速度・応力型差分法での固体・流体境界の扱いについて

Abstract

We must consider the seismo-acoustic problem at the irregular fluid-solid boundary in various acoustical and seismological subjects: e. g., the ocean-acoustics problem, the analysis of the data from ocean-bottom seismometers, the analysis of the scattered waves from the irregular core-mantle boundary, and the strong motion prediction at sites close to the ocean. In this paper, we study why and how the fluid-solid or the acoustic-seismic boundary conditions are satisfied in the velocity-stress staggered grid finite-difference method (FDM) which has been widely used for seismo-acoustics problems. A simple analysis of the 2D finite-difference equations together with the comparisons between finite-difference seismograms and the discrete wavenumber method (DWM) seismograms show that the conditions are satisfied provided that (1) the boundary is placed through the shear stress (τxz) grid points (i. e., the rigidity of the shear stress grid points on the boundary must be set to zero), (2) the 2nd-order centered equation with averaged density is applied to the normal velocity component of the grid points right on the boundary, and (3) inadequate higher order finite-difference equations (e. g. the standard 4th-order centered equation) is not applied to grid points neighboring the boundary so that the values with discontinuity at the boundary are not included in the finitedifference equations for grid points off from the boundary. The accuracy of the spatial finitedifference equation at the boundary is O(h) with h being the space increment despite its apparent form of “2nd-order” equation. Violations to these conditions deteriorate the results of computation, especially for the interface wave such as Stoneley wave. In FDM inclined interface is approximated by stairsteps made from the grid. We find that these stairsteps can generate scattered waves with amplitudes up to 20% of the Stoneley wave. We need about 60 grid points per wavelength of the Stoneley wave at the dominant frequency in order to suppress such scattered wave. In the case of basin-like fluid layer with free surface and with “shoreline” we need large number of grid points (about 120 grid points) per wavelength at the dominant frequency in order to model the secondary Rayleigh wave generated (scattered) at the basin edges.