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Proceedings of the National Academy of Sciences of the United States of America
The first term within the curly brackets corresponds to ϕs and the second, with the convolution integral, corresponds to ϕd.
We deal with stress and displacement at values of x, z corresponding to FFT sample points, and these values are constrained to be related to one another to meet the constitutive law. In the 3D spectral implementation, δ(x, z, t) is expanded as a Fourier series in z and x, with vertical period 2Hm and mirror symmetries as above, and with horizontal period along strike as noted in Fig. 1. Other formulations (4–6, 8, 12, 25) have been based on a cellular basis set for the slip distribution, making it spatially uniform within rectangular cells and enforcing the constitutive law in terms of the stresses at cell centers.
The mirror symmetry condition of the spectral formulation causes the shear-free conditions τzx(x, y, 0, t)=τzy(x, y, 0, t) =0 to be met exactly at the Earth’s surface, and also at z= −Hm. However, the symmetry condition replaces the condition of vanishing normal stress, τzz(x, y, 0, t)=0, at the surface with vanishing displacement, uz(x, y, 0, t)=0. One condition implies the other in the 2D cases but not in 3D. A number of 3D trial runs with the quasidynamic procedures, comparing spectral results with uz=0 to those for the cellular slip basis in a formulation meeting τzz=0, have shown no qualitative differences for results with the different boundary conditions. That may be less so with full elastodynamics since the former case disallows Rayleigh waves on z=0.
The fully elastodynamic results so far available on modeling repeated sequences of earthquakes have used coarse meshes of 64 or 128 FFT sample points through the 24-km thickness. These are zero-padded below 24 km and repeated as images above the earth’s surface such that the total number of FFT points is 4 or 8 times greater than the basic 64 or 128 points for Hm=2Hc or Hm=4Hc, respectively. The studies of repeated earthquakes have been done on SPARC10 workstations, whereas the basic spectral elastodynamic methodology (20, 21) was also implemented on the massively parallel Connection Machine 5 (CM5, available with up to 512 processors). We anticipate that the version of the fully elastodynamic program with ϕs extracted and ϕd truncated, to deal with long earthquake histories, can be similarly implemented on the CM5 and extended to 3D fault modeling.