Earthquake triggering, stress changes

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Earthquake triggering, stress changes
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Outline
Seismicity in Southern California
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Earthquake clustering

• Seismicity in Southern California with m2
• Symbol size rupture area 10m
• Color represents longitude -120 -116

Northridge M6.7
Joshua- Tree, M6.1
Hector Mine M7.1
Landers M7.3
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Earthquake clustering

• Seismicity in Southern California with m2
• Symbol size rupture area 10m
• Color represents longitude -120 -116

Joshua- Tree, M6.1
Landers M7.3
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Foreshocks, mainshocks, aftershocks
? average over many sequences ---- 1 typical
sequence
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!! Non-conventional definition of
mainshocksand aftershocks

• Mainshock any isolated EQ
• Aftershock EQ within the influence zone
  (T,R) of the mainshock
• can be LARGER than the mainshock
• Motivation can we explain the triggering of a
  large EQ by a smaller one using the same laws
  as for a small EQ triggered by a larger one?

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When Relaxation of number of aftershocks with
timeand scaling with mainshock magnitude M
P(m) GR law

• Aftershock rate decays with time as N1/t0.9 for
  all M (Omori law)
• Only the number of aftershocks increases with M,
  N rupture area 10M

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How are large EQs from smaller ones?

• theyre larger L100.5m
• but same stress drop
• static coulomb stress change ?(r/L,?)
• so the number of aftershocks fault area L2
  10m as observed
• small and large EQS collectively as important
  for triggering and for stress transfers between
  EQS if P(m)10m (GR law)

M5, L3 km
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How big are triggered events Distribution of
aftershocks magnitudes for different mainshock
size M
Southern California 1980-2005

• Same mag distribution for all M
• Only the number of aft increases with M

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Where Spatial distribution of aftershocks
mainshocks M7-7.5
? aftershocks --- background

• relocated catalog for Southern Califorina
• Shearer et al., 2004
• Triggering distance increases with M
• Max triggering distance
• R 2 rupture length
• 0.02x10m/2 km

Number of aftershocks
mainshock M2-2.5
Distance from mainshock (km)
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Where Triggering distance as a function M

• triggering distance d(m) 0.01x100.5m km
  rupture length

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Summary earthquake triggering, observations

• Aftershock rate decays as N1/t0.9, for t
  between a few sec and several yrs, independently
  of the mainshock magnitude M.
• The number of aftershocks increases as N 10M
  L2 , for 0ltMlt9.
• Small EQs collectively as important as larger
  ones for EQ triggering
• The size of a triggered event is not constrained
  by the mainshock size
• The typical triggering distance L 0.01x10M/2
  km.
• The maximum triggering distance is 2L
• These observations can be explained by several
  models of earthquake triggering by static stress
  changes, such as the rate-and-state model of
  Dieterich 1994.

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How

• static (permanent) coseismic stress change
  stress weakening
• - rate-and-state friction Dieterich, 1994
• subcritical crack growth Das Scholz, 1981
  Shaw, 1993
• damage rheology Ben-Zion Lyakhovsky, 2005
• or stress relaxation with time
• - fluid diffusion Nur Booker, 1972
• - postseismic slip asperities Schaff et al.,
  1998
• - viscous relaxation (?) Mikumo and Miyatake,
  1979
• dynamic stress changes (seismic waves) ?
• - does not explain long-time triggering
• Gomberg, 1997 Brodsky et al., 1998, 2003

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Models for EQ triggering and forecasting

• Epidemic Type EQ Sequence (ETES)
• Kagan and Knopoff, 1981 Ogata, 1988
• Empirical model, based on Omori law GR law
• Seismicity rate depends on time, location and
  mag of past EQs
• Rate-and-State
• Dieterich, 1994
• Physical model based on rate-and-state friction
  law
• Seismicity rate as a function of stress history
  rate-and-state friction

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Epidemic Type Earthquake Sequence (ETES) model

• Input
• Probability that an EQ (t,x,m) triggers another
  (t,x,m)
• Results
• Multiple interactions between EQs

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Definition of ETES model

• Seismicity rate "background"
  "aftershocks"
• Magnitude distribution G.R. law
• Aftershocks Omori law and increase of
  aftershock with magnitude

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ETES model, foreshocks and aftershocks

