Rupture Velocity

ERI Notes: RJA

86

Importance of Fitting theSpectrum at the Corner Frequency

ERI Notes: RJA

Tumarkin & Archuleta, Natural Hazards, 199787

• Fault Geometry:Length, Width• SlipSlip

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

Time
Slip Rate• Rupture Velocity
Rise TimeTime

V

ERI Notes: RJA

88

Rupture Directivity

ERI Notes: RJA

Benioff, 1955

89

Rupture Velocity (1)From A. Udias, Principlesof Seisimology⎛r0 − ξ cos θ ⎞u(xi ,t) = ∫∫ Δu ⎜ ξ,t −dξ dη⎟A⎝⎠βwhere ξ is along L and η is along W
the slip is constant and occurs instantaneously across W.
⎛r0 − ξ cos θ ⎞u(xi ,t) = W ∫ Δu ⎜ ξ,t −dξ⎟0⎝⎠βLERI Notes: RJA

90

Rupture Velocity (2)⎛r0 − ξ cos θ ⎞u(xi ,t) = W ∫ Δu ⎜ ξ,t −dξ
⎟0
⎝
⎠β
L

If the rupture velocity v is constant, Δu(ξ ,t)=Δu(t − ξ / v)
⎛ r0 ξ ⎛ β⎞⎞u(xi ,t) = W ∫ Δu ⎜ t − − ⎜ − cos θ ⎟ ⎟ dξ0⎠⎠⎝β β⎝v
Lr ξ⎛β⎞Let ψ = 0 + ⎜ − cos θ ⎟⎠β β⎝v
Then the Fourier transform of u(xi ,t) isL∞0−∞ −ψ )e−iω (t−ψ ) dt
U(xi , ω ) = W ∫ d ξ ∫ Δu(t∫∞−∞ −ψ )e−iω (t−ψ ) dt = iωΔU(ω )e−iωψΔu(twhere ΔU(ω ) is the Fourier transform of Δu(t) and iωΔU(ω ) is the FT of Δu(t)U(xi , ω ) = WiωΔU(ω )e
ERI Notes: RJA−iω r0 / β⎡ ωξ ⎛ β⎞⎤exp−i−cosθ⎟⎠ ⎥ d ξ∫0 ⎢⎣ β ⎜⎝ v⎦
L

A. Udias, Principles of Seismology 91

Rupture Velocity (3)U(xi , ω ) = WiωΔU(ω )elet γ = −−iω r0 / β⎡ ωξ ⎛ β⎞⎤∫0 exp ⎢⎣ −i β ⎜⎝ v − cosθ ⎟⎠ ⎥⎦ dξ
L

ω⎛β
⎞
−cos
θ
⎜
⎟⎠β⎝v

L⎡ ωξ ⎛ β
2 ⎛ γ L⎞⎞⎤⎛ γ L⎞iγξ
exp−i−cos
θ
d
ξ
=ed
ξ
=sinexp⎟⎠ ⎥⎜⎝⎟⎠⎜⎝ i⎟⎠∫0 ⎢⎣ β ⎜⎝ v
∫0γ22⎦
2 ⎛ γ L⎞
γLωL ⎛ β⎛ γ L⎞⎛ sin Χ ⎞ iΧ
⎞sin ⎜expi=LewhereΧ==−−cosθ⎜⎝⎟⎜⎝⎟⎜⎟⎠⎝ 2 ⎟⎠γ2 ⎠Χ ⎠22β ⎝ v
L⎡ ⎛ ω r0sin Χπ⎞⎤U(xi , ω ) = WLΔU(ω )exp ⎢ −i ⎜−Χ− ⎟⎥Χ2⎠⎦⎣ ⎝ β

