Ray perturbation theory for traveltimes and ray paths in 3‐D heterogeneous media

Both paraxial ray tracing and two‐point ray tracing are powerful tools for solving wave propagation problems. When a slowness model is mildly perturbed from a reference model, one can use perturbation theory for the determination of the ray positions and the traveltimes. An extension of Fermat's theorem is presented, which states that the traveltime is stationary with respect to the perturbations in the ray position provided that the endpoints of the ray are perturbed along the wavefront of the unperturbed ray. It is shown that when the ray perturbation satisfies this condition the second‐order traveltime perturbation can be computed from the first‐order ray perturbation. A perturbation analysis of the equation of kinematic ray tracing leads to a simple second‐order differential equation for the ray deflection expressed in ray coordinates. This constitutes a perturbation method based on a Lagrangian formulation, and leads to a first‐order expression for the ray deflection and a second‐order expression for the traveltime perturbation. This is of relevance to non‐linear traveltime tomography because it leads to an efficient method for evaluating the lowest order ray deflection and the non‐linear effect this has on the traveltimes. The theory is applicable both to two‐point ray tracing and to the determination of paraxial rays. The derivations in this paper are completely self‐contained. All expressions, including the transformation to ray coordinates, are derived from first principles. In this way one obtains insight in the approximations that are actually made. A scale analysis leads to dimensionless numbers that give an indication whether the theory is applicable to a specific problem. For the special case of a layered reference medium the final equations are particularly simple. Plane discontinuities in the reference model and the slowness perturbation are incorporated in the theory. The final expressions for the ray deflection and the traveltime perturbation can be implemented numerically in a simple way. It is indicated how applications to very large‐scale problems can be achieved. Several examples, including the propagation of waves through a quasi‐random model of the earth's mantle illustrate the theory.