Afterslip, slow earthquakes and aftershocks:Modeling using the rate & state friction law

Days after Nias earthquake

Cumulative number of aftershocks

Ex: 2005 m=8.7 Nias earthquake [Hsu et al, Science 2006]

Agnès Helmstetter (LGIT Grenoble) and Bruce Shaw (LDE0 Columbia Univ)

Main questions:

• relation between coseismic and postseismic slip?

• relation between afterslip and aftershocks?

• can we use afterslip to constrain the rheology of the crust

(stable/unstable)?

• mechanisms for aftershock triggering?

Outline

• intro: observations of afterslip

• modelling afterslip with rate & state friction laws

• modelling aftershocks triggered by afterslip

Postseismic deformation

• observed after most « large »  earthquakes

• duration : sec to decades

• postseismic displacement ≈ 10% of coseismic slip

but with huge fluctuations between earthquakes (0-100%!)

• physical processes:

friction: afterslip on mainshock fault

diffuse deformation: poro-elasticity or viscous deformation

superposition of # mechanisms, ex: Denali [Freed et al 2006]

Spatial distribution of afterslip and aftershocks

2003 m=8 Tokachi2005 m=8.7 Nias

[Hsu et al, Science 2006] [Miyazaki et al, GRL 2004]

Temporal distribution of afterslip and aftershocks

2004 m=6.0 Parkfield earthquake

[Peng and Vidale, 2006]

[Langbein et al., 2006]

beforemainshok

aftermainshok

1 hour

• afterslip occurs mostly around coseismic slip

• overlap between coseismic and afterslip, and aftershock areas

• displacement ≈ log(1+t/t*), with t* ≈ hrs to days

• similar time-dependence for afterslip and aftershocks (“Omori’)but with # characteristic times (“c « t*”)

Observations

Rate-and-state friction law

B>A unstable µ with V “velocity weakening”

B<A: stable µ with V “velocity-hardening”

!

"= µ = µ

0+ A ln

V

V*

# $

% & + Bln

'

'*# $

% &

• friction law [Dieterich, 1979]

• state variable θ≈ age of contacts

dθ/dt = 1 - Vθ/Dc

• parameters values in the lab :

- A≈B≈0.01, depend on T,σ, gouge thickness, strain…

- Dc≈1-100 µm, depends on roughness and gouge thickness

Vµ=τ/σV1

slip

V2 >V1fr

ictio

n co

effici

ent µ

A

Dc

B

Rate-and-state friction lawFrom stick-slip to stable sliding

• transition controlled by stiffness k/kc (or rupture length) and ratio B/A,

with kc = (B-A)σn/Dc [Rice and Ruina, 1983]

• stable sliding: k>kc or B<A

• stick-slip: k<kc and B>A

mk

µ(V,θ)

tectonic loading VL

fault slip rate V

timedi

spla

cem

ent

fault

loading point

slip

stick

post-seismic slip

nucleation

Rate-and-state friction law and afterslip

• slip speed for a slider-block with a fixed loading point

• relaxation or nucleation of a slip instability after a stress step

• initial condition: slip rate V0 and stress τ0

• inertia and tectonic loading negligible

• tectonic deformation « slip rate « cosesimic slip rate

fixed loading point(locked part of the fault)

mk

µ0(V,θ)

V0

First model: steady-state approximation

• Scholz [1989], Marone et al [1991] and many others assume

- slip-strengthening friction (stable) A>B

- steady state θ=constant

• R&S equations becomes:

µ = µ0 +(A-B) log(V/V0) = µ0 -kδ/σn

V= θ/Dc

⇒ V= V0/(1+t/t*)

t* = σn(A-B)/kV0

☺ good fit to afterslip data, and its distribution with depth:

mostly above and bellow the seismogenic zone, where A>B

☹ constant θ ⇒ constant V: OK when V varies between m/s to mm/yr?

☹ overlap between coseismic rupture, aftershocks and afterslip areas?

mk

µ0(V,θ)

V0

δ

Rate-and-state friction law and afterslip

• velocity weakeningregime B>A :transition betweenpostseismic relaxation,slow earthquake, and

aftershock as τ0

• velocity strengthening

regime B<A :transition betweenpostseismic relaxation and

slow earthquake as τ0, no

aftershock

• power-law relaxationV~1/tp, with p≈1

k=0.8|kc|

• initial acceleration dV/dt>0 if stress is large enough : µ>µa

⇔ if state variable θ decreases rapidly with time

⇔ dθ/dt < dθa/dt and dθa/dt<0

⇒ no acceleration if dθa/dt>0 ⇔

(in-)stability after a stress step

• if V after the stress step, the system can evolve toward:

slip-instability (« aftershock »)

or transcient slip event (« slow earthquake »)

• condition for instability: V and dθ/dt ("weakening" of friction interface)

