Statistical similarity between high energy charged particle fluxes in near-earth space and earthquakes

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Introduction
Recently, rapid short-term variations of high energy charged particle fluxes, which were referred to as "particle bursts" (PBs), in near-Earth space were found more frequently occur several hours before the main shock of strong earthquakes (Galper et al., 1995;Aleksandrin et al., 2003;Sgrigna et al., 2005;Fidani et al., 2010;Sidiropoulos et al., 2011;Anagnostopoulos et al., 2012).This research suggests a new possible precursor of strong earthquakes and shows potential advantage over other precursors.Further study (Wang et al., 2012) indicated positive correlations between earthquake magnitude and occurrence frequency of PBs while approaching strong earthquakes.
Although great efforts have been dedicated to the governing mechanism of the complex spatiotemporal behavior of earthquakes, it is far away to get a reliable earthquake precursor.However, concepts and methods from statistical physics have provided deep insight into the understanding of the universal behavior of earthquakes.The number of earthquakes N with a magnitude M(≥ m) is given by the Gutenberg-Richter law (Ishimoto and Iida, 1939;Gutenberg and Richter, 1944) 1993; Kagan, 1999).The number of earthquakes per unit time decreases as a power law since the sudden rise of activity provoked by a main shock.Short-range temporal correlations between earthquakes are expressed by Omori-Utsu law (Omori, 1894;Utsu, 1971).
In this paper, we show that PBs within short time window around seismic events and earthquakes share almost the same universal statistical properties.

DEMETER data
To study this analogy, we analyze the PBs frequency fluctuations.The PBs frequency fluctuation is defined as Z ∆n = k n+∆n − k n , where k n is the total number of occurrence counted in time window centered around the nth seismic event.Here seismic events are arranged in order of time and ∆n is a time scale of discrete interval between two seismic events.
DEMETER is the first of its kind in the micro satellite series, which aims at research on space electromagnetic and high energy particle precursors associated with strong seismic events (Sauvaud et al., 2006).We select DEMETER data with electron energy E = 0.97-2.deviations from average value of the background fluxes.Furthermore, higher counting rates region of satellite orbits (lat : −90 • ∼ 0 • , lon : −100 • ∼ 45 • ) are excluded from consideration.
As in survey mode all orbit-related parameters are provided every 28 s, so counting rates are averaged every 28 s and each PB at least experiences a period of 28 s.For one PB lasting over 28 s and still staying in the region 1.3 ≤ L ≤ 1.4, it will be regarded as two or more PBs depending on its time duration (see Fig. 1a).Successive records of PBs and their frequency fluctuations within a time window of ±0.5 day are shown in Fig. 1b and c.It should be noted that there is possibility that one PB is regarded as two events if one PB occurs just within the overlapping of two time windows at the same time which belong to two successive seismic events.However, this case is rarely occurred and would not introduce evident systematic errors.

Statistical similarity
The probability density function (PDF) of PBs frequency fluctuations within time window ±0.5 day, from P (z 1 ) to P (z 100 ), collapses on nearly the same curve (Fig. 2a) and exhibits nearly symmetric behavior with a fat tail.Here, z ∆n is rescaled (z ∆n = Z ∆n /σ) by the standard deviation.For comparison, we also present the PDF of PBs frequency fluctuations within time window ±4 days (Fig. 2b).It is clearly shown that the PDF is no longer symmetric around the center and deviates evidently from the former.
PBs frequency fluctuations for more time interval scales are also investigated.We find that the P (z ∆n ) within time window ±0.5 day exhibits a common functional form for various time interval scales in the range 1 ≤ ∆n ≤ 100.When superimposed, all the data fall on nearly the same curve (Fig. 2c).The PDFs with more time interval scales within time window ±4 days are plotted in Fig. 2d.The energy dissipated in an earthquake relates with the earthquake magnitude Z ∆n = S n+∆n − S n and their PDFs are plotted with ∆n = 1, 10, and 100 (solid symbols) in Fig. 3b, respectively.The results show fat tails and non-Gaussian behavior.It is consistent with previously reported results both for the worldwide seismic catalog and the Northern California catalog (Caruso et al., 2007).For comparison, we fit P (S 1 ), P (S 10 ) and P (S 100 ) to a q-Gaussian distribution, referred as Tsallis statistics (Tsallis, 1988;Tsallis et al., 1998) where both β and q are parameters.For q = 1, Eq. ( 1) reduces to Gaussian distribution.As shown in Fig. 3b, a good agreement with q-Gaussian distribution may find for q = 1.98 (red line).In spite of local fluctuation, both PBs frequency fluctuations (open symbols) and corresponding earthquakes energy fluctuations (solid symbols) fall on nearly the same curve.

