Slowness vector versus ray direction in polar anisotropic media
Authors: Ravve Igor , Koren Zvi 
Published: February 2021

SUMMARY
The inverse problem of finding the slowness vector from a known ray direction in general anisotropic elastic media is a challenging task, needed in many wave/ray-based methods, in particular, solving two-point ray bending problems. The conventional resolving equation set for general (triclinic) anisotropy consists of two fifth-degree polynomials and a sixth-degree polynomial, resulting in a single physical solution for quasi-compressional (qP) waves and up to 18 physical solutions for quasi-shear waves (qS). For polar anisotropy (transverse isotropy with a tilted symmetry axis), the resolving equations are formulated for the slowness vectors of the coupled qP and qSV waves (quasi-shear waves polarized in the axial symmetry plane), and independently for the decoupled pure shear waves polarized in the normal (to the axis) isotropic plane (SH). The novelty of our approach is the introduction of the geometric constraint that holds for any wave mode in polar anisotropic media: The three vectors—the slowness, ray velocity and medium symmetry axis—are coplanar. Thus, the slowness vector (to be found) can be presented as a linear combination of two unit-length vectors: the polar axis and the ray velocity directions, with two unknown scalar coefficients. The axial energy propagation is considered as a limit case. The problem is formulated as a set of two polynomial equations describing: (i) the collinearity of the slowness-related Hamiltonian gradient and the ray velocity direction (third-order polynomial equation) and (ii) the vanishing Hamiltonian (fourth-order polynomial equation). Such a system has up to twelve real and complex-conjugate solutions, which appear in pairs of the opposite slowness directions. The common additional constraint, that the angle between the slowness and ray directions does not exceed ${90^{\rm{o}}}$, cuts off one half of the solutions. We rearrange the two bivariate polynomial equations and the above-mentioned constraint as a single univariate polynomial equation of degree six for qP and qSV waves, where the unknown parameter is the phase angle between the slowness vector and the medium symmetry axis. The slowness magnitude is then computed from the quadratic Christoffel equation, with a clear separation of compressional and shear roots. The final set of slowness solutions consists of a unique real solution for qP wave and one or three real solutions for qSV (due to possible triplications). The indication for a qSV triplication is a negative discriminant of the sixth-order polynomial equation, and this discriminant is computed and analysed directly in the ray-angle domain. The roots of the governing univariate sixth-order polynomial are computed as eigenvalues of its companion matrix. The slowness of the SH wave is obtained from a separate equation with a unique analytic solution. We first present the resolving equation using the stiffness components, and then show its equivalent forms with the well-known parametrizations: Thomsen, Alkhalifah and ‘weak-anisotropy’. For the Thomsen and Alkhalifah forms, we also consider the (essentially simplified) acoustic approximation for qP waves governed by the quartic polynomials. The proposed method is coordinate-free and can be applied directly in the global Cartesian frame. Numerical examples demonstrate the advantages of the method.