Transcript Earthquake triggering, stress changes Relation between

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
Landers, aftershocks and Hernandez et al. [1999] slip model
“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
-- Omori law R~1/t
Aftershock duration ta
• A≈ 0.01 (friction exp.)
• n≈100 MPa (P at 5km)
«min» time delay c()
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
Modified «k2» slip model: u(k)~1/(k+1/L)2.3 [Herrero & Bernard, 94]
aftershock map
shear stress
slip
synthetic R&S catalog
stress drop 0 =3 MPa
0
0
stress distrtibution
P()≈Gaussian
Stress heterogeneity and aftershock decay with time
Aftershock rate on the fault with R&S model for modified k2 slip model
-- Omori law
R(t)~1/tp
with p=0.93
∫ R(t,)P()d
ta
Rr
Short times t‹‹ta : apparent Omori law with p≤1
Long times t≈ta : stress shadow R(t)<Rr
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/An
• But long-time shadow difficult to detect
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
d
L
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
d/L
d/L=0.1
• For an exponential pdf P()~e-/o with >0
• R&S gives Omori law R(t)~1/tp with p=1- An/o
log P()
Stress heterogeneity and Omori law
0

• black: global EQ rate,
p=0.8
heterogeneous :
R(t) = ∫ R(t,)P()d
with o/An=5
• colored lines:
EQ rate for a unif :
R(t,)P()
p=1
from =0 to =50 MPa
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
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
*=20 MPa
- invert for ta, 0 and *
- Gaussian P(): - fixed An , 0 and Rr
- invert for ta and *
p=0.93
Parkfield 2004 M=6 aftershock sequence
• Fixed:
An = 1 MPa
0 = 3 MPa
data, aftershocks
data, `foreshocks’
fit R&S model Gaussian P()
fit Omori law p=0.88
• Inverted:
* = 11 MPa
ta = 10 yrs
ta
foreshock
Rr
Landers, 1992, M=7.3, aftershock sequence
• Fixed:
An = 1 MPa
0 = 3 Mpa
Data, aftershocks
Fit R&S model Gaussian P()
Fit Omori law p=1.08
• Inverted:
* = 2350 MPa
ta = 52 yrs
• Loading rate
d/dt = An / ta
= 0.02 MPa/yr
foreshocks
ta
• « Recurrence time »
tr= ta 0/An
= 156 yrs
Rr
Superstition Hills 1987 M=6.6
• Fixed:
An = 1 MPa
0 = 3 MPa
(South of Salton Sea 33oN)
Data, aftershocks
Fit R&S model Gaussian P()
Fit Omori law p=1.3
Elmore
Ranch
M=6.2
R
foreshocks
r
Morgan Hill, 1984 M=6.2, aftershock sequence
• Fixed:
An = 1 MPa
0 = 3 Mpa
data, aftershocks
Fit R&S model Gaussian P()
Fit Omori law p=0.68
• 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
ta
foreshocks
R
r
Stacked aftershock sequences, Japan (80, 3<M<5, z<30)
Data, aftershocks
Fit R&S model Gaussian P()
Fit Omori law p=0.89
• 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
foreshocks
ta
Rr
[Peng et al., in prep, 2006]
Inversion of P() from R(t) for real aftershock sequences
* (MPa)
Sequence
p
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
* [Peng et al., in prep 2005]
ta (yrs)
(inversion gives *=inf)
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
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