Stress and StateDependence of Earthquake Occurrence

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Stress- and State-Dependence of Earthquake
Occurrence
Jim Dieterich, UC Riverside
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Formulation for earthquake rates

• Unified and quantitative framework for analysis
  of effects of stress changes on earthquake
  occurrence
• Some applications
• Aftershocks
• Foreshocks
• Complexity of earthquake events
• Triggering of earthquakes by seismic waves
• Tidal triggering (why the effect is so weak)
• Earthquake probabilities following stress change
• Solutions for stress changes from observations of
  earthquake rates
• Stress relaxation by seismic processes for
  geometrically complex faults

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Earthquake rate formulation Model

• Earthquake occurrence is controlled by earthquake
  nucleation processes
• Earthquake nucleation as given by rate- and
  state-dependent friction is time dependent and
  highly non-linear in stress and gives the
  following
• Coulomb stress function as
  where
• At steady state
• The characteristic time to reach steady-state

Dieterich, JGR (1994), Dieterich, Cayol, Okubo,
Nature, (2000), Dieterich and others, USGS
Professional Paper 1676 (2003)
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Earthquake rates following a stress step
Earthquake rate (R/r )
Time (t / ta )
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Aftershock model
Stress
Pict of Coulomb change
Time
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Stress Shadow
Stress
Time
Earthquake rate Background rate
Time
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Method Stress time series
STEPS 1) Select region and magnitude threshold 2)
Smooth earthquake rate R(t) 3) Obtain time
series for g 4) Solve evolution equation for
Coulomb stress S. For example
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Synthetic Data
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Synthetic Data
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Cross section through east rift of Kilauea
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Inversion of EQ rates for stress (1996-1994)
Dieterich and others, 2000, Nature
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Method to obtain stress changes from earthquake
rates
STEPS 1) From earthquake rates obtain time series
for g at regular grid points 2) Solve
evolution equation for Coulomb stress S as a
function of time at each grid point
Dieterich, Cayol, and Okubo, Nature
(2000) Dieterich and others, USGS Prof Paper(2003)
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1976-1983
gt1983 Deformation
0.5MPa/yr (0.1MPa) Seismicity
0.30.6 MPa/yr 0.1 Mpa/yr Rift
intrusion rate 0 .18km3/yr 0
.06km3/yr NS extension 25cm/yr
4cm/yr (Summit region)
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Stress changes before 1983 eruption
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Dieterich, Cayol, and Okubo, Nature
(2000) Dieterich and others, USGS Prof Paper(2003)
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Stress changes at the time of the 1983 intrusion
eruption
Deformation model
Dieterich and others, USGS Prof Paper(2003)
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Stress changes at the time of the 1977 intrusion
eruption
Deformation model
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Kalapana Earthquake M7.3 1975
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Earthquakes M4.6 1976-1979
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Earthquakes M4.6 1980-1989
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Earthquakes M4.6 1990-1999
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M 5 Earthquakes following Sept. 13, 1977 eruption
M4.6 9/27/79
5.4 9/21/79
M4.6 9/27/79
5.4 9/21/79
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M 5 Earthquakes following Jan. 1, 1983 eruption
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1/3/83
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Geometrically complex faults
USGS, 2003
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Fault geometry
Individual faults exhibit approximately
self-similar roughness (fractal dimension1).
Fault in the Monterrey Formation
Fault systems also appear to be scale-independent
San Francisco Bay Region
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Random Fractal Fault Model
Solve for slip using boundary elements. Simple
Coulomb friction with ? 0.6 Periodic B.C, or
slip on a patch
? 0.3
? 0.1
? 0.03
? 0.01
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Slip of a fault patch
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Fault slip and stress changes
Smooth fault
Fractal fault H1, ?0.01
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Fault slip and stress changes
Smooth fault
Fractal fault H1, ?0.01
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Non-linear scaling of slip with fault length
Hurst exponent H 1.0 Roughness amplitude ?
0.05
Region of linear scaling of Slip with fault
length
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Non-linear scaling of slip with fault
length Average of slip for n?100 simulations
dMAX 85
FAULT LENGTH
Hurst exponent H 1.0 Roughness amplitude ?
0.1
Region of linear scaling of slip with fault
length
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? 0.01
? 0.03
? 0.1
? 0.3
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Non-linear scaling and system size-dependence
Geometric complexity forms barriers to
slip. Barrier stress increases with total slip
and sequesters strain energy that would
otherwise be released in slip.
The barrier stress acts as an elastic
back-stress, which opposes slip. Back-stress
increases linearly with slip. Slip saturates at
when the back-stress equals the applied
stress.
SA Applied stress
Back stress, SBACK
Slip, d
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Non-linear scaling of slip with fault length
Hurst exponent H 1.0 Roughness amplitude ?
0.1
Average slip on non-planar faults n100
Planar fault model with elastic back stress
dMAX 85
FAULT LENGTH
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Yielding and Stress Relaxation

• Stresses due to heterogeneous slip cannot
  increase without limit - some form of
  steady-state yielding and stress relaxation must
  occur
• Slope of 0.01 ? shear strain ???.01,
  ?????? brittle failure
• In brittle crust, stress relaxation may occur by
  faulting and seismicity off of the major faults.
• Instantaneous failure and slip during earthquake
• Post-seismic aftershocks and long-term
  seismicity
• Yielding will couple to the failure process, by
  relaxing the back-stresses

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Landers and Hector Mine Earthquakes
Data from Sowers and others (1992) , US
Geological Survey
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Steady-state yielding by earthquakes EQ rate
µ Coulomb stress rate µ Long-term slip rate
????????
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Average long-term earthquake rate by
distance from fault with random fractal roughness

• Stressing due to fault slip at constant long-term
  rate
• Model assumes steady-state seismicity at the
  long-term stressing rate, in regions where

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Average long-term earthquake rate by
distance from fault with random fractal roughness
Scaling
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Initial Aftershock Rate / Background Rate
????????
???????
????????
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????????
????????
???????
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Aftershock rates as function of distance
0.01 0.03 L 0.03 0.05 L 0.05 0.07 L 0.07
0.09 L 0.09 0.10 L
Lrupture length
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Conclusions Seismicity stress solutions

• Stress shadows are seen for all earthquakes M4.7
  
• Quantitative agreement of deformation
  measurements and seismicity stress solutions
• Stress changes 1976-1983
• Stress changes related to 1983 eruption
• Stressing rate change 1980-1983
• Provides greater detail of stress changes at
  depth than can be obtained from deformation
  modeling
• Resolve stress patterns for earthquakes M5 at
  depths of 10km. This includes stress shadows
• Useful for guiding deformation modeling, by
  eliminating alternative models
• Reveals stress interactions between magmatic and
  earthquake processes at Kilauea volcano

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Conclusions - Seismicity and non-planar faults

• Fault complexity ? heterogeneous slip and stress
• Fault complexity elasticity ? non-linear
  scaling and system size-dependence
• Heterogeneous stresses increase with slip ?
  yielding stress relaxation
• ?Slope 0.01 ? shear strain ???.01 ??????
  brittle failure
• Instantaneous failure and slip during earthquake
• Fall-off of background seismicity by distance ?
• Post-seismic aftershocks within stress
  shadow
• Stress relaxation process will couple to slip on
  major faults by relaxing the back stresses.
  Speculation
• Restore linear scaling
• Restore independence of system size