Stress- and State-Dependence of Earthquake Occurrence Jim Dieterich, UC Riverside

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

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)

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

Aftershock model Pict of Coulomb change Stress Time Earthquake rate Background rate Time Coulomb stress change

Stress Shadow Stress Time Earthquake rate Background rate Time

Method: Stress time series STEPS 1) Select region and magnitude threshold 2) Smooth earthquake rate: R(t) 3) Obtain time series for  : 4) Solve evolution equation for Coulomb stress S. For example:

Synthetic Data

Figure: Barry Eakins Rift zones modified from Fiske and Jackson (1972) Bathimetry: USGS JAMSTEC

Cross section through east rift of Kilauea

Inversion of EQ rates for stress ( ) Dieterich and others, 2000, Nature

Method to obtain stress changes from earthquake rates 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 Dieterich, Cayol, and Okubo, Nature (2000) Dieterich and others, USGS Prof Paper(2003) 3) Prepare maps (or cross sections) of stress changes over specified time intervals

>1983 Deformation 0.5MPa/yr (0.1MPa) Seismicity 0.3–0.6 MPa/yr ≤0.1 Mpa/yr Rift intrusion rate 0.18km 3 /yr 0.06km 3 /yr NS extension 25cm/yr 4cm/yr (Summit region)

Stress changes before 1983 eruption Seismicity solution, MPa/year Deformation model, MPa/year 2 Dieterich, Cayol, and Okubo, Nature (2000) Dieterich and others, USGS Prof Paper(2003)

Stress changes at the time of the 1983 intrusion & eruption Deformation model Seismicity solution Dieterich and others, USGS Prof Paper(2003)

Stress changes at the time of the 1977 intrusion & eruption Deformation model

Kalapana Earthquake M

Earthquakes M≥

Earthquakes M≥

Earthquakes M≥

M~ 5 Earthquakes following Sept. 13, 1977 eruption M4.6 9/27/ /21/79

M~ 5 Earthquakes following Jan. 1, 1983 eruption M5.0 3/20/83 M5.2 9/9/83

1/3/83

Geometrically complex faults USGS, 2003

Fault geometry Individual faults exhibit approximately self-similar roughness (fractal dimension~1). Fault in the Monterrey Formation San Francisco Bay Region Fault systems also appear to be scale-independent

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

Slip of a fault patch 

Fault slip and stress changes Smooth fault Fractal fault: H=1,  =0.01

Global slip Fault slip and stress changes Smooth fault Fractal fault: H=1,  =0.01

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

Non-linear scaling of slip with fault length: Average of slip for n  100 simulations Hurst exponent: H = 1.0 Roughness amplitude:  = 0.1 d MAX = 85 Region of ~ linear scaling of slip with fault length FAULT LENGTH

 = 0.01  = 0.03  = 0.1  = 0.3

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. Non-linear scaling and system size-dependence Slip, d Back stress, S BACK S A = Applied stress

 = 0.01  = 0.03  = 0.1  = 0.3

Average slip on non-planar faults n~100 Planar fault model with elastic back stress Hurst exponent: H = 1.0 Roughness amplitude:  = 0.1 d MAX = 85 Non-linear scaling of slip with fault length FAULT LENGTH

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

Data from Sowers and others (1992), US Geological Survey Landers and Hector Mine Earthquakes

 Steady-state yielding by earthquakes: EQ rate  Coulomb stress rate  Long-term slip rate

Average long-term earthquake rate by distance from fault with random fractal roughness Earthquake rate Stressing due to fault slip at constant long-term rate Model assumes steady-state seismicity at the long-term stressing rate, in regions where

Average long-term earthquake rate by distance from fault with random fractal roughness Earthquake rate Scaling:

   Initial Aftershock Rate / Background Rate

  

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 L=rupture length

Conclusions: Seismicity stress solutions Stress shadows are seen for all earthquakes M≥4.7 Quantitative agreement of deformation measurements and seismicity stress solutions –Stress changes –Stress changes related to 1983 eruption –Stressing rate change Provides greater detail of stress changes at depth than can be obtained from deformation modeling –Resolve stress patterns for earthquakes M~5 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

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