1 / 21

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

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.

lorand
Download Presentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 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

  2. -- 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()

  3. 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

  4. 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

  5. 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

  6. ∫ 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

  7. 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

  8. d L Modified k2 slip model, off-fault stress change • distance d<L from the fault: (k,d) ~ (k,0) e-kd for d«L • fast attenuation of high frequency  perturbations with distance coseismic shear stress change (MPa)

  9. 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

  10. 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

  11. 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)?

  12. 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

  13. 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

  14. 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

  15. 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

  16. 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

  17. 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

  18. 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]

  19. 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]

  20. 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

  21. 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

More Related