• Input
• Results

Aftershocks Aftershocks of aftershocks
 global Omori law Rg(t) 1 / t p
with p p Rg(t) Rl(t)
Foreshocks Inverse Omori law R(t) 1 / t p
with p p
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Foreshock, mainshocks , aftershocks
LM

X
X
LF
x
X
X

• same properties F?M and M?A
• - in time Omori, aftershock duration and p
  exponent indep. of mM
• - in space rM-rF LF rA-rM LM
• - in magnitude P(mA) G.R. law indep. of mM
• NA(mM) 10?m
• same physical mechanisms for F?M and M?A
• from nucleation, critical point model, or
  rupture experiments

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Rate-and-state friction law

• Friction law Dieterich, 1979 Ruina, 1983
• ? ?0 A ln (V) B ln(?)
• d?/dt 1 - ?V/Dc

• steady state
• ? const. (A-B) ln (V)
• stable if AgtB EQs AltB

Dieterich Kilgore, 1996
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Rate-and-state friction law and EQ triggering
by a static stress change

• without perturbation V1/(tc-t)
• (nucleation phase)
• Initial population of faults with uniform pdf of
  rupture time
• ?? increases V and moves the fault closer to
  failure
• Clock change ?tc depends on initial state
• Increase of seismicity rate and Omori law decay

Dieterich, 1994
?tc
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Rate-and-state model of seismicity Dieterich
1994

• Relation between seismicity rate R and (Coulomb)
  stress history
• and
• For a static stress change constant tectonic
  loading rate

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Coseismsic slip, stress change, and aftershocks
Planar fault, uniform stress drop
Slip Shear stress Seismicity rate RS
model
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Spatial distribution of aftershocks
Map aftershocks Landers, 1992, M7.3 S.
California tlt1yr
Aftershocks within 1 km from the fault plane
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Spatial distribution of aftershocks

• Most aftershocks occur on or close to the
  fault plane, where the shear stress change
  decreases on average
• Shear stress change must be very heterogeneous
  to explain on fault aftershock triggering with
  the rate-and-state model
• - fault roughness Dieterich, 2005
• - slip heterogeneity

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Shear stress heterogeneity due to fault geometry

• Dieterich, 2005

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Slip and shear stress heterogeneity

• Stochastic scale-invariant kinematic slip model
  Herrero and Bernard, 1994
• Large kgt1/L U(k) 1/k2
• Large EQ sum of small ones
• Small klt1/L U(k) const.
• Random phase
• Reproduces the 1/f2 power spectrum seismogramms
  (displacement) for fgtfc
• Shear stress power-spectrum ?(k) 1/k for kgt1/L
• - infinite standard deviation!
• - small scale cutoff
• - or U(k) 1/kn with ngt2

U(k)1/k2
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Slip and shear stress heterogeneity

• Modified  k2  slip model U(k) 1/(k1/L)2.3
• Stress drop ?0 3 MPa

?0
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Slip and shear stress heterogeneity RS model

• Synthetic aftershock catalog generated using
  Dieterich 1994 model
• (without multiple interactions between
  aftershocks)

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Stress heterogeneity and aftershock decay with
time

• Aftershock rate from RS model with modified k2
  slip model
• R(t) ?fault R(t,?) P(?) d?
• assuming A?n 1 MPa

Stress distribution Gaussian
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Modified k2 slip model, Off-fault aftershocks

• On fault
• - for kgt1/L
• - slip
• U(k) 1/k2.3
• - shear stress change
• ?(k) 1/k1.3
• Distance dltL
• from the fault
• ?(k) exp(-kd)/k1.3

Distance from the fault d/L
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Stress heterogeneity and aftershock decay with
time

• RS model produces Omori law with p1 for an
  exponential pdf P(?)
• P(?) exp(- ?/?o)
• R(t) ? R(t,?) P(?) d? 1/tp for tta
  with p 1- A?n/?o

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Stress heterogeneity and aftershock decay with
time

• Aftershock rate for P(?) exp(-?/?o)
• p 1- A?n/?o for t ta
• p decreases if  heterogeneity 
• ?o increases

p0.9 Stacked aft sequences in S. California
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Aftershock decay for a Gaussian stress
distribution

• stress distribution
• P(?) exp-(?-?0)2 /2?2
• aftershock rate close to Omori law
• with effective exponent
• p 1- A ?n?0 - A2?n2 log(t/ta) /?2

