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Relation between stress heterogeneity and aftershock rate in the “rate-and-state” model

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Agnès Helmstetter1 and Bruce Shaw2

1,2 LDEO, Columbia University

1 now at LGIT, Univ Grenoble, France

Landers, aftershocks and Hernandez et al. [1999] slip model

-- Omori law R~1/t

c

Seismicity rate R(t) after a unif stress step (t) [Dieterich, 1994]

- ∞ population of faults with R&S friction law
- constant tectonic loading ’r
Aftershock duration ta

- A≈ 0.01 (friction exp.)
- n≈100 MPa (P at 5km)
«min» time delay c()

Planar fault, uniform stress drop, and R&S model

slipshear stressseismicity rate

Real data:

most aftershocks occur on or close

to the rupture area

Slip and stress must be heterogeneous to produce an increase of and thus R on parts of the fault

Seismicity rate triggered by a heterogeneous stress change on the fault

- R(t,t) : R&S model, unif stress change [Dieterich 1994]
- P(t) : stress distribution (due to slip heterogeneity or fault roughness)
- instantaneous stress change; no dynamic t or postseismic relaxation
Goals

- seismicity rate R(t) produced by a realistic P(t)
- inversion of P(t) from R(t)
- see also Dieterich 2005; and Marsan 2005

0

0

aftershock map

synthetic R&S catalog

shear stress

stress drop 0 =3 MPa

slip

Modified «k2» slip model: u(k)~1/(k+1/L)2.3 [Herrero & Bernard, 94]

stress distrtibution

P()≈Gaussian

∫ R(t,)P()d

-- Omori law

R(t)~1/tp

with p=0.93

ta

Rr

Aftershock rate on the fault with R&S model for modified k2 slip model

Short timest‹‹ta : apparent Omori law with p≤1

Long timest≈ta : stress shadow R(t)<Rr

- Early time rate controlled by large positive
- Huge increase of EQ rate after the mainshock
even where u>0 and where <0 on average

- Long time shadow for t≈tadue to negative
- Integrating over time: decrease of EQ rate
∆N = ∫0∞[R(t) - Rr] dt ~ -0 Rrta/An

- But long-time shadow difficult to detect

d

L

- distance d<L from the fault: (k,d) ~ (k,0) e-kdfor d«L
- fast attenuation of high frequency perturbations with distance

coseismic shear

stress change (MPa)

stress (MPa)

standard deviation

d/L=0.1

d/L

average stress change

- stress change and seismicity rate as a function of d/L
- quiescence for d >0.1L

log P()

0

- For an exponential pdf P()~e-/owith >0
- R&S gives Omori law R(t)~1/tpwith p=1- An/o

- black: global EQ rate,
heterogeneous :

R(t) = ∫ R(t,)P()d

with o/An=5

- colored lines:
EQ rate for a unif :

R(t,)P()

from=0 to=50 MPa

p=0.8

p=1

- smooth stress change, or large An
Omori exponent p<1

- very heterogeneous stress field, or small An
- Omori p≈1

- p>1 can’t be explained by a stress step (r)
postseismic relaxation (t)?

Deviations from Omori law with p=1 due to:

- (r) : spatial heterogeneity of stress step [Dieterich, 1994; 2005]
- (t) : stress changes with time [Dieterich, 1994; 2000]
We invert for P() from R(t) assuming (r)

- solve R(t) = ∫R(t,)P()d for P()
does not work for realistic catalogs (time interval too short)

- fit of R(t) by∫R(t,)P()d assuming a Gaussian P()
- invert for ta and * (standard deviation)

- stress drop fixed (not constrained if tmax<ta)

- good results on synthetic R&S catalogs

Synthetic R&S catalog:- input P()

N=230- inverted P(): fixed An ,Rr and ta

An=1 MPa- Gaussian P(): - fixed An and Rr

0 = 3 MPa - invert for ta, 0 and *

*=20 MPa - Gaussian P(): - fixed An , 0 and Rr

- invert for ta and *

p=0.93

data, aftershocks

data, `foreshocks’

fit R&S model Gaussian P()

fit Omori law p=0.88

ta

foreshock

Rr

- Fixed:
An = 1 MPa

0 = 3 MPa

- Inverted:
*= 11 MPa

ta = 10 yrs

Data, aftershocks

Fit R&S model Gaussian P()

Fit Omori law p=1.08

foreshocks

ta

Rr

- Fixed:
An = 1 MPa

0 = 3 Mpa

- Inverted:
*= 2350 MPa

ta = 52 yrs

- Loading rate
d/dt = An / ta

= 0.02 MPa/yr

- « Recurrence time »
tr= ta 0/An

= 156 yrs

- Fixed:
An = 1 MPa

0 = 3 MPa

Data, aftershocks

Fit R&S model Gaussian P()

Fit Omori law p=1.3

Elmore

Ranch

M=6.2

Rr

foreshocks

data, aftershocks

Fit R&S model Gaussian P()

Fit Omori law p=0.68

foreshocks

ta

Rr

- Fixed:
An = 1 MPa

0 = 3 Mpa

- Inverted:
*= 6.2 MPa

ta = 26 yrs

- Loading rate
d/dt = An / ta

= 0.04 MPa/yr

- «Recurrence time»
tr= ta 0/An

= 78 yrs

Data, aftershocks

Fit R&S model Gaussian P()

Fit Omori law p=0.89

foreshocks

ta

Rr

- Fixed:
An = 1 MPa

0 = 3 Mpa

- Inverted:
*= 12 MPa

ta = 1.1 yrs

- Loading rate
d/dt = An / ta

= 0.9 MPa/yr

- «Recurrence time»
tr= ta 0/An

= 3.4 yrs

[Peng et al., in prep, 2006]

Sequence p * (MPa)ta (yrs)

Morgan Hill M=6.2, 19840.68 6.278.

Parkfield M=6.0, 2004 0.88 11.10.

Stack, 3<M<5, Japan*0.89 12. 1.1

San Simeon M=6.5 20030.93 18. 348.

Landers M=7.3, 1992 1.08 ** 52.

Northridge M=6.7, 19941.09 ** 94.

Hector Mine M=7.1, 19991.16 **80.

Superstition-Hills, M=6.6,19871.30 ** **

** : we can’t estimate * because p>1(inversion gives *=inf)

* [Peng et al., in prep 2005]

R&S model with stress heterogeneity gives:

- “apparent” Omori law with p≤1 for t<ta, if * ›0 ,

p1 with «heterogeneity» *

- quiescence:
- for t≈ta on the fault,
- or for r/L>0.1 off of the fault
- in space : clustering on/close to the rupture area

Inversion of stress drop not constrained if catalog too short

trade-off between ta and 0

trade-off between space and time stress variations

can’t explain p>1 : post-seismic stress relaxation?

or other model?

An?

- 0.002 or 1MPa??

- heterogeneity of An could also produce change in p value

secondary aftershocks?

renormalize Rr but does not change p ? [Ziv & Rubin 2003]

submited to JGR 2005, see draft at www.ldeo.columbia.edu/~agnes