stress- and state-dependence of earthquake occurrence jim dieterich, uc riverside
TRANSCRIPT
Stress- and State-Dependence of Earthquake Occurrence
Jim Dieterich, UC Riverside
Formulation for earthquake rates
• Unified and quantitative framework for analysis of effects of stress changes on earthquake occurrence
• Some applications:– Aftershocks– Foreshocks– Complexity of earthquake events– Triggering of earthquakes by seismic waves– Tidal triggering (why the effect is so weak)– Earthquake probabilities following stress change– Solutions for stress changes from observations of
earthquake rates– Stress relaxation by seismic processes for
geometrically complex faults
Earthquake rate formulation: Model
• Earthquake occurrence is controlled by earthquake nucleation processes
• Earthquake nucleation as given by rate- and state-dependent friction is time dependent and highly non-linear in stress and gives the following
• Coulomb stress function as where
• At steady state
The characteristic time to reach steady-state
R =r
γ ˙ S r
€
dγ =1
Aσdt −γ dτ −γ
τ
σ−α
⎛
⎝ ⎜
⎞
⎠ ⎟dσ
⎡
⎣ ⎢
⎤
⎦ ⎥
€
dS = dτ −μ eff σ
€
μeff =τ
σ−α ≈ 0.35
R =r
γ ˙ S rdγ =
1Aσ
dt−γdS[ ]
€
γ=1˙ S
€
ta =Aσ
˙ S
Dieterich, JGR (1994), Dieterich, Cayol, Okubo, Nature, (2000), Dieterich and others, USGS Professional Paper 1676 (2003)
Earthquake rates following a stress stepE
art
hq
uak
e r
ate
(R
/r )
Time (t / ta )
€
˙ S = 0 :
R =rAσ ˙ S r
Aσ ˙ S r exp−ΔS
Aσ
⎛
⎝ ⎜
⎞
⎠ ⎟ + t
R =a
c + 1 Omori's Law
where a = rAσ ˙ S r ,
and c = Aσ ˙ S r( )exp−ΔS
Aσ
⎛
⎝ ⎜
⎞
⎠ ⎟
€
˙ S ≠ 0 :
R =r
1+ exp−ΔS
Aσ
⎛
⎝ ⎜
⎞
⎠ ⎟−1
⎡
⎣ ⎢
⎤
⎦ ⎥exp
−t
ta
⎛
⎝ ⎜
⎞
⎠ ⎟
Aftershock model
Pict of Coulombchange
Str
ess
Time
Ea
rth
qu
ake
ra
teB
ack
gro
un
d ra
te
Time
€
R =r
1+ exp−ΔS
Aσ
⎛
⎝ ⎜
⎞
⎠ ⎟−1
⎡
⎣ ⎢
⎤
⎦ ⎥exp
−t
ta
⎛
⎝ ⎜
⎞
⎠ ⎟
Coulomb stress change
Coulomb stress change
Stress ShadowS
tres
s
Time
Ea
rth
qu
ake
ra
teB
ack
gro
un
d ra
te
Time
€
R =r
1+ exp−ΔS
Aσ
⎛
⎝ ⎜
⎞
⎠ ⎟−1
⎡
⎣ ⎢
⎤
⎦ ⎥exp
−t
ta
⎛
⎝ ⎜
⎞
⎠ ⎟
€
With increasing stress drop the time toreturn to fixed fraction of background rateincreases asymtotically to
tc =−ΔS
˙ S
(simple stress - shadow clock offset)
Method: Stress time series
dγ =1
Aσdt−γdS[ ]R=
rγ˙ S r
,
STEPS
1) Select region and magnitude threshold
2) Smooth earthquake rate: R(t)
3) Obtain time series for γ:
4) Solve evolution equation for Coulomb stress S. For example:
γ(t) =r
R(t)˙ S r
ΔS=Aσ lnγ1 +
Δt2Aσ
γ2 −Δt
2Aσ
⎡
⎣
⎢ ⎢
⎤
⎦
⎥ ⎥
S
time
Δt
(t1,γ1)
(t2,γ2)
ΔS
Synthetic Data
Input stressSimulated seismicityStress Inversion
€
R =r
1+ exp−ΔS
Aσ
⎛
⎝ ⎜
⎞
⎠ ⎟−1
⎡
⎣ ⎢
⎤
⎦ ⎥exp
−t
ta
⎛
⎝ ⎜
⎞
⎠ ⎟
Input stressSimulated seismicityStress Inversion
Synthetic Data
€
R =r
1+ exp−ΔS
Aσ
⎛
⎝ ⎜
⎞
⎠ ⎟−1
⎡
⎣ ⎢
⎤
⎦ ⎥exp
−t
ta
⎛
⎝ ⎜
⎞
⎠ ⎟
Figure: Barry EakinsRift zones modified from Fiske and Jackson (1972)Bathimetry: USGS JAMSTEC
Cross section through east rift of Kilauea
Inversion of EQ rates for stress (1996-1994)
Dieterich and others, 2000, Nature
Method to obtain stress changes from earthquake rates
STEPS1) From earthquake rates obtain time series
