Abstract Benford’s analysis is applied to the recurrence times of approximately17,000 seismic events in different geological contexts of Italy over the last 6 years,including the Mt. Etna volcanic area and the seismic series associated with the de-structive Mw 6.3, 2009 L’Aquila earthquake. A close conformity to Benford’s law anda power-law probability distribution for the recurrence times of consecutive events isfound, as typical of random multiplicative processes. The application of Benford’slaw to the recurrence event times in seismic series of specific seismogenic regionsrepresents a novel approach, which enlarges the occurrence and relevance of Benford-like asymmetries, with implications on the physics of natural systems approaching apower law behaviour. Moreover, we propose that the shift from a close conformityof Benford’s law to Brownian dynamics, observed for time separations among non-consecutive events in the study seismic series, may be ruled by a periodical noisefactor, such as the effects of Earth tides on seismicity tuning.
Keywords Seismic hazard · Tidal frequencies · Random multiplicative process ·Random additive process
G. Sottili (�) · B. Giaccio · P. MessinaIstituto di Geologia Ambientale e Geoingegneria-CNR, Area della Ricerca di Roma 1, Via Salariakm 29,300 - Montelibretti, 00016 Monterotondo Stazione, Rome, Italye-mail: [email protected]
Fig. 1 Locations of the nineseismogenic areas of Italyrelated to the investigatedseismic time series. Theoutcropping main seismogenicfaults (Galli et al. 2008) and theearthquakes with M greater than5.5 recorded during historicaltimes in Italy (after CPTI042004, modified) are alsoreported
1 Introduction
Time series analysis of seismic clusters represents a fundamental clue to investigatetrigger mechanisms, controlling factors and timing of major earthquakes, with impli-cations on hazard assessment. In this paper, we adopt a Benford’s law based statisticalanalysis (Benford 1938; Pietronero et al. 2001; Sambridge et al. 2010) to event recur-rence times during seismic swarms with the aim of detecting the possible influence ofperiodical noise factors (for example, Earth tides). Specifically, we analyse a data setof approximately 17,000 earthquakes related to nine seismogenic areas in differentgeological contexts of Italy (Fig. 1; Table 1). The distribution of the recurrence timesamong consecutive seismic events results in a power law distribution and a close con-formity to Benford’s law (Fig. 2). These constitute two typical characteristics of phe-nomena involving random multiplicative stochastic processes (Pietronero et al. 2001;Kitada 2006). The distribution of the recurrence times among non-consecutive eventsis also tested for the same data set, yielding either Gaussian-like or multi-peakedprobability distributions and a departure from Benford’s law distribution. The changefrom scale-invariant to uniform probability distributions may reveal the transitionfrom a random multiplicative process which conforms to Benford’s law, to Browniandynamics which are ruled by an independent (for example, additive) noise term pos-sibly related to some external parameters, such as the tuning of seismicity by Earthtides.
