Time Scaling Analysis of Lightning in Italy

Communications in Nonlinear Science and Numerical Simulation 13 (2008) 1384–1396

www.elsevier.com/locate/cnsns

Time-scaling analysis of lightning in Italy

Luciano Telesca a,*, Marina Bernardi b, Cinzia Rovelli b

a Institute of Methodologies for Environmental Analysis, National Research Council (IMAA-CNR), C.da S.Loja,

I-85050 Tito Scalo (PZ), Italyb CESI, Via Ribattino 54, 20134 Milano, Italy

Received 3 February 2006; received in revised form 30 January 2007; accepted 1 February 2007Available online 13 February 2007

Abstract

The series of lightning measured during 2002 and 2003 in two areas of Italy, Lombardia (northern Italy) and Campania(southern Italy), has been deeply investigated to detect and quantify time-clustering structures by using several methods.Our results suggest the existence of time-clustering in the sequence of lightning time-occurrences, characterized by the pres-ence of two scaling regions, respectively related with clustering effect inside a single thunderstorm and among successivethunderstorms.� 2007 Elsevier B.V. All rights reserved.

Keywords: Lightning; Time-clustering; Allan factor; Fano factor; Count-based periodogram; Coefficient of variation; Interevent times

1. Introduction

Lightning as a high and transitory electric current is generally generated by electrostatic charge separationoccurring in thunderstorm clouds [1,23–25]. The most accepted explanation for this process is the friction andcollision among ice and water particles traveling in convective drafts in the cloud. Charged particles tend tomigrate to far regions of the cloud, typically the top and bottom, to reach a configuration of a dipole or multi-pole; earth polarizes opposite to the bottom cloud charge becoming another pole. When the electric field inten-sity between dipoles reaches the breakdown level, a lightning discharge is originated [2,26].

The total discharge is called flash and is a complex process involving one or more subsequent current flows;each of these currents is called a stroke. A flash lasts for about half a second, while each stroke takes aroundone millisecond [3].

A flash can thus happen between two clouds (CC) or into the same cloud (IC) or from cloud to ground(CG). This last category, even if the less numerous, is by far the more known and studied [4]. CG flashes couldbe subdivided in positive and negative depending on the current polarity, or in upward and downward

1007-5704/$ - see front matter � 2007 Elsevier B.V. All rights reserved.

doi:10.1016/j.cnsns.2007.02.001

* Corresponding author. Tel.: +39 0971 427277; fax: +39 0971 427271.E-mail address: [email protected] (L. Telesca).

mailto:[email protected]

L. Telesca et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 1384–1396 1385

depending on the direction of motion. Positive CG discharges are rare, around 10% of the CG total; they cantransfer to ground a considerable quantity of charge, compared to the negative lightning, and can have highercurrent intensity than negative ones. Usually positive flashes are much more common in winter thunderstorms[5].

Thunderstorms vary from region to region on the Earth, due to different latitude, meteorological conditionsand orography [6]. A mean of 100 s�1 lightning strikes the earth, of which much more fall over land than onthe oceans; tropical thunderstorms present a higher number of flashes [7]. In Italy the thunderstorms are ofthree main kinds:

(1) large fronts: these are formed by collision between incoming cold air mass and pre-existent hot air mass.These thunderstorms are joined to atmospheric disturbance and they are dynamic thunderstorms. InItaly large front thunderstorms come generally from North-West;

(2) local convective cells: these are produced by convective cells that develop usually during afternoon onmountains and during late-afternoon or evening on plain, when the unstable weather conditions aremore favorable. This thunderstorm kind is typical in summer season;

(3) orographic thunderstorms: they are generated by raising air mass due to the presence of a slope. Theydevelop typically in Alpine and Appenninic regions.

Statistical analyzes of space–time distributions of cloud-to-ground (CG) lightning have been performed,addressing several questions. A statistical study on the interstroke time intervals in lighting activity in Brazilhas been performed by de Miranda et al. [17], with values consistent with those obtained in other regions ofthe world. Influence of the land–sea effect was detected analyzing the diurnal variation of the lightning activityon the sea of offshore regions around Java, which showed an almost out-of-phase pattern to the variation onthe land [18]. Detailed analysis of the spatial density of flashes in Brazil was performed in [19]. Thunderstorm–solar activity relationship have been studied correlating lightning frequency with parameters characterizingsolar activity [20]. Dependence on the latitudes on lightning density has been investigated in [21].

A complete and detailed analysis of the temporal characteristics of lightning is still lacking. Are lightning acorrelated process or they are just a realization of a temporal Poisson process, completely uncorrelated andgiven by independent events? And if lightning are a correlated process, is it possible to identify and quantifythe degree of correlation? Therefore, are lightning a clusterized process, where the flashes occur organized intemporal clusters, obeying to a structure law, or are they randomly located on the time axis, without any tem-poral structure? To answer to these questions, further analyzes have to be performed. In [22] partial investi-gations about the temporal characterization of the lightning process have been carried out. In the presentpaper, a complete and exhaustive analysis of the temporal properties of lightning has been performed, in orderto characterize their time dynamics.

2. Lightning data

In this study we analyze the temporal properties of the CG lightning activity in two areas of Italy, Lom-bardia (northern Italy) and Campania (southern Italy). The data have been provided by the lightning locationsystems (LLS) of Sistema Italiano Rilevamento Fulmini (SIRF) [8], which is a network detecting CG lightningactivity over all of Italy, composed by 16 IMPACT (Improved Accuracy from Combined Technology) sensors[9] located throughout Italian territory. The mean accuracy of the system is 500 m for both co-ordinates of thepoint of impacts. SIRF allows the detection of single strokes with peak current value greater than about2.5 kA. The nominal flash detection efficiency is roughly 90% over all the area [8]. The SIRF system providesa set of information for each cloud-to-ground lightning: date, time of occurrence, lightning impact coordi-nates, peak current, stroke number, polarity and statistic parameters. A sequence of 34,502 (during 2002)and 29,200 (during 2003) lightning flashes was recorded in Campania, and 55,496 (during 2002) and 48,540(during 2003) occurred in Lombardia. Fig. 1 shows the time distribution of the CG lightning occurred in2002 and 2003 in the two areas investigated. Fig. 2 shows the distribution of the current intensity for negativeand positive lightning; the positive lightning are more frequent than negative at large intensity, although theyare the 10% of the total lightning.

Fig. 1. Lightning data occurred in Campania and Lombardia during 2002 and 2003.

1386 L. Telesca et al. / Communications in Nonlinear Science and Numerical Simulation 13 (2008) 1384–1396

3. Formulation of the statistical methods

3.1. Representations of the lightning process

A lightning sequence can be viewed as a realization of a stochastic point process. A stochastic point processdescribes events that occur at some random locations in time [10] and is completely defined by the set of theevent times.

A point process may be called fractal when a number of its relevant statistics exhibit scaling with relatedscaling exponents, that indicate the represented phenomenon contain clusters of points over a relatively largeset of timescales [11]. The standard method to investigate the presence of clustering in a time series is the powerspectral density, which furnishes information on how the power of the process is concentrated at various fre-quency bands. The power spectral density is calculated by Fourier transforming the data, and thus allowsdetecting periodic, multi-periodic or no-periodic behaviors. The power spectral density of a process with scal-ing properties follows a