Farfield Point Source

ERI Notes: RJA

49

Haskell & Kostrov Slip Rate
The farfield (which contributes most of the ground motion even in thenear-source region) displacement amplitude is proportional to the slip
rate on the fault.
Slip Rate Functions!
6!

Haskell!

5!

Kostrov!

Slip Rate!

4!

3!

2!

1!

0!
0!

0.2!

0.4!

0.6!

0.8!

1!

Time (s)!
ERI Notes: RJA

1.2!

1.4!

1.6!

1.8!

2!

50

Near Fault Ground Motion

ERI Notes: RJA

51

Peak Ground Motions

ERI Notes: RJA

McGarr, et al., 198152

PGA and PGV Yucca Mountain

Figure 3.!Probability of exceedance calculations for horizontal PGA at Site A, a hypothetical,
reference rock outcrop site at the repository horizon (Stepp et al., 2001). From the 1998 Yucca
Mountain PSHA extrapolated to 10-8/yr by Stepp and Wong (2003).

Figure 4. Same as Figure 3, but for horizontal PGV.

While not formally published until 2006, the Workshop Report was available in draft
form in June 2005, laying out the basic framework of the Extreme Ground Motion
In response to these concerns, the Science and Technology Program (S&T) in the Office(ExGM) research program of Physical Limits to Ground Motion, Unexceeded Groundof Civilian Waste Management (OCRWM) of the United States department of EnergyMotions, and Frequency of Occurrence of ground-motion amplitudes such as PGA and(DOE) formed a committee in September 2003 to investigate the nature and plausibilityPGV or source properties related to them, such as stress drops and faultingof these extreme ground motions. The Extreme Ground Motion Committee (ExGMCom,displacements. DOE then launched the ExGM research program as a 5-year program toTable 1) took on as its the first order of business a comprehensive assessment of extremebe funded at $1M/yr.
ground motions through many disciplines of the earth sciences, achieved through theWorkshop on Extreme Ground Motions at Yucca Mountain, held in Menlo Park,DOE required that ExGM engage the broad earth-science community, which was easily
California, August 23-25, 2004. The heart of this Workshop was the 29 presentationsaccomplished through the Southern California Earthquake Center (SCEC). In September
made over three days, although the commentary and criticism of all the invited2005, approximately 100 SCEC scientists met for three days (September 8-10) after theparticipants figured prominently in the outcome. The report on the Workshop (Hanks etSCEC Annual Meeting to discuss and review the Workshop draft report. The SCECal., 2006) includes a written summary of all the presentations together with findings andwritten review, prepared by Paul Somerville, of the Workshop report was taken into
ERIrecommendations;
Notes: RJA it also includes, in a lengthy appendix, short papers written by eachaccount in preparing the final report (Hanks et al., 2006). Approximately 1/3 of the53
ExGM budget has been placed at SCEC, to support its scientists engaged in the ExGM
invited speaker(s) on the topic of their presentations.
research program, as well as to support the highly successful SCEC/ExGM annual

Slip and Stress at a Pointon the Fault

σE =Δσ =Δσ(static) = σ0 –σ1

ERI Notes: RJA

54

Stress Drop and Moment

ERI Notes: RJA

55

LinearElasticity

ERI Notes: RJA

56

Stress Drops: Southern CaliforniaCountnsta ! 5100

10Count$S (MPa)nsta ! 1010.1

nsta ! 201/4CountM0
1.6

1.82.0

2.2

2.4

2.6

2.8

3.0

Mw

ERI Notes: RJA

Shearer, Prieto, Hauksson, 2007 57$S (MPa)
Figure 10

countdata available at that time.
150
For these reasons,and because our earthquake depths may not be
accurate enough to100
be sure we are improving our corner frequency
estimates, we do not apply corrections for depth phases in the results presented here.50However, it is possible that depth phases may
have a stronger effect on some of the second-order features in the0spectra, such as the parameter
to1unity2in equation2) or an−2−1 γ (set03
intermediate frequency range with a different fall-off rate.

