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

A new 3-dimensional spectral method for séismic modelling in general anisotropic media is presented. An important feature of the method is its ability to handle the free surface boundary conditions very accurately . Moreover the algorithm is able to account for complete material variability, since it is a grid method, 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, but does not make use of staggered grids. For the horizontal directions spatial derivatives are carried out by the Fourier method which has periodic boundary conditions, whereas differencing with respect to the vertical direction is performed by a Chebyshev derivative operator, which is non-periodic and therefore can account for specified boundary conditions. Time integration is achieved by a fourth order Taylor expansion of the formal solution .