ERI Notes: RJA

A. Udias, Principles of Seismology 92

Rupture Velocity (4)U(xi , ω ) = WLΔU(ω )⎡ ⎛ ωrsin Χπ⎞⎤exp ⎢ −i ⎜ 0 − Χ − ⎟ ⎥Χ2⎠⎦⎣ ⎝ β

sin ΧωL ⎛ β⎞
is the sinc function Χ = −−cosθ⎜⎝⎟⎠ thus the ruptureΧ2β v
1contributes ato the decay of the spectrum.
ωSuppose Δu(t) = ΔuH (t), then ΔU(ω ) = Δu iω
For Haskell's model: Δu(t) = Δut / τ where τ is the rise time of the ramp
Note that−iω tΔu(1−e)ΔU(ω ) =: What is the spectral shape?
ω 2τ
For Brune's dislocation Δu(t) = ΔuH (t)(1− e−t /τ )ΔU(ω ) = Δu
: What is the spectral shape?
(1+ iωτ ) iω

ERI Notes: RJA

A. Udias, Principles of Seismology 93

Double-Couple Radiation
Affected by Rupture Speed

Note change in scale when Vr/Vs= 0.9
ERI Notes: RJA

94

1966 Parkfield
Earthquake

Horizontal Particle Velocity (cm/s)

60

Parkfield 1966
Component N65E

40200-20-40-60

Time (s)ERI Notes: RJA

95

Imperial Valley Slip and Rupture

ERI Notes: RJA

From Archuleta, 1984, JGR

96

Rupture Directivity: Imperial Valley
115 40'33˚00'

115 20'BrawleyN

Brawley Fault Zone30!
!
!

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lt

aulF
ria

10!

32 40'!

pe!

!

Holtville!

Im!
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! El Centro!

Calexico
United States!
Mexico!

0"

1.0
S waves forward and back directions of rupture
0.80.6

Velocity (normalized)

0.4
0.2
0-0.2
-0.4
-0.6
-0.8

ERI Notes: RJA-1.005101520
Time (s)25303540

Archuleta, 1984 97

Fault Normal Components

ERI Notes: RJA98

Bam Earthquake;259'18<!">?;3@3H%GC%B>?/3@3EEC%D,-,-,&'(+&'()*+>?43@3A%BCDE>?43@3FEGCH%>?;3@3GFCJB>?/3@3HBC%B./././&'(+>?/3@3FCBA>?;3@3J%CHD>?43@3IAACGD301(23&*+01(23&*+FGFG,""!"#!"#D&'()*+FG.
-3!

FG&'()*+D!"#!"#

D&'()*+3,//&'()*+&'()*+D

01(23&*+.

3!
-ERI Notes: RJA!"&'(+&'()*+!"/1*=57'2(2:8&'()*+#!$%&'()*%+#!$%&'()*)%+#!$%&'()*%+4''25267819:

FG

D!"#!"#D&'()*+

FG

Natalia Poaita, 201099

Distribution of Seismic Energy:
Effect of Directivity

40% of the seismicenergy passes
through 6.4% ofthe sphere (darkred and purple).

Contours of seismic flux with each shade being 20%.

ERI Notes: RJA

Favreau and Archuleta, 2003 GRL 100

Supershear Rupture Velocity• Tradition, supported by most data, suggests earthquakes
have subshear rupture velocities V~0.85cs• Ground motion estimates and building designcodes are based on this assumption• However, several recent earthquakes might haveoccurred at supershear speeds (V > Vs)- 1979 M6.5 Imperial Valley, California- 1999 M7.4 Izmit, Turkey- 2001 M8.1 Kunlun, China- 2002 M7.9 Denali Fault, Alaska• The consequences of supershear rupture velocity cansignificantly alter the landscape of predicting the severity ofthe ground motion
ERI Notes: RJA101

Supershear Rupture

ERI Notes: RJA

Dunham and Bhat, 2008 102

Rupture Directivity

ERI Notes: RJA

Dunham, 2005

103

Experimental Observation
Laboratory experiments of Xia and Rosakis (2004)

explodingwire

ERI Notes: RJA

104

Experimental Observation
Laboratory experiments of Xia and Rosakis (2004)

explodingwire

ERI Notes: RJA

105

Experimental Observation
Laboratory experiments of Xia and Rosakis (2004)

explodingwire

ERI Notes: RJA

Rayleighwave?

