Stress- and State-Dependence of Earthquake Occurrence: Tutorial 2

Jim Dieterich

University of California, Riverside

Log

Coe

ffici

ent o

f fric

tion €

=0 + A lnV

V *

⎛

⎝ ⎜

⎞

⎠ ⎟+ Bln

θ

θ *

⎛

⎝ ⎜

⎞

⎠ ⎟

€

ss = const.+ (B− A) lnθss

Constant V (high)

Constant V (lo

w)

ssB

B-A

x V1

€

ss =Dc

V

Log

Coe

ffici

ent o

f fric

tion €

=0 + A lnV

V *

⎛

⎝ ⎜

⎞

⎠ ⎟+ Bln

θ

θ *

⎛

⎝ ⎜

⎞

⎠ ⎟

€

ss = const.+ (B− A) lnθss

Constant V (high)

Constant V (lo

w)

ssB

B-A

€

ss at V1

x

During slip evolves toward ss

V1

€

ss =Dc

V

Log

Coe

ffici

ent o

f fric

tion €

=0 + A lnV

V *

⎛

⎝ ⎜

⎞

⎠ ⎟+ Bln

θ

θ *

⎛

⎝ ⎜

⎞

⎠ ⎟

€

ss = const.+ (B− A) lnθss

Constant V (high)

Constant V (lo

w)

ssB

B-A

€

ss at V1

x

During slip evolves toward ss

V1

€

ss =Dc

V

Log

Coe

ffici

ent o

f fric

tion €

=0 + A lnV

V *

⎛

⎝ ⎜

⎞

⎠ ⎟+ Bln

θ

θ *

⎛

⎝ ⎜

⎞

⎠ ⎟

€

ss = const.+ (B− A) lnθss

Constant V (high)

Constant V (lo

w)

ssB

B-A

€

ss at V1

x

During slip evolves toward ss

V1

€

ss =Dc

V

1011101010910810710610510410310210110010-

1

10-

2

6

4

2

0

-2

-4

-6

-8

-10

-12

-14

-16

1 yr10 yr20 yr

Time to instability (seconds)

Log

(sl

ip s

pee

d)

m

/s

Effect of stress change on nucleation time

€

ti =Aσ

˙ τ ln

˙ τ

Hσ ˙ δ 0+1

⎡

⎣ ⎢

⎤

⎦ ⎥

€

˙ τ = 0.05 MPa/yr

1011101010910810710610510410310210110010-

1

10-

2

6

4

2

0

-2

-4

-6

-8

-10

-12

-14

-16

5min

1 yr10 yr20 yr

~1hr~5hr

Time to instability (seconds)

Log

(sl

ip s

pee

d)

m

/s

Effect of stress change on nucleation time

€

ti =Aσ

˙ τ ln

˙ τ

Hσ ˙ δ 0+1

⎡

⎣ ⎢

⎤

⎦ ⎥

€

˙ τ = 0.05 MPa/yr

€

˙ δ = ˙ δ 0σ

σ 0

⎛

⎝ ⎜

⎞

⎠ ⎟

α / A

exp τ

Aσ−

τ 0

Aσ 0

⎛

⎝ ⎜

⎞

⎠ ⎟

= 11.6

Earthquake rate formulation: Model

• Earthquake occurrence is represented as a sequence of earthquake nucleation events.

• Dependence of nucleation times on stressing history is given by nucleation solutions derived for rate- and state-dependent fault strength.

• Model assumes 1) The population of nucleation sources is

spontaneously renewed as stress increases2) Reference steady-state seismicity rate r at the

constant stressing rate .

€

˙ S r

Use the solution for time to nucleation an earthquake

(1) , where

and assume steady-state seismicity rate r at the stressing rate

This defines the distribution of initial conditions

(slip speeds) for the nucleation sources

(2)

The distribution of slip speeds (2) can be updated at successive time steps

for any stressing history, using solutions for change of slip speed as a

function of time and stress.

