### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. The Non-Critical Precursory Accelerating Seismicity Theory (N-C PAST)
- 3. Real Data Analysis
- 4. Precursory Paths
- 5. Conclusions
- Acknowledgments
- References
- Supporting Information

[1] Various seismicity patterns before large earthquakes have been reported in the literature. They include foreshocks (medium-term acceleration and short-term activation), quiescence, doughnut patterns and event migration. The existence of these precursory patterns is however debated. Here, we develop an approach based on the concept of stress accumulation to unify and categorize all claimed seismic precursors in a same physical framework. We first extend the Non-Critical Precursory Accelerating Seismicity Theory (N-C PAST), which already explains most precursors, to additionally include short-term activation. Theoretical results are then compared to the time series observed prior to the 2009*M*_{w} = 6.3 L'Aquila, Italy, earthquake. We finally show that different precursory paths are possible before large earthquakes, with possible coupling of different patterns or non-occurrence of any. This is described by a logic tree defined from the combined probabilities of occurrence of the mainshock at a given stress state and of precursory silent slip on the fault. In the case of the L'Aquila earthquake, the observed precursory path is coupling of quiescence and accelerating seismic release, followed by activation. These results provide guidelines for future research on earthquake predictability.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. The Non-Critical Precursory Accelerating Seismicity Theory (N-C PAST)
- 3. Real Data Analysis
- 4. Precursory Paths
- 5. Conclusions
- Acknowledgments
- References
- Supporting Information

[2] Although numerous seismicity precursors have been proposed in the past decades (see, e.g., reviews by *Kanamori* [1981], *Wyss and Habermann* [1988], and *Mignan* [2011]), our ability to predict large earthquakes remains highly doubted [*Jordan et al.*, 2011]. A number of studies have shown that claimed precursory anomalies are not observed systematically, which questions their existence as valid precursors [e.g., *Zechar and Zhuang*, 2010]. Others have shown that observed patterns are undistinguishable from the normal behavior of seismicity [e.g., *Hardebeck et al.*, 2008; *Marzocchi and Zhuang*, 2011], which is well described by the Epidemic-Type Aftershock Sequence (ETAS) model where multiple aftershock sequences are superposed on a constant background rate*μ* [*Ogata*, 1988]. Although such models based on aftershock statistics [e.g., *Gerstenberger et al.*, 2005] are promoted for Operational Earthquake Forecasting (OEF) [*Jordan and Jones*, 2010], they provide probabilities too low for operational forecasting of large mainshocks [*van Stiphout et al.*, 2010]. Further investigation of potential precursors is therefore crucial.

[3] The aim of this paper is to extend the Non-Critical Precursory Accelerating Seismicity Theory (N-C PAST) [*Mignan et al.*, 2007; *Mignan*, 2008, 2011] to explain, in a unified framework, all basic seismicity precursors proposed in the literature and to provide new guidelines for earthquake predictability research. Considered precursory patterns are: quiescence [e.g., *Wyss and Habermann*, 1988], Accelerating Seismic Release (ASR) [e.g., *Bufe and Varnes*; 1993; *Mignan*, 2011], short-term activation [e.g.,*Jones and Molnar*, 1979], Mogi doughnut and event inward migration [e.g., *Kanamori*, 1981], outward migration [e.g., *Jones et al.*, 1982], decelerating seismicity [e.g., *Papadimitriou*, 2008], and *b-*value decrease (i.e., increase in event size [e.g., *Rundle et al.*, 2000]). We will show in this study that the N-C PAST makes specific predictions for the space-time-rate characteristics and couplings of these various patterns.

### 2. The Non-Critical Precursory Accelerating Seismicity Theory (N-C PAST)

- Top of page
- Abstract
- 1. Introduction
- 2. The Non-Critical Precursory Accelerating Seismicity Theory (N-C PAST)
- 3. Real Data Analysis
- 4. Precursory Paths
- 5. Conclusions
- Acknowledgments
- References
- Supporting Information

[4] The N-C PAST [*Mignan et al.*, 2007; *Mignan*, 2008, 2011] is a theoretical framework based on the concepts of elastic rebound [*Reid*, 1910] and of static stress transfer [e.g., *King*, 2007] which predicts that various precursory patterns may be observed days to months before a large earthquake due to loading on the main fault [*King and Bowman*, 2003]. A large earthquake is defined as any event due more likely to tectonic loading than to nearby triggering [*Mignan et al.*, 2006; *King*, 2007], which corresponds to faults with a length/width ratio greater than one, i.e., about *M*_{w} ≥ 6 [*Mignan*, 2008]. In the N-C PAST, the background seismicity rate is non-stationary, the variations of which are function of the stress state surrounding the fault [e.g.,*Ogata*, 2005]. This hypothesis is supported by independent geodetic measurements [*Ogata*, 2007]. Aftershocks can be superposed on the background rate *μ*(*t*) following the ETAS model. This process is however considered as noise that masks precursory anomalies, in agreement with *Ogata* [1988, 2005, 2007].