Fig. 2 Probability distributions(%) of the first digits of theinter-event time intervals(expressed in seconds) for thenine seismic time seriesinvestigated (total approximately17,000 events; see Table 1 fordetails). The distributions arestrongly asymmetric and fit well(see φ values in parentheses, asobtained from Eq. (5) in thetext) to Benford’s law prediction(black bars)
2 Theoretical Background and Analytical Methods
Benford’s law states that the frequency of first digits in a specific series of numbersshows an uneven distribution characterised by a marked asymmetry in favour of smalldigits, as follows
PD = log10(1 + 1/D) (1)
where PD is the probability of the first (non-zero) digit, D, to occur (D = 1, . . . ,9).This means that, for a PD distribution, numbers with 1 as a first digit occur approxi-mately 30 % of the total, numbers with 2 as first digit occur approximately 17 %, andso on. It has been demonstrated that if all numbers in a Benford’s data set (a groupof numbers that conform to Benford’s law) are multiplied by a non-zero constant,the resulting data set will continue to follow Benford’s law (Pinkham 1961). For ex-ample, the practical consequence of Pinkham’s theorem is that if a time series dataset expressed in seconds is restated in minutes or hours, the resulting data set willstill hold a Benford-like distribution. Based on a statistical analysis of Benford’s law,
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Pietronero et al. (2001) found that some physical quantities or objects will followBenford’s law if their time evolution is ruled by multiplicative fluctuations. Hence,given a variable N that changes with time t
N(t + 1) = ξN(t) (2)
the intensity of the fluctuations of N produced by a multiplicative noise term,the stochastic variable ξ , is dependent on and related to the values of N itself(for example stock price, which is found to follow a Benford-like distribution, hasfluctuations that depend on the price itself). Notably, many physical phenomenaruled by multiplicative fluctuations are characterised by scale-invariant behaviourand power law distributions due to cooperative effects with the onset of criticalpoints and other nonlinear processes (Kobayashi et al. 2006; Mandelbrot 1982;Pietronero et al. 2001). Therefore, the uneven distribution of the first digits for differ-ent data sets reveals a tendency of various phenomena to self-organise into scale-invariant processes, corresponding to a uniform distribution in logarithmic space(Pietronero et al. 2001). On the other hand, many stochastic dynamical phenomenathat follow Brownian dynamics are ruled by a noise term whose amplitude does notdepend on N
N(t + 1) = ξ + N(t). (3)
In this case, the additive noise term ξ shows a Gaussian probability distributionP(N, t) and a variance σ ∼ t0.5.
In this paper, Benford’s law is applied to the analysis of seismic time series in dis-crete seismogenic regions. If the seismic events have a regular and/or periodical pat-tern, then the occurrence of an event should be a function of the time interval relativeto the previous ones. If n is the number of the measured arrival times (t1, t2, . . . , tn) ofthe events defining a seismic cluster, the initial data set can be converted into anothertime series by measuring the intervals among the nearest neighbour events to obtaina set of interval data (n − 1)
The first digits of the obtained time intervals, expressed in seconds, are then comparedwith Benford’s distribution (Fig. 2). The conformity of the time series with Benford’slaw is evaluated through the following relationship (Sambridge et al. 2010)
φ =[
1 −(
9∑D=1
(nD − nPD)2
nPD
)0.5]× 100, (5)
where nD is the number of data with first digit D, PD is the proportion of first digitD predicted by Benford’s distribution, and n is the total number of data. In Fig. 2,the fitness to Benford’s law, φ, is reported for each time series. To test the aboveself-similarity hypothesis of inter-events time lags among successive pairs of seismicevents, the cumulative number of inter-seismic intervals vs. time intervals amongconsecutive events is fitted to a power law distribution. Then the possible influence ofan external noise factor on seismic occurrences is considered by analysing the timeintervals among non-consecutive events. More specifically, we convert the initial dataset, given by the measured arrival times, t1, t2, . . . , tn, of the events, into a number of
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time series, each originated by the time intervals among non-consecutive events sep-arated by a discrete number of events. The subscripts 1,2, . . . , n are shifted forwardof 1,2, (n − 1) steps by using a lag operator, B
Xj = tj+B − tj , (6)
for j = 1,2, . . . , (n−1) and B = 1,2, . . . , (n−1), where B is constant for each timeseries. In this analysis, we adopt B values from 1 to 100, thus obtaining 100 timeseries from each of the nine original data sets. Then we analyse the distribution ofthe recurrence times among non-consecutive events, separated by fixed B steps. Thetheoretical changes of the sampled arrival times among non-consecutive events, aswell as the related periodograms, are sketched in Fig. 3 as a function of B . In theadditive case, for B spanning the 1–100 interval, N values fluctuate around a near-constant central value and the Probability Density Function (PDF) takes a Gaussianform. In the multiplicative case, N values drift from the central value with increas-ing B . In this case, the plot of the cumulative numbers of recurrence intervals amongconsecutive events (B = 1) displays a power-law distribution (Fig. 3).
