The Role of Asperities in Aftershocks

Transcription

1 The Role of Asperities in Aftershocks James B. Silva Boston University April 7, 2016 Collaborators: William Klein, Harvey Gould Kang Liu, Nick Lubbers, Rashi Verma, Tyler Xuan Gu

2 OUTLINE Introduction The Olami-Feder-Christensen Model Adding asperities Looking for Omori s law Conclusions / Further Work

3 WHY STUDY EARTHQUAKES? 1906 San Francisco Earthquake Earthquakes have resulted in a very large amount of financial and human damage. Earthquakes are a quintessential example of a complex system that is difficult to predict.

4 EMPIRICAL LAWS FOR EARTHQUAKES Port-au-Prince 2010 Earthquake Gutenberg-Richter scaling Power law for the frequency of events of a given seismic moment. Omori s law Number of aftershocks follows power law with constant offset. Bath s law Mainshock and aftershocks have constant magnitude difference.

5 PROPERTIES OF AFTERSHOCKS Aftershock locations in Haiti after 2010 earthquake Aftershocks after 2011 Great Tohoku earthquake Aftershocks are localized. Aftershocks follow Omori s law: n(t) = k (t + C) 1 Aftershocks are not completely understood.

6 OMORI S LAW Proposed by F. Omori in 1894 while researching aftershocks in Japan. k n(t) = (t + C) 1 Further work by Utsu suggested generalized form. n(t) = k (t + C) p Recent work by Rundle et al. suggests extending the form by having C depend on the initial shock magnitude M 0.

7 MOTIVATION: IS IT POSSIBLE TO BUILD A SIMPLE MODEL TO OBTAIN THIS BEHAVIOR? n(t) = k [t + C(M 0 )] p What are the key elements in understanding Omori s law? What role does the structure of the earthquake fault play in the aftershock properties? Will introduce the Olami-Feder-Christensen model. Hard Rocks: Introduce asperities to the model.

8 THE OLAMI FEDER CHRISTENSEN MODEL Model fault system by a lattice of sites which hold a stress σ and have a stress threshold at which they fail (slip). Analogous to block-spring models used to model earthquake fault systems. Cellular automaton model created to study self-organized criticality.

9 THE OLAMI FEDER CHRISTENSEN MODEL Parameters: α is the stress dissipation σ i is stress on site i σ res is the residual stress σ f is the stress to failure η is the noise in the residual stress. During simulation the system is loaded using the zero velocity limit min(σ f σ i ).

10 THE OLAMI FEDER CHRISTENSEN MODEL A lattice of sites with a stress at each site.

11 THE OLAMI FEDER CHRISTENSEN MODEL System is loaded until a single site reaches the stress failure threshold.

12 THE OLAMI FEDER CHRISTENSEN MODEL Stress at failed site is lowered to residual stress with excess stress transferred equally to neighbors. Neighbors can be extended to accommodate long range stress transfer case.

13 THE OLAMI FEDER CHRISTENSEN MODEL Process continues. Load system until a new site fails.

14 SUCCESS AND FAILURE OF THE OFC MODEL Events τ = Probability Events Model with long-range stress transfer exhibits power law scaling for the number of earthquakes of size s (Gutenberg-Richter scaling). Model does not exhibit Omori s law behavior. OFC model does not have a well defined energy. A well defined energy helps in applying the toolbox of equilibrium physics.

15 SUCCESS AND FAILURE OF THE OFC MODEL Events τ = Probability Events Model with long-range stress transfer exhibits power law scaling for the number of earthquakes of size s (Gutenberg-Richter scaling). Model does not exhibit Omori s law behavior. Make model more realistic. Introduce asperities. OFC model does not have a well defined energy. A well defined energy helps in applying the toolbox of equilibrium physics.

16 INTRODUCTION TO ASPERITIES Asperity: unevenness of surface, roughness, ruggedness. Uneven surface results in increased difficulty of a slip event. An asperity will be modeled as a site with a stress failure threshold above the non-asperity failure threshold.

17 INTRODUCTION TO ASPERITIES Asperity locations will be randomly uniformly distributed. Asperity strength will be drawn from a Gaussian distribution. Variance of asperity strength expected for a realistic model. Common OFC parameters L = 120 R = 10 α = 0.01 σf = 100,σ res = 0 η = 5

18 SINGLE ASPERITY SLIP RATE Failures of asperity are nearly periodic: ± plate updates between failures for a single asperity 500 times the regular stress threshold.