⇔ µ>µl and kc/k<1

⇔ dθ /dt < dθl/dt <0

⇒ instability possible only if k<|kc| and B>A

• if V but dθ/dt (« healing »), then latter V will : « slow EQ »

(in-)stability after a stress step

(in-)stability after a stress step

(1) (2)

(3)

(4)

(5)

• behavior controlledby both frictionparameters k/kc, B/Aand stress

(1) and (2)postseismic relaxation

(3) and (4)transition frompostseismic relaxation

to slow EQ as µ0

(5) transition frompostseismic relaxationto slow EQ and

aftershock as µ0

(in-)stability after a stress step

• behavior as a function of distance from steady-state and B/A for k=0.8kc

θθ

steady-state no steady state

regime

• θ ("healing")

• or θ ("weakening")

• steady state approx

only valid for B<A and

k«kc

slip rate history - 1D model

Simulations with parameters

• top: B=1.5A and k=0.8kc

• bottom: B=0.5A and

k=2.5|kc|

# power-law afterslip

regimes, with

# slope exponents: B/A or 1

# characteristic times t*

µ0>µlµl>µ0>µaµ0<µa

µ0>µlssµ0=µlssµ0<µss

Slip history - 1D model

Simulations with:

• top: B=1.5A and k=0.8kc

• bottom: B=0.5A and k=2.5|kc|

afterslip ≈ Dc

for both B<A and B>A cases

Slip with stress

µ0>µlµl>µ0>µaµ0<µa

µ0>µlssµ0=µlssµ0<µss

Slip rate history

• Superstition Hills EQ [Wennerberg and Sharp, 1997]

• surface displacement for 6 points along the fault

• model:

(1) R&S friction law in the steady-state regime :

bad fit because in the data V ~1/(1+t/t*)p with p<1 or p>1

(2) initial form of the law [D., 1979] with θ=const.

• good fit, but :

- variations of A and B?

- p<1A<0!?

• R&S with 2 state variables?

µ=µ0+(A -B)logV

« data »fit B>Afit B<A

Slip history - 1D model and afterslip data

• fit (fit of) afterslip data Wennerberg and Sharp [1997] for Superstition Hills

• 6 points along the fault

• invert for A, B, k, Dc, V0 and µo, using broad range of initial values

• data can be fitted with:

V=V0/(1+t/t*)p

inversion not

constrained!

Slip history - 1D model and afterslip data

• for most points, we

can’t distinguish

between B > or < A

• we don’t need A<0

or more complex

friction law

• limits of 1D model?

Conclusions: aferslip and slow EQs

☺ R&S friction law can be used to model afterslip data or slow EQs

• including deviations from log slip history (p< or >1)

• in both slip-weakening B>A and slip strengthening regimes B<A

• we don’t need along strike variations or temporal variations in B or A to

explain overlap between coseismic and afterslip areas

• we don’t need A<0 or more complex friction laws

• but requires relatively large Dc ≈ afterslip

• R&S friction law produces triggered slow EQs, both for B> or <A

☹ R&S friction can’t be used to invert for the model parameters

• 6 model parameters, but 3 are enough to fit the data

• except if good spatial resolution of co- and postseismic slip and using

homogeneous or smooth friction parameters ?

• similar time dependence of afterslip rate and aftershock rate (‘Omori’)

afterslip due to aftershocks, or aftershocks triggered by afterslip?

• coseismic slip induces:

stress increase (on and) around the rupture

afterslip

stress release in sliping areas

reloading on locked parts of the faults

aftershocks triggered by afterslip (in unstable areas)

[Dieterich 1994, Schaff et al 1998, Perfettini and Avouac 2004, Wennerberg

and Sharp 1997, Hsu et al 2006, Savage 2007, …]

• we use the R&S model of Dieterich [1994] to model the effect of stress

changes on seismicity rate, instead of assuming seismicity rate ~ stress rate

Afterslip and aftershocks

• Dieterich [2004] model is equivalent to

• for a stress step τ(0)=Δτ : R(t) ≈ ‘Omori law’ with p=1

Relation between stress changes andseismicity in the R&S model

R: seismicity rateR0= R(t=0)N=∫

0

tR dtr: ref seismicity rate for τ’=τ’r τ: coulomb stress change (=0 at t=0)ta: nucleation time = Aτn/τr’

long-times regime

for T»ta

N~τ(tectonic loading, …)

short-times regime

for T«ta

R~R0exp(τ/Aτn)

(tides [Cochran et al

2005])

