Afterslip, slow earthquakes and aftershocks: Modeling using the rate

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Afterslip, slow earthquakes and
aftershocksModeling using the rate state
friction law
Agnès Helmstetter (LGIT Grenoble) and Bruce Shaw
(LDE0 Columbia Univ)
http//www-lgit.obs.ujf-grenoble.fr/ahelmste/index
.html
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Observations example for 2005 m8.7 Nias EQ
Afterslip and of aftershocks
Co- and after- slip
Afterslip (time)
Hsu et al, Science 2006
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Main questions

• relation between coseismic and postseismic
  slip?
• can we use afterslip to constrain the rheology
  of the crust (stable/unstable)?
• relation between afterslip and aftershocks?
• mechanisms for aftershock triggering?

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Rate-and-state friction law and afterslip

• friction law Dieterich, 1979
• µ µ0 A log(V/V0) B log(?/?0) µ0 -k???n
• d?/dt 1 - V?/Dc
• relaxation or nucleation of a slip instability
  after a stress step
• inertia and tectonic loading negligible
• tectonic deformation V coseismic slip rate

fixed loading point (locked part of the fault)
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First model steady-state approximation

• Scholz 1989, Marone et al 1991 and many
  others assume
• - slip-strengthening friction (stable) AgtB
• - steady state ?constant
• ???solution of RS equations
• ? V V0/(1t/t)
• t ?n(A-B)/kV0
• ? good fit to afterslip data V(t)
• ? afterslip should be restricted aseismic zones

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First modelRS friction and fault behavior
In the lab B and A are functions of normal
stress and temperature In the earth B and A
should be functions of depth
depth
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Observations of postseismic behavior
2003 m8 Tokachi Miyazaki et al, GRL 2004
2005 m8.7 Nias Hsu et al, 2006
? overlap between co- and after- slip, and with
aftershock area ? variations of A or B in space
and/or time?
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Observations of postseismic behavior
Parkfield 2004, M6 Langbein et al 2006
Izmit 1999, M7.4 Burgman, 2002
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Numerical analytical analysis Fault behavior
after a stress step
Fault behavior controlled by friction parameters
B/A, stiffness k/kc (or fault length L/Lc) and
stress µ Aftershock Slip instability triggered
by stress change if µgtµl, kltkc and BgtA Slow
EQ Slip rate increase followed by relaxation if
µaltµltµl Afterslip Relaxation toward background
rate if µltµa
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Fault behavior after a stress step
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Slip rate history 1D model

• Unstable case
• B1.5A
• k0.8kc
• µ0gtµl aftershock
• µlgtµ0gtµa slow EQ
• µ0ltµa afterslip

behaviors Aftershocks, slow EQ, and
afterslip afterslip regimes, with slope
exponents B/A or 1 characteristic times
t
Stable case B0.5A k2.5kc afterslip µ0gtµss µ0µ
ss µ0ltµss
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Slip history - 1D model and afterslip data

• fit afterslip data of Wennerberg and Sharp
  1997 for Superstition Hills event (South
  California 1987 mw6.6)
• invert for A, B, k, Dc,
• V0 and µo
• data can be fitted with
• VV0/(1t/t)p
• ??inversion not
• constrained!
• Both BgtA and BltA
• can fit the data?

- - data _ fit BgtA _ fit BltA
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Conclusions afterslip and slow EQs

• ? RS friction law can be used to model afterslip
  data or slow EQs
• deviations from log slip history (plt or gt1)
• afterslip and slow EQS for BgtA and for BltA
• no need for variations in B or A in time or space
• behaviors may be explained by stress
  heterogeneity
• no need for Alt0 or more complex friction laws
• requires Dc afterslip
• ? 1D model with RS friction cant be used to
  estimate the parameters
• 6 model parameters, but 3 are enough to fit the
  data
• Afterslip and slow EQs zones may no always be
  aseismic

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Afterslip and aftershocks

• similar time dependence of afterslip rate and
  aftershock rate (Omori)
• ? afterslip due to aftershocks, or aftershocks
  triggered by afterslip?
• coseismic slip ? stress increase (on and) around
  the rupture
• afterslip
• reloading on locked parts of the faults
• aftershocks triggered by afterslip
• Dieterich 1994, Schaff et al 1998, Perfettini
  and Avouac 2004, 2007 Wennerberg and Sharp 1997,
  Hsu et al 2006, Savage 2007a,b,
• we use the RS model of Dieterich 1994 to
  model the effect of stress changes on seismicity
  rate, instead of assuming seismicity rate
  stress rate

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Relation between stress changes and seismicity
in the RS model

• Dieterich 2004 model is equivalent to

R seismicity rate R0 R(t0) N?0tR dt r ref
seismicity rate for ttr t coulomb
stress change (0 at t0) ta nucleation time
Atn/tr
short-times regime for Tta RR0exp(t/Atn) (tides
, )
long-times regime for Tta Rdt/dt (tectonic
loading, )
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Aftershocks triggered by afterslip

• numerical solution of R-t relation assuming
  reloading due to afterslip is of the form dt/dt
  V (elastic stress transfer) t0/(1t/t)p with
  p0.8

t
t

• R stress rate for tt when plt1
• t for EQ rate lt for t for stress rate

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Temporal distribution of afterslip and aftershocks
2004 m6.0 Parkfield earthquake
100 sec
1 hour
Peng and Vidale, 2006
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Conclusions aftershocks triggered by afterslip

• RS friction law can be used to model aftershock
  rate
• afterslip is likely a significant mechanism for
  aftershock triggering
• BUT only for large times gt day
• EQ rate does not scale with stress rate

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Conditions for acceleration and instability

• initial acceleration dV/dtgt0 ??µgtµa and
  k/kclt(1-A)/B
• condition for instability V? and d?/dt ????µgtµl
  , kltkc and BgtA
• intermediate behavior µ altµlt µl
•  slow earthquake  acceleration followed by
  relaxation

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(in-)stability after a stress step

• behavior as a function of distance from
  steady-state and B/A for k0.8kc

??
steady-state

• ?? ("healing")
• or ?? ("weakening")
• steady state approx only valid for BltA and kkc

??