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