VISCOSITY SOLUTIONS AND UPWIND SCHEMES

In this section we introduce the key ideas of nonclassical solution and viscosity solution of eikonal equations and related problems, which underlie our treatment of traveltime calculations. Our approach is based on this conjecture: The first-arrival time field is a viscosity solution of the eikonal equation. The conjecture is supported by some theoretical results and by a great deal of numerical ediffvidtimeence. The notion of viscosity solution was developed first for hyperbolic conservation laws, particular instances of which are satisfied by the gradient components of the time field. Numerical methods for these conservation laws use upwind finite differencing, and may be adapted to produce viscosity solutions of the eikonal equation. Here we introduce these ideas in a non-technical, heuristic way. Formal mathematical treatment can be found in subsequent sections, or in the references.