Stress- and State-Dependence of Earthquake Occurrence: Tutorial 2 Jim Dieterich University of California, Riverside

Log  Coefficient of friction  Constant V (high) Constant V (low)  ss B B-A x V1V1

Log  Coefficient of friction  Constant V (high) Constant V (low)  ss B B-A x During slip  evolves toward  ss V1V1

Log  Coefficient of friction  Constant V (high) Constant V (low)  ss B B-A x During slip  evolves toward  ss V1V1

Log  Coefficient of friction  Constant V (high) Constant V (low)  ss B B-A x During slip  evolves toward  ss V1V1

yr 10 yr 20 yr Time to instability (seconds) Log (slip speed)  m/s Effect of stress change on nucleation time

min 1 yr 10 yr 20 yr ~1hr ~5hr Time to instability (seconds) Log (slip speed)  m/s Effect of stress change on nucleation time  = 11.6

Earthquake rate formulation: Model Earthquake occurrence is represented as a sequence of earthquake nucleation events. Dependence of nucleation times on stressing history is given by nucleation solutions derived for rate- and state-dependent fault strength. Model assumes 1) The population of nucleation sources is spontaneously renewed as stress increases 2) Reference steady-state seismicity rate r at the constant stressing rate.

Use the solution for time to nucleation an earthquake (1), where and assume steady-state seismicity rate r at the stressing rate This defines the distribution of initial conditions (slip speeds) for the nucleation sources (2) The distribution of slip speeds (2) can be updated at successive time steps for any stressing history, using solutions for change of slip speed as a function of time and stress., n is the sequence number of the earthquake source Model for earthquake occurrence Log (time to instability) Log (slip speed)

For example changes of with time are given by the nucleation solutions and change of with stress are given directly from the rate- and state- formulation In all cases, the final distribution has the form of the original distribution where Evolution of distribution of slip speeds

Earthquake rate is found by taking the derivative dn/dt = R For any stressing history Evolution of distribution of slip speeds

Coulomb stress formulation for earthquake rates Earthquake rate, Coulomb stress Assume small stress changes (treat as constants), Note:. Hence, Earthquake rate, Dieterich, Cayol, Okubo, Nature, (2000), Dieterich and others, US Geological Survey Professional Paper (2003)

Some useful solutions Earthquake rate Evolution with time Stress step

Example S t (steady state)

Example - Secondary aftershocks t S t =0

Earthquake rates following a stress step Earthquake rate (R/r ) Time (t / t a )

Earthquake rates following a stress step

Aftershocks by time and distance Time at which x is at edge of aftershock zone

Factors affecting rate of aftershock decay (p) In this model, intrinsic value p=1 The following factors result in p≠1 Spatial heterogeneity of stress change (  S/A  ) p < 1 Stress relaxation by log(t) after the stress step if u>0.2  p > 1 Secondary aftershocks p > 1 (short-term effect)

Time (t/t a ) Earthquake rate (R/r 0 ) Slope p=0.8 Net aftershock rate for region surrounding a circular shear rupture

Over short time intervals (  t <<t a ) Triggering by seismic waves

Stress (S/A  ) Cumulative Number Time (seconds) Triggering by seismic waves r = 5.0x10 -5 /s

Triggering by seismic waves Peak stress  S/A 

Triggering by seismic waves Peak amplitude (  S) to trigger EQ (A  ) Time to EQ with no triggering (sec) 10 year

Triggering by seismic waves Peak amplitude (  S) to trigger EQ (A  ) Time to EQ with no triggering (sec) 10 year Threshold stress model x

Change of earthquakes rates, tidal stresses Over short time intervals (  t <<t a ) For small stress changes (  S << A  ) this becomes

Minimum number of events to see tidal influence  S = S S 0-25  S~ bar A  = 1 bar:  R/r = A  = 2 bar:  R/r =

Method to obtain stress time series from earthquake rates STEPS 1) Select region and magnitude threshold 2) Obtain time series for  : 4) Solve evolution equation for Coulomb stress S. For example:

Synthetic Data

STEPS 1) From earthquake rates obtain time series for  at regular grid points: 2) Solve evolution equation for Coulomb stress S as a function of time at each grid point 3) Prepare maps (or cross sections) of stress changes over specified time intervals Dieterich, Cayol, and Okubo, Nature (2000) Dieterich and others, USGS Prof Paper(2004) Maps of stress changes from earthquake rates

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Synthetic Data

Model A Foreshocks Advance the time of Mainshock Mainshocks following foreshocks and aftershocks have similar origins. The stress change of a prior earthquake results in increased nucleation rates at all magnitudes. Extrapolation of aftershock rates to larger magnitudes gives rate of foreshocks  S is a function of distance from the prior earthquake. Net earthquake rate following a stress step is obtained by integrating over the region affected by the stress change. Foreshock models

Foreshocks in Southern California

Foreshock models Model B Mainshock Nucleation Causes Foreshocks Premonitory creep of a large nucleation zone causes rapid stressing on nearby smaller nucleation sources

Foreshock model B Nucleation on fractal fault 2 hours before instability

Foreshocks in Southern California