Traveltime computation through isotropic media via the eikonal equation

Traveltimes can be computed through a complex velocity structure by a finitedifference approximation to the eikonal equation. The main methods investigated here consist of the same finite-difference scheme yet differ in the order in which the points were computed. The first method estimates traveltimes along an expanding square while the second uses an expanding wavefront. Theoretically, it is expected that the expanding wavefront algorithm will be more accurate, however it takes more computational resources and time for computing the traveltimes. The expandingsquare method is fast but is expected to suffer accuracy problems in complex structures. Deciding between the two methods is a trade-off between accuracy and computational time. INTRODUCTION Traveltime computations of waves propagating through different layers of isotropic media have been estimated with two basic methods. The first is raytracing, in which the path that the energy travels is computed from the invariance of the ray parameter. Many raytracing methods have been suggested, yet these all have problems that can be divided into two categories. The first problem is that in strongly varying velocity fields, there may be more than one raypath possible to reach a certain point. To ascertain the minimum traveltime, all these paths should be computed and compared, something that is quite tedious. The second is the problem of shadow zones. Shadow zones are areas in which rays cannot be found that pass through it. There is a specific angle, or finite number of angles, at which each ray must depart from the source to reach a certain point. At times, the position at which the ray ends is extremely sensitive to the angle of departure and finding the appropriate angle can be difficult. Another method, which will be the focus of this discussion, calculates traveltimes through a finite-difference approximation to the eikonal equation. The methods that will be investigated here are those proposed by Vidale (1988) and by Qin et al. (1992). These methods do not have a shadow zone problem and their intent is only to search for the first-arrival traveltimes. A drawback of this method is that the path travelled may not always be obvious. The nature of the eikonal equation does not allow for travel path information to be propagated and hence is somewhat at a disadvantage compared to the raytracing methods. Considerations discussed within the report include the computational time taken to calculate the traveltimes and accuracy. The raytracing methods can be quite expensive in comparison to the eikonal method because it does not have the problems encountered in the raytracing method. This report does not intend to advocate one