In this section we describe numerical results corresponding to the response of the San Fernando valley and provide timings that characterize the performance of the parallel explicit wave propagation code on the Cray T3D.
The San Fernando simulations involve meshes of up to 77 million
tetrahedra and 40 million equations. As mentioned before, the largest
mesh corresponds to a shear-wave velocity range of 220 m/s (softest
layer) to 4500 m/s (rock) and a maximum frequency of 1.6 Hz; the code
requires nearly 16 Gb of memory and takes 7.2 hours (5.0 hours
excluding I/O) to execute for 16,667 times steps on 256 processors of
the Cray T3D at the Pittsburgh Supercomputing Center (PSC). The
simulated time is 40 seconds, with a time step of 0.0024s. The
simulated seismic event is a 1994 Northridge Earthquake aftershock,
with its epicenter denoted by a white 
on
Fig. 2. The characteristics of the source were obtained
from [23] and are listed in Table 1.
Table 1: Source characteristics.
Figures 7 and 8 show the E-W and N-S surface
velocity components respectively, along the d-d 
axis shown in
Fig. 2; the color column on the left of the seismograms
depicts the shear-wave velocity profile of the basin along the same
axis. While it is clear that longer durations are associated with the
deeper parts of the valley, it also seems that the constructive
interference of surface and trapped body waves in the shallower
regions of the valley is responsible for the stronger motion
amplification observed on the surface overlying those regions. It is
also noteworthy that no spurious wave reflections seem to be generated
at the artificial boundaries. Figure 9 shows the
distribution of maximum surface horizontal displacements throughout
the valley; it can be seen that the motion in the softer parts of the
basin is amplified five to six times when compared with the motion on
rock. Naturally, this is suggestive of greater damage in these
regions. The solid dark line in the same figure is the outline of key
topological features of the valley. By comparing the distribution of
the shear-wave velocity depicted in Fig. 2 with the
response shown in Fig. 9, the correlation between stronger
amplification and softer layers becomes even clearer. Notice also how
well the response distribution follows even the finest of the
topological features; as expected, stronger response is also
concentrated along different material interfaces within the valley
itself.
Figure 7:
Horizontal surface velocity seismogram of the E-W component along
the d-d 
axis shown in Fig. 2.
Figure 8:
Horizontal surface velocity seismogram of the N-S component along
the d-d 
axis shown in Fig. 2.
Figure 9: Distribution of maximum horizontal surface displacement.
Once the response in the time domain is obtained from the simulation, the record at every point in the valley in the frequency domain can be obtained through Fast Fourier Transforms. In Fig. 10 we plot the distribution of the amplitude of one such Fourier transform for the E-W component of the surface displacement and for a fixed frequency of 1.45Hz. This is helpful for assessing the response of the valley at that frequency and identifying resonant regions. Indeed, the narrow stripes depicted in Fig. 10 are indicative of strong modal response; this is a property of the geological structure, which is expected to be nearly independent of the particular seismic scenario.
Figure 10: Surface distribution of the amplitude of the Fourier Transform of the E-W displacement component for a frequency of 1.45Hz.
It is important for the design process to be able to assess the
response of a hypothetical structure to a given seismic event. To this
end we construct response spectra for two distinct
single-degree-of-freedom oscillators. As an example, we place a simple
oscillator oriented along the E-W direction at every point on the
surface of the valley. We assume a natural frequency 
for each
oscillator and 
critical damping. The valley's response along
the same E-W direction is used as the excitation for the oscillator;
we obtain its response in the time domain and plot the maximum
relative displacement at every point in the valley.
Figures 11 and 12 depict the response spectra for
two distinct oscillators with natural frequencies of 0.3Hz and
1.45Hz. The figures clearly assist in identifying regions where the
oscillators will experience large responses; we note that the chosen
values of natural frequencies are typical of tall (30-story) to
moderately short (6-story) building structures.
Figure 11:
Displacement response spectrum for a simple oscillator with 
Hz and 5% critical damping.
Figure 12:
Displacement response spectrum for a simple oscillator with 
Hz
and 5% critical damping.
