Wavefronts and solutions of the eikonal equation

S U M M A R Y In this paper, several methods for the solution of the eikonal equation are investigated. Using the method of characteristics, traveltimes are computed along rays. Approximate solutions of the eikonal equation can also be obtained. From Fermat’s principle, first-order changes in the traveltime can be found by integrating along initial ray trajectories. Alternatively, linearized ray equations can be used to obtain perturbed rays for which traveltimes can be found. A comparison of the ray solution with approximate solutions for the traveltime is presented. The eikonal equation can also be solved directly without rays. This includes recent finitedifference methods for traveltimes. These methods avoid the ambiguity of multivalued time-fields by tracking only minimum traveltimes. The direct, numerical solution of the eikonal equation is illustrated by several examples using forward and reverse propagation of ’curved wavefronts. From these examples, it is shown that first-arrival wavefronts are generally non-reversible. This is relevant for tomographic applications which often use only first-arrival information. In order to obtain unique reverse propagation, multi-valued traveltimes must be accounted for. This is similar to difficulties encountered in traveltime inversion for 1-D structure when only first-arrivals are used.