3-D finite-difference traveltimes in complex media

First-arrival traveltime information is important in many seismic processes. This information is often needed on a 2D or 3D grid. Typically this is found by interpolating traveltimes calculated by raytracing. Ray-tracing can become difficult and expensive when the velocity field becomes complicated. In addition the ray-trace may be incapable of calculating the actual first arrival, for example If it is a head or guided wave. Alternative approachs to calculating traveltimes based upon finite-difference solutions of the eikonal equation have recently been proposed (Vldale (1988, 1990), Qin et al. (1990), Van Trier and Symes (1990). These methods calculate traveltime along wavefronts rather than raypaths. Wavefronts penetrate shadow zones and will generate head and guided wave arrivals (if they are the first arrival at the observation point.). Vidale's methods have been shown to be unstable in models with moderate to large velocity contrasts. Qin et al's 2D method is robust but is penalized by being computationally slow and expensive. A new method for calculating 3D first-arrival traveltimes by solving the eikonal equation with finite differences is described. It combines aspects of Vidale's 3D method with ideas developed by Qin et al. The method is robust without becoming computationally expensive. Examples of its application are presented.