for γ at regular grid points:
2) Solve evolution equation for Coulomb stress S as a function of time at each grid point€
R(t) =r
γ (t) ˙ S r
Dieterich, Cayol, and Okubo, Nature (2000)Dieterich and others, USGS Prof Paper(2003)
€
dγ =1
Aσdt − γdS( )
3) Prepare maps (or cross sections) of stress changes over specified time intervals
0.6 MPa/year
-0.6 MPa/year
Aσ = 0.6MPa
1976-1983 >1983Deformation 0.5MPa/yr (0.1MPa)Seismicity 0.3–0.6 MPa/yr ≤0.1 Mpa/yrRift intrusion rate 0 .18km3/yr 0 .06km3/yrNS extension 25cm/yr 4cm/yr (Summit region)€
˙ S
€
˙ S
Stress changes before 1983 eruption
Seismicity solution, 1980-1983
0.2 MPa/year
Deformation model, 1976-1983
0.5 MPa/year
2
Dieterich, Cayol, and Okubo, Nature (2000)Dieterich and others, USGS Prof Paper(2003)
Stress changes at the time of the 1983 intrusion & eruptionDeformation model
Seismicity solution
Dieterich and others, USGS Prof Paper(2003)
Stress changes at the time of the 1977 intrusion & eruptionDeformation model
Kalapana Earthquake M7.3 1975
AM4.9 12/18/76M4.6 1/14/77M4.7 2/3/77M4.7 6/5/77M4.7 6/5/77M5.4 9/21/79M4.6 9/27/79M4.6 9/27/79M5.4 9/21/795 km5 km5 km5 km5 km5 km5 km5 kmM4.9 12/18/76M4.6 1/14/77155°20’155°10’155°00’19°15’19°25’19°25’5 km155°20’155°10’155°00’19°15’19°25’19°25’5 km155°20’155°10’155°00’19°15’19°25’19°25’5 kmBCa1a1’a2a2’a1a1’M4.7 2/3/775 km5 kma2a2’bb’c1c1’c2c2’bb’c1c1’c2c2’0.3 MPa-0.3 MPa0 MPaCoulomb
stress0.3 MPa-0.3 MPa0 MPaCoulomb
stress0.3 MPa-0.3 MPa0 MPaCoulomb
stress
Earthquakes M≥4.6 1976-1979
M5.0 3/20/83M5.2 9/9/83M5.2 9/9/83M5.0 3/20/83M6.2 6/25/89M6.2 6/25/895 km5 km5 km5 km5 km5 km19°15’19°25’19°25’5 km19°15’19°25’19°25’5 km19°15’19°25’19°25’5 kmDEFdd’ee’ff’dd’ee’ff’0.3 MPa-0.3 MPa0 MPaCoulomb stress
0.3 MPa-0.3 MPa0 MPaCoulomb stress
0.6 MPa-0.6 MPa0 MPaCoulomb stress
Earthquakes M≥4.6 1980-1989
M5.4 8/8/90M5.1 6/30/97M5.1 6/30/975 km5 km5 km5 km19°15’19°25’19°25’5 km155°20’155°10’155°00’19°15’19°25’19°25’5 kmGHgg’hh’M5.4 8/8/90h’gg’hh’0.3 MPa-0.3 MPa0 MPaCoulomb stress
0.3 MPa-0.3 MPa0 MPaCoulomb stress
Earthquakes M≥4.6 1990-1999
M5.4 9/21/79M4.6 9/27/79M4.6 9/27/79M5.4 9/21/795 km5 km5 km5 km19°15’19°25’19°25’5 kmCc1c1’c2c2’c1c1’c2c2’0.3 MPa-0.3 MPa0 MPaCoulomb stress
155°20’155°10’155°00’aa’Eruption of 9/13/77A Zone of eruptive fissures
20026838519°15’19°25’19°25’5 kmaa’0.6 MPa-0.6 MPa0 MPaCoulomb stress
1.25 MPa-1.25 MPa0 MPaCoulomb stress
M~ 5 Earthquakes following Sept. 13, 1977 eruption
M4.6 9/27/79M4.6 9/27/79
5.4 9/21/795.4 9/21/79
M~ 5 Earthquakes following Jan. 1, 1983 eruption
M5.0 3/20/83
M5.2 9/9/83
1/3/83
5 kmM4.7 6/5/77M5.4 9/21/79M4.6 9/27/79M6.2 6/25/89M5.1 6/30/97
Geometrically complex faults
USGS, 2003
Fault geometry
Individual faults exhibit approximately self-similar roughness (fractal dimension~1).
Fault in the Monterrey Formation
San Francisco Bay Region
Fault systems also appear to be scale-independent
Random Fractal Fault Model
€
Ampl.∝ βl H
H = Hurst exponent
At reference length l =1 ,
rms (slope) = β
Solve for slip using boundary elements.
