Tomographic traveltime inversion is a process of reconstructing slowness models using the traveltimes of seismic events. In recent years, this technique has been successfully used in exploration geophysics for unraveling complex geological structures. Many new methods have been developed to improve the results of the slowness model reconstruction (Paige and Saunders, 1982; Nolet, 1985; Scales, 1987). Traveltime inversion is usually formulated as a nonlinear optimization problem and is solved iteratively by using a gradient method. The largest portion of the computational effort in each iteration of the gradient method is the computation of the traveltimes and the back-projections of the traveltime residuals through ray tracing. Although ray tracing can produce the correct answer, but it is computationally intensive, frequently encounters shadow zones, and sometimes picks raypaths other than the first arrival, which is miserable for the first-arrival traveltime inversion. Using appropriate basis functions for representing the slowness model not only reduces the computational effort but also constrains the inversion process (Harlan, 1989; Van Trier, 1989; Michelena and Harris, 1991). It allows us to incorporate the information from the interpretation of geological structures into the inversion process. However, implementing such an algorithm with ray tracing is awkward, for ray trajectories generally have irregular geometries.
The finite difference calculation of traveltimes has been proved to be an accurate and efficient way to compute, on a regular grid, the traveltimes of first-arrival seismic waves through virtually any velocity structure (Vidale, 1988; Van Trier and Symes, 1991; Podvin and Lecomte, 1991). This new approach removes several difficulties imposed by ray tracing. When Vidale first published this method, he claimed that the method promises to aid in tomographic inversion. Van Trier (1990) applied this method in his tomographic inversion after depth migration, and obtain promising results.
In this paper, I present an application of the finite difference method to cross-well tomographic traveltime inversion. I show that we can completely abandon ray tracing and do the required computations with the finite difference method. The new method greatly reduces the computational cost imposed by the conventional ray tracing method. It also ensures the proper treatment of the first arrival traveltimes. Furthermore, the method handles the slowness model expanded with arbitrary basis functions. The first part of this paper reviews general concepts in the tomography of cross-well data and identifies the three major computational tasks that need to be done. Readers who are familiar with this subject can skip this part. The second part explains how to use the finite difference method to compute the traveltimes, gradient vectors, and conjugate gradient vectors that are required in the conjugate gradient algorithm, and derives the corresponding partial differential equations and initial conditions. This paper emphasizes the descriptions of theories. The details of the algorithms can be found in some of my other papers (Zhang, 1991a and 1991b).