A spectral method for seismic wave propagation in elastic media

Abstract

We present am algorithm forward modelling of seismic waves in a 2D isotropic linearly elastic medium with arbitrary density, compressional (or P-wave) velocity and shear (or S-wave) velocity variations in both horizontal and vertical directions. The modelling algorithm directly solves the elastic wave equation with the displacements as unknowns through highly accurate approximations of spatial and temporal derivatives. Spatial derivatives are calculated by the pseudospectral Fourier method on a rectangular grid. Time stepping is performed with a spectral method based on a Chebyshev expansion of the formal evolution operator to the spatially discretized wave equation. The modelling scheme is implemented on an Amdahl VP 1100 vector processor. The large main memory of the Amdahl (128 MByte) and extensive use of vectorization enable calculations on meaningfully sized models within minutes. Results from a model with plane interfaces are tested against analytical solutions, showing that the combination of spectral methods in both space and time produces very accurate results. Application of the logarithm is demonstrated on a geologic model with curved interfaces by combining global snapshots of wavefields and time series of displacement velocities in selected points.