of the studies
compiledin Figure
8 used different
sampling
over 100
iterations
and computing
standardsource
errorsparamefor eachters, such assource radius [Mori and Frankel, 1990; Abercrombie,
magnitude
bin.
1995]
and stress[Hough,1996;ofVenkataraman
Kanamori,
Becauseof thedrop
finitebandwidth
our data with andthe lower
limit2004].
We
have
rescaled
thesetocornerfrequency
using
our modelof our analysis at 0.02 Hz, we display the same data in a cross-plotofassumptionsaccordingequation
3. Taken
together,
we seearea
nocornerfrequency
versus tomoment(Figure8). The
gray shaded
dependenceof stress
dropThewithdata
seismic
overlimited
13 ordersmarks
the resolution
limit.
in ourmoment
catalog are
to Mofw
magnitude,whichisstrongevidenceforearthquake
self-similarity
greater than 5.2. Within the resolution bounds of the data we findonsignificant
a global scale.
novariation of stress drops with moment. Figure 8 compares our result with some previous studies of scaling parameters[Archuleta et al., 1982; Mori and Frankel, 1990; Humphrey and An3
derson, 1994;
1994; Abercrombie, 1995; Hough, 1996;
10 Boatwright,
Venkataraman and Kanamori, 2004; Tajima and Tajima, 2007] andwe find them to be remarkably consistent with our results. Someof the studies
10 2compiled in Figure 8 used different source parameters, such as source radius [Mori and Frankel, 1990; Abercrombie,1995] and stress drop [Hough, 1996; Venkataraman and Kanamori,2004]. We have10 1 rescaled these to corner frequency using our model
assumptions according to equation 3. Taken together, we see nodependence of stress drop with seismic moment over 13 orders of0magnitude, 10whichis strong evidence for earthquake self-similarity
on a global scale.

b)mean: 3.36 MPa
median: 3.34 MPa# events: 875
mean: 3.85 MPa
median: 3.7 MPa# events: 1759

250
100counta)

150

count

200
15050
100
050 −2−1012
log10 stress drop [MPa]30Figure 6. Histogramof logarithmic−2−10
1 stress-drop
2
3 estimates for adifferent minimum number of required stations. a) For at leastb)
150
three stations
per event. b) For at least 20 stations per event.
mean: 3.36 MPa
median: 3.34 MPa# events: 875

10 -1

10 3count100

stress drop [MPa]Global Stress Drops

3. Global average source properties

10 -2

10 2 5.0

We first investigate the average scaling properties of our resultsin order to check for consistency with previous results, as well as for
0indicators of possible
bias in the stress-drop estimates. A histogram−2−1012
3of the whole data set (Figure6)stress
shows
that
the stress-drop estimates
log10drop[MPa]vary over more than three orders of magnitude with a median global
Figure
6. Histogramof logarithmic
estimatesfor astress dropof 3 to 4 MPa.
Requiringstress-drop
more stations
per earthquake
different
numberofthe
requireda) For at least
does not minimum
significantly
reducescatterstations.
in the distribution,
so wecontinuerequiring
only three
to obtain
threestations
per event.
b) Forstationsat leastso20asstations
perresults
event.for the

5.5

6.0

6.5

7.0

7.5

8.0

Mw

stress drop [MPa]50

Figure 10
7. 1 Stress-drop versus moment. The mean of 100
bootstrap-resampled median stress drops for bins of 0.4 in moment magnitude is shown by the white squares. Error bars denotethe standard
10 0 errors from bootstrap resampling. Note the general
independence of stress drop and moment over the magnitude
range of the data.
10 -1

3. Global average source properties

10 -2
5.0

We first
investigate the average scaling properties of our results
ERI Notes:
RJA
in order to check for consistency with previous results, as well as for

5.5

6.0

6.5

7.0

7.5

8.0

Allmann andShearer, 2009 58
Mw

Stress Drop World Wide Eqks

ERI Notes: RJAKanamori and Anderson, 1975 59

Global Stress Drops
B. ALLMANN AND P. SHEARER: GLOBAL STRESS-DROP VARIATIONS

X-7Mw
1245678100 MPa

10 2

10 1

fc [Hz]30.1 MPa

this study
Hough (1996)

10 0

Boatwright (1994)
Mori and Frankel (1990)
Tajima and Tajima (2007)
10 -1

Humphrey and Anderson (1994)
Archuleta et al. (1982)
Venkataraman and Kanamori (2004)
Abercrombie (1995)

10 -2
10 9

10 10

10 11

10 12

10 13

10 14

10 15 10 16
m0 [Nm]

10 1710 18

10 19

10 20

10 21

10 22

Figure 8. Corner frequency versus seismic moment (lower scale) and moment magnitude (upper scale). The reddashed lines show constant stress drops of 0.1, 1, 10, and 100 MPa. The gray shaded area shows the resolution limit ofour data. The vertical dashed line marks the lower magnitude cutoff of our data. The results of this study are plotted
as open black circles. All other different shaped and colored symbols show data from various other studies. The datasuggest self-similarity over a wide moment range.