3 Data Set
The data set includes ca. ∼17,000 earthquakes that occurred in nine seismogenic ar-eas from April 2005–April 2011 in different geological contexts of Italy (Fig. 1, Ta-ble 1), obtained from the Italian Seismological Instrumental and parametric Database(ISIDe) of the Istituto Nazionale di Geofisica e Vulcanologia (INGV 2010). Allthese areas were characterised by the occurrence in the investigated time intervalof single or multiple seismic swarms (each with at least 30 shocks during 24 hours,with a mainshock ML ≥ 3.0, within broadly circular areas 10–40 km across). Forindividual areas, Table 1 summarises the main characteristics of the reported seis-mic records in terms of peak seismicity rate (maximum number of events per day),average number of events per day, average recurrence time among events, maxi-mum time interval between two successive events and maximum magnitude. Mostof the investigated seismic series occurred in close proximity to the epicentresof historical earthquakes with M greater than 5.5 and/or well-known outcroppingseismogenic faults focused along the axis of the Apennine Chain (CPTI04 2004;Galli et al. 2008; Fig. 1). This seismicity is mainly related to the North East (NE)–South West (SW) extensional processes that have taken place since late Pliocenetimes, after the compressive fold and thrust structuring of the Apennine Chain (CNR-PFG 1987). Global Positioning System (GPS) measurements indicate that the currentextensional rate is on the order of 2–5 mm/yr (D’Agostino et al. 2011). The long-term geological expression of these extensional processes is represented by systemsof SW-dipping normal faults, clustered along the axis of the chain, driving the forma-tion of intermontane, half-graben basins and representing the seismogenetic sourcesof the main historical earthquakes (Galadini and Galli 2000; Boncio et al. 2004;Roberts and Michetti 2004; Galli et al. 2008).
In particular, times series 4 is related to the causative fault responsible for thedestructive April 6th, 2009, Mw 6.3 L’Aquila earthquake (Falcucci et al. 2009;Galli et al. 2010), including 6044 events with ML greater than 1.8 recorded in the
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Fig. 3 Representative maps of Probability Density Functions (PDF), expressed as a function of the vari-able N vs. the sampling interval B , for a random additive (a; Eq. (3)) and a random multiplicative(d; Eq. (2)) stochastic process. The normal distribution of the recurrence intervals among consecutiveevents (B = 1) is also shown (b and c). In (d) the fitting curve and R2 values are also reported
August 2008 to June 2009 time span within a distance of 20 km from the epicentre(350 foreshocks, ML max 4.0, and 5694 aftershocks, ML max 5.8). The time series 9(Forlimpopoli) and 1 (Zafferana Etnea) make exception to this general seismotec-tonic framework, being related to the compressive tectonic regime of the NorthernApennine Arc thrust system (Gorshkov et al. 2002) and the active volcanic setting ofMt. Etna, respectively.
4 Results
For each seismogenic area, the inter-events time series (Fig. 1), obtained from therecurrence times among successive events, shows an asymmetric distribution of the
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first digits, with a decreasing occurrence of the first digits from 1 to 9, as predicted byBenford’s law. The fit of the probability distributions to Benford’s law, φ, evaluatedthrough Eq. (5), ranges from 72.59 % (L’Aquila foreshock time series) to 94.42 %(Foligno time series). The distribution of the first digits shows departures from Ben-ford’s law prediction from 0.02 % (Montereale time series) to 5.64 % (L’Aquilaforeshock time series). Of note, φ values increase during the L’Aquila aftershocktime series, presumably due to the availability of a more complete data set follow-ing the installation of new seismic stations of the mobile network (D’Alessandro etal. 2011) soon after the April 6th, 2009, mainshock. The inter-events times typicallyshow power law cumulative distributions with fitting values to power law distribu-tions ranging from 0.8–1.0 (Figs. 4, 5, and 6), as already noticed for other discreteseismogenic regions and/or when averaged over multiple regions (Molchan 2005;Saichev and Sornette 2006, 2007).
