Subcritical and supercritical regimes in epidemic models of earthquake aftershocks

By Agnes Helmstetter, Didier Sornette. 2001

303 citations recorded by the source corpus.

[1] We present an analytical solution and numerical tests of the epidemic-type aftershock (ETAS) model for aftershocks, which describes foreshocks, aftershocks, and main shocks on the same footing. In this model, each earthquake of magnitude m triggers aftershocks with a rate proportional to 10αm. The occurrence rate of direct aftershocks triggered by a single main shock decreases with the time from the main shock according to the “local” modified Omori law K/(t + c)p with p = 1 + θ. Contrary to the usual definition, the ETAS model does not impose an aftershock to have a magnitude smaller than the main shock. Starting with a main shock at time t = 0 that triggers aftershocks according to the local Omori law, which in turn trigger their own aftershocks and so on, we study the seismicity rate of the global aftershock sequence composed of all the secondary and subsequent aftershock sequences. The effective branching parameter n, defined as the mean aftershock number triggered per event, controls the transition between a subcritical regime n 1. A characteristic time t*, function of all the ETAS parameters, marks the transition from the early time behavior to the large time behavior. In the subcritical regime, we recover and document the crossover from an Omori exponent 1 − θ for t t* found previously in the work of Sornette and Sornette for a special case of the ETAS model. In the supercritical regime n > 1 and θ > 0, we find a novel transition from an Omori decay law with exponent 1 − θ for t t*. The case θ < 0 yields an infinite n-value. In this case, we find another characteristic time τ controlling the crossover from an Omori law with exponent 1 − |θ| for t < τ, similar to the local law, to an exponential increase at large times. These results can rationalize many of the stylized facts reported for aftershock and foreshock sequences, such as (1) the suggestion that a small p-value may be a precursor of a large earthquake, (2) the relative seismic quiescence sometimes observed before large aftershocks, (3) the positive correlation between b and p values, (4) the observation that great earthquakes are sometimes preceded by a decrease of b-value, and (5) the acceleration of the seismicity preceding great earthquakes.

Subfields: Geophysics

View Paper: https://www.semanticscholar.org/paper/eef3b8d3eecfd3ee8ebb2b9b2412b676ae5ef1aa

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In Geophysics

Citations in the corpus

Comparative performance of time‐invariant, long‐range and short‐range forecasting models on the earthquake catalogue of Greece (2006, methodology): “The ETAS model has been used in many studies to describe or predict the spatiotemporal distribution of seismicity and reproduces many properties of real seismicity (see Ogata [1998], Console and Murru [2001], Console et al. [2003], and Helmstetter and Sornette [2002, 2003] for JOURNAL OF GEOPHYSIC”
A Common Origin for Aftershocks, Foreshocks, and Multiplets (2002, methodology): “This single-process triggering model has been used to build statistical models of seismicity (Vere-Jones, 1966; Kagan and Knopoff, 1981; Ogata, 1988; Helmstetter and Sornette, 2002), to analyze earthquake clustering (Console and Murru, 2001), to estimate the prevalence of secondary aftershocks…”
Earthquake scaling relations for mid‐ocean ridge transform faults (2004, background): “The continental data in Figure 4 yield a 0.8, which agrees with previous studies [Utsu, 1969; Yamanaka and Shimazaki, 1990; Guo and Ogata, 1997; Helmstetter and Sornette, 2003a], and Dmafter 0.9, consistent with the data for southern California [Felzer et al., 2002; Helmstetter, 2003].”
HAWKES PROCESSES ON LARGE NETWORKS (2016, unknown): “These include: seismology for modelling earthquake replicas (Helmstetter-Sornette [21], Kagan [27], Ogata [36], Bacry-Muzy [5]), neuroscience for modelling spike trains in brain activity (Gr¨un et al. [16], Okatan et al. [37], Pillow et al. [38], Reynaud et al. [40, 41]), genome analysis…”
Detecting fluid signals in seismicity data through statistical earthquake modeling (2005, background): “some magnitude-dependent ability to trigger its own Omori law type aftershocks [Ogata, 1988, 1993; Helmstetter and Sornette, 2002].”
Prediction (2010, unknown): “Forn < 1 (sub-critical regime), the rate of events triggered by a giv en shock decays according to an effective Omori power law, characterized by a crossove r from an Omori exponent p = 1−θ for t < t∗ to a larger exponent p = 1+θ for t > t∗ [128, 51], wheret∗ is a characteristic time t∗ ∼ c/(1 − n)1”
Seismicity rate immediately before and after main shock rupture from high-frequency waveforms in Japan (2007, methodology): “The epidemic-type aftershock sequence (ETAS) model [Kagan and Knopoff, 1981; Ogata, 1988; Helmstetter and Sornette, 2002] can also explain our observed foreshock and aftershock”
Mainshocks are aftershocks of conditional foreshocks: How do foreshock statistical properties emerge from aftershock laws (2002, unknown): “3”
Foreshocks explained by cascades of triggered seismicity (2002, methodology): “…clustered around the mainshock at times just before it may be taken as an apparent inverse diffusion of the seismicity rate when using standard quantifiers of diffusion processes (see [Helmstetter and Sornette, 2002b] for a discussion of a similar effect for the apparent diffusion of aftershocks).”
Estimating value-at-risk: a point process approach (2005, methodology): “Related papers which have independently explored similar ideas are Ogata (1988), Helmstetter and Sornette (2002) and Helmstetter and Sornette (2003a), who discuss several classes of stochastic models for the origin times and magnitudes of earthquakes.”
Earthquake clusters in southern California I: Identification and stability (2013, result): “591 These results are consistent with the existing knowledge about properties of aftershocks and 592 foreshocks [Utsu, 1961; Papazachos, 1973; Jones and Molnar, 1979; Kisslinger and Jones, 593 1991; Utsu et al., 1995; Helmstetter and Sornette, 2002; Helmstetter et al., 2003].”
Estimating Background Activity Based on Interevent-Time Distribution (2006, methodology): “Additionally, we analyze simulations of the epidemictype aftershock sequences (ETAS) model, which is a stochastic point process, where each earthquake has some magnitude-dependent ability to trigger its own Omori-lawtype aftershocks (Ogata, 1988; Helmstetter and Sornette, 2002).”
Space–time ETAS models and an improved extension (2006, background): “See Utsu and Ogata (1997) for computational codes and technical aspects, and Helmstetter and Sornette (2002), for example, for some discussions of the statistical features of the ETAS model.”
Short-Term Properties of Earthquake Catalogs and Models of Earthquake Source (2004, methodology): “…decay have been used for a long time (Kagan and Knopoff, 1980, 1987b; Ogata, 1988, 1998, 1999), there is now widespread interest in applying such models (Console and Murru, 2001; Felzer et al., 2002, Gerstenberger et al., 2002; Helmstetter and Sornette, 2002; Sornette and Helmstetter, 2002).”
Dragon-kings: Mechanisms, statistical methods and empirical evidence (2012, unknown): “…from amplification mechanisms at smaller hierarchical levels in the system, they might be preceded by a precursory activity allowing their prediction, or forecasted using time-dependent algorithms (such as the ETAS model (see Helmstetter and Sornette (2002) for a review and references therein).”
Båth's law derived from the Gutenberg‐Richter law and from aftershock properties (2003, background): “…aftershocks) has the same dependence with the mainshock magnitude hNaftðmM Þi ¼ K 1 n 10 aðmM m0Þ ð2Þ as the number of direct aftershocks r(mM) given above [Helmstetter and Sornette, 2002]. n is the branching ratio, defined as the average number of directly triggered earthquakes per earthquake,…”