Omori law with p0.80
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Inversion of stress changes from aftershock rate

• Deviations from Omori law with p1 due to
• stress decrease / increase with time
• Dieterich, JGR 1994 Nature 2000
• or heterogeneity of coseismic stress change, due
  to
• - fault geometry Dieterich, 2005
• - heterogeneity of coseismic slip
• hard to distinguish between small scale stress
  heterogeneity and temporal variation
• We invert for P(?) from aftershocks rate R(t)
• Solve R(t) ? R(t,?) P(?) d?
• assuming stress does not change with time
• problem RS model with instantaneous stress
  change cant explain pgt1

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Inversion of stress pdf from aftershock rate
Test on synthetic RS catalogs
modified k2 model
slip
Shear stress change
Rate State Dieterich, 1994
Aftershock rate (model)
EQ catalog ti , i1,..,N tminltt lttmax
Aftershock rate (simul)
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Inversion of stress pdf from aftershock rate

• Complete distribution P(?)
• - solution of R(t) ? R(t,?) P(?) d?
• - fixed ta, Rr, A?n
• Gaussian P(?)
• - fixed A?n, Rr
• - invert for ?, ?0 and ta

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Inversion of stress pdf from aftershock rate
Synthetic RS catalog - input P(?) N150000 -
inverted P(?), fixed A?n , Rr and ta A?n1
MPa - Gaussian P(?) ?0 3 MPa ?20 MPa
fixed A?n and Rr , invert for ta , ?0 and ?
p0.93
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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 , invert for ta,
?0 and ? ?0 3 MPa ?20 MPa - Gaussian
P(?), fixed A?n , ?0 and Rr , invert for ta and ?
p0.93
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Inversion of stress pdf from aftershock rate
Synthetic RS catalog - input P(?) N3857 -
inverted P(?) not constrained A?n0.1 MPa -
Gaussian P(?), fixed A?n and Rr , invert for ta,
?0 and ? ?0 3 MPa ?20 MPa - Gaussian
P(?), fixed A?n , ?0 and Rr , invert for ta and ?
Omori p0.993
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Parkfield 2005 m6 aftershock sequence

• Fixed
• A?n 1 MPa
• ?0 3 Mpa
• Inverted
• ? 11 MPa
• ta 10 yrs
• Loading rate
• d?/dt A?n / ta
• 0.1 MPa/yr
•  Recurrence time 
• tr ta ?0/A?n
• 30 yrs

Data, aftershocks Fit RS model Gaussian P(?) Fit
Omori law p0.88
ta
foreshock
Rr
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Stacked aftershock sequences, Japan (80, 3ltMlt5,
zlt30)
Data, aftershocks Fit RS model Gaussian P(?) Fit
Omori law p0.89

• 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
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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

Data, aftershocks Fit RS model Gaussian P(?) Fit
Omori law p1.08
foreshocks
ta
Rr
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Hector Mine 1999 M7.1 aftershock sequence

• Fixed
• A?n 1 MPa
• ?0 3 MPa
• Inverted
• ? 438 MPa
• ta 80 yrs
• Loading rate
• d?/dt A?n / ta
• 0.012 MPa/yr
•  Recurrence time 
• tr ta ?0/A?n
• 240 yrs

Data, aftershocks Fit RS model Gaussian P(?) Fit
Omori law p1.16
ta
foreshocks
Rr
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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

data, aftershocks Fit RS model Gaussian P(?) Fit
Omori law p0.68
foreshocks
ta
Rr
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Inversion of stress - Conclusion

• Stress drop not constrained if catalog too short
• We can estimate ? (width of P(?)) for a
  limited catalog if plt1
• And if we know A?n
• - A?n 1 MPa ? A0.01 (Lab experiments,
  Dieterich and Kilgore 1996)
• ?n100 MPa (lithostatic pressure at
  5km)
• - A?n 0.1 MPa ? (relation between ta and
  recurrence time Dieterich, 1994)
• Effect of secondary aftershocks?
• - increase ref EQ rate Rr Ziv and Rubin, 2003,
  but does not change p or ? ?
• Heterogeneity of A?n? (does not change R(t) if
  A?n less heterogeneous than ??)
• Post-seismic stress relaxation?