On the computation of ray-velocity vectors for seismic body waves in viscoelastic anisotropic media

Computation of the ray-velocity vector is crucial in seismic ray tracing for the three body waves (qP, qSV, qSH) in viscoelastic anisotropic media. The primary challenge is dealing with the likely cusps or triplications of the qSV wavefronts, which makes it theoretically difficult to track qSV raypaths and the reflection and transmission of these body waves in such media. We review three traditional methods, namely, g-Hamiltonian, p-Hamiltonian, and explicit c-derivative, and then present two new approaches called implicit c-derivative and g*-Hamiltonian to tackle the challenges of seismic ray tracing. We theoretically prove the equivalence of these five methods to calculate the group-velocity vector in a viscoelastic anisotropic medium, apply these methods to some rock samples, and investigate the applicability of each method. Our results indicate that if the body wave is homogeneous (i.e., its propagation and attenuation wavefronts are parallel to each other) or if the body wavefront has no cusps or triplications, then all the methods offer a consistent solution of the ray-velocity vector. If the body wave is inhomogeneous (its propagation and attenuation wavefronts are at different angles or cross each other) and cusps and triplications occur in the wavefronts, then all the methods but the g*-Hamiltonian one fail to give the proper solution of the ray-velocity vector. This work demonstrates that the innovative g*-Hamiltonian method is the only approach to overcome the theoretical difficulty of seismic ray tracing in viscoelastic anisotropic media.