Staggered-Grid Finite Difference Method for First-Order Elastic Wave Equations

The staggered-grid finite difference method for first-order elastic wave equations is a widely used numerical simulation technique, particularly suitable for solving seismic wave propagation problems. This method significantly improves simulation accuracy and stability by arranging wavefield variables (such as velocity and stress components) at different grid positions (i.e., staggered grid configuration). In the staggered-grid layout, velocity and stress components are offset by half a grid spacing spatially, which provides a more natural representation of the differential relationships inherent in elastic wave equations. Temporally, a second-order finite difference approximation is employed to ensure precision in time evolution. Spatially, a fourth-order finite difference scheme is implemented to effectively reduce numerical dispersion and enhance simulation accuracy. Boundary condition handling is a critical aspect of this method. Common boundary conditions include free-surface boundaries and absorbing boundaries. Free-surface boundaries simulate wave reflections at the Earth's surface, while absorbing boundaries (such as Perfectly Matched Layers - PML) model infinite media to prevent artificial boundary reflections from contaminating simulation results. In code implementation, boundary conditions typically require specialized treatment through conditional statements or auxiliary arrays. Wavefield snapshots serve as intuitive visualization tools for this method. By outputting wavefield distributions at specific timesteps, one can clearly observe subsurface wave propagation processes. This visualization technique is particularly valuable for understanding wave propagation mechanisms and validating simulation correctness. Programmatically, this involves implementing snapshot saving routines at predetermined intervals using file I/O operations. Due to its computational efficiency and excellent numerical properties, this method finds extensive applications in seismic exploration, geological structure imaging, and geophysical research. Key implementation aspects include: 1) Memory allocation for staggered grid arrays 2) Time-stepping loops with stability condition checks 3) Spatial differentiation using centered finite-difference operators 4) Boundary condition application before each timestep update 5) Wavefield output modules for visualization and analysis