The Snowball Effect: Statistical Evidence that Big Earthquakes are Rapid Cascades of Small Aftershocks Karen Felzer U.S. Geological Survey

Earthquakes have often been modeled as avalanches of small subevents Housner (1955) Vere-Jones (1976) Kagan and Knopoff (1981) Zeng et al. (1994) Rydelek and Sacks (1996) Lavallee and Archuleta (2003) …….

How do the small subevents organize into a rapid, propagating chain? Kagan and Knopoff (1981): Subevents trigger each other via the aftershock process Mainshock fault plane Small subevent

Fundamental Idea of Kagan and Knopoff (1981) Omori’s Law If the time gap between subevents comes from here, we see a single, larger earthquake If the time gap comes from here we see a separate mainshock and aftershock Here we propose an earthquake propagation model based on this idea, called the Aftershock Avalanche Model Mainshock propagation = Aftershock Triggering

Aftershock Avalanche Model Earthquake remains small An aftershock subevent is triggered; earthquake grows A subevent nucleates at a point of weakness On average, the subevent is stopped by strength heterogeneities when it reaches area A If waves from the subevent quickly trigger an aftershock subevent, propagation may continue Fault plane Points of high strength OR

Aftershock Avalanche Model Characteristics 1.The smallest “unit” subevents, have an equal area A (Different from Kagan & Knopoff model) 2.Mainshock propagation and aftershock triggering are the same process => time lag between a triggering and triggered unit subevent follows Omori’s Law for aftershock decay (Kagan and Knopoff (1981)). 3.Each unit subevent triggers the same number of aftershocks, independent of current earthquake size (Vere-Jones, 1976) 4.Over all time, the average number of aftershocks triggered per subevent = 1 (Vere-Jones, 1976)

Two tests of the Aftershock Avalanche Model 1.Monte Carlo Simulations (preliminary, rough) 2.Comparison of Aftershock Avalanche model predictions with earthquake catalog statistics (well developed, robust)

1st Test of Aftershock Avalanche Model Do Monte Carlo model realizations reproduce the Gutenberg-Richter relationship (with b = 1) and realistic earthquake time series?

Monte Carlo Simulation Basics Model starts with a few random subevents Locations of triggered subevents: Chosen randomly from distribution cr -1.4 (Felzer and Brodsky, 2005) Time of triggered subevents: S wave travel time + minimum wait time (t min ) + Omori’s Law (at -p )

Monte Carlo Simulation Parameters ParameterValueReason p1.34 Felzer et al. (2003) a Reproduces Båth’s Law for > 20 s bet. eqs t min seconds Each subevent produces ~ 1 aftershock

Monte Carlo Simulation: Combining subevents into earthquakes tt Unit subevent Composite earthquake If  t Subevent is added to the preceding earthquake Composite Magnitude= log(Earthquake Area) time

Simulation results: Unit subevents in space and time

Simulation results: G-R relationship, b = 1

Simulation results: Time series of coalesced earthquakes

Simulation results: Omori’s Law after M 3.7 mainshock

Results of 1st Test Monte Carlo Simulations of the model (with approximate parameters) reproduce a power law distribution of earthquake magnitudes & Omori’s Law => The model is consistent with the fundamental empirical earthquake statistics

2nd Test of Aftershock Avalanche Model Do predictions of the model agree with observed data?

Aftershock Avalanche Model Prediction #1 The nucleation of unit subevents = high frequency bursts -- should be seen during an earthquake, on the mainshock fault. The initiation times of the bursts should follow Omori’s law from the arrival time of the wave front

1999 Chi Chi earthquake: High frequency bursts seen on mainshock fault Filtering the mainshock seismogram reveals many high frequency bursts (Chen et al., in press) Chen et al. (in press) and Fischer and Sammis (2005)

Chi-Chi high frequency bursts follow Omori’s Law Chen et al. (in press) and Fischer and Sammis (2005)

Aftershock Avalanche Model Prediction #2 Mainshock Propagation = Dynamic Stress If mainshock propagation = aftershock triggering Aftershock triggering = dynamic stress

Test for Dynamic Stress Aftershock Triggering Triggering by the Denali Earthquake; Gomberg et al. (2004) Dynamic stress change accomplishes distant triggering Static and dynamic stress changes decay at different rates with distance If far & near field aftershock densities decay with distance at the same rate => Dynamic triggering throughout

Test for Static or Dynamic Aftershock Triggering: What we should see To get a wide distance range with respect to mainshock size, we use a range of mainshock magntiudes Exact expectation Approximate expectation

Dynamic Triggering Results Aftershock density decay with distance is constant from 0.05 to 280 Fault Lengths => Dynamic triggering throughout ( Felzer and Brodsky, 2005)

Avalanche Aftershock Model Prediction #3 The aftershocks produced by each unit subevent are statistically constant, independent of the number of nearby subevents (from Vere-Jones (1976) and Kagan and Knopoff (1981)) All aftershock statistical properties (temporal distribution, spatial distribution, magnitude distribution) are independent of mainshock magnitude.

Data Set for Testing Prediction #3 We use mainshocks from three different magnitude ranges: M 2--3, M 3--4, and M Mainshocks selected are all mainshocks well separated in space/time from larger mainshocks. Aftershocks data sets are all of the M≥2 aftershocks triggered by the mainshocks in a given unit magnitude range Random aftershocks are then removed from each set such that the final data sets contain equal number of aftershocks.

Aftershock spatial distribution is independent of mainshock magnitude

Aftershock temporal distribution is independent of mainshock magnitude

Aftershock magnitude distribution is independent of mainshock magnitude

Results of 2nd Test Several observational predictions of the Aftershock Avalanche model are satisfied by the data. These include: 1.High frequency bursts, following Omori’s law, are observed during mainshocks 2.Aftershocks appear to be dynamically triggered 3.Aftershock properties are independent of mainshock magnitude

Aftershock Avalanche model implications for Earthquake Probability Forecasting Traditional renewal model: Slip only initiates when stress is high enough to guarantee through going rupture. Prediction potential. Aftershock Avalanche Model: Slip may initiate whenever the weakest point is ready. Any earthquake may be large (or not) - - no special conditions required. No predictability.

Conclusions We propose that earthquakes propagation = rapid aftershock chain of unit subevents of equal area: Aftershock Avalanche Model. Monte Carlo model simulations reproduce G-R and Omori’s Laws. Model predictions of aftershock physics and scaling relationships are satisfied by the data. In the model no special circumstances are required to produce a large earthquake => Earthquake magnitude cannot be predicted