FD modeling with topography using 3D viscoelastic parameter-modified free-surface condition

The free-surface boundary condition is a crucial aspect in numerical modeling of (visco)elastic wave equations, especially when using a finite-difference (FD) method in the presence of surface topography. The parameter-modified method is a widely used approach to solve this problem. In this regard, the adaptive Poisson's ratio parameter-modified (APM) method has proven to be effective in accurately simulating seismic surface waves within the FD discretization framework. Based on the equivalent medium theory, vacuum approximation, and mathematical limit, we developed a viscoelastic parameter-modified (VPM) method for the implementation of the free-surface boundary condition in the three-dimensional (3D) viscoelastic wave equation. Our approach modifies the viscoelastic constitutive relation and density at the free surface and provides a formulation in terms of displacement and stress. We demonstrated that our VPM method is more general than the viscoelastic stress-image (VSI) method as it includes the latter as a limit case when Poisson's ratio equals zero. The presented free-surface method is represented implicitly within the FD grid, and we provide implementation details when simulating surface waves with topography in the standard staggered-grid FD scheme. We support our theory' feasibility and accuracy through numerical examples.