Seismic anisotropy is not a niche curiosity. It arises naturally in shale, fractured reservoirs, and layered sedimentary sequences, and it measurably distorts the paths and arrival times of seismic waves. When anisotropy is ignored, calculated traveltimes and ray paths become biased, which degrades earthquake localization, velocity analysis, and the seismic imaging workflows that underpin oil and gas exploration. Transversely isotropic media provide a workable mathematical representation of many such formations. Geophysicists classify them by the orientation of their symmetry axis: vertically transversely isotropic models describe horizontally layered strata, horizontally transversely isotropic models approximate vertical fracture sets, and tilted variants capture dipping beds and tectonically deformed rocks whose symmetry axes vary in both inclination and azimuth.

Ray tracing itself is a mature technique. Under the high-frequency approximation, valid when wavelengths are much smaller than the scale of medium variations, the wave equation reduces to first-order ordinary differential equations that describe how rays propagate. Compared with full-wavefield simulation, ray tracing demands far fewer computational resources while delivering exactly the kinematic quantities, ray paths, propagation directions, and traveltimes, that many seismic-processing workflows require. The trouble is that in anisotropic media the phase velocity, the speed of wavefront motion along its normal, generally differs from the group velocity, the speed and direction of energy propagation. Any credible anisotropic ray tracer must treat phase velocity, group velocity, slowness, and symmetry-axis orientation consistently.

Conventional Hamiltonian ray tracing in anisotropic media is built on elastic stiffness parameters. The Christoffel equation supplies the eigenvalues and eigenvectors associated with the quasi-P, quasi-SV, and SH wave modes, and the eigenvalue of the selected branch is used to construct the Hamiltonian and its derivatives. This machinery becomes especially cumbersome for tilted transversely isotropic media, because the stiffness tensor is defined in a local symmetry-axis coordinate system and must be rotated into global coordinates through a Bond transformation. After rotation, most entries of the global stiffness matrix become nonzero, and the algorithm must repeatedly assemble, differentiate, and decompose the Christoffel matrix at every integration step. Wave-mode singularities and near-degenerate directions can further complicate the numerics. Exploration applications, moreover, usually describe anisotropy through symmetry-axis velocities and Thomsen parameters rather than a complete stiffness tensor, so conversion between parameterizations adds yet more algebraic overhead.

The new method, developed by Jianen Xiao and Ning Qin, takes a different route. Instead of constructing the Hamiltonian from a Christoffel eigenvalue, it builds the ray equations directly from the analytical quasi-P-wave phase velocity of the transversely isotropic medium and its spatial and directional derivatives. The phase angle is computed from the wavefront-normal direction and the local symmetry axis, so the same equations represent isotropic, vertically transversely isotropic, and tilted transversely isotropic media within a single framework. In the isotropic limit, the directional phase-velocity derivative vanishes and the system reduces to the conventional isotropic ray equations, an elegant continuity that reflects the underlying physics. Because the analytical phase-velocity expression directly connects commonly used anisotropic parameters to directional speed, the method avoids repeated Bond transformations and Christoffel eigensystem calculations altogether during ray integration.

Mathematically, the formulation retains the canonical Hamiltonian structure. The slowness vector points along the wavefront normal with a magnitude equal to the reciprocal of phase velocity, and its evolution along the trajectory determines the ray path and accumulated traveltime. In anisotropic propagation, the derivative of the Hamiltonian with respect to slowness contains a first-order directional derivative of phase velocity, a term that captures the geometry of the phase-velocity surface and distinguishes anisotropic from isotropic behavior. The spatial evolution can equivalently be written using group-velocity components, which account for the fact that energy propagation direction differs from the wavefront normal. The resulting first-order equations are integrated with a fourth-order Runge-Kutta scheme, updating ray position and slowness simultaneously at each step.

