Finite-difference solution of the eikonal equation along expanding wavefronts

We show that a scheme to solve the 2-D eikonal equation by a finite-difference method can violate causality for moderate to large velocity contrasts (V₂/V₁>√2). As an alternative, we present a finite-difference scheme in which the solution region progresses outward from an ``expanding wavefront'' rather than an ``expanding square,'' and therefore honors causality. Our method appears to be stable and reasonably accurate for a variety of velocity models with moderate to large velocity contrasts. The penalty is a large increase in computational cost and programming effort.