Determination of the phase-velocity directions in anisotropic media using a direction vector

Determination of the phase-velocity direction (PVD) 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 (GVD), and we thus refer to it as the direction vector for the GVD. Similarly, the quantity in the numerator of this slowness vector provides a new direction vector which points in the PVD, and we thus refer to it as the direction vector for the PVD. This new direction vector provides a convenient way to directly compute the PVD by using the extrapolated vector wavefields in general anisotropic media. We validate the effectiveness of the proposed direction vector for the PVD by using an analytical relation in a homogeneous transversely isotropic medium. Next, three anisotropic models are used to indicate the differences between the GVD and PVD, which are estimated by using the direction vectors for the GVD and PVD. The proposed direction vector also is suitable for the computation of the PVD in low-symmetry anisotropic media, including orthorhombic, monoclinic, or triclinic media.