The authors validated the approach through three groups of numerical experiments. In homogeneous isotropic, vertically transversely isotropic, and tilted models on a 101 by 101 by 101 grid with 10-meter spacing, ray-traced traveltime contours agreed closely with analytical solutions, fast-marching reference traveltimes, and wavefronts simulated with a pure quasi-P-wave finite-difference equation. In the anisotropic cases, wavefronts became elliptical rather than circular because phase velocity varied with direction, and traveltime errors remained below 5 milliseconds. Additional tests with symmetry-axis tilt angles of 30 and 60 degrees confirmed that the method tracks wavefronts accurately as the symmetry axis rotates. Notably, the study did not include a matched comparison against an independently implemented Christoffel-based ray tracer, which the authors acknowledge would allow runtime and accuracy differences to be quantified under identical conditions.

A layered model then tested the transition from isotropic to anisotropic layers on a 301 by 301 by 301 grid. Rays traced in isotropic and anisotropic versions of the model coincided within the isotropic uppermost layer and gradually diverged after entering the anisotropic region, exactly as physics dictates. The ray-traced traveltime field matched the simulated wavefronts across layer interfaces and within the anisotropic region, supporting the method’s validity in heterogeneous layered media. The final and most demanding test used the SEAM Foothills benchmark, a geologically realistic model with irregular structures and strong velocity variation, discretized with 439 by 180 by 180 grid points at 25-meter spacing. The local tilt angle was derived from the velocity field using plane-wave destruction filters, which estimate local structural slopes, while the azimuth angle was fixed at zero so symmetry-axis variation was confined to the inline-depth plane. Despite the strong heterogeneity and complex interfaces, ray-traced traveltime contours closely followed the simulated quasi-P wavefronts throughout the model.

The distinction between ray tracing and full-wavefield modeling is worth emphasizing. A finite-difference method propagates the complete wavefield on a spatial grid through successive time steps, capturing waveforms, interference, reflections, diffractions, and finite-frequency effects at a cost that scales with grid points and time steps. The new ray tracer, by contrast, integrates high-frequency quasi-P-wave characteristics only along selected rays, so its cost depends primarily on the number of rays and integration steps rather than the entire space-time grid. It does not reproduce complete waveforms or finite-frequency scattering, and the finite-difference simulations in the study served only as independent references for the calculated wavefronts. The authors also note that sequential integration can accumulate numerical errors from Runge-Kutta truncation, parameter interpolation, and phase-velocity derivative evaluation, so long ray paths, sharp interfaces, and rapidly varying symmetry axes may require smaller steps or adaptive integration.

Within these boundaries, the phase-velocity-based formulation offers a transparent and practical framework for quasi-P-wave traveltime modeling, tomography, and ray-based imaging in heterogeneous transversely isotropic media. By replacing repeated stiffness-tensor rotations and eigensystem decompositions with a single analytical velocity expression and its derivatives, the method lowers the algebraic barrier that has kept fully three-dimensional tilted anisotropy out of routine ray-based workflows. As exploration moves into increasingly complex foothills and fractured terrains, tools that compute anisotropic traveltimes accurately and efficiently will only grow in importance, and this study provides a concrete step in that direction.

Subject of Research: A phase-velocity-based 3D ray-tracing method for qP waves in transversely isotropic media that avoids repeated stiffness-tensor rotations and Christoffel eigensystem calculations

Article Title: A phase-velocity-based 3D ray tracing method for transversely isotropic media

Article References: Xiao, J., & Qin, N. (2026). A phase-velocity-based 3D ray tracing method for transversely isotropic media. Frontiers in Earth Science, 14, Article 1967527. https://doi.org/10.3389/feart.2026.1967527

Image Credits: AI Generated

DOI: 10.3389/feart.2026.1967527

Keywords: ray tracing, transversely isotropic media, seismic anisotropy, qP waves, phase velocity, eikonal equation, Hamiltonian ray theory, Runge-Kutta integration, traveltime modeling, SEAM Foothills model, geophysics, Frontiers in Earth Science