FINITE-DIFFERENCE SCHEMES FOR ELASTIC-WAVES BASED ON THE INTEGRATION APPROACH

The authors present a second order explicit finite-difference scheme f or elastic waves in 2-D nonhomogeneous media. These schemes are based on integrating the equations of motion and the stress-free surface con ditions across the discontinuities before discretizing them on a grid. As an alternative for the free-surface treatment, a scheme using zero density above the surface is suggested. This scheme is first order an d is shown to be a natural consequence of the integrated equations of motion and is called a vacuum formalism. These schemes remove instabil ities encountered in earlier integration schemes. The consistency stud y reveals a close link between the vacuum formalism and the integrated /discretized stress-free condition, giving priority to the vacuum form alism when a material discontinuity reaches the free surface. The two presented free-surface treatments coincide in the sense of the limit ( grid size --> 0) for lateral homogeneity at or near the free surface.