A FREE-SURFACE BOUNDARY-CONDITION FOR INCLUDING 3D TOPOGRAPHY IN THE FINITE-DIFFERENCE METHOD

A flexible and simple way of introducing stress-free boundary conditio ns for including three-dimensional (3D) topography in the finite-diffe rence method is presented. The 3D topography is discretized in a stair case by stacking unit material cells in a staggered-grid scheme. The s hear stresses are distributed on the 12 edges of the unit material cel l so that only shear stresses appear on the free surface and normal st resses always remain embedded within the solid region. This configurat ion makes it possible to implement stress-free boundary conditions at the free surface by setting the Lame coefficients lambda and mu to zer o without generating any physically unjustified condition. Arbitrary 3 D topographies are realized by changing the distribution of lambda and mu in the computational domain. Our method uses a parsimonious stagge red-grid scheme that requires only 3/4 of the memory used in the conve ntional staggered-grid scheme in which six stress components and three velocity components need to be stored. Numerical tests indicate that 25 grids per wavelength are required for stable calculation. The finit e-difference results are compared with those of the boundary-element m ethod for the two-dimensional (2D) semi-circular canyon model, We also present the responses of a segment of semi-circular canyon and hemisp herical cavity to vertically incident plane P, SV, and SH waves and di scuss the response of a Gaussian hill to an isotropic point source emb edded in the hill. in the segment of semi-circular canyon, the later p ortions of the synthetics are characterized by phases scattered from t he two vertical side walls. The hemispherical cavity and 2D semi-circu lar canyon both show focusing of energy at the bottom of the cavity, a lthough the focusing effect is stronger in the former geometry. Focusi ng and defocusing effects due to the strong topography of the Gaussian hill produce a strong amplification of displacements at a spot locate d on the flank opposite to the source. Backscattering from the top of the hill is also clearly seen.