Ray velocity derivatives in anisotropic elastic media. Part II—polar anisotropy

Abstract

SUMMARY Considering general anisotropic (triclinic) media and both, quasi-compressional (qP) and quasi-shear (qS) waves, in Part I of this study, we obtained the ray (group) velocity gradients and Hessians with respect to the ray locations, directions and the elastic model parameters along ray trajectories. Ray velocity derivatives for anisotropic elastic media with higher symmetries were considered particular cases of general anisotropy. In this part, Part II, we follow the computational workflow presented in Part I, formulating the ray velocity derivatives directly for polar anisotropic media (transverse isotropy with tilted axis of symmetry, TTI) for the coupled qP waves (quasi-compressional waves) and qSV waves (quasi-shear waves polarized in the ‘axial’ plane) and for SH waves (shear waves polarized in the ‘normal’ plane). The acoustic approximation for qP waves is considered a special case. In seismology, the medium properties, normally specified at regular 3-D fine gridpoints, are the five material parameters: the axial compressional and shear wave velocities, the three (unitless) Thomsen parameters and two geometric parameters: the polar angles defining the local direction (the tilt) of the medium symmetry axis. All the parameters are assumed spatially (smoothly) varying, so that their spatial gradients and Hessians can be reliably numerically computed. Two case examples are considered; the first represents compacted shale/sand rocks (with positive anellipticity) and the second, unconsolidated sand rocks with strong negative anellipticity (manifesting a qSV triplication). The ray velocity derivatives obtained in this part are first tested by comparing them with the corresponding numerical (finite difference) derivatives. Additionally, only for validation purpose, we show that exactly the same results (ray velocity derivatives) can be obtained if we transform the given polar anisotropic model parameters (five material and two geometric) into the 21 stiffness tensor components of a general anisotropic (triclinic) medium, and apply the theory derived in Part I. Since in many practical wave/ray-based applications in polar anisotropic media only the spatial derivatives of the axial compressional wave velocity are taken into account, we analyse the effect (sensitivity) of the spatial derivatives of the other parameters on the ray velocity and its derivatives (which, in turn, define the corresponding traveltime derivatives along the ray).