Eigenray in 3d Heterogeneous General Anisotropic Media: Dynamics

Abstract

Summary Considering 3D heterogeneous media and general anisotropy, and a parameterized stationary ray path obtained by the kinematic Eigenray, we analyse the second traveltime variation and derive the linear, second-order, vector-form Jacobi dynamic ray tracing (DRT) ODE. It delivers paraxial rays defined by shift vectors normal to the ray direction in its vicinity. The solution is obtained by applying the same finite element scheme (with the Hermite polynomial interpolation) used in the kinematic Eigenray. The resolving matrix of the linear algebraic DRT equation set coincides with the global traveltime Hessian already computed in the kinematic Eigenray stage. Two different solutions of the Jacobi DRT ODE with their corresponding point-source initial conditions, related to the chosen ray coordinates (RC), are needed to compute the ray Jacobian representing to the signed cross-area of the corresponding ray tube. For the chosen RC, we derive an original relationship between the ray Jacobian and the relative geometric spreading. The proposed Eigenray method has been tested using a benchmark numerical example with orthorhombic elliptic factorized inhomogeneous anisotropy. This model has an analytic solution for the ray path configuration, traveltime, arclength, and parameter sigma. The results demonstrate the high accuracy obtained by the proposed method.