19 WHAT HAPPENS TO THE STRESS NEAR AN ASPERITY AFTER AN ASPERITY SLIP? < σ > Post-Asperity Fail r R Single asperity creates a stress memory once an asperity fails. A single asperity 100 times larger than regular failure threshold.

20 INCREASE ASPERITY STRENGTH < σ > Post-Asperity Fail r R Increasing asperity strength increases frequency of stress maxima and minima. A single asperity 400 times larger than regular failure threshold.

21 DOES THE LOCAL STRESS MEMORY DISSIPATE? < σ > Post-Asperity Fail r R 500 plate updates after an asperity failure stress distribution still present. A single asperity 400 times larger than regular failure threshold.

22 DOES THE LOCAL STRESS MEMORY DISSIPATE? < σ > Post-Asperity Fail t:0 Post-Asperity Fail t: r R A single asperity 400 times larger than regular failure threshold. Propagation of stress memory is minimal. Large event rate will be used to determine dissipation time.

23 LOOKING FOR OMORI S LAW Measure large event rate n(t): Measured average event size for OFC model without any asperities. Large event is defined as an event 25 times the pure system average. Time is measured with respect to the last asperity failure. Fit to modified version of Omori s Law. n(t) = k (t + C) p + Ae λt sin ( t T ) Empirical range of exponent p [ ] for earthquake faults.

24 SINGLE ASPERITY LARGE EVENT RATE 10 4 Large Event Rate Data Fit Omori s Law + Sine : p= Fit Omori s Law : p= Real Time Since Failure Single asperity does not give Omori s law type behavior.

25 INTRODUCE MORE ASPERITIES Introducing 125 asperities 500 times than regular failure threshold. Start with a nearly constant value for the asperity failure threshold. Focus on measuring the large event rate.

26 MULTI-ASPERITY LARGE EVENT RATE: SMALL ASPERITY VARIANCE 10 5 Large Event Rate 10 4 Data Fit Omori s Law + Sine : p=36.7 Fit Omori s Law : p= Real Time Since Failure Asperity Strength 500σ f with standard deviation 0.005σ f Large event rate characterized by complex non-omori s law behavior.

27 MULTI-ASPERITY LARGE EVENT RATE: LARGE ASPERITY VARIANCE 10 5 Large Event Rate 10 4 Data Fit Omori s Law + Sine : p=0.881 Fit Omori s Law : p= Real Time Since Failure Asperity Strength 500σ f with standard deviation 0.5σ f Multiple asperities results with a larger variance in observed Omori s law with reasonable values for exponents. Similar results obtained for different number of multiple asperities.

28 MULTI-ASPERITY LARGE EVENT RATE: INCREASE ASPERITY STRENGTH Large Event Rate Data Fit Omori s Law + Sine : p=1.75 Fit Omori s Law : p= Real Time Since Failure 125 Asperities 5000σ f with standard deviation 50σ f Increasing asperity strength maintains Omori s law with reasonable values for exponents.

29 MULTI-ASPERITY LARGE EVENT RATE: INCREASE ASPERITY VARIANCE FURTHER Data Fit Omori s Law + Sine : p=28.9 Fit Omori s Law : p=26.0 Large Event Rate Plate Updates Since Failure Increasing variance in asperity strength results leads to noisier large event rate.

30 REVIEW OF EMPIRICAL LAWS Need to check Gutenberg-Richter scaling behavior is preserved. Sornette et al. have shown the combination of Omori s law and Gutenberg-Richter scaling results in Bath s law.

31 GUTENBERG-RICHTER SCALING WITH ASPERITIES Scaling still preserved despite the introduction of multiple asperities.

32 CONCLUSIONS A single asperity can arrange the stress locally to have a temporary memory of low and high stress areas of the form of a damped sinousodal. Asperity-asperity interactions play a key role in modeling Omori s law. Asperities must have a non-negligible amount of variance in strength. Gutenberg-Richter scaling is preserved for multi-asperities. The inclusion of many asperities to the Olami-Feder-Christensen model leads to crucial improvements in modeling real earthquake fault systems.

33 FURTHER WORK How does drawing the asperity failure strength from multiple distinct Gaussian distributions change the observed noise in the large event rate? Does the noise in the large event rate infer information obout the nature of the earthquake fault? Do very large events in a system without asperities leads to Omori s law-like behavior?

