Subcritical and supercritical regimes in epidemic models of earthquake aftershocks

Helmstetter, Agnès; Sornette, Didier

doi:10.1029/2001jb001580

Cited by 303 publications

(409 citation statements)

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“…Note that the condition that θ be small is fully compatible with many empirical studies in the literature for the Omori law reporting an observable (renormalized) Omori law decay ∼ 1/t 0.9−1 corresponding to θ = 0 − 0.1 [17]. Fig.…”

supporting

confidence: 82%

“…1 extracted from [11] on which we have superimposed our theoretical prediction (11). Note that expression (11) exhibits a slight departure from the data for small x's (defined in (9)), which can be attributed to the linearization of (7), which amounts to neglecting the renormalization of the Omori law by the cascade of triggered aftershocks [17]. Taking into account this renormalization effect by the higher-order terms in the expansion (7) improves the fit to the data shown in Fig.…”

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confidence: 83%

“…Under very weak and general conditions, Molchan proved mathematically that the only possible form for f (x), if universality holds, is the exponential function [14], in strong disagreement with observations. Recently, from a re-analysis of the seismicity of Southern California, Molchan and Kronrod [15] have shown that the unified scaling law (1) is incompatible with multifractality which seems to offer a better description of the data.Here, our goal is to provide a simple theory, which clarifies the status of (1), based on a largely studied benchmark model of seismicity, called the EpidemicType Aftershock Sequence (ETAS) model of triggered seismicity [16] and whose main statistical properties are reviewed in [17]. The ETAS model treats all earthquakes on the same footing and there is no distinction between foreshocks, mainshocks and aftershocks: each earthquake is assumed capable of triggering other earthquakes according to the three basic laws (i-iii) mentioned above.…”

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confidence: 99%

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“Universal” Distribution of Interearthquake Times Explained

2006

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We propose a simple theory for the "universal" scaling law previously reported for the distributions of waiting times between earthquakes. It is based on a largely used benchmark model of seismicity, which just assumes no difference in the physics of foreshocks, mainshocks and aftershocks. Our theoretical calculations provide good fits to the data and show that universality is only approximate. We conclude that the distributions of inter-event times do not reveal more information than what is already known from the Gutenberg-Richter and the Omori power laws. Our results reinforces the view that triggering of earthquakes by other earthquakes is a key physical mechanism to understand seismicity. PACS numbers: 91.30.Px ; 89.75.Da; Understanding the space-time-magnitude organization of earthquakes remains one of the major unsolved problem in the physics of the Earth. Earthquakes are characterized by a wealth of power laws, among them, (i) the Gutenberg-Richter distribution ∼ 1/E 1+β (with β ≈ 2/3) of earthquake energies E [1]; (ii) the Omori law ∼ 1/t p (with p ≈ 1 for large earthquakes) of the rate of aftershocks as a function of time t since a mainshock [2]; (iii) the productivity law ∼ E a (with a 2/3) giving the number of earthquakes triggered by an event of energy E [3]; (iv) the power law distribution ∼ 1/L 2 of fault lengths L [4]; (v) the fractal (and even probably multifractal [5]) structure of fault networks [6] and of the set of earthquake epicenters [7]. The quest to squeeze novel information from the observed properties of seismicity with ever new ways of looking at the data goes unabated in the hope of better understanding the physics of the complex solid Earth system. In this vein, from an analysis of the probability density functions (PDF) of waiting times between earthquakes in a hierarchy of spatial domain sizes and magnitudes in Southern California, Bak et al. discussed in 2002 a unified scaling law combining the Gutenberg-Richter law, the Omori law and the fractal distribution law in a single framework [8] (see also ref.[9] for a similar earlier study). This global approach was later refined and extended by the analysis of many different regions of the world by Corral, who proposed the existence of a universal scaling law for the PDF H(τ ) of recurrence times (or inter-event times) τ between earthquakes in a given region S [10, 11]:The remarkable finding is that the function f (x), which exhibit different power law regimes with cross-overs, is found almost the same for many different seismic regions, suggesting universality. The specificity of a given region seems to be completely captured solely by the average rate λ of observable events in that region, which fixes the only relevant characteristic time 1/λ.The common interpretation is that the scaling law (1) reveals a complex spatio-temporal organization of seismicity, which can be viewed as an intermittent flow of energy released within a self-organized (critical?) system [12], for which concepts and tools from the theory of critical phenomena...

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“Universal” Distribution of Interearthquake Times Explained

2006

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“…To account for this secondary triggering, the Epidemic Type Aftershock Sequence (ETAS) model has been developed. It is a stochastic point process model that builds on the Omori-Utsu law and also takes stationary background seismicity and secondary aftershocks into Duration of aftershock sequences account (Ogata 1988;Helmstetter & Sornette 2002). In the ETAS model, each earthquake has a magnitude-dependent ability to trigger aftershocks with an intensity proportional to K 10 α (M−Mmin) , where α and K are constants and M min is the lower magnitude cut-off of the earthquakes under consideration.…”

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Statistical estimation of the duration of aftershock sequences

Hainzl¹,

Christophersen

Rhoades

et al. 2016

Geophys. J. Int.