ERI Notes: RJA

Allmann and Shearer, 2009 60

Finite Fault Parameters

ERI Notes: RJA61

σ (x, t ) = σ H (t − x / β )σ (x, t ) = µ∂u / ∂xu=0t<0u(t ) = (σ / µ )β tt>0The Fourier spectrum of u(t) is∞Brune’sSpectral Model( ω )(Ω(ω ) = ∫ (σ / µ )β t exp − iω t = − 102σβµ)The initial particle velocity is:( )σu(t)=µ β
The inverse Fourier transform of u(t ) over a finite frequency band( )( µ ) ∫ ( 1ω ) exp(iω t )dωσβ ) = ( 1u(t)2π (µ ) ∫ exp(iω t ) dω ) = ( 1 ) (σβ ) ω ( sin ω tu(tπµω t)u(t ) = 1 2πσβωs2−ω sωs−ω ssss f = 10)  2g
For ω s = 10Hz and σ = 10MPa, u(ERI Notes: RJA

62

u(x = 0,t) = (σ / µ )βτ (1− e−t /τ )−t /τ = 0,t) = (σ / µ )β eu(xThe Fourier transform of displacement is
Ω(ω ) = (σ / µ )βω −1 (ω 2 + τ −2 )−1/2

Brune’sSpectral Model

where τ = O(r / β )The farfield displacement is proportional to slip rate
and the effect of diffraction from the finite size (r) of
the fault is approximated by multiplying by e−α t .
Of course we now have to consider the reduced timet ′′=t-R/ β and we will multiply by a factor f i(r/R)
u(R,t) = f i( r / R ) (σ / µ )β t ′′e−α t ′′
u(R,t)= f i( r / R ) (σ / µ )βat t ′′= 0

The Fourier transform of the displacement u(R,t) is
Ω(ω ) = f i( r / R ) (σβ / µ )(ω 2 + α 2 )−1f and α are determined by requiring the long-period
limit of the spectral density agrees with that from adouble-couple determined from a dislocation.
Ω s (ω ) = Rθϑ M 04πρβ 3 R
ERI Notes: RJA()

63

Brune’s Spectral Model
The average spectrum is given by:Ω s (ω ) = Rθϑ (σβ / µ ) ( r / R ) (ω 2 + (2.34 β / r)2 )−1The corner frequency fc = (1 2π )(2.34 β / r) = 0.37 β / r
At zero frequency, the spectrum must be the same as that fromthe double-couple dislocation.
Ω s (0) =Rθϑ M 0( 4πρβ R )3or4πρβ R ) Ω (0)(M =3s0Rθϑ

Brune : fc = 0.37 β / r
Madariaga(1976) fc = 0.21β / r
Madariaga(1979) fc = 0.28 β / r
ERI Notes: RJA

64

Brune Spectrum
Spectra!
1.E+24!

Spectral Value!

1.E+23!

1.E+22!

1.E+21!

1.E+20!

Magnitude 6.7!
1.E+19!
0.1!

ERI Notes: RJA

1!

10!

Frequency (Hz)!

100!

65

Computing Corner Frequency andSeismic Moment from Spectrum
X-5

B. ALLMANN AND P. SHEARER: GLOBAL STRESS-DROP VARIATIONS

azimuth = 1.4o0.5 depth = 39 km

P0pP

−0.5

sP
−1−500time [s]relative log amplitudec)

50

0−0.05−0.1−101010frequency [Hz]0best fitobserved−1−2−3

fc = 0.480 Hz∆σ = 14.5 MPa−101010frequency [Hz]d)

0.05

−0.15

relative log amplitudeb)

relative log amplitude

normalized amplitudea)

18

best fit
synthetic

17.5
1716.516fc = 0.489 Hz∆σ = 15.3 MPa−101010frequency [Hz]Figure 5. a) Windowed normalized time series of one example trace with strong depth phases (black). The computed
stick seismogram is shown in bold grey. b) Computed EGF-corrected source spectra for the example trace (black).
The best-fitting theoretical model (dashed grey) has a corner frequency of 0.48 Hz and a stress drop of 14.5 MPa. c)
Demeaned spectrum of the stick seismogram. d) Sum of best-fitting theoretical model and spectrum of stick seismogram (black). The dashed grey line shows the new best-fitting model. Note the small difference in fc compared tob).
ERI2.2.
Notes:
RJAdrop estimation
Stress66
AllmannandShearer,tion limit that may
bias stress-drop
estimates
for large 2009
magnitude
earthquakes. This resolution limit is discussed in Appendix A.

