Variable grid Finite-difference modeling including surface topography

by
Koichi Hayashi

Submitted to the Department of Earth, Atmospheric, and Planetary Sciences on August 6, 1999 in partial fulfillment of the requirements for the degree of Master of Science

ABSTRACT

We have developed a two-dimensional viscoelastic finite-difference modeling method for highly complex surface topography and subsurface structures. Realistic modeling of seismic wave propagation in the near surface region is complicated by many factors, such as strong heterogeneity, topographic relief and large attenuation. In order to account for these complications, we use a velocity-stress staggered grid and employ an O(2,4) accurate viscoelastic finite-difference scheme. The implementation includes an irregular free surface condition for topographic relief and a variable grid technique in the shallow parts of the model.

Several methods of free surface condition are bench marked, and an accurate and simple condition is proposed. In the proposed free surface condition, stresses are calculated so that the shear and normal stresses perpendicular to the boundary are zero. The calculation of particle velocities does not involve any specific calculations, and the particle velocities are set to zero above the free surface. A stable variable grid method is introduced, where we use a three times finer grid in the near surface or low velocity region compared to the rest of the model. In order to reduce instability, we apply averaging or weighting to the replacement of the coarse grid components within the fine grid. The method allows us to avoid any limitation of the shape of the grid size boundary. Numerical tests indicate that approximately ten grid-points per shortest wavelength with the variable grid method results in accurate calculations. The method requires a stair-shaped discretization of a free surface. We investigated the stair-shaped structures, and found that the cause of the dispersion from irregular free surface is mainly a numerical error due to the large grid sizes rather than the Rayleigh waves scattering due to the stair-shaped boundary.

The finite-difference modeling is applied to the investigation of near surface wave propagation. Several numerical simulations are performed to show the characters of wave propagation in the near surface region. The simulations show that the low velocity thin layers just below the surface and anelastic attenuation have significant effect on surface seismic record. The 2-D modeling of near surface structure beneath a 2-D refraction survey line is carried out. The comparison of the observed data with theoretical waveforms is performed. The characters in the observed data can be explained by a subsurface model constructed by P-wave traveltime tomography.

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Updated: August, 1999

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