3-D seismic modelling of general material anisotropy in the presence of the free surface by a Chebyshev spectral method

Summary A 3-D spectral method for seismic modelling in general anisotropic media is presented. An important property of the method is its ability to handle the free-surface boundary conditions accurately. Moreover, since it is a grid method, the algorithm is able to account for complete material variability where every grid node can represent different material properties. The modelling scheme is based on the velocity-stress formulation of the equations of dynamic elasticity. An important feature is that it does not make use of staggered grids, which are problematic for anisotropy because not all the strain components are defined at the same grid node. Spatial derivatives in the horizontal directions are carried out by the Fourier method, which has periodic boundary conditions. Differentiation with respect to the vertical direction is performed by a Chebyshev derivative operator. The free-surface and transparent bottom boundary conditions are introduced into the modelling scheme by using characteristics. Time integration is achieved by a fourth-order Taylor expansion of the formal solution. Numerical examples show the method's ability to produce synthetic seismograms and snapshots of the wavefield under the influence of the free surface in vertically and laterally inhomogeneous anisotropic media.

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