Relation between stress heterogeneity and aftershock rate in the rate and state model
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Agnès Helmstetter 1 and Bruce Shaw 2 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.

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Relation between stress heterogeneity and aftershock rate in the rate and state model

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


Rate and state model of seismicity dieterich 1994

-- 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
Coseismsic slip, stress change, and aftershocks:

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 and stress heterogeneity
Seismicity rate and stress heterogeneity

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


Slip and shear stress heterogeneity aftershocks

0

0

Slip and shear stress heterogeneity, aftershocks

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


Stress heterogeneity and aftershock decay with time

∫ R(t,)P()d

-- Omori law

R(t)~1/tp

with p=0.93

ta

Rr

Stress heterogeneity and aftershock decay with time

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


Stress heterogeneity and aftershock decay with time1
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≈tadue to negative 

  • Integrating over time: decrease of EQ rate

    ∆N = ∫0∞[R(t) - Rr] dt ~ -0 Rrta/An

  • But long-time shadow difficult to detect


Modified k 2 slip model off fault stress change

d

L

Modified k2 slip model, off-fault stress change

  • 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)


Modified k 2 slip model off fault aftershocks

stress (MPa)

standard deviation

d/L=0.1

d/L

average stress change

Modified k2 slip model, off-fault aftershocks

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

  • quiescence for d >0.1L


Stress heterogeneity and omori law

log P()

0

Stress heterogeneity and Omori law

  • For an exponential pdf P()~e-/owith >0

  • R&S gives Omori law R(t)~1/tpwith p=1- An/o

  • black: global EQ rate,

    heterogeneous :

    R(t) = ∫ R(t,)P()d

    with o/An=5

  • colored lines:

    EQ rate for a unif :

    R(t,)P()

    from=0 to=50 MPa

p=0.8

p=1


Stress heterogeneity and omori law1
Stress heterogeneity and Omori law

  • smooth stress change, or large An

     Omori exponent p<1

  • very heterogeneous stress field, or small An

    • Omori p≈1

  • p>1 can’t be explained by a stress step (r)

     postseismic relaxation (t)?


Inversion of stress distribution from aftershock rate
Inversion of stress distribution from aftershock rate

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 pdf from aftershock rate

Synthetic R&S catalog:- input P()

N=230- inverted P(): fixed An ,Rr and ta

An=1 MPa- Gaussian P(): - fixed An and Rr

0 = 3 MPa - invert for ta, 0 and *

*=20 MPa - Gaussian P(): - fixed An , 0 and Rr

- invert for ta and *

Inversion of stress pdf from aftershock rate

p=0.93


Parkfield 2004 m 6 aftershock sequence

data, aftershocks

data, `foreshocks’

fit R&S model Gaussian P()

fit Omori law p=0.88

ta

foreshock

Rr

Parkfield 2004 M=6 aftershock sequence

  • Fixed:

    An = 1 MPa

    0 = 3 MPa

  • Inverted:

    *= 11 MPa

    ta = 10 yrs


Landers 1992 m 7 3 aftershock sequence

Data, aftershocks

Fit R&S model Gaussian P()

Fit Omori law p=1.08

foreshocks

ta

Rr

Landers, 1992, M=7.3, aftershock sequence

  • Fixed:

    An = 1 MPa

    0 = 3 Mpa

  • Inverted:

    *= 2350 MPa

    ta = 52 yrs

  • Loading rate

    d/dt = An / ta

    = 0.02 MPa/yr

  • « Recurrence time »

    tr= ta 0/An

    = 156 yrs


Superstition hills 1987 m 6 6 south of salton sea 33 o n
Superstition Hills 1987 M=6.6(South of Salton Sea 33oN)

  • Fixed:

    An = 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


Morgan hill 1984 m 6 2 aftershock sequence

data, aftershocks

Fit R&S model Gaussian P()

Fit Omori law p=0.68

foreshocks

ta

Rr

Morgan Hill, 1984 M=6.2, aftershock sequence

  • Fixed:

    An = 1 MPa

    0 = 3 Mpa

  • Inverted:

    *= 6.2 MPa

    ta = 26 yrs

  • Loading rate

    d/dt = An / ta

    = 0.04 MPa/yr

  • «Recurrence time»

    tr= ta 0/An

    = 78 yrs


Stacked aftershock sequences japan 80 3 m 5 z 30

Data, aftershocks

Fit R&S model Gaussian P()

Fit Omori law p=0.89

foreshocks

ta

Rr

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

  • Fixed:

    An = 1 MPa

    0 = 3 Mpa

  • Inverted:

    *= 12 MPa

    ta = 1.1 yrs

  • Loading rate

    d/dt = An / ta

    = 0.9 MPa/yr

  • «Recurrence time»

    tr= ta 0/An

    = 3.4 yrs

[Peng et al., in prep, 2006]


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

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]


Conclusion
Conclusion

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


Problems future work
Problems / future work

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?

An?

- 0.002 or 1MPa??

- heterogeneity of An 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


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