Relation between stress heterogeneity and aftershock rate in the rateandstate model

1
Relation between stress heterogeneity and
aftershock rate in the rate-and-state model
Landers, aftershocks and Hernandez et al. 1999
slip model
2
Rate-and-state model of seismicity Dieterich
1994

• Seismicity rate R(t) after a unif stress step
  ?(t) Dieterich, 1994
• 8 population of faults with RS friction law
• constant tectonic loading ?r
• Aftershock duration ta
• A 0.01 (friction exp.)
• ??n100 MPa (P at 5km)
• min time delay c(?)

3
Coseismsic slip, stress change, and aftershocks
Planar fault, uniform stress drop, and RS 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
4
Seismicity rate and stress heterogeneity

• Seismicity rate triggered by a heterogeneous
  stress change on the fault
• R(t,t) RS 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
Slip and shear stress heterogeneity, aftershocks

• Modified k2 slip model u(k)1/(k1/L)2.3
  Herrero Bernard, 94

shear stress stress drop ?0 3 MPa
aftershock map synthetic RS catalog
slip
stress distrtibution P(?)Gaussian
6
Stress heterogeneity and aftershock decay with
time
Aftershock rate on the fault with RS model for
modified k2 slip model
Short times tta apparent Omori law with
p1 Long times tta stress shadow R(t)ltRr
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 ugt0 and where ? lt0 on average
• Long time shadow for tta due to negative ?
• Integrating over time decrease of EQ rate
• ?N ?08 R(t) - Rr dt -?0 Rrta/A?n
• But long-time shadow difficult to detect

8
Modified k2 slip model, off-fault stress change

• distance dltL from the fault ?(k,d) ?(k,0)
  e-kd for dL
• fast attenuation of high frequency ?
  perturbations with distance

coseismic shear stress change (MPa)
9
Modified k2 slip model, off-fault aftershocks

• stress change and seismicity rate as a function
  of d/L
• quiescence for d gt0.1L

10
Stress heterogeneity and Omori law

• For an exponential pdf P(?)e-?/?o with ?gt0
• RS gives Omori law R(t)1/tp with p1- A?n/?o

• black global EQ rate,
• heterogeneous ?
• R(t) ? R(t,?)P(?)d?
• with ?o/A?n5
• colored lines
• EQ rate for a unif ?
• R(t,?)P(?)
• from ?0 to ?50 MPa

p0.8
p1
11
Stress heterogeneity and Omori law

• smooth stress change, or large A?n
• ? Omori exponent plt1
• very heterogeneous stress field, or small A?n
• Omori p1
• pgt1 cant be explained by a stress step ?(r)
• ? postseismic relaxation ?(t) ?

12
Inversion of stress distribution from aftershock
rate

• Deviations from Omori law with p1 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
  tmaxltta)
• - good results on synthetic RS catalogs

13
Inversion of stress pdf from aftershock rate
Synthetic RS catalog - input P(?) N230 -
inverted P(?) fixed A?n , Rr and ta A?n1
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 ?
p0.93
14
Parkfield 2004 M6 aftershock sequence

• Fixed
• A?n 1 MPa
• ?0 3 MPa
• Inverted
• ? 11 MPa
• ta 10 yrs

15
Landers, 1992, M7.3, 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 M6.6 (South of Salton
Sea 33oN)

• Fixed
• A?n 1 MPa
• ?0 3 MPa

Data, aftershocks Fit RS model Gaussian P(?) Fit
Omori law p1.3
Elmore Ranch M6.2
Rr
foreshocks
17
Morgan Hill, 1984 M6.2, 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
Stacked aftershock sequences, Japan (80, 3ltMlt5,
zlt30)

• 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
M6.2, 1984 0.68 6.2 78. Parkfield M6.0,
2004 0.88 11. 10. Stack, 3ltMlt5,
Japan 0.89 12. 1.1 San Simeon M6.5
2003 0.93 18. 348. Landers M7.3,
1992 1.08 52. Northridge M6.7,
1994 1.09 94. Hector Mine M7.1,
1999 1.16 80. Superstition-Hills,
M6.6,1987 1.30
we cant estimate ? because pgt1 (inversion
gives ?inf) Peng et al., in prep 2005
20
Conclusion

• RS model with stress heterogeneity gives
• - apparent Omori law with p1 for tltta, if ?
  ?0 ,
• p ?1 with heterogeneity ?
• quiescence
• for tta on the fault,
• or for r/Lgt0.1 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 cant explain pgt1 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