[5] The regional stress state (Figure 1 and see 360° rotation animation in the auxiliary material) depends on the following static stress field

function of time *t* and distance *r* along the stress field gradient with boundary conditions *σ*(*r* ∞, *t*) = *σ**_{0} and *σ*(*r* = 0, *t*) = *σ*_{0} + *t* [*Mignan et al.*, 2007]. *σ*_{0} is the stress drop associated to a hypothetical silent slip occurring at time *t*_{0} on the fault [*Wyss and Habermann*, 1988; *Ogata*, 2005, 2007], *h* the depth of the fault segment base, the secular stress loading rate on the fault and *σ**_{0} the crustal background stress. Only horizontal variations in *σ*(*r*,*t*) are considered with changes with depth *d* assumed negligible (*d* ≪ *r*). In the original N-C PAST, two categories of background events were defined with density*δ*_{bm} and *δ*_{b0} for quiescence regime and normal background regime, respectively [*Mignan et al.*, 2007; *Mignan*, 2008]. Here we introduce a third category, with density *δ*_{bp}for activation regime, to include short-term foreshocks in the proposed framework, with*δ*_{bm} ≪ *δ*_{b0} ≪ *δ*_{bp}. Each event category is associated with a given stress range [e.g., *Ogata*, 2005; *King*, 2007]. Two stress thresholds are defined: *σ**_{0} − Δ*σ** the limit between quiescence and normal activity, and *σ**_{0} + Δ*σ** the limit between normal activity and activation. The boundary layer *σ**_{0} ± Δ*σ** corresponds to the net stress change in the crust with *σ**_{0} = 0 [e.g., *King*, 2007]. Δ*σ** is expected to be small since changes of the order of millibars can trigger relative quiescence and activation [e.g., *Ogata*, 2005].

[6] The spatiotemporal extent of the quiescence envelope (i.e., stress shadow, blue surface in Figure 1) is described by

by applying the conditions *σ*(*r*_{Q}, *t*) = *σ*(0, *t**_{m}) = *σ**_{0} − Δ*σ** to equation (1) [*Mignan et al.*, 2007]. It is the decrease of *r*_{Q} with time *t* due to stress accumulation on the fault, which explains the phenomenon of ASR, as well as inward migration of events and the Mogi Doughnut. The abrupt appearance of quiescence at *t*_{0} is in agreement with observations [*Wyss and Habermann*, 1988]. The spatiotemporal coupling of quiescence and ASR predicted by the N-C PAST has been observed prior to several large earthquakes [*Mignan and Di Giovambattista*, 2008; *Mignan*, 2011]. It should be emphasized that observation of these different patterns is not systematic but that the N-C PAST can also explain the possible non-occurrence of specific patterns (seeSection 4).

[7] The activation envelope (red surface in Figure 1) is similarly described by

where *σ*(*r*_{A}, *t*) = *σ*(0, *t**_{p}) = *σ**_{0} + Δ*σ**. Δ*σ** ∼ 0 yields *t**_{m} ≈ *t**_{p} ≈ *t**, the time limit between quiescence/ASR and short-term activation. While the original N-C PAST predicts the occurrence of the mainshock when the stress on the fault reaches back*σ**_{0} [*Mignan et al.*, 2007], a delay of the rupture would lead to more stress accumulation, potentially up to *σ* > *σ**_{0} + Δ*σ** (see Section 4). If the fault is in a state of overloading, the delay is expected to be short. The N-C PAST also predicts that these short-term foreshocks concentrate in the epicentral area, as typically observed [e.g.,*Jones and Molnar*, 1979]. The observation of repetitive seismic bursts in short-term foreshock sequences [*Bouchon et al.*, 2011] is also compatible with N-C PAST activation due to overloading. Event inward migration and activation in the epicentral area can also explain the potential increase in event size (i.e.,*b*-value decrease) prior to a large earthquake if the*b-*value is lower near the main fault than in outer regions [*Page et al.*, 2011]. Noteworthy, the N-C PAST also predicts an outward migration of events during activation, in agreement with observations in relocated foreshock sequences [e.g.,*Jones et al.*, 1982].

[8] The non-stationary background rate (Figure 2) then takes the form

for a fixed volume of maximum extent *r*_{max} = max(*r*_{Q}), with *k* a geometrical parameter, *d* the spatial dimension and *r*_{Q} and *r*_{A} described respectively by equations (2) and (3) (*Mignan et al.* [2007] only described the case *t* ≤ *t**_{m}). Figure 2a represents a stochastic iteration of equation (4) obtained by thinning [*Lewis and Shedler*, 1979] as well as the theoretical curve. ASR follows a power-law as originally proposed by*Bufe and Varnes* [1993]. Activation however differs from the simple *t*^{−1} type behavior originally proposed by *Jones and Molnar* [1979]. The inverse Omori law is however empirical and the N-C PAST leads to a behavior visually close to the*t*^{−1} type behavior. Noteworthy, the phenomenon of decelerating seismicity [e.g., *Papadimitriou*, 2008] is consistent with the appearance of quiescence, as can be seen in Figure 2.