34 SPECIAL THANKS TO COLLABORATORS Bill Klein Harvey Gould Kang Liu Tyler Xuan Gu Rashi Verma Nick Lubbers

35 REFERENCES B. Gutenberg and C. F. Richter, Magnitude and energy of earthquakes, Annali di Geofisica 9 (1956). F. Omori, On the aftershocks of earthquakes, Journal of the College of Science, Imperial University of Tokyo 7 (1894). M. Bath, Lateral inhomogeneities in the upper mantle, Tectonophysics 6 (1965). R. Shcherbakov, D. L. Turcotte, and J. B. Rundle, A generalized omori s law for earthquake aftershock decay, Geophysical Research Letters 31, L11613, n/a n/a (2004). R. Burridge and L. Knopoff, Model and theoretical seismicity, Bull. Seismol. Soc. Am. 57 (1967). Z. Olami and K. Christensen, Self-organized criticality in a continuous, nonconservative cellular automaton modeling earthquakes, Phys. Rev. Lett. 62 (1992).

36 A. Helmstetter and D. Sornette, Bath s law derived from the gutenberg-richter law and from aftershock properties, Geophys Res. Lett, (2003). J. B. Rundle, W. Klein, S. Gross, and D. L. Turcotte, Rundle et al reply, Physical Review Letters 78 (1997). J. B. Rundle and D. Jackson, Numerical simulations of earthquake sequences, Bull. Seismol. Soc. Am. 67(5) (1977). D. Sornette and H. J. Xu, Non-boltzmann fluctuations in numerical simulations of nonequilibrium lattice threshold systems, Physical Review Letters 78 (1997). T. Utsu, Y. Ogata, and R. S. Matsuura, The centenary of the omori formula for a decay law of aftershock activity, Journal of Physics of the Earth 43, 1 33 (1995).

37 BUILDING A MODEL FOR AN EARTHQUAKE FAULT SYSTEM Block spring model: Linear springs connecting media on a rough surface that is loaded with a moving plate. [5] Loader plate drags the blocks until a block slips (fails) initiating an avalanche event. K c : Spring constant between a block and another block. K l : Spring constant between the blocks and the loader plate.

38 MODEL DYNAMICS Olami-Feder-Christensen Rundle - Jackson - Brown Lattice of sites with stress σ System is loaded until a single site fails (zero velocity limit). Stress in failing site returns to residual level. Excess stress is dissipated to sites using long range stress transfer. σ diss = (1 α)[σ i σ f ] System of blocks with displacements φ (slip deficit) Stress is computed using: σ i = K L φ i + K c (φj φ i ) Loader plate in system is loaded until a single site fails (zero velocity limit). Failing block displacement is reassigned: φ f = φ i + σ i σ R K L +qk c

39 MODEL DYNAMICS Dynamics evolve identically for properly chosen set of parameters K L α = K L + qk c α - OFC stress dissipation KL - Loader plate - block spring constant Kc - Block - block spring constant Total energy of RJB model is given by the following: E RJB = 1 2 [K lφ 2 i + j Range(R) K c (φ i φ j ) 2 ]

40 STRESS MEMORY DISSIPATION TIME Block spring model: Linear springs connecting media on a rough surface that is loaded with a moving plate. [5] Loader plate drags the blocks until a block slips (fails) initiating an avalanche event. K c : Spring constant between a block and another block. K l : Spring constant between the blocks and the loader plate.

41 LARGE EVENT RATE : 125 ASPERITIES 500σ f WITH STANDARD DEVIATION 0.005σ f Data Fit Omori s Law + Sine : p=36.7 Fit Omori s Law : p=0.326 Large Event Rate Real Time Since Failure

42 LARGE EVENT RATE : 50 ASPERITIES 500σ f WITH STANDARD DEVIATION 0.5σ f 10 6 Large Event Rate 10 5 Data Fit Omori s Law + Sine : p=1.93 Fit Omori s Law : p= Real Time Since Failure

43 LARGE EVENT RATE : 125 ASPERITIES 500σ f WITH STANDARD DEVIATION 0.01σ f Data Fit Omori s Law + Sine : p=2.46 Fit Omori s Law : p=3.86 Large Event Rate Real Time Since Failure

44 LARGE EVENT RATE : 125 ASPERITIES 5000σ f WITH STANDARD DEVIATION 250σ f Data Fit Omori s Law + Sine : p=28.9 Fit Omori s Law : p=26.0 Large Event Rate Plate Updates Since Failure