Brune Spectrum with Kappa
κ= t* = travel time/Q!"#$%#&'(&')*%+,%-./.!0'+123.244560'
1.E+24!

1.E+23!

Spectral Amplitude!

1.E+22!

1.E+21!

1.E+20!

1.E+19!

Kappa 0.040!

1.E+18!

1.E+17!

1.E+16!
0.1!

ERI Notes: RJA

1.0!

Frequency (Hz)!

10.0!

100.0!

67

Acceleration: Gold MineERI Notes: RJA

McGarr et al., 1981

68

Brune Spectrum with Kappa
κ= t* = travel time/Q!"#$%#&'(&')*%+,%-./.!0'+123.244560'
1.E+24!

1.E+23!

Spectral Amplitude!

1.E+22!

1.E+21!

1.E+20!

1.E+19!

Kappa 0.020!

1.E+18!

1.E+17!

1.E+16!
0.1!

ERI Notes: RJA

1.0!

Frequency (Hz)!

10.0!

100.0!

69

Slip and Stress at a Pointon the Fault

σE =Δσ =Δσ(static) = σ0 –σ1

ERI Notes: RJA

70

Brune’s Spectral Model
M 0 = (16 7 ) σ r 3
fc = 0.37 β / r

M 0 = (16 7 ) σ (0.37 β / fc )3σ = ⎡⎣(7 /16)(1 / 0.37 β )3 ⎤⎦ M 0 fc3
Brune : fc = 0.37 β / r
Madariaga(1976) fc = 0.21β / r
Madariaga(1979) fc = 0.28 β / r
Thus the difference in stress drop between Brune and Madariaga
is (0.37 / 0.21)3 = 5.5

ERI Notes: RJA71

Stress and Energy (1)Es
σa = µ: Apparent StressM0
σ a ≤ Δσ /2

Es = 12 (σ 0 + σ 1) DA − σ f DA : Elastic energy - Frictional Energy

Orowan's model: σ f = σ1
Es = 12 (σ 0 − σ1) DA : Only the stress difference is measured
Es = 12 ( Δσ ) DA
DA = M 0 / µ1 Δσ = µ E s)
2(M0ERI Notes: RJA72

Stress and Energy (2)Following Randall, BSSA, 1973: The spectral theory of seismic
sourcesΩS (0) = Rθϑ M 0 / (4πρβ 3 R) = Rθϑ M 0 / (4πµβ R)∞

ΩS (ω ) = ∫ u(t)e−iω t−∞∞

dt ≤ ∫ u(t) e−iω t dt = ΩS (0) provided that−∞the displacement pulse does not change sign.
Following Randal (1972)
let x=ω /2π fc ΩS (ω ) = ⎡⎣ Rθϑ M 0 / (4πρβ 3 R) ⎤⎦ F(x)where F(x)=(1+x 2 )−1 for Brune's ModelNote: for P waves Ω P (ω ) = ⎡⎣ Rθϑ M 0 / (4πρα 3 R) ⎤⎦ F(x)and fc = 0.37α / r∞

E R (PorS) = (1 / 2)( R M f ) / ( ρc ) ∫ x 2 F 2 (x) where c = α or β2θϑ

E R (S)

3α=E (P)RERI Notes: RJA

203c5022β 273

Stress and Energy (3)Thus the total radiated energy is∞

E R = (11 / 9)(4π / 5)( R M f ) / ( ρβ ) ∫ x 2 F 2 (x)2θϑ203c50

This is valid for any choice of spectral model.
Assuming the stress drops from σ 0 to σ 1 : σ = σ 0 − σ 1
The energy available for seismic radiation from a circular fault
W = (1 / 2)(σ M 0 / µ )Using M 0 = (7 /16)σ r 3 ⇒ σ = (16 / 7)(M 0 / r 3 )Substitute σ this into the expresssion for W(W= 7π 3 M 02 f c3) 4 ρβ k where 2π f = kβ / r53c

