Determination of the Phase-Velocity Direction in Anisotropic Media Using a Direction Vector

Determination of the phase-velocity direction in anisotropic media is important in the general research of wave phenomena and in the construction of seismic modeling, imaging, and inversion algorithms for practical applications. By taking the dot-product of a scaled displacement vector with both sides of the elastic wave equation and rearranging terms, we identify two relevant computational formulas for the group-velocity and slowness vectors. The quantity in the numerator of this group-velocity vector is linked to the conventional Poynting vector which points in the group-velocity direction, and we thus refer to it as the direction vector for the group-velocity direction. Similarly, the quantity in the numerator of this slowness vector provides a new direction vector which points in the phase-velocity direction, and we thus refer to it as the direction vector for the phase-velocity direction. This new direction vector provides a convenient way to directly compute the phase-velocity direction by using extrapolated vector wavefields in general anisotropic media. We vali- date the effectiveness of the proposed Poynting vector for the phase-velocity direction by using an analytical relation in a homogeneous transversely isotropic medium. After that, three anisotropic models are used to show the differences between the group-velocity and phase-velocity directions, which are estimated by using the direction vectors for the group- velocity and phase-velocity directions. The proposed direction vector is also suitable for the computation of the phase-velocity direction in low-symmetry anisotropic media, including orthorhombic, monoclinic, or triclinic media.