Symes

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ABSTRACT

Fluid flow is often described by conservation laws that define the conservation of mass, momentum, or energy in a fluid. In fluid mechanics, a standard technique for solving such laws is upwind finite differencing, a numerical method that uses different finite-difference operators depending on the direction of the fluid flow. Upwind finite-difference methods are more stable than centered finite-difference techniques because they mimic the behavior of fluid flow by only using information taken from upstream in the fluid.

Seismic traveltimes can be computed with upwind finite differences by solving a transformed eikonal equation. The transformed equation is a conservation law that describes the changes in the gradient components of the traveltime field among points on the computational grid.

A first-order upwind finite-difference scheme proves accurate enough for seismic applications. The method calculates single-valued traveltime functions (i.e. first arrival times) efficiently on a regular grid, and is useful both in Kirchhoff migration and modeling and in seismic tomography.