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S U M M A R YIt is well known that large earthquakes generally trigger aftershock sequences. However, the duration of those sequences is unclear due to the gradual power-law decay with time. The triggering time is assumed to be infinite in the epidemic type aftershock sequence (ETAS) model, a widely used statistical model to describe clustering phenomena in observed earthquake catalogues. This assumption leads to the constraint that the power-law exponent p of the Omori-Utsu decay has to be larger than one to avoid supercritical conditions with accelerating seismic activity on long timescales. In contrast, seismicity models based on rate-and statedependent friction observed in laboratory experiments predict p ≤ 1 and a finite triggering time scaling inversely to the tectonic stressing rate. To investigate this conflict, we analyse an ETAS model with finite triggering times, which allow smaller values of p. We use synthetic earthquake sequences to show that the assumption of infinite triggering times can lead to a significant bias in the maximum likelihood estimates of the ETAS parameters. Furthermore, it is shown that the triggering time can be reasonably estimated using real earthquake catalogue data, although the uncertainties are large. The analysis of real earthquake catalogues indicates mainly finite triggering times in the order of 100 days to 10 years with a weak negative correlation to the background rate, in agreement with expectations of the rate-and state-friction model. The triggering time is not the same as the apparent duration, which is the time period in which aftershocks dominate the seismicity. The apparent duration is shown to be strongly dependent on the mainshock magnitude and the level of background activity. It can be much shorter than the triggering time. Finally, we perform forward simulations to estimate the effective forecasting period, which is the time period following a mainshock, in which ETAS simulations can improve rate estimates after the occurrence of a mainshock. We find that this effective forecasting period is only in the order of 100 days for moderate mainshocks and in the order of a few years for large events, even if the underlying triggering process lasts much longer.

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“…This integral converges to a finite value n < ∞ for θ > 0 (local Omori's law decay faster than 1/t) and for a < µ (not too large a growth of the number of daughters as a function of the energy of the mother). The resulting average rate N (t) of seismicity is the solution of the Master equation [26] …”

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Occurrence of Finite-Time Singularities in Epidemic Models of Rupture, Earthquakes, and Starquakes

Sornette

Helmstetter

2002

Phys. Rev. Lett.

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We present a new kind of critical stochastic finite-time-singularity, relying on the interplay between long-memory and extreme fluctuations. We illustrate it on the well-established epidemic-type aftershock (ETAS) model for aftershocks, based solely on the most solidly documented stylized facts of seismicity (clustering in space and in time and power law Gutenberg-Richter distribution of earthquake energies). This theory accounts for the main observations (power law acceleration and discrete scale invariant structure) of critical rupture of heterogeneous materials, of the largest sequence of starquakes ever attributed to a neutron star as well as of earthquake sequences.A large portion of the current work on rupture and earthquake prediction is based on the search for precursors to large events in the seismicity itself. Observations of the acceleration of seismic moment leading up to large events and "stress shadows" following them have been interpreted as evidence that seismic cycles represent the approach to and retreat from a critical state of a fault network [1]. This "critical state" concept is fundamentally different from the long-time view of the crust as evolving spontaneously in a statistically stationary critical state, called self-organized criticality (SOC) [2]. In the SOC view, all events belong to the same global population and participate in shaping the self-organized critical state. Large earthquakes are inherently unpredictable because a big earthquake is simply a small earthquake that did not stop. By contrast, in the critical point view, a great earthquake plays a special role and signals the end of a cycle on its fault network. The dynamical organization is not statistically stationary but evolves as the great earthquake becomes more probable. Predictability might then become possible by monitoring the approach of the fault network towards the critical state. This hypothesis first proposed in [1] is the theoretical induction of a series of observations of accelerated seismicity [3,4] which has been later strengthened by several other observations [5,6,7,8]. Theoretical support has also come from simple computer models of critical rupture [9] and experiments of material rupture [10], cellular automata, with [11] and without [12] long-range interaction, and from granular simulators [13]. Models of regional seismicity with more faithful fault geometry have been developed that also show accelerating seismicity before large model events [14,15,16].There are at least five different mechanisms that are known to lead to critical accelerated seismicity of the formending at the critical time t c , where N (t) is the seismicity rate (or acoustic emission rate for material rupture). Such finite-time-singularities are quite common and have been found in many well-established models of natural systems, either at special points in space such as in the Euler equations of inviscid fluids, in vortex collapse of systems of point vortices, in the equations of General Relativity coupled to a mass field leading to...

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Subcritical and supercritical regimes in epidemic models of earthquake aftershocks

Cited by 303 publications

References 82 publications

“Universal” Distribution of Interearthquake Times Explained

“Universal” Distribution of Interearthquake Times Explained

Statistical estimation of the duration of aftershock sequences

Occurrence of Finite-Time Singularities in Epidemic Models of Rupture, Earthquakes, and Starquakes

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