Agnès Helmstetter1 and Bruce Shaw2

1,2 LDEO, Columbia University 1 now at LGIT, Univ Grenoble, France

Relation between stress heterogeneity and

aftershock rate in the “rate-and-state” model

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

-- Omori law R~1/t

c

“Rate-and-state” model of seismicity [Dieterich 1994]

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()

Coseismsic slip, stress change, and aftershocks:

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

slip shear stress seismicity 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 and stress heterogeneity

Seismicity rate triggered by a heterogeneous stress change on the fault

• R(t,) : R&S model, unif stress change [Dieterich 1994]

• P() : stress distribution (due to slip heterogeneity or fault roughness)

• instantaneous stress change; no dynamic or postseismic relaxation

Goals

• seismicity rate R(t) produced by a realistic P()

• inversion of P() from R(t)

• see also Dieterich 2005; and Marsan 2005

Slip and shear stress heterogeneity, aftershocks

slip shear stressstress drop 0 =3 MPa

aftershock map synthetic R&S catalog

0

0

stress distrtibution P()≈Gaussian

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

Stress heterogeneity and aftershock decay with time

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

Short times t‹‹ta : apparent Omori law with p≤1

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

-- Omori lawR(t)~1/tp

with p=0.93

Rr

ta

∫ R(t,)P()d

Stress heterogeneity and aftershock decay with time

• 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≈ta due 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

• distance d<L from the fault: (k,d) ~ (k,0) e-kdfor d«L

• fast attenuation of high frequency perturbations with distance

Modified k2 slip model, off-fault stress change

Ld

coseismic shear stress change (MPa)

Modified k2 slip model, off-fault aftershocks

• stress change and seismicity rate as a function of d/L

• quiescence for d >0.1L

standard deviation

average stress change

stre

ss (

MP

a)

d/L

d/L=0.1

Stress heterogeneity and Omori law

• For an exponential pdf P()~e-/o with >0

• R&S gives Omori law R(t)~1/tp with p=1- An/o

p=0.8

p=1

• 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

log

P(

)

0

Stress heterogeneity and Omori law

• 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

Inversion of stress distribution from aftershock rate

Inversion of stress pdf from aftershock rate

p=0.93

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 *

Parkfield 2004 M=6 aftershock sequence

• Fixed:

An = 1 MPa

0 = 3 MPa

• Inverted:

* = 11 MPa

ta = 10 yrs

data, aftershocks

data, `foreshocks’

fit R&S model Gaussian P()

fit Omori law p=0.88

foreshockRr

ta

Landers, 1992, M=7.3, aftershock sequence

Data, aftershocksFit R&S model Gaussian P()Fit Omori law p=1.08

foreshocks

Rr

ta

• 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

Superstition Hills 1987 M=6.6 (South of Salton Sea 33oN)

Data, aftershocksFit R&S model Gaussian P()Fit Omori law p=1.3

foreshocksRr

• Fixed:

An = 1 MPa

0 = 3 MPa

Elmore Ranch M=6.2

Morgan Hill, 1984 M=6.2, aftershock sequence

data, aftershocksFit R&S model Gaussian P()Fit Omori law p=0.68

foreshocks

Rr

ta

• 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

Stacked aftershock sequences, Japan (80, 3<M<5, z<30)

Data, aftershocksFit R&S model Gaussian P()Fit Omori law p=0.89

foreshocks

Rr

ta

• 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]

Inversion of P() from R(t) for real aftershock sequences

Sequence p * (MPa) ta (yrs)

Morgan Hill M=6.2, 1984 0.68 6.2 78.

Parkfield M=6.0, 2004 0.88 11. 10.

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

San Simeon M=6.5 2003 0.93 18. 348.

Landers M=7.3, 1992 1.08 ** 52.

Northridge M=6.7, 1994 1.09 ** 94.

Hector Mine M=7.1, 1999 1.16 ** 80.

Superstition-Hills, M=6.6,1987 1.30 ** **

** : 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 ,

p 1 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

Conclusion

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]

Problems / future work

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