A dynamic programming approach to first arrival traveltime computation in media with arbitrarily distributed velocities

Abstract

Curved‐ray tomographic traveltime inversion, reverse‐time migration and various other seismic modeling applications require the calculation of traveltime and raypath information throughout a two‐ or three‐dimensional medium. When arbitrary velocity distributions and curved rays are involved, traditional ray shooting or bending procedures can be time consuming and error prone. A two‐dimensional dynamic programming traveltime computation technique, based upon Fermat’s principle, uses simple calculus techniques and a systematic mapping scheme to determine first‐arrival times on a uniform grid, given an arbitrary, discrete velocity distribution. It accurately handles large contrast, discontinuous velocity distributions, including those that generate caustics. First arrival seismic energy can travel either as transmitted waves, diffracted waves, or headwaves, and this technique models all types. The traveltime computations begin with starting values computed near the source location. Then, the mapping systematically steps through the grid, where each new arrival time is calculated using two previously computed “neighbor” traveltimes. We present two mapping procedures, a brute force approach that advances across the grid one column (or row) at a time and a more natural approach that computes times along expanding rectangles. Isotime lines within the traveltime grid represent wavefronts, and seismic raypaths can be numerically computed through the traveltime grid as orthogonal trajectories to the wavefronts. Such a forward model was used in a curved‐raypath tomographic inversion program to successfully invert a physical model crosshole data set. The traveltime computation algorithm also performed successfully on several difficult velocity models. On a discontinuous, large contrast velocity model, traveltimes agreed to within 0.12 percent with those of an accurate ray‐tracing program.

Shear wave velocity and Poisson's ratio models across the southern Chile convergent margin at 38°15′S

“…The forward part of the traveltime tomography was solved by tracing rays through the medium using a finite differences approximation of the eikonal equation (Vidale 1988;Podvin & Lecomte 1991;Schneider et al 1992). Partial derivatives of the calculated traveltimes with respect to the velocity nodes were derived using the techniques described by Lutter et al (1990) and Zelt & Smith (1992).…”

A fast sweeping method for Eikonal equations

“…Other related methods include a dynamic programming approach and post sweeping idea in [15] and a group marching method in [13].…”

A simple anisotropy correction procedure for acoustic wood tomography

“…Some authors proposed algorithms based on ray tracing (Cerveny 1989), while others considered finite-difference approximations to the Eikonal equation (Faria and Stoffa 1994). In this study we followed Faria and Stoffa (1994) and modified the algorithm described by Schneider et al (1992) to yield anisotropic travel times.…”

“…Curved ray paths may be computed using two-point ray tracing schemes (Zelt and Smith 1992), but in the presence of strong velocity contrasts this may lead to numerical instabilities. A more robust alternative is to compute travel times using a finite-difference approximation to the governing Eikonal equation (Schneider et al 1992) and to reconstruct the ray paths by following the steepest descent of the resulting travel-time fields (Aldrige and Oldenburg 1993).…”