Bulletin of the Seismological Society of America

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. INTRODUCTION A problem commonly encountered in wave propagation problems may be formulated as follows: given the positions of a source and a receiver point, and a model of the medium in which they are embedded, estimate the time required for a wave group to propagate from the former to the latter. For seismological purposes, the medium is the Earth, the source is generally an earthquake, an explosion or a rockburst, and the receiver is usually an accelerometer. In most seismic problems, several simplifying assumptions may be made. First, travel time may be defined as the time required for an infinite frequency wave front to propagate from the source point to the receiver point. That is, t h e geometrical optics approximation is adequate. Second, as the Earth is spherically symmetric to zeroth order and the receivers are always positioned very near the free surface, the source-receiver geometry may conveniently be specified by a hypocentral depth (or radius) and an epicenter-receiver angular distance (or range). The best-known seismological application of travel times occurs in the problem of locating earthquakes using Geiger's method (Geiger, 1910). In Geiger's algorithm, the location is determined by successive nonlinear least-squares corrections to an assumed source position. Traditionally, travel time is computed either by ray tracing (or equivalently by direct evaluation of the range and travel-time integrals) (Lee 1271 1272 RAY BULAND AND C. H. CHAPMAN and Lahr, 1975) or by interpolating tables of travel time in both source depth and range (Engdahl and Gunst, 1966). The former technique is computationally unreasonable for all but the simplest Earth models. Not only are the rays (or range and travel-time integrals) expensive to evaluate, but they are functions of a third variable, ray parameter, which is related to ray take-off angle at the source. Thus, for each arrival of interest, one must trace the ray (or evaluate the range integral) a number of times in order to determine the proper take-off angle before the travel time may be evaluated. The latter (table) method generally requires a great deal of mass storage since travel time as a function of epicentral distance can be discontinuous and multi-valued making interpolation awkward. This trouble is aggravated by the fact that the ranges delimiting travel-time branches (segments where travel time is a continuous, monotonic, single-valued function of range) vary with hypocentral depth. In practice, these difficulties serve to limit the flexibility of traveltime tables as each phase of each model may require special treatment. In this paper, we propose a new method of estimating travel times. Computational efficiency is gained by retaining a table-driven approach. However, ease of table generation is gained by saving neither travel time nor range, but the theoretically more desirable delay or intercept time (tau function) as a function of source depth and ray parameter. The simple behavior of the tau function is well known and has been exploited in inversion studies (Johnson and Gilbert, 1972; Bessonova et al., 1976). Not only is each delay-time branch monotonic and single-valued (even when there is a caustic in the corresponding travel time), but also ray parameters delimiting each branch are simple, invariant functions of the model. Further, tau is simply related to the Earth model and the singular behavior of tau and its derivatives is known analytically. These facts have been used to derive "natural" schemes for interpolating delay time in both source depth and range resulting in sparse ray parameter and model depth sampling and so in relatively compact tables. RAY INTEGRALS Consider waves propagating in the half-space z =< 0, in which velocity, v(z), is a function of depth alone, v (z) may be taken to be either compressional velocity, a (z), or shear velocity, ~(z), depending upon the type of ray desired. For convenience, assume the ray of interest is propagating in the x z plane and in the +x direction. Define medium slowness u( z ) = v-l(z); (1) horizontal ray slowness (or ray parameter) sin(i(z)) (2) P = v(z ) ' where i (z ) is the (acute) angle the direction of ray propagation makes with the vertical and vertical ray slowness q(p , z) = (u2(z) -p2)1/2. (3) It is well known that ray parameter is invariant along the ray path under these circumstances. The ray may propagate where p <= u ( z ) and will turn (change from THE COMPUTATION OF SEISMIC TRAVEL TIMES 1273 downto up-going or vice-versa) when p = u(z) (vertical slowness is zero). No geometrical ray may propagate where p > u (z). Thus, both p and q (p, z) are always real and (taking the principal value of the square root) nonnegative. The ray slowness vector is then p(x, z) = p~ +_ qi (4) where + is taken for an up-going ray segment and for a down-going segment, p(x, z) has magnitude u (z) and points in the direction of ray propagation. Travel time is given by ~ u2(z) dz T ( p ) = q ( p , z) ' (5) where ~ indicates the integral over the complete ray path. The contribution of every segment of the ray path must be taken as positive. The range integral is given by P dz X(p) = "~ q(p, z)" ,,,. 2 (6) The delay-time function, r(p) , is related to T(X) by the Legendre transformation (Goldstein, 1964, section 7.1) r(p) = T(p) pX(p) (7)