Earthquake Engineering
GE / CEE - 479/679
Topic 15. Character of Strong Motion on Rock
and Ground Motion Prediction Equations
John G. Anderson
Professor of Geophysics
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Scaling of strong motion in
Guerrero, Mexico
Corner Frequency
(approx)
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Begin the study of strong motion with an examination of the
character of strong motion on rock. A subsequent step will be to
consider the perturbation to the rock motions caused by surface
geology.
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Point Source
• Much can be learned from the equation
giving the motion in an infinite medium
resulting from a small (mathematically, a
point) seismic source.
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Point Source: Discussion
• Both u and x are vectors.
• u gives the three components of
displacement at the location x.
• The time scale t is arbitrary, but it is
most convenient to assume that the
radiation from the earthquake source
begins at time t=0.
• This assumes the source is at location
x=0. The equations use r to represent
the distance from the source to x.
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Point Source: Discussion
• Near-field term
• Intermediatefield P-wave.
• Intermediatefield S-wave.
• Far-field Pwave.
• Far-field Swave.
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Point Source: Discussion
• A* is a radiation
pattern.
• A* is a vector.
• A* is named after
the term it is in.
• For example,
AFS is the “farfield S-wave
radiation
pattern”
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Radiation Pattern Terms
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Point Source: Discussion
• ρ is material
density
• α is the P-wave
velocity
• β is the S-wave
velocity.
• r is the sourcestation distance.
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Point Source: Discussion
• M0(t), or it’s first derivative,
controls the shape of the radiated
pulse for all of the terms.
• M0(t) is introduced here for the
first time.
• Closely related to the seismic
moment, M0.
• Represents the cumulative
deformation on the fault in the
course of the earthquake.
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Point Source: Discussion
Fault perimeter at
different times in
the rupture process.
1s
2s
3s
4s
5s
• Imagine an earthquake source which is growing
with time.
• At each instant in time, one could define the
moment that has been accumulated so far.
• That would involve the area A(t) and the average
slip D(t) at each point in time.
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Point Source: Discussion
M 0 t    At Dt 
• M0(t)=0 before the earthquake begins.
• M0(t)= M0, the final seismic moment, after
slip has finished everyplace on the fault.
• M0(t) treats this process as if it occurs at a
point, and ignores the fault finiteness.
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Consider:
M0(t)
M0
0
t
This is the shape of M0(t). It is zero before the earthquake
starts, and reaches a value of M0 at the end of the
earthquake.
This figure presents a “rise time” for the source time
function, here labeled T. Do not confuse with the period
of a harmonic wave.
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Point Source: Discussion
• 1/r4
• 1/r2
• 1/r2
• 1/r
• 1/r
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Point Source: Discussion
• The far field terms decrease as r-1. Thus, they
have the geometrical spreading that carries energy
into the far field.
• The intermediate-field terms decrease as r-2. Thus,
they decrease in amplitude rapidly, and do not
carry energy to the far field. However, being
proportional to M0(t) , these terms carry a static
offset into the region near the fault.
• The near-field term decreases as r-4. Except for
the faster decrease in amplitude, it is like the
intermediate-field terms in carrying static offset
into the region near the fault.
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Point Source: Discussion
• Signal between
the P and the S
waves.
• Signal for
duration of
faulting, delayed
by P-wave speed.
• Signal for
duration of
faulting, delayed
by S-wave speed.
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Consider these relations:
M 0 t 
M0(t)
From M0(t), this suggests that the simplest
possible shape of the far-field displacement
pulse is a one-sided pulse.
The simplest possible shape of M0(t) is a very smooth ramp.
Thus the simplest intermediate-field term is a smooth ramp.
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Consider these relations:
M 0 t 
M0(t)
 t 
M
0
 t 
M
0
•Differentiating again, the simplest
possible shape of the far-field velocity
pulse is a two-sided pulse.
•Likewise, the simplest possible shape of the far-field
acceleration pulse is a three-sided pulse.
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Consider these relations:
M 0 t 
M0(t)
Far-field:
displacement
 t 
M
0
velocity
 t 
M
0
acceleration
If the simplest possible far-field displacement
pulse is a one-sided pulse, the simplest velocity
pulse is two-sided, and the simplest acceleration
pulse is three sided.
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Simple P-pulse
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Simple S-pulse
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Point Source: Discussion
• These results for the shape of the seismic
pulses will always apply at “low”
frequencies. They will tend to break down
at higher frequencies.
• They have important consequences for the
shape of the Fourier transform of the
seismic pulse.
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Fourier spectrum: Definition
• For any time series g(t), the Fourier
spectrum is:
G( ) 

 gt exp i tdt

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Parseval’s Theorem
Td

 gt  dt    Gd
0
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2
1
0
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Point Source: Discussion
fc
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• The Fourier transform of a
one-sided pulse is always
flat at low frequencies, and
falls off at high
frequencies.
• The corner frequency is
related to the pulse width.
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Point Source: Discussion
• A high corner frequency corresponds to a
short pulse duration.
• A low corner frequency corresponds to a
long pulse duration.
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To get more from the spectrum
• We will calculate the Fourier transform of a
“boxcar” function.