W = ER( 7π M f ) 4 ρβ k = (11 / 9)(4π / 5)( R M f ) / (ρβ )∫ x F (x)3203c532θϑ

203c5∞

220which leads to an expression for k∞k  17.3 ∫ x 2 F 2 (x)0

ERI Notes: RJA74

Stress and Energy (4)For Brune's spectral model one findsk  2.80 as opposed to what Brune derived 2.34 (correction, 1971)
fc = 0.44 β / 4 which is 20% larger.
Many different models including simple ω −2 (k=2.35) andeven simple ω −3 models (k=2.96) or spherical models (ω −2 ) with valuesof k~2.7 allow one to estimate E R⎛ M 02 fc3⎞E R  3.0 ⎜ρβ 5 ⎟⎠⎝
Recall that we started with an energy for a circular fault⎛ M 02 fc3⎞
W=(1/2)(σ M 0 / µ ) = E R = 3.0 ⎜5⎟ρβ ⎠⎝(σ = 2 µ ER M0) twice the "apparent stress" (Wyss and Molnar, 1972)⎛ M 0 fc3 ⎞σ  6.0 ⎜⎝β 3 ⎟⎠ERI Notes: RJA75

Fraction of Energy in Source Spectrum

1

Fraction of Source Energy as a Function ofNormalized Corner Frequency

f/fc %
1! 0.18
2! 0.45
3! 0.60
6! 0.79
10! 0.85
20! 0.94

0.1

0.01
0.1

ERI Notes: RJA

From Singh and Ordaz, BSSA, 1994

110
Fraction of Corner Frequency (f/fc)

100

76

Representation Theorem∞

un ( x,t ) = ∫ dτ ∫∫ [ ui (ξ , τ ] cijpqν j ∂Gnp (x,t − τ ; ξ , 0) / ∂ξq dΣ−∞Σ

Aki and Richards, (3.2)

s⎡⎤
⎛
⎞
⎛
⎞
dsdG
dcs2
s2sra
 = â • ufr ∗ ∫ ⎢ c ⎜• Ga ⎟ + c ⎜• sr ⎟ +s r • G a ⎥dl⎝ dq⎠⎝ dq⎠ dty ( t ,x ) ⎣⎦(Slip Rate
Time Function
Stress Drop(spatial derivative of slip)

ERI Notes: RJA)

Change in isochrone
Velocity--accelerationof the the rupture front

Spudich and Frazer, BSSA, 1984 77

“Seismic Displacements Near a Fault”
Keiiti Aki ( J. Geophys. Res., 1968⎡⎤
sin Xasin XbU l (Q, ω ) = DΔhΔl ⎢ A (ω )exp ( −iω r a − iXa −iω l v ) + B (ω )exp ( −iω r b − iXb −iω l v ) ⎥XaXb⎣⎦

Xc = ω (Δl 2) ⎡⎣(1 v ) − ( cosθ c ) ⎤⎦h −Z )(D=D202h120<Z <hU(Q,t) = FFT −1 ⎡⎣ ∑ l U l (Q, ω ) ⎤⎦ERI Notes: RJA78

• Fault Geometry:
Length, Width• SlipSlip

Finite Fault—Kinematic (Haskell, 1964)• Rise Time

TimeSlip Rate

• Rupture Velocity
Rise TimeTime

ERI Notes: RJA

79

Haskell Model
Haskell Model: A rupture front radiates out from thehypocenter at constant speed. Each point on thefault has the same slip and the same rise time. Thetime at which a point starts to slip is determined by
the distance of that point from the hypocenter
divided by the rupture speed.

ERI Notes: RJA

80

Imperial Valley Slip and Rupture

ERI Notes: RJA

From Archuleta, 1984, JGR

81

Kinematic Slip Functions

ERI Notes: RJA

Jordanovski, 198682

Displacements from
Kinematic Ruptures withDifferent Slip Functions

Station 3
ERI Notes: RJA

Station 1
Jordanovski, 1986

83

Displacements from
Kinematic Ruptures withDifferent Slip Functions

Station 2
ERI Notes: RJA

Jordanovski, 1986

84