Conclusions
We found that the PDFs for PBs frequency fluctuations and corresponding earthquakes energy fluctuations share similar invariant structure of non-Gaussian form with fat tails.
It is valid for very different time interval scales 1 ≤ ∆n ≤ 100.As for earthquakes energy fluctuations, their PDFs are consistent with previously reported results which are compatible with the fact that the occurrence of earthquakes is spatiotemporal correlated (Corral, 2004).Compared with Gaussian processes, the fat tails character of the distribution indicates that there are rare but relatively large magnitude fluctuations.Clusters of small fluctuations are connected by these rare but large magnitude fluctuations.Specially, this invariant structure for PBs frequency fluctuation is valid only within a relatively shorter time window ±0.5 day.This time window dependent behavior around earthquakes shows consistence with previously reported results.These results show that sharp variations of particle counting rates occur more frequently when approaching the earthquakes (Anagnostopoulos et al., 2012;Wang et al., 2012).Introduction

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Full In this paper we focused on universal statistical behavior both for PBs frequency fluctuations and earthquakes energy fluctuations.Furthermore, we emphasized the similarity of the PDFs between PBs frequency fluctuations and earthquakes energy fluctuations.Both their PDFs share the similar time scale invariance and exhibit symmetry around center with fat tails.The scaled PDFs for different time intervals ∆n collapsed on nearly the same curve.However, this invariant character is time windows dependent and valid within ±0.5 day in contrast to ±4 days.Introduction

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Full . Gutenberg-Richter law states that the N (for M ≥ m) satisfies the formula log N ∝ −bm, where b 1 (Frohlich and Davis, Discussion Paper | Discussion Paper | Discussion Paper | 3 MeV under survey mode ranged in time from 2005 to 2010.The seismic data are supplied by the IPGP (Institut de Physique du Globe de Paris).Both seismic events and trapped particles are selected with McIIwain L parameter 1.3 ≤ L ≤ 1.4.The PBs are identified by current counting rates which exceed 4 Discussion Paper | Discussion Paper | Discussion Paper | Figure 3a shows that the frequency distributions of earthquakes are power-law form y ∝ S −3.0 (S ≡ exp(M)).The energy fluctuations for earthquakes are defined as Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | nition of PBs and their occurrence frequency fluctuations.(a) Typical trapped particles fluxes and seismic events from selected e range.The solid blue line is the average counts of background particles.The dotted blue line is the sum of the average counts e green squares are PBs within a time window ±1 day.Seismic events are represented by red triangles.(b) Number k n (integer) rrence frequency according to successive seismic events n.(c) Successive frequency fluctuation, Z 1 = k n+1 − k n obtained from own in (b).

Fig. 1 .
Fig. 1.Definition of PBs and their occurrence frequency fluctuations.(a) Typical trapped particles fluxes and seismic events from selected area and time range.The solid blue line is the average counts of background particles.The dotted blue line is the sum of the average counts and 4σ.The green squares are PBs within a time window ±1 day.Seismic events are represented by red triangles.(b) Number k n (integer) of PBs occurrence frequency according to successive seismic events n.(c) Successive frequency fluctuation, Z 1 = k n+1 −k n obtained from the series shown in (b).

Fig. 2 .
Fig. 2. Invariant structure of PBs frequency fluctuations.(a) The PDFs of PBs frequency fluctuations, P (z1), P (z10) and P (z100), for all records within a time window ±0.5 day.z∆n = Z∆n/σ, where σ is the standard deviation.(b) The PDFs of PBs frequency fluctuations, P (z1), P (z10) and P (z100), for all records within a time window ±4 days.(c) Superposition of P (z∆n) at different time intervals within a time window ±0.5 day.(d) Superposition of P (z∆n) at different time intervals within a time window ±4 days.

Fig. 2 .
Fig. 2. Invariant structure of PBs frequency fluctuations.(a) The PDFs of PBs frequency fluctuations, P (z 1 ), P (z 10 ) and P (z 100 ), for all records within a time window ±0.5 day.z ∆n = Z ∆n /σ, where σ is the standard deviation.(b) The PDFs of PBs frequency fluctuations, P (z 1 ), P (z 10 ) and P (z 100 ), for all records within a time window ±4 days.(c) Superposition of P (z ∆n ) at different time intervals within a time window ±0.5 day.(d) Superposition of P (z ∆n ) at different time intervals within a time window ±4 days.