Analytical Solutions of Ray-Tracing Equations in Generalized Elliptical Anisotropy

Ray-tracing in anisotropic media is pivotal for interpreting observed seismic data and creating high-resolution images of subsurface structures, which are crucial in exploration geophysics. Elliptical anisotropy, a simplified model that approximates a transversely isotropic medium, is particularly relevant for geologic settings like shale formations or stressed sedimentary layers where directional dependencies of seismic velocities are pronounced. This paper presents an analytical solution of the ray-tracing equations for a two-dimensional inhomogeneous and anisotropic medium, where velocities depend elliptically on direction and increase linearly with depth – a scenario frequently encountered in stratified geologic formations. Unlike previous studies that assume constant ellipticity throughout the medium, our approach allows for variations in ellipticity, providing a more flexible and realistic representation of subsurface anisotropy. The phase velocities along the x- and z-axis are not necessarily multiples of each other at every point, offering a generalized version of the elliptical anisotropy. This enhancement may enable more accurate predictions and interpretations of observed seismic data, particularly in complex exploration scenarios. The analytical solution yields expressions for both the ray paths and the wavefront normals. By setting the normals of the wavefront at the seismic source point and the location of the seismic source as the initial conditions in phase space, we explore the evolution of these wavefront normal curves across different types of the elliptical anisotropy. Our innovative approach includes plotting the evolution of wavefront normal curves on the generalized momentum coordinate plane of the phase space – that commonly overlooked in traditional models focused only on position coordinates.