Traveltime computation with the linearized eikonal equation

INTRODUCTION

Traveltime computation is an important part of seismic imaging algorithms. Conventional implementations of Kirchhoff migration require precomputing traveltime tables or include traveltime calculation in the innermost computational loop . The cost of traveltime computations is especially noticeable in the case of 3-D prestack imaging where the input data size increases the level of nesting in computational loops.

The eikonal differential equation is the basic mathematical model, describing the traveltime (eikonal) propagation in a given velocity model. Finite-difference solutions of the eikonal equation have been recognized as one of the most efficient means of traveltime computations (Vidale, 1990; van Trier and Symes, 1991; Popovici, 1991). The major advantages of this method in comparison with ray tracing techniques include an ability to work on regular model grids, a complete coverage of the receiver space, and a fair numerical robustness. The most common implementations of the finite-difference eikonal equation compute the first-arrival traveltimes, though frequency-dependent enhancements (Nichols, 1994; Biondi, 1992) can extend the method to computing the most energetic arrivals. The major numerical complexity of the finite-difference eikonal computations arises from the fundamental non-linearity of the eikonal equation. The numerical complexity is related not only to the direct cost of the computation, but also to the accuracy and stability of finite-difference schemes.

It is important to note that the current practice of seismic imaging is not limited to a single migration. Moreover, it is repeated migrations, with velocity analysis and refinement of the velocity model at each step, that take most of the computational effort. When the changes in the velocity model at each step are small compared to the initial model, it is appropriate to linearize the eikonal equation with respect to the slowness and traveltime perturbations. Mathematically, the linearized eikonal equation corresponds precisely to the linearization assumption, commonly used in traveltime tomography.

In this paper, I propose an algorithm of finite-difference traveltime computations, based on an iterative linearization of the eikonal equation. The algorithm takes advantage of an implicit finite-difference scheme with superior stability and accuracy properties. I test the algorithm on a simple synthetic example and discuss its possible applications in residual traveltime computation, interpolation, and tomography.

Traveltime computation with the linearized eikonal equation