Real ray tracing in viscoelastic anisotropic media with homogeneous ray and energy velocities

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The very recently developed g*-Hamiltonian (g*H) method has demonstrated strong potential in handling qSV waves in real ray tracing (RRT). But its investigation and application has so far been limited to homogeneous media or at an interface. In this study, the RRT equations based on the g*H method are derived and solved by the fourth-order Runge-Kutta method for four inhomogeneous viscoelastic, vertical transversely isotropic (VTI) velocity models. The ray tracing equations are solved for specified real incident slowness vectors and specified ray directions, highlighting their practical significance. However, raypaths traced for specified incident ray directions exhibit a broad shadow zone with sparsely distributed rays for the qSV wave. This phenomenon is caused by triplication (wavefront crossings) even though it is small. We also demonstrate that the g*H method can successfully trace rays even if the VTI material has large triplications on the qSV wave fronts. The calculations demonstrate the critical importance of verifying the homogeneity of ray velocities at every ray tracing step in these RRT methods, as homogeneous ray velocity vectors with parallel real and imaginary components cannot always be guaranteed. By applying the concepts of time average energy velocity and quality factor, we demonstrate that a complex energy flux velocity vector is consistently homogeneous without the need to search for stationary slowness vectors and matches the corresponding homogeneous ray velocity well. For the first time, the complex energy flux velocity vector is used for ray tracing by applying the modified Newton algorithm based on Fermat's Variational Principle.