Simple Coulomb friction with μ = 0.6
Periodic B.C, or slip on a patch
α = 0.3α = 0.1α = 0.03α = 0.01Faults in Nature = 0.3
= 0.1
= 0.03
= 0.01
Slip of a fault patch
SlipShear stressPlanar faultSlipShear stressPlanar faultFractal fault profile α=.05 Fractal fault
Fault slip and stress changes
Smooth fault Fractal fault: H=1, =0.01
Global slip Global slip
Fault slip and stress changes
Smooth fault Fractal fault: H=1, =0.01
Non-linear scaling of slip with fault length
Hurst exponent: H = 1.0Roughness amplitude: = 0.05Region of ~ linear scaling of
Slip with fault length
Non-linear scaling of slip with fault length:Average of slip for n100 simulations
Hurst exponent: H = 1.0Roughness amplitude: = 0.1
dMAX = 85
Region of ~ linear scaling ofslip with fault length
FAULT LENGTH
€
dmax
€
dmax = cα −2
€
dmax = cβ −2 = 0.01
= 0.03
= 0.1
= 0.3
Geometric complexity forms barriers to slip.
Barrier stress increases with total slip and sequesters strain energy that would otherwise be released in slip.
€
SBACK = Kd
The barrier stress acts as an elastic back-stress, which opposes slip. Back-stress increases linearly with slip.
Slip saturates at when the back-stress equals the applied stress.
€
dMAX
€
K =SA
dmax
= cβ 2SA
Non-linear scaling andsystem size-dependence
Slip, d
€
dMAX
Ba
ck s
tre
ss, S
BA
CK
SA = Applied stress
€
K =SA
dmax
= cβ 2SA
€
dmax
€
dmax = cβ −2
SBACK = cβ 2SA( )d
= 0.01
= 0.03
= 0.1
= 0.3
Average slip on non-planar faults n~100
Planar fault model with elastic back stress
Hurst exponent: H = 1.0Roughness amplitude: = 0.1
dMAX = 85
Non-linear scaling of slip with fault length
FAULT LENGTH
Yielding and Stress Relaxation
• Stresses due to heterogeneous slip cannot increase without limit - some form of steady-state yielding and stress relaxation must occur
Slope of 0.01 shear strain 0.01, brittle failure
• In brittle crust, stress relaxation may occur by faulting and seismicity off of the major faults.
Instantaneous failure and slip during earthquake Post-seismic – aftershocks and long-term seismicity
• Yielding will couple to the failure process, by relaxing the back-stresses
€
RMS Slope∝ βl H −1
Data from Sowers and others (1992) , US Geological Survey
Landers and Hector MineEarthquakes
010Steady-state yielding by earthquakes:
EQ rate Coulomb stress rate Long-term slip rate
€
R∝ d−1.5
€
R∝ d−1.0
Average long-term earthquake rate by distancefrom fault with random fractal roughnessE
arth
quak
e ra
te
Ear
thqu
ake
rate
•Stressing due to fault slip at constant long-term rate•Model assumes steady-state seismicity at the long-
term stressing rate, in regions where
€
˙ S > 0
€
R∝ d−1.5
€
R∝ d−1.0
Average long-term earthquake rate by distancefrom fault with random fractal roughnessE
arth
quak
e ra
te
Ear
thqu
ake
rate
€
R∝ d−n , where n = D − H
D = 2 for 2D systems
D = 3 for 3D systems
Scaling:
001
00
01
Initial Aftershock Rate / Background Rate
001
00
01
Aftershock rates as function of distance
0.01 – 0.03 L0.03 – 0.05 L0.05 – 0.07 L0.07 – 0.09 L0.09 – 0.10 L
€
Omori's AS decay law : R∝ t p
L=rupture length
Conclusions: Seismicity stress solutions
• Stress shadows are seen for all earthquakes M≥4.7 • Quantitative agreement of deformation measurements
and seismicity stress solutions – Stress changes 1976-1983– Stress changes related to 1983 eruption– Stressing rate change 1980-1983
• Provides greater detail of stress changes at depth than can be obtained from deformation modeling– Resolve stress patterns for earthquakes M~5 at depths of 10km.
This includes stress shadows– Useful for guiding deformation modeling, by eliminating
alternative models
• Reveals stress interactions between magmatic and earthquake processes at Kilauea volcano
Conclusions - Seismicity and non-planar faults• Fault complexity heterogeneous slip and stress
• Fault complexity + elasticity non-linear scaling and system size-dependence
• Heterogeneous stresses increase with slip yielding & stress relaxation
ΔSlope = 0.01 shear strain 0.01 brittle failure
Instantaneous failure and slip during earthquake
Fall-off of background seismicity by distance
Post-seismic – aftershocks within “stress shadow”
• Stress relaxation process will couple to slip on major faults by relaxing the back stresses. Speculation:
Restore linear scaling Restore independence of system size
€
R∝ d−n , where n = D − H