€

t =Aσ

˙ τ ln

˙ τ

Hσ ˙ δ 0+1

⎡

⎣ ⎢

⎤

⎦ ⎥

˙ τ =const≠0˙ σ =0

H =B

DC

−Kσ

€

˙ τ r

t = n r , n is the sequence number of the earthquake source

€

˙ δ 0(n) =1

Hσ˙ τ r

exp˙ τ r n

Aσ r

⎛

⎝ ⎜

⎞

⎠ ⎟−1

⎡

⎣ ⎢

⎤

⎦ ⎥

Model for earthquake occurrence

Log (time to instability)

Lo

g (

slip

sp

ee

d)

For example changes of with time are given by the nucleation solutions

and change of with stress are given directly from the rate- and state-

formulation

In all cases, the final distribution has the form of the original distribution

where

€

˙ δ 0(n) =1

Hσ

γexp

˙ τ r n

Aσ r

⎛

⎝ ⎜

⎞

⎠ ⎟−1

⎡

⎣ ⎢

⎤

⎦ ⎥

Evolution of distribution of slip speeds

€

˙ δ 0(n)

€

˙ δ 0(n)

€

˙ δ = ˙ δ 0σ

σ 0

⎛

⎝ ⎜

⎞

⎠ ⎟

α / A

expτ

Aσ−

τ 0

Aσ 0

⎛

⎝ ⎜

⎞

⎠ ⎟

€

dγ =1

Aσdt − γdτ + γ

τ

σ−α

⎛

⎝ ⎜

⎞

⎠ ⎟dσ

⎡

⎣ ⎢

⎤

⎦ ⎥

€

˙ δ =1˙ δ 0

+Ht

A

⎡

⎣ ⎢

⎤

⎦ ⎥

-1

, ˙ τ = 0

€

˙ δ =1˙ δ 0

+Hσ

˙ τ

⎡

⎣ ⎢

⎤

⎦ ⎥exp

− ˙ τ t

Aσ

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥−

Hσ˙ τ

⎧ ⎨ ⎩

⎫ ⎬ ⎭

-1

, ˙ τ ≠ 0

Earthquake rate is found by taking the derivative dn/dt = R

For any stressing history

€

dγ =1

Aσdt − γdτ + γ

τ

σ−α

⎛

⎝ ⎜

⎞

⎠ ⎟dσ

⎡

⎣ ⎢

⎤

⎦ ⎥

€

R =r

γ ˙ τ r

Evolution of distribution of slip speeds

Coulomb stress formulation for earthquake rates

Earthquake rate ,

Coulomb stress

Assume small stress changes (treat as constants) ,

Note: . Hence,

Earthquake rate ,

R =r

γ ˙ τ rdγ =

1Aσ

dt−γdτ +τσ

−α⎛ ⎝

⎞ ⎠ dσ

⎡ ⎣ ⎢

⎤ ⎦ ⎥

dS=dτ −μdσ

€

−α

⎛ ⎝ ⎜

⎞ ⎠ ⎟ (Aσ )

R =r

γ ˙ S rdγ =

1Aσ

dt−γdS[ ]

Dieterich, Cayol, Okubo, Nature, (2000), Dieterich and others, US Geological Survey Professional Paper - 1676 (2003)

€

eff =τ

σ−α

⎛

⎝ ⎜

⎞

⎠ ⎟≈ 0.3−0.4

€

= and 0 ≤ α ≤ μ

Some useful solutions

Earthquake rate

Evolution with time

Stress step

R =r

γ ˙ S rdγ =

1Aσ

dt−γdS[ ]

€

γ=γ0 +t

Aσ , ˙ S = 0 γ = γ 0 −

1˙ S

⎡ ⎣ ⎢

⎤ ⎦ ⎥exp

−t ˙ S

Aσ

⎛

⎝ ⎜

⎞

⎠ ⎟+

1˙ S

, ˙ S = const

€

γ=γ0 exp−ΔS

Aσ

⎛

⎝ ⎜

⎞

⎠ ⎟

Example

€

R =r

γ ˙ S r

€

γ1 = γ 0 exp−ΔS

Aσ

⎛

⎝ ⎜

⎞

⎠ ⎟ =

1˙ S 1

exp−ΔS

Aσ

⎛

⎝ ⎜

⎞

⎠ ⎟

€

γ2 =1˙ S 2

+ γ1 −1˙ S 2

⎡

⎣ ⎢

⎤

⎦ ⎥exp

−t ˙ S 2Aσ

⎛

⎝ ⎜

⎞

⎠ ⎟

€

γ0

€

γ1

€

γ2

S

t€

˙ S 1 €

˙ S 2

€

S

€

γ0 =1˙ S 1

(steady state)

