Introduction

Traveltime calculations play an important role in many methods in seismic data processing. For example, in the Kirchhoff implementation of migration and modeling of seismic data Green's functions need to be calculated that describe the traveltime between survey points on the surface and depth points in the velocity model. These traveltimes are often calculated by ray tracing, and, since depth and surface points are normally distributed on a regular grid, the traveltimes along the rays are then interpolated onto the grid. For complicated velocity models, rays may cross or not penetrate shadow zones, making the interpolation cumbersome and computationally expensive.

Vidale (1988) recently described a method that calculates traveltimes directly on a regular grid. His approach is as follows: given the initial traveltime value on a certain grid point, the traveltimes for neighbouring points are extrapolated using a planar or circular wavefront approximation. Although Vidale's method represents an elegant alternative to interpolating traveltimes between rays, it has some drawbacks. First, to ensure stability, extrapolation can only be initiated from certain points on the grid (the so-called ``relative minimum points''). Second, to achieve accuracy, at each grid point the curvature of the wavefront has to be determined, and, if necessary, an expensive circular wavefront extrapolation step has to be carried out. These drawbacks not only increase the computational cost, but also make it impossible to vectorize the computations.

W. Symes of Rice University introduced me to Vidale's method during a recent visit to Europe, and suggested an alternative scheme that overcomes the problems described above. Symes suggests writing the Eikonal equation as a flux equation, and then solving it with a standard upwind finite-difference scheme. This leads to a method that is computationally more efficient than Vidale's scheme. Just like Vidale's method, the finite-difference calculations correctly estimate traveltimes of first arrivals in caustics and of diffractions in shadow zones. One disadvantage however, is that traveltimes can only be calculated for rays traveling in one direction, and that the method does not calculate multi-valued traveltime curves. Although this may be a serious limitation for some applications, many Kirchhoff algorithms use single-valued traveltime functions anyway, and the method is readily applicable to those algorithms.

In this paper I first describe the method and the finite-difference implementation. Then I demonstrate the calculations with some examples, where I compare the results both with analytical solutions and with Vidale's method. Finally, I discuss some of the limitations and applications of the method.