Novel methods to determine the slowness and ray-velocity vectors in viscoelastic anisotropic media

Determination of the slowness vector and the homogeneous ray-velocity vector is critical for seismic ray tracing in a viscoelastic anisotropic medium. Three formulae, the traditional g-Hamiltonian, newly developed conjugate real ray tracing (C-RRT) and innovative g∗-Hamiltonian, are employed to calculate the ray-velocity vectors with the determined slowness vectors in a viscoelastic anisotropic medium. We demonstrate the forward and reverse searching procedures to determine the ray-velocity vectors’ slowness vectors. The former implements either a linear search or an optimization method to find the slowness vectors that lead to homogeneous complex ray-velocity vectors (its real and imaginary parts are parallel). The latter is based on a new generalized cost function and applies an optimization method to find the slowness vector for a known ray direction. Using sandstone as an example material, we compare the accuracies and efficiencies of the three formulae and the two searching procedures. Our examples show that the forward searching procedure with the traditional g-Hamiltonian formula and the linearly searching method may generate unphysical solutions for qSV wave due to its cusps or triplication, but using the optimization method may not only mitigate the influence of the cusps and triplication but also significantly improve the accuracies and efficiencies almost two orders higher. For the reverse searching procedure, we propose a general form of the cost function valid for all the formulae of the ray-velocity vector and easily solved by an optimization method. The examples demonstrate that the solutions yielded by the forward and reverse searching procedures coincide well for all three body waves (qP, qSV and qSH), except for the triplication of the qSV wave. In particular, the optimization method combined with the novel g^∗-Hamiltonian formula may completely overcome the issues of spurious solutions and the qSV-wave cusp and triplication.