€

R =r

γ ˙ S r =

r˙ S 1˙ S 2

+ exp−ΔS

Aσ

⎛

⎝ ⎜

⎞

⎠ ⎟−

˙ S 1˙ S 2

⎡

⎣ ⎢

⎤

⎦ ⎥exp

−t ˙ S 2Aσ

⎛

⎝ ⎜

⎞

⎠ ⎟+

, where ˙ S r = ˙ S 1€

= 1˙ S 2

+1˙ S 1

exp−ΔS

Aσ

⎛

⎝ ⎜

⎞

⎠ ⎟−

1˙ S 2

⎡

⎣ ⎢

⎤

⎦ ⎥exp

−t ˙ S 2Aσ

⎛

⎝ ⎜

⎞

⎠ ⎟

10610510410310210110010-110-210-4

10-3

10-2

10-1

100

101

102

103

104

Time (t/ t )

Earthquake rate (R/r)

Example - Secondary aftershocks

t

S

t =0

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

⎛

⎝ ⎜

⎞

⎠ ⎟

Earthquake rates following a stress step

€

te = Aσ ˙ S r( )exp−ΔS

Aσ

⎛

⎝ ⎜

⎞

⎠ ⎟= c

€

R0 r( ) = expΔS

Aσ

⎛

⎝ ⎜

⎞

⎠ ⎟

Aftershocks by time and distance

€

te = ta exp−ΔS

Aσ1−

c 3

x 3

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

−1/ 2

−1 ⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

⎡

⎣

⎢ ⎢

⎤

⎦

⎥ ⎥

Time at which x is at edge of aftershock zone

Factors affecting rate of aftershock decay (p)

In this model, intrinsic value p=1

The following factors result in p≠1

• Spatial heterogeneity of stress change (S/A) p < 1

• Stress relaxation by log(t) after the stress step

if u>0.2 p > 1

• Secondary aftershocks p > 1 (short-term effect)€

˙ S = −u ln wt +1( )

Time (t/ta)

Ea

rth

qu

ake

ra

te (

R/r

0) Slope p=0.8

Net aftershock rate for region surroundinga circular shear rupture

Over short time intervals (t <<ta)

€

γ=γ0 exp−ΔS

Aσ

⎛

⎝ ⎜

⎞

⎠ ⎟=

1˙ S

exp−ΔS

Aσ

⎛

⎝ ⎜

⎞

⎠ ⎟

R

r= exp

ΔS

Aσ

⎛

⎝ ⎜

⎞

⎠ ⎟ ,

ΔR

r= exp

ΔS

Aσ

⎛

⎝ ⎜

⎞

⎠ ⎟−1

Triggering by seismic waves

100806040200-20

-10

0

10

20

30

0

5

10

Str

ess

(S

/A

)

Cu

mu

lativ

eN

um

be

r

Time (seconds)

Triggering by seismic waves

r = 5.0x10-5 /s

Triggering by seismic waves

140120100806040200010-7

10-6

10-5

10-4

10-3

10-2

10-1

100

101

102

103

104

PEAK STRESS = 10.15

PEAK STRESS = 10.14

PEAK STRESS = 9.00

Initial earthquake nucleation time - 100,000s

Time (seconds)

Slip speed (Dc/s)

Peak stress S/A

10.1510.14 9.00

1091081071061051041031020

2

4

6

8

10

12

14

16

18

20

Time to EQ with no triggering (sec)

Peak amplitude required to trigger EQ

1 year30 days1 day

peak amplitude of stess waves needed toinstantaneously trigger an EQ. Calculated using Excel program "Dyn triggering using slip rates"

Triggering by seismic waves

Pe

ak a

mpl

itud

e (S

) to

trig

ge

r E

Q (

A

)

Time to EQ with no triggering (sec)

10 year

1091081071061051041031020

2

4

6

8

10

12

14

16

18

20

Time to EQ with no triggering (sec)

Peak amplitude required to trigger EQ

1 year30 days1 day

peak amplitude of stess waves needed toinstantaneously trigger an EQ. Calculated using Excel program "Dyn triggering using slip rates"

Triggering by seismic waves

Pe

ak a

mpl

itud

e (S

) to

trig

ge

r E

Q (

A

)

Time to EQ with no triggering (sec)

10 year

Threshold stress model

€

˙ S =1

x

Change of earthquakes rates, tidal stresses

Over short time intervals (t <<ta)

