Seismic Ground Motion - Dynamic Behavior of Concrete and Seismic Engineering - page 112

3.4.2.2. Multiple pathways

As it meets heterogenities in the crust during propagation, the wave emitted from
the source will be partially transmitted, partially reflected, and possibly partially
diffracted. There are also possible conversions between radial P and S waves.
Overall, this results in the existence of many additional waves with different paths
that will prolong the signal, which decrease faster with the distance of direct waves.

For certain strong earthquakes, energy phases corresponding to reflections of
direct waves at the base of the crust appear at distances ranging from 50 to 100 km
This was evident in San Francisco in 1989 during the Loma Prieta earthquake
(located 80-100 km away).

Effects like these are almost impossible to predict with simple models. However,
their numerical modeling is relatively simple, providing the structure to the Earth's
crust is known (location and characteristics of the main discontinuities).

3.4.2.3. Inelastic attenuation

As is the case in any real material, these waves undergo some energy loss
(mainly via aggregate friction), which is characterized in seismology by a quantity
factor Q that is related to the damping coefficient ] by the relationship Q = 0.5/].
The definition of Q is linked to the relative energy loss on a wavelength O, chosen to
be equal to 'E/E = - 2S/Q. For a frequency f, plane waves propagating in the R
direction with a speed c, have an associated spatial dependency given by:

A(R) = A (R). exp ( i 2ʌf t) exp ( ʌf r/Q c)

0

for which A 0 (R) corresponds to the geometric expansion decrease:

ª AR+n O A(R)exp ʌf n O /Q c A R exp ʌ n/Q º

¬ ¼

This expression shows that for a given distance R, higher frequencies will be
attenuated more than lower frequencies, assuming Q has the same value at high and
low frequencies. This effect is often used to account for the high frequency decrease
of the Fourier spectrum.

Typically, inside the Earth's crust, Q values are a few hundred (generally higher
for P waves than for S waves), and this effect only becomes evident when the
distance corresponds to a number n of value of Q wavelengths. However, in the
most superficial materials, Q can reach values equal to only a few dozen: if we take