• numerical solution of R-τ relation assuming reloading due to afterslip is of

the form dτ/dt ~ V (elastic stress transfer) ~ τ’0/(1+t/t*)p with p=0.8

Aftershocks triggered by afterslip

• R ~ stress rate for t»t* when p<1

• short time cut-off of Omori law for EQ rate is always larger than for stress rate

t*

— dτ/dt– – τ

t*

• numerical solution of R-τ relation assuming reloading due to afterslip is of

the form dτ/dt ~ V (elastic stress transfer) ~ τ’0/(1+t/t*)p with p=1.3

Aftershocks triggered by afterslip

• short times: R≈0

• intermediate times: R ~ dτ/dt ~ 1/tp with p>1

• large times: R ~1/t = aftershock rate for a stress step of the same anplitude

t*

— dτ/dt– – τ

t* stress stepafterslip

Δτ= 2MPa

τ’r

Conclusions: afershocks triggered by afterslip

☺ R&S friction law can be used to model aftershock rate

• aftershock rate decreases as a modified Omori law with p< or >1

• apparent exponent changes with time

• because afterslip is comparable to coseismic slip, the number of

aftershocks triggered by afterslip must be significant

☹ but…

• t* for afterslip ≈ days » c of aftershocks ≈ sec

⇒ short times aftershocks can’t be explained by afterslip

• many models reproduce Omori law, not very helpful

• complex relation between stress and seismicity history, so be careful

when using seismicity to identify possible triggering processes

The end …

For details see:

Helmstetter, A., and B. E. Shaw, Afterslip and aftershocks

in the rate-and-state friction law, submitted to J.

Geophys. Res. (March 2007)

Draft availble at:

http://www.arxiv.org/abs/physics/0703249).

Analytical approximations

afterslip, p=B/Aafterslip p=1afterslipp=1

aftershocknucleation[Dieterich1992, 1994]

slow EQ

Rate-and-state friction law and EQs

Slip speed for a slider-block with a constant loading rate m Vl

k

time

log

slip

spe

ed

nucleation

post-seismicslip

triggered EQ delayed

EQ

<0 or >0τ change slow

EQ

EQ

ta = Aσ/τ’

B<A

B>A

B≈A

Vl

µ(V,θ)

Rate-and-state friction lawFrom stick-slip to stable sliding

A-B

stableunstable

A-B

(A-B)σ

z (km)z (km) [Scholz, 1998]

Spatial distribution of afterslip and aftershocks

2002 m=7.8 Denali[Freed et al, JGR 2006]

Temporal distribution of afterslip and aftershocks

2005 m=8.7 Nias earthquake[Hsu et al, Science 2006]

Days after Nias earthquake

Cumulative number of aftershocks

• log variation of afterslip and number of M>0 aftershocks with time

~log(1+t/t*) with same characteristic time t* ≈ 1 day (!?)

• but looking at M>5 events gives t* ≈ 10-3 days

Temporal distribution of afterslip and aftershocks

1999 m=7.6 ChiChi earthquake [Perfettini and Avouac 2004]

Time decay of afterslip

• log variation of afterslip with time

• no surface afterslip if coseismic rupture reaches the surface (Landers)

Afterslip measured using

creep-meters

Marone et al., [JGR 1991];

Marone [AREPS, 1998]

• friction µ depends on slip rate V and state variable θ• derived from friction experiment [Dieterich, 1979] and theoretical

models

• applied to earthquakes, landslides, material sciences, aseismic slip

• models the transition between stable and unstable slip in the crust,

afterslip, slow earthquakes, and earthquake triggering due to stress

changes

Rate-and-state friction law

First model of afterslip

• Model of Scholz [1990],

Marone et al., [JGR 1991];

Marone [AREPS, 1998]

• Shallow afterslip due to slip

deficit during the EQ

• behavior as a function of distance from steady-state and B/A for k=2.5|kc|

heal

ing θ

w

eake

ning

θ

steady-state

(in-)stability after a stress step

no steady state

regime

• θ ("healing")

• or θ ("weakening")

• steady state approx

only valid for B<A and

k«kc

Friction as a function of slip rate - 1D model

(1) (2) (3) (4) (5) (6)

µ0>µlµl>µ0>µaµ0<µa

µ0>µlssµ0=µlssµ0<µss

• Simulations with parameters

top: B=1.5A and k=0.8kc

bottom: B=0.5A and k=2.5|kc|

history of µ(V) can’t be used

to measure A or B-A, because slope

depends on model parameters

Friction as a function of slip rate2005 M=8.7 Nias EQ [Hsu et al., 2006]• GPS displacement

afterslip map

slip rate and stress

• Model:[Perfettini 2004]µ = µ0 +A logV/V0

A is <0 !!??

Friction as a function of slip rate

2003 M=8.0 Tokachi EQ [Miyazaki et al., 2004]

• model with R&S friction

‘A’,‘B’,‘C’: steady-state friction

µ=µ0+(A -B)logV

(A-B)σ ≈ 0.6 MPa ?

‘D’ : Velocity weakening B>A ?