### 3. Real Data Analysis

- Top of page
- Abstract
- 1. Introduction
- 2. The Non-Critical Precursory Accelerating Seismicity Theory (N-C PAST)
- 3. Real Data Analysis
- 4. Precursory Paths
- 5. Conclusions
- Acknowledgments
- References
- Supporting Information

[9] We fitted equation (4) to the time series observed prior to the April 2009 *M*_{w} = 6.3 L'Aquila earthquake (Figure 2b), for which the existence of seismic precursors has been ardently debated [*Hall*, 2011]. Earthquake data was retrieved from the Bollettino Sismico (http://bollettinosismico.rm.ingv.it) for magnitudes *m* ≥ 1.8 (i.e., conservative completeness magnitude *M*_{c} estimate) in a cylinder of radius *r*_{fix} = 80 km centered on the mainshock epicenter (13.380°E, 42.342°N). We assume the role of aftershocks to be negligible, as 95% of events are expected not to produce any aftershock (using the parameters of *Marzocchi and Zhuang* [2011]). The largest event of the time series is expected to produce 18 aftershocks in normal conditions, but its occurrence during the activation stage makes it difficult to access the role of triggering in a state of overloading. To fit the data, we used *k* = *π* and *d* = 2 assuming a stress field of circular extent, *h* = 12.9 km [*Anzidei et al.*, 2009], = 2.5/300 = 0.0083 MPa/yr [*Maercklin et al.*, 2011; *Walters et al.*, 2009], the bin width Δ*t* = 0.02 yr (∼1 week) and Δ*σ** = 4 10^{−5} MPa such that *t**_{m} ≈ *t**_{p} (i.e., ). Parameters derived from the dataset are *t*_{0} = 2008.1, *t**_{m} ≈ *t**_{p} ≈ *t** = 2009.234, *t*_{f} = 2009.264, *δ*_{bm} = 0.0075, *δ*_{b0} = 0.0160 and *δ*_{bp} = 7.4403 with *δ* in events/yr/km^{2}. Based on the computation of the temporal joint log-likelihood, we found that the theoretical time series provides a good match to the observations (Figure 2b) with a corrected Akaike Information Criterion AICc(N-C PAST, 6 parameters) = 709 versus AICc(Poisson with rate*μ* = 6.29 events/Δ*t*, 1 parameter) = 856. Only for a minimum magnitude cutoff *M*_{co} ≥ 3.4 is the time series better fitted by a Poisson process. It shows that a better evaluation of microseismicity is crucial [e.g., *Mignan et al.*, 2011; *Mignan*, 2012]. Studies that claim that precursory patterns are unreliable commonly use *M*_{co} = 4.0 [*Hardebeck et al.*, 2008; *Marzocchi and Zhuang*, 2011].

### 4. Precursory Paths

- Top of page
- Abstract
- 1. Introduction
- 2. The Non-Critical Precursory Accelerating Seismicity Theory (N-C PAST)
- 3. Real Data Analysis
- 4. Precursory Paths
- 5. Conclusions
- Acknowledgments
- References
- Supporting Information

[10] The occurrence of the different precursory patterns in the N-C PAST depends on the probability of occurrence of silent slip on the fault 1-Pr(*σ*_{0} = *σ**_{0} = 0) with Δ*σ** ∼ 0, and on the probability of occurrence of the mainshock at a given stress state Pr(*σ*_{f}) = Beta(*α*, *β*) bounded on the stress interval [*σ*_{0}, *σ*_{max}] with *σ*_{f} the failure stress on the fault (Figure 3a). The Beta distribution, chosen for its flexibility, represents the combination of aleatoric and epistemic uncertainties on the process of rupture. It follows that the probability of occurrence of quiescence, ASR and activation can be roughly described by Pr(Q) = 1-Pr(*σ*_{0} = 0), Pr(ASR) = Pr(Q)Pr(*σ*_{f} ≥ 0) and Pr(A) = Pr(*σ*_{f} > 0), respectively. Based on this set of combinations, we defined a logic tree (referred to as N-C PAST precursory tree) that describes all possible precursory paths before a large earthquake due to stress accumulation on the fault in the range [*σ*_{0}, *σ*_{f}] (Figure 3b). It is an analogue to the established probabilistic volcanic hazard assessment based on event trees [*Newhall and Hoblitt*, 2002], both earthquake and volcanic eruption processes being based on the concept of stress accumulation. The logic tree includes five different precursory paths including one where no precursor is observed. The L'Aquila earthquake precursory time series corresponds to path #1 (coupling of quiescence and ASR, followed by activation).