The computation of seismic travel times

We have developed a method for estimating travel times as a function of epicentral range and hypocentral depth which is faster than ray tracing or direct evaluation of ray integrals yet more compact and general than the interpolation of traditional travel-time tables. In this method, delay or intercept time (tau) as a function of ray parameter and source depth is tabulated. We show that direct manipulation of delay time yields travel time as an explicit function of range eliminating the iteration required to determine the proper ray parameter in ray tracing or the evaluation of ray integrals. Travel-time versus range tables exhibit the same explicit dependence by virtue of having eliminated the common independent variable, ray parameter. However, delay-time branches are monotonic, single-valued, and have fixed ray parameter limits. In contrast, travel-time branches may be nonmonotonic and multi-valued and generally have range limits which vary with source depth. Thus, delay-time tables are simpler to generate and interpolate than are travel-time tables and so result in travel-time estimates which are both more reliable and more precise for a given table size. Further, by retaining the explicit ray parameter dependence and hence a more direct relationship with the ray integrals, the delay-time method offers more flexibility in organizing suites of related phases and in tailoring the algorithm to meet specific computational requirements than does the travel-time table approach. Additionally, we show how the delay-time method can be extended to laterally inhomogeneous media.