Stable Iterative Reconstruction Algorithm for Nonlinear Traveltime Tomography

Reconstruction of acoustic, seismic, or electromagnetic wave speed distribution from rst arrival traveltime data is the goal of traveltime tomography. The reconstruction problem is nonlinear, because the ray paths that should be used for tomographic backprojection techniques can depend strongly on the unknown wave speeds. In our analysis, Fermat's principle is used to show that trial wave speed models which produce any ray paths with traveltime smaller than the measured traveltime are not feasible models. Furthermore, for a given set of trial ray paths, nonfeasible models can be classiied by their total number of \feasibility violations", i.e., the number of ray paths with traveltime less than that measured. Fermat's principle is subsequently used to convexify the fully nonlinear traveltime tomography problem. In principle, traveltime tomography could be accomplished by solving a multidimensional nonlinear constrained optimization problem based on counting the number of ray paths that exactly satisfy the measured traveltime data. In practice, this approach would be too compu-tationally intensive without the use of massive parallel computing architecture. Nevertheless, the insight gained from from this new point of view leads to a stable iterative reconstruction algorithm. The new algorithm is a modiied version of damped least-squares (also known as \ridge regression"). The correction step at each iteration is in the direction of the damped least-squares solution, but the size of the step is determined by the location of the point having the minimum number of feasibility violations in the direction of the step. The computational burden of computing the number of feasibility violations is virtually negligible. Examples of the results produced by this algorithm are given.