0

bt    B0

0

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D
t
2
D
D
 t 
2
2
D
0 t
2
B0
0

D
2
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D
2
We derived …
 D
sin  

2

G ( )  B0 D
D

2
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Next, a plot
• This uses D=1.0 and B0=1.0.
• The assymptotic limit for frequency -->0 is
B0D.
D
• The first zero is at:   
2
D
2 f

2
1
f 
D
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Corner
frequency
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First
zero
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Discussion
• The spectrum is flat at low frequencies, then starts
to decrease at a corner frequency.
• We will treat the corner frequency as half of the
frequency of the first zero in this case, i.e.
fc=1/(2D)
• Above the corner frequency, the spectrum falls off
as f-1, with some fine structure superimposed.
• The corner frequency is inversely related to the
duration of slip on the fault.
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Point Source: Discussion
• The duration of the pulse gives information about
the size of the source.
• Expect that rupture will cross the source with a
speed (vr) that does not depend much, if at all, on
magnitude.
• Thus, the duration of rupture is ~L/vr. We thus
expect the pulse width (T in the last figure) is
T~L/vr.
• If we measure T, we can estimate the fault
dimension. The uncertainty may be a factor of 2
or so.
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Point Source: Discussion
• For a circular fault with radius rb, Brune (1970,
1971) proposed the relationship:
2.34
rb 
2 f c
• This is widely used in studies of small
earthquakes.
• Uncertainties in rb due to the approximate nature
of Brune’s model are probably a factor of two or
so.
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Static Stress Drop
• In general, there is no way to measure the absolute
stress in the Earth at depths of earthquakes.
• Seismologists do measure a static stress drop,
commonly written as Δτs.
• The static stress drop is estimated from the slip in
D
the earthquake. In general,
 s  C
W
• C is a dimensionless constant.
• W is the small dimension of the fault.
• This is called a “W-model”, since for constant
stress drop slip is proportional to W.
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The constant C depends on the fault type.
For a small circular rupture that does not reach the surface, replace
W with aE, the radius of the fault.
Then…
Rupture Type
Stress Drop
7 D

16 aE
Circular
 S 
Strike Slip
 S 
D
 W
 S 
    D

4  2 W

Normal, thrust

(assume
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
2

  )
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Point Source: Discussion
• Thus, seismologists can estimate the stress
drop of the earthquake using the estimate of
the radius.
• The equation is:
7 M0
 s 
16 rb3
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Fourier transform
• An important property is how the Fourier transform of a
derivative of a time series is related to the Fourier
transform of the time series itself.
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Fourier Amplitude Spectrum (gals*sec)
103
Consequences: Fourier
spectrum:
102
Increases at low
frequencies,
101
100
Flattens at middle
frequencies
10-1
Need to explain roll off at
high frequencies.
10-2
10-3
10-2
10-1
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100
Frequency (Hz)
101
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High frequency spectral behavior.
•Over a fairly broad band on large events,
the acceleration spectrum is flat,
implying that the source displacement
spectrum falls off as f-2.
•Above about 5 Hz, the acceleration
spectrum also falls off, as seen on these
plots of the same spectrum on log and
semilog axes.
Anderson and Hough (1984)
defined a parameter κ to
characterize the high-frequency
slope
e
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 f
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This figure suggests that the high
frequency behavior is due to attenuation at
the site. Parameter kappa is larger,
consistent with lower Q, for deep
sediments than for rock sites, as suggested
in the model below.
Anderson (1986) suggested that κ
results from both a site term and a path
term:
 R, S    0 S   ~R 
Anderson (1986)
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Scaling law of the seismic
spectrum. First described by
Aki (1967).
This figure based on Brune
(1970), modified to include
the effect of attenuation
through the parameter κ .
Figure is from Anderson
(1986).
0.85M 0 2 f 
A f  
exp  f 
2
3
4 r  f 
1  
 f c 
2
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Scaling of strong motion in
Guerrero, Mexico
Corner Frequency
(approx)
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Representation Theorem
un x,t  
 d  u ,  c

i

j

d
Gnp x,t   ;, 0
ijpq

 q
Green’s function
Elastic constants
Slip on the fault
Integral over the fault surface
Convolution over time
Displacement at the station at location x
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7.
7
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Slip functions
D(t)
vr

E
L
F
r2
r1
O
S-wave pulse duration at O:
L
r2  r1
 
vr

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