Slowness vectors in viscoelastic anisotropic media and their physical interpretation and significance

Under the high-frequency assumption, the slowness vector in a viscoelastic anisotropic medium is often defined by a complex-valued vector, whose direction is given by a complex unit vector that can hardly ever be explained by the intuitive physical reasoning related to the actual seismic wave propagation direction. We extend the existing conjugate real ray-tracing (C-RRT) method to compute the ray velocity vectors and then apply it to determine the slowness vectors for three body waves (qP, qSV, and qSH) in a viscoelastic anisotropic medium. Moreover, we dissect the slowness vector with two physical specifications - traveltime gradient and inhomogeneity components, to reveal the physical significance of the slowness vector and give illustrative examples of these components and the phase propagation (real traveltime) and wave-energy attenuation (imaginary traveltime) wavefronts of seismic waves. We also display the homogeneous and inhomogeneous components, as well as the inhomogeneity angles (or deviation angles between these two wavefronts) for the three body waves (qP, qSV, and qSH) in shale with different quality factors (Q-factors). These results reveal the elusive link between the homogeneous slowness vector and the Q-factors of the medium. The theoretical and numerical dissections of the slowness vector and the extended C-RRT method are extremely helpful in understanding seismic wave propagation and offer a ray-tracing method for wave-energy compensation in seismic migration and reconstruction of subsurface images through seismic ray tomography in viscoelastic anisotropic media.