The Pseudospectral Method for Simulating Wave Propagation.

This thesis presents the results of several efforts to make discrete numerical solutions of seismic wave simulation problems more accurate and efficient, and therefore to make the solution of complicated seismic wave propagation problems more tractable. The pseudospectral method for the numerical solution of partial differential equations is emphasized throughout this thesis, primarily because it potentially offers tremendous savings in required computer memory relative to techniques such as finite-differences or finite-elements. Many of the techniques developed in this thesis are, however, quite general, and need not be limited to pseudospectral applications. A staggered-mesh pseudospectral method is presented as an alternative to centered-mesh methods, for problems with orthorhombic or greater symmetry. Boundary and initial conditions are discussed, as are aspects of simulating seismic sources. Sampling theory suggests that pseudospectral meshes need not be as fine as finite-difference meshes. This issue is examined, with the conclusion that the pseudospectral method is significantly superior to finite-differences for smoothly varying media, but that the advantage disappears as the "roughness" of the media increases. Several methods for reducing artificial edge reflections are presented. In particular, methods based on the representation theorem and paraxial perturbations to the wave equation appear most attractive. Anelastic effects typically should be included in realistic seismic simulations for paths greater than several wavelengths. Rather than using a Pade approximant technique that optimally approximates the complex modulus over all frequencies, a method is presented that optimally reproduces a weighted sum of the attenuation and velocity dispersion over a limited frequency band. Both methods are relatively expensive to implement. A new technique is developed for coupling discrete schemes to other discrete or analytic methods. This hybrid method offers the opportunity for decomposing a complicated seismic problem into separate, simpler problems which may have efficient solution algorithms, thus making larger problems more tractable. Finally, several of the newly developed techniques are used to perform simulations of wave propagation in subducting lithospheric slabs. The simulations suggest that amplitude diminution and pulse broadening are ubiquitous features of the basic slab geometry, and should be discernible in seismic data.