106Transition of Subshear to Supershear

Visualization of the sub-Rayleigh–to–supershear rupture transition (ϕ = 25° and P= 9 MPa). ( Ato C) were taken at 18 µs, 30 µs, and 38 µs, respectively. In the rupture-tip history plot ( D),lines corresponding to P, S , and Rayleigh waves as reference.

ERI Notes: RJA

107
Xia, Rosakis, Kanamori, Science 2004

Subshear & Supershear Velocities( A) Purely sub-Rayleigh (ϕ = 25° and P= 7 MPa) and( B) purely supershear (ϕ = 25° and P= 15 MPa)
rupture at the same time (28 µs) after triggering.
ERI Notes: RJA

Xia, Rosakis, Kanamori, Science 2004

108

Imperial Valley Dynamic Rupture

ERI Notes: RJA

Favreau and Archuleta, GRL, 2003 109

Velocity Field for Intersonic Rupture

Mach front

ERI Notes: RJA

Dunham, 2005

110

Multiple Diffractions Between Crack Tips

Diffractions generated
during nucleation

4. Rayleigh wave
left behind
1. Original subRayleigh rupture

ERI Notes: RJA

3. Supershear
rupture becomes
self-sustaining

Predicted
theoretically byBurridge (1973)
and observed
numerically byAndrews (1976)

2. PS diffraction
triggers slip

Dunham, 2005 111

Transient Diffractions:
Step-function Stress Drop Behind Stationary Crack

Slip velocity

Shear traction
SSRS

negativePS

PP

R SP

ERI Notes: RJA

traction-free

no slip

Dunham, 2005 112

Velocity Field for Sub-Rayleigh Rupture

Slip zone

ERI Notes: RJA

Breakdownzone

Dunham&Archuleta, GRL113

Velocity Field for Intersonic Rupture

Mach front(radiatingS waves)

ERI Notes: RJA

Dunham&Archuleta, GRL 114

Observational Signature
V=1.3csERI Notes: RJA

Dunham, 2005115

Velocity Field for Intersonic Rupture

Reversal of Swave motion

ERI Notes: RJA

Dunham&Archuleta, GRL116

Observational Signature
V=1.6csERI Notes: RJA

Dunham, 2005

117

Ground Motion Carried Off the Fault

ERI Notes: RJA

Dunham, 2005

118

2002 Denali Fault Earthquake
P-waveOrigin time：
22:12:41.5, 03Nov.,
2002(UTC)CMTMS： 7.9 ／ MW: 7.9
Aftershock distribution(24hrs), AEIC
(Ratchkovski et al., 2003)
Strong motion stations:
USGS, Alaska Univ.,
Alyeska Pipeline Service
Company
ERI Notes: RJA

Iwata, 2004

119

The Puzzle

FP=1.5FN?

ERI Notes: RJA

three pulses?

120

Directivity: Denali Fault

D. Eberhart-Phillips et al., Science 300, 1113 -1118 (2003)
ERI Notes: RJA

Dunham & Archuleta, 2004, BSSA

121

Rupture History: Model 2
PS10

Model 2
With healing

q=0.45
Ds=14MPa
qa=0.77
Dsa=24.3 MPa
dc=100 cm
th=1.47s

RSP

Dx=125m
Dt= 4.09 ms
Snapshots
every 1.83s
ERI Notes: RJA

Dunham & Archuleta, 2004, BSSA

122

Ground Motion: Model 2
Model 2
With healing
Fixes FP width

S=1.22
Ds=14MPa
Sa=0.28
Dsa=24.3MPa
dc=78cmth=1.47s
Dx=125m
Dt=7.1ms
2nd Rayleigh pulse isearly

ERI Notes: RJA

Dunham & Archuleta, 2004, BSSA

123

Movie-Fault Parallel: Model 2

ERI Notes: RJA

Dunham & Archuleta, 2004, BSSA

124

Movie-Fault Normal: Model 2

ERI Notes: RJA

Dunham & Archuleta, 2004, BSSA

125

Movie Supershear Rupture

ERI Notes: RJA

126

Ground Motion

Strong dependency of ground motion on autocorrelation of initialstress. Smooth model yields larger ground motion

ERI Notes: RJA

127