Efficient Methods to Simulate Planar Free Surface in the 3D 4th-Order Staggered-Grid Finite-Difference Schemes

Abstract

We numerically tested accuracy of two formulations of Levander's (1988) stress-imaging technique for simulating a planar free surface in the 4th-order staggered-grid finite-difference schemes. We have found that both formulations (one with normal stress-tensor components at the surface, the other with shear stress-tensor components at the surface) require at least 10 grid spacings per minimum wavelength (λ min÷h = 10) if Rayleigh waves are to be propagated without significant grid dispersion in the range of epicentral distances up to 15λ dom S. Because interior 4th-order staggered-grid schemes usually do not require more than 6 grid spacings per minimum wavelength, in the considered range of epicentral distances, it was desirable to find alternative techniques to simulate a planar free surface, which would not require denser spatial sampling than λ min÷h = 6. Therefore, we have developed and tested new techniques: 1. Combination of the stress imaging (with the shear stress-tensor components at the surface) with Rodrigues' (1993) vertically refined grid near the free surface. 2. Application of the adjusted finite-difference approximations to the z-derivatives at the grid points at and below the surface that uses no virtual values above the surface and no stress imaging. The normal stress-tensor components are at the surface in one formulation, while the shear stress-tensor components are at the surface in the other formulation. The three developed formulations give for the spatial sampling λ min÷h = 6 results very close to those obtained by the discrete-wavenumber method. Because, however, the technique with the vertically refined grid near the free surface requires 3 times smaller time step (due to the refined grid), the technique with adjusted finite-difference approximations is the most accurate and efficient technique from the examined formulations in the homogeneous halfspace.