Viscoelastic wave finite-difference modeling in the presence of topography with adaptive free-surface boundary condition

An accurate free-surface boundary condition is significant for seismic forward modeling and inversion. The finite-difference method (FDM) is widely used for its simplicity and efficiency. However, unlike the finite-element method (FEM) satisfying naturally the stress-free condition at the free surface, FDM needs additional treatment, particularly in the presence of irregular topography. In the elastic wave finite-difference modeling, the adaptive parameter-modified free-surface boundary condition has the advantages of high accuracy and simple operation. The viscoelastic wave equation can describe the nature of seismic waves more realistically. Based on the staggered-grid FDM, we extend the adaptive free-surface boundary condition to the viscoelastic medium with topography. This approach involves a combination of the average medium theory, vacuum approximation and limit idea. It is realized by modifying the viscoelastic constitutive relation. This method is simple enough, because three types of grid elements and in fact only two kinds of expressions are enough in the presence of topography. We only need to deal with the Lamé parameters and the density at the free surface without reconstructing the existing algorithm. Viscoelastic analysis of different quality factor settings shows the viscous effect. Numerical examples display that the results of the presented method agree well with the reference solutions of spectral-element method both in crest- and trough-like model and in simplified Foothill model with irregular topography. The simulation of original Foothill model demonstrates the feasibility of our method.