We now consider each time series generated by taking into account the time inter-vals among non-successive events. As the sampling steps, B , increase, the fitting val-ues to Benford’s law, φ, show significant fluctuations, thus indicating a departure ofthe system from a random multiplicative behaviour (Eq. (2)) toward a different prob-ability distribution possibly induced by an external dynamical parameter (Figs. 4,5, and 6). In this regard, the Central Limit theorem (CLT) states that a sufficientlylarge set of independent random events, characterised by finite mean and variance inrecurrence times, tends toward a Gaussian distribution regardless of their individualPDF. Hence, one may argue that non-consecutive seismic events will tend to follow aGaussian PDF merely because they are the sum of consecutive inter-event times, withno need to invoke an external noise factor to explain the change from log-uniform toGaussian PDFs. In order to identify a significant effect of a periodical noise factorbeyond the obvious consequence of the CLT (Fig. 3), we focus on a few key exam-ples of the dependence of the inter-event frequency peaks on the B values (Figs. 4, 5,and 6). The polymodal patterns of the PDFs of inter-event times, which invariably fallon hourly scale periods, rule out the CLT as the reason for the observed correlations.For each data set, the most significant departures of the time series from Benford’s-like distributions are evidenced graphically by the negative flections points (such asthe local minimum φ values; Figs. 4, 5, and 6). It appears that the departures frompower law distributions parallel uniform distributions of the inter-event times, as theyare ruled by a noise term, ξ , whose intensity is independent of the variable value N
(random additive process, Eq. (3)). We find significant peaks in the probability distri-butions of the inter-event times corresponding to clustered hourly variations (Figs. 4,5, and 6). For example, in the Cittareale case (6749 events), the distribution of therecurrence time intervals corresponding to one of the main negative fluctuations ofφ values for B = 38, shows a good match with the ter-diurnal (8.15 ± 0.25 h) andquasi-diurnal (24.5 ± 0.5 h) tidal periods and a moderate fit to a log-normal prob-ability distribution of consecutive events. The L’Aquila aftershock recurrence timesfor B = 77 show an additional correlation to the semi-diurnal tidal period and a log-normal PDF of consecutive events (Fig. 4).
In addition, the time series characterised by recurrence intervals in the order ofa few tens of hours are clustered around polymodal, multi-days variations (Figs. 4and 6). In particular, PDF’s in Fig. 6 enlighten the correlation of the frequency peaks
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Fig. 4 Departures from Benford’s law distribution (φ; Eq. (5)) resulting from increasing inter-seismicstep intervals, B (left column) associated with the probability distributions of the inter-events recurrencetimes shorter than 40 hours (frequency %; right column), reported for selected seismic time series. Themain tidal constituents with ter-, semi- and quasi-diurnal periodicities (vertical grey lines) are reportedfor comparison. The log–log plots of the cumulative number of inter-seismic intervals vs. time intervaldurations among consecutive events (B = 1) are also reported, along with the fitting curves and R2 valuesfor log-normal distributions (central column). The distributions of the inter-seismic intervals in the rightplots are calculated for time windows of 15 min. The corresponding interpolating functions of the meanvalues are calculated on running windows of 5 samples with one sample step
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Fig. 5 Departures from Benford’s law distribution (φ, left column; Eq. (5)), associated with the proba-bility distributions of the inter-events recurrence times shorter than 40 days (frequency %; right column),compared to the tidal constituents with fortnightly and lunar month periodicities (vertical grey lines), forseismic time series 4, 7, 9 (Table 1). The log–log plots of the cumulative number of inter-seismic intervalsvs. time intervals among consecutive events, the fitting curves and R2 values for power law distributionsare also reported (central column). The distributions of the inter-seismic intervals in the right plots arecalculated for time windows of 2 hours. The corresponding interpolating functions of the mean values arecalculated on running windows of 3 samples with one sample step
of inter-seismic intervals to the ter-diurnal and quasi-diurnal main tidal periods (andsubordinately to 16 and 32 h), for B values corresponding to negative fluctuations ofφ values (low fitness to Benford’s law). To note, concomitant vertical sub-trends ofPDF’s consistently indicate the superposition of a random additive process, possiblyrelated to an external controlling factor like Earth tides.
