Seismic Wave Equation, Ray Tracing, and Wavefront Tracking

The seismic wave equation describes the propagation laws of seismic waves in subsurface media, serving as the core theoretical foundation in seismology and exploration geophysics. Ray tracing and wavefront tracking are two primary numerical simulation methods for solving the seismic wave equation, each with distinct characteristics and application scenarios.

Ray tracing method is based on high-frequency approximation assumptions, simplifying the wave equation into ray theory. This approach treats seismic wave propagation as energy transmission along specific paths (rays) and offers high computational efficiency, making it suitable for rapid simulation of large-scale models. In complex media, traditional ray tracing encounters issues like caustics. To address this challenge, the constant-gradient ray tracing method assumes constant velocity gradients in the medium, achieving analytical solutions within each computational unit. This implementation typically involves calculating ray paths using Snell's law iteratively while maintaining constant gradient interpolation between grid points, improving computational accuracy and stability under complex geological conditions.

Wavefront tracking method directly simulates the expansion process of wavefronts, providing more accurate descriptions of wave phenomena such as diffraction. It is particularly suitable for media containing complex scatterers. Compared to ray tracing, wavefront tracking requires greater computational resources but delivers richer wavefield information including amplitude and phase variations. The algorithm often utilizes front propagation techniques like the fast marching method or level-set methods, where the Eikonal equation is solved to track wavefront progression through heterogeneous media.

Both methods play crucial roles in seismic data interpretation and velocity modeling. Ray tracing is commonly used for first-arrival traveltime tomography, while wavefront tracking is more suitable for applications requiring complete wavefield information such as full-waveform inversion. Selecting appropriate methods for specific problems, or combining both approaches, represents the current frontier in seismic wave simulation research. Modern implementations often integrate these techniques through modular code architecture, where ray tracing modules handle initial path estimation and wavefront tracking modules refine the solution near complex geological features.