Heterogeneous finite-difference (FD) modeling assumes that the boundary conditions of the elastic wavefield between material discontinuities are implicitly fulfilled by the distribution of the elastic parameters on the numerical grid. It is widely applied to weak elastic contrasts between geologic formations inside the earth. We test the accuracy at the free surface of the earth. The accuracy for modeling Rayleigh waves using the conventional standard staggered-grid (SSG) and the rotated staggered grid (RSG) is investigated. The accuracy tests reveal that one cannot rely on conventional numerical dispersion discretization criteria. A higher sampling is necessary to obtain acceptable accuracy. In the case of planar free surfaces aligned with the grid, 15 to 30 grid points per minimum wavelength of the Rayleigh wave are required. The widely used explicit boundary condition, the so-called image method, produces similar accuracy and requires approximately half the sampling of the wavefield compared to heterogeneous free-surface modeling. For a free-surface not aligned with the grid (surface topography), the error increases significantly and varies with the dip angle of the interface. For an irregular interface, the RSG scheme is more accurate than the SSG scheme. The RSG scheme, however, requires 60 grid points per minimum wavelength to achieve good accuracy for all dip angles. The high computation requirements for 3D simulations on such fine grids limit the application of heterogenous modeling in the presence of complex surface topography.

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