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 “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 t a 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 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 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 stress stress drop  0 =3 MPa aftershock map synthetic R&S catalog 00 00 stress distrtibution P(  )≈Gaussian Modified «k 2 » 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 k 2 slip model Short times t‹‹t a : apparent Omori law with p≤1 Long times t≈t a : stress shadow R(t)<R r -- Omori law R(t)~1/t p with p=0.93 RrRr tata ∫ 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≈t a due to negative  Integrating over time: decrease of EQ rate ∆N = ∫ 0 ∞ [R(t) - R r ] dt ~ - 0 R r t a /A  n But long-time shadow difficult to detect

distance d<L from the fault: (k,d) ~ (k,0) e -kd for d«L fast attenuation of high frequency  perturbations with distance Modified k 2 slip model, off-fault stress change L d coseismic shear stress change (MPa)

Modified k 2 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 stress (MPa) 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/t p with p=1- A n / o p=0.8 p=1 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  log P() 00

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) ?

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 t a and * (standard deviation) - stress drop    fixed (not constrained if t max <t a ) - 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 A n, R r and t a A n =1 MPa- Gaussian P(): - fixed A n and R r  0 = 3 MPa - invert for t a,  0 and  *  * =20 MPa - Gaussian P(): - fixed A n,  0 and R r - invert for t a and  *

Parkfield 2004 M=6 aftershock sequence Fixed: A  n = 1 MPa  0 = 3 MPa Inverted:  * = 11 MPa t a = 10 yrs data, aftershocks data, `foreshocks’ fit R&S model Gaussian P(  ) fit Omori law p=0.88 foreshock RrRr tata

Landers, 1992, M=7.3, aftershock sequence Data, aftershocks Fit R&S model Gaussian P(  ) Fit Omori law p=1.08 foreshocks RrRr tata Fixed: A  n = 1 MPa  0 = 3 Mpa Inverted:  * = 2350 MPa t a = 52 yrs Loading rate d  /dt = A  n / t a = 0.02 MPa/yr « Recurrence time » t r = t a  0 /A  n = 156 yrs

Superstition Hills 1987 M=6.6(South of Salton Sea 33 o N) Data, aftershocks Fit R&S model Gaussian P(  ) Fit Omori law p=1.3 foreshocks RrRr Fixed: A  n = 1 MPa  0 = 3 MPa Elmore Ranch M=6.2

Morgan Hill, 1984 M=6.2, aftershock sequence data, aftershocks Fit R&S model Gaussian P(  ) Fit Omori law p=0.68 foreshocks RrRr tata Fixed: A  n = 1 MPa  0 = 3 Mpa Inverted:  * = 6.2 MPa t a = 26 yrs Loading rate d  /dt = A  n / t a = 0.04 MPa/yr «Recurrence time» t r = t a  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 RrRr tata Fixed: A  n = 1 MPa  0 = 3 Mpa Inverted:  * = 12 MPa t a = 1.1 yrs Loading rate d  /dt = A  n / t a = 0.9 MPa/yr «Recurrence time» t r = t a  0 /A  n = 3.4 yrs [Peng et al., in prep, 2006]

Inversion of P() from R(t) for real aftershock sequences Sequence p  * (MPa) t a (yrs) Morgan Hill M=6.2, Parkfield M=6.0, Stack, 3<M<5, Japan* San Simeon M= Landers M=7.3, ** 52. Northridge M=6.7, ** 94. Hector Mine M=7.1, **80. Superstition-Hills, M=6.6, ** ** ** : 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<t a, if  * ›  0, p  1 with «heterogeneity»  * - quiescence: - for t≈t a 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 t a and  0 trade-off between space and time stress variations can’t explain p>1 : post-seismic stress relaxation? or other model? A  n ? or 1MPa?? - heterogeneity of A  n could also produce change in p value secondary aftershocks? renormalize R r but does not change p ? [Ziv & Rubin 2003] Problems / future work submited to JGR 2005, see draft at