Seismic rays in real rocks bend, split and fade as they pass through layers that both absorb energy and transmit waves differently in different directions. That combination makes it hard to predict the first seismic arrivals that imaging methods rely on. In this paper, the authors build a ray-tracing method that uses a more rigorous local formula, called the g*-Hamiltonian, inside a grid-based shortest-path solver to better capture first-arrival travel times and attenuation in layered, dissipative anisotropic media.

At each grid node the team computes complex energy velocities and three ray quantities: ray velocity, ray attenuation, and a ray quality factor. They compare three ways of computing those local quantities. The first is the original complex energy velocity (OEV). The second is a conjugate real-ray-tracing kernel (C-RRT). The third is the g*-Hamiltonian formulation (also referred to as MEV in the paper). Those local kernels are then plugged into a Dijkstra-like modified shortest-path (MSPM-type) algorithm that builds travel-time fields and backtraces ray paths on a discretized grid.

Why use the g*-Hamiltonian locally? In brief, the g*-Hamiltonian is a more rigorous formulation for complex energy velocity. It accounts for an asymmetry in the complex, density-normalized elastic properties that the other approximations neglect. The grid-based shortest-path solver avoids the need to smooth sharp contrasts or prescribe precise initial ray angles. That makes it better suited to handle abrupt interfaces and refraction, which are common in layered subsurface models.

The numerical comparisons give clear patterns. For qSH and qP body waves the three kernels agree. But for qSV rays — the mode that often shows cusps and wavefront triplications — differences appear. The OEV kernel can create pseudo-reflections or numerical instability and produce unphysical attenuation along qSV paths. The C-RRT kernel sometimes yields large errors, including non-physical negative attenuation for some qP and qSV travel times and biased ray quality factors under the tested parameter choices. Among the three, the g*-Hamiltonian (MEV) driven shortest-path approach produced the most computationally stable first-arrival behavior in the experiments reported.