9706169 Tromp This research involves the use of a pseudospectral or spectral collocation method with which techniques will be developed to simulate the propagation of elastic and anelastic seismic waves in complex local, regional, and global Earth models. This method requires significantly fewer grid points than spatial finite- differencing techniques to achieve comparable accuracy, exhibits no grid anisotropy and limited grid dispersion, and requires less CPU time because of the use of Finite Fourier Transforms (FFT). On a local scale, it will be used to simulate seismic wave propagation in the Boston, Los Angeles and Tokyo basins based upon detailed geological models. A spherical version will be used to simulate regional wave propagation across Australia where data from the dense SKIPPY array are available for comparison and inversion. On a global scale, the goal is to replace waveforms calculated based upon the path-average approximation with pseudospectral waveforms. These much more accurate waveforms will be used in the Harvard centroid moment-tensor project and in our global tomographic inversions. This research is a component of the National Earthquake Hazard Reduction Program. ***