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In this part, we apply the same finite-element approach, used in Part III forthe vanishing first traveltime variation (to obtain the stationary rays), forthe second traveltime variation, in order to compute the dynamiccharacteristics along the stationary ray. The finite-element solver involvesapplication of the weak formulation and the Galerkin method to the linearsecond-order Jacobi ordinary differential equation (derived in Part V),yielding an original linear algebraic equation set for dynamic ray tracing. Inour formulation, the resolving matrix of the linear equation set coincides withthe global traveltime Hessian computed for the kinematic ray tracing, makingthe solution of the dynamic problem straightforward. The proposed method isunconditionally stable (the solution does not explode when the intervalsbetween the nodes are increased) and is more accurate than the commonly usednumerical integration (e.g., Runge-Kutta) methods, in particular, forstationary rays passing through heterogeneous anisotropic models with complexwave phenomena.

Eigenrays in 3D heterogeneous anisotropic media: Part VII -- Dynamics, Finite-element implementation