For small stress changes (S << A) this becomes

ΔRr

=ΔSAσ

€

γ=γ0 exp−ΔS

Aσ

⎛

⎝ ⎜

⎞

⎠ ⎟=

1˙ S

exp−ΔS

Aσ

⎛

⎝ ⎜

⎞

⎠ ⎟

R

r= exp

ΔS

Aσ

⎛

⎝ ⎜

⎞

⎠ ⎟ ,

ΔR

r= exp

ΔS

Aσ

⎛

⎝ ⎜

⎞

⎠ ⎟−1

Minimum number of events to see tidal influence

S = S75-100 - S0-25

S~ 0.01 - 0.02 bar

ΔRr

=ΔSAσ

A = 1 bar: R/r = 0.01 - 0.02

A = 2 bar: R/r = 0.005 - 0.01

Method to obtain stress time series from earthquake rates

dγ =1

Aσdt−γdS[ ]R=

rγ˙ S r

,

STEPS

1) Select region and magnitude threshold

2) Obtain time series for γ :

4) Solve evolution equation for Coulomb stress S. For example:

γ(t) =r

R(t)˙ S r

€

S = Aσ lnγ1 +

Δt

2Aσ

γ 2 −Δt

2Aσ

⎛

⎝

⎜ ⎜ ⎜

⎞

⎠

⎟ ⎟ ⎟

S

time

t

(t,γ)

(t2,γ2)

S

Synthetic Data

Input stressSimulated seismicityStress Inversion

€

R =r

1+ exp−ΔS

Aσ

⎛

⎝ ⎜

⎞

⎠ ⎟−1

⎡

⎣ ⎢

⎤

⎦ ⎥exp

−t

ta

⎛

⎝ ⎜

⎞

⎠ ⎟

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

3) Prepare maps (or cross sections) of stress changes over specified time intervals

€

R(t) =r

γ (t) ˙ S r

Dieterich, Cayol, and Okubo, Nature (2000)Dieterich and others, USGS Prof Paper(2004)

€

dγ =1

Aσdt − γdS( )

Maps of stress changes from earthquake rates

1/3/83

Input stressSimulated seismicityStress Inversion

Synthetic Data

€

R =r

1+ exp−ΔS

Aσ

⎛

⎝ ⎜

⎞

⎠ ⎟−1

⎡

⎣ ⎢

⎤

⎦ ⎥exp

−t

ta

⎛

⎝ ⎜

⎞

⎠ ⎟

Model AForeshocks Advance the time of Mainshock

Mainshocks following foreshocks and aftershocks have similar origins. The stress change of a prior earthquake results in increased nucleation rates at all magnitudes. Extrapolation of aftershock rates to larger magnitudes gives rate of foreshocks

S is a function of distance from the prior earthquake. Net earthquake rate following a stress step is obtained by integrating over the region affected by the stress change.

Foreshock models

R =r

1+ exp−ΔSAσ

⎛ ⎝

⎞ ⎠

−1⎡ ⎣ ⎢

⎤ ⎦ ⎥ exp

−tta

⎛

⎝ ⎜ ⎞

⎠ ⎟

Foreshocks in Southern California

b = -0.5ta = 10.2 yr

Background rate (M≥3) =93.4/yr

Foreshock models

R =r ˙ S / ˙ S r

1+˙ S ˙ S r

⎛

⎝ ⎜ ⎞

⎠ ⎟ exp

−tta

⎛

⎝ ⎜ ⎞

⎠ ⎟

Model BMainshock Nucleation Causes Foreshocks

Premonitory creep of a large nucleation zone causes rapid stressing on nearby smaller nucleation sources

˙ S =C ˙ δ =C1˙ δ 0

+Hσ˙ S 0

⎡

⎣ ⎢

⎤

⎦ ⎥ exp

−˙ S 0t

Aσ

⎛

⎝ ⎜ ⎞

⎠ ⎟

⎡

⎣ ⎢ ⎤

⎦ ⎥ −Hσ˙ S 0

⎧ ⎨ ⎩

⎫ ⎬ ⎭

−1

Nucleation zoneMainshock

Premonitory creep

N-zonesforeshocks(Lc small)

Lc

Foreshock model B

Nucleation on fractal fault2 hours before instability

Foreshocks in Southern California

b = -0.5Scale factor for numberis adjusted to fit observations