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Fig. 6 Maps of PDF’s for selected seismic time series (top: Cittareale; bottom: Montaquila; Table 1) andcorresponding plots of B vs. φ (the degree of fitness to Benford’s law), showing the correlation of thefrequency peaks of inter-seismic intervals to the ter-diurnal (ca. 8 h) and quasi-diurnal (ca. 24 h) main tidalperiods (and subordinately to 16 and 32 h)
Figure 7 reports a conceptual overview of the model, assuming that a periodicalnoise factor (for example a tidal stress) with period T may modulate the initiationof seismic events. The effects of a sinusoidal process can be detected by consideringthe initiation times of the sole events synchronised with the stress maxima (blacksquares; Fig. 7a), while they can be overshadowed by any other events with increasingresponse times (tR = 0.25T , 0.5T , 0.75T ) and by the presence of a random noise(grey stars; Fig. 7a). Time series generated by consecutive events (inter-event step
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Fig. 7 Influence of a sinusoidal noise factor with period T on the initiation of seismic events. See text forexplanation
B = 1) display a probability distribution of recurrence times approaching a powerlaw distribution (Fig. 7b) and a Benford’s like distribution of the first digits (Fig. 1).For consecutive events, the effects of the sinusoidal process in the time domain arecompletely masked by the response times of the events and by the random noise.Non-consecutive events (B > 1) result in the sampling of an increasing number ofinter-events steps of duration T , so that the effects of the sinusoidal process in thetime domain are revealed by a Gaussian probability distribution of the inter-eventstime intervals (Fig. 7c) and a departure from Benford’s law distribution.
Interestingly, the August 2008–June 2009 L’Aquila seismic series highlights achanging sensitivity of the inter-event recurrence times to the ter- and quasi-diurnaltidal periodicities with time. While approaching the April 6th, 2009, Mw 6.3 main-shock, we note the appearance of the ter- and quasi-diurnal tidal frequencies inthe inter-event periodicities ca. one week before the mainshock, concurrent with anabrupt, marked departure from Benford’s law (sharp decrease of φ values).
5 Discussion and Conclusions
Benford’s law applied to natural sciences has implications on the knowledge of thephysics of a vast class of natural systems approaching a power law behaviour (Geyerand Martì 2012; Nigrini and Miller 2007; Sambridge et al. 2010; Sambridge et al.
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2011). Although the origin of scale-invariant processes is still a matter of debate, Ben-ford’s law may help describe quantitatively the behaviour of scale-invariant phenom-ena governed by multiplicative fluctuations (Kobayashi et al. 2006; Mandelbrot 1982;Pietronero et al. 2001). In particular, in the seismology literature, the probability dis-tribution of inter-seismic time intervals is known to approach a power law distribu-tion, although conflicting interpretations exist in terms of a universal law vs. a mereconsequence of the Gutemberg–Richter law of earthquake magnitudes (Molchan2005; Saichev and Sornette 2006, 2007).
The application of Benford’s law to the recurrence event times in seismic seriesof individual seismogenic regions represents a novel approach, which enlarges theoccurrence and relevance of Benford-like asymmetries and provides insights in theseismic behaviour of specific hazardous areas. Overall, in the study data set the timeintervals among consecutive seismic events reveal a good correlation to Benford’slaw and a probability distribution approaching a power-law. In particular, the ex-cellent correlation of the L’Aquila aftershock inter-events time series to Benford’slaw—and to a power-law probability distribution—likely results from the availabil-ity of a more complete data set following the April 6th, 2009 mainshock. From apractical perspective, this strongly suggests that earthquake time series in specificseismogenic areas tend to conform to Benford’s law. In this regard, a nonconformitymay result from (a) an incomplete data set, (b) an excessive rounding of the data, or(c) data errors, as also found for hydrology data sets and possibly relevant to othergeophysical phenomena (Nigrini and Miller 2007). On the other hand, the recurrencetimes among non-consecutive seismic events reveal the presence of significant depar-tures from Benford’s law, in the form of negative fluctuations of φ values. These areaccompanied by Gaussian and/or multi-peaked PDF’s of the recurrence times, whichremarkably overlap with the well-known periodicities of the main tidal constituents atlow (fortnightly and monthly) and high (ter-diurnal, semi-diurnal, and quasi-diurnal)frequencies.
The influence of tidal stresses on geodynamic processes at different scales, includ-ing plate tectonics, seismicity, and volcanism, is widely discussed in the scientificliterature (Beeler and Lockner 2003; Neuberg 2000; Riguzzi et al. 2010; Sottili andPalladino 2012; Tanaka 2012). The investigation of the tidal triggering mechanismsof earthquakes is beyond the aim of this paper. Usually, to detect possible correla-tions, the rates of seismic occurrences are compared with the quasi-diurnal and/orsemi-diurnal tidal phases (Beeler and Lockner 2003; Lockner and Beeler 1999;Vidale et al. 1998; Wilcock 2001). It appears that seismicity may correlate with theamplitude and frequency of small periodic stresses, comparable to tidal ones in mag-nitude, if the stress period exceeds the duration of the earthquake nucleation (the timerequired for rock failure). This is the case of seismic clusters, which often have beenfound to correlate with solid Earth tides (Kasahara 2002) as a result of local stress-ing rates leading to earthquake nucleation times shorter than tidal periods (Scholz2003). For lower tidal frequencies (example fortnightly or monthly) with amplitudevariations even 5 times lower than quasi-diurnal ones (Hartzell and Heaton 1989),the seismicity-tidal synchronicity is more uncertain (Cochran and Vidale 2007;Crockett et al. 2006). According to Beeler and Lockner (2003), the poorly corre-lated response of Earth seismicity to tidal stress maxima might be a consequence
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of time-dependent or delayed rock failure. Assuming a simple behaviour of a seis-mogenic source, in which the stress build-up increases up to failure, then the cyclicstress variations induced by Earth tides in the crust (1–4 kPa; Melchior 1983; upto 0.001 MPa h−1; Glasby and Kasahara 2001) may largely exceed the tectonicstress rates in specific geological settings, including volcanic contexts (McNutt andBeavan 1981; Sottili et al. 2007). On these grounds, although tidal stresses can-not be considered as a prime factor for earthquake initiation, the tidal control onthe timing of seismicity may be effective when superimposed favourably to localgeological factors of the earthquake nucleation area (such as relatively slow tec-tonic stress build-up, presence of stress and rock anisotropies, and of overpres-surised fluid reservoirs) in crust domains already in a critical state (Emter 1997;Hirata and Imoto 1991; Klein 1976; Tanaka 2012; Tsuruoka et al. 1995).
Concerning the boundary conditions that may promote tidal control of seismicity,we remark that individual seismogenic areas affected by seismic swarms are likelycharacterised by high stressing rates and earthquake nucleation times shorter thantidal periods (Scholz 2003). In fact, the ter-, semi-, and quasi-diurnal tidal frequenciesare revealed only for inter-seismic recurrence times in the order of minutes to a fewtens of minutes, while longer periodicities (fortnightly and monthly) are revealed onlyfor recurrence times in the order of tens of hours to days. Moreover, the anisotropicnature of tidal stresses with different frequencies seems to play a role as important asmagnitude and frequency. For example, the semi-diurnal constituent, which is com-monly reported as the second main tidal frequency after the quasi-diurnal one, isseldom found in the analysed data set. By contrast, the ter-diurnal constituent whichhas a horizontal component much higher than the semi-diurnal one, is often found inthe recurrence time analysis. This suggests a complex interaction of the anisotropictidal stresses with the local geological stresses (Sottili et al. 2007), which deservesdeeper investigation. In this regard, the detection of changing degree and pattern ofseismic response to tidal stresses in foreshock time series while approaching a majorearthquake (as also noticed by Tanaka 2012) may provide a promising statisticallybased approach for short-term hazard assessment in seismic areas.
Acknowledgements We are grateful to the Editor-in-Chief, R. Dimitrakopoulos, and two anonymousreferees for their helpful comments to the manuscript.
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