Eigenray Tracing in 3D Heterogeneous Media

Conventional two-point ray tracing in a general 3D heterogeneous medium is normally performed by a shooting method. The location and the slowness components are specified at the start point, and the ray path is the solution of ODEs with the initial conditions. The ray arrives to some proximity of the destination, and the start direction is then successively refined, so that the ray path eventually includes the destination point. Eigenray tracing, however, is a boundary-value problem, rather than an initial-value problem. The boundary conditions are two endpoint locations, and the ray trajectory satisfies Fermat’s principle of least traveltime. In this study, we apply the non-linear Finite Element Analysis to find the least-time ray path. The ray trajectory is split into a number of three-nodal segments with quadratic interpolation of trajectory points, traveltime and other functions between the nodes. For each segment, we compute the traveltime, and its first and second derivatives with respect to the nodal locations. The local derivatives (related to a single segment) are combined into the global derivatives of the entire path. For the least time, the first derivatives vanish. Knowledge of the second derivatives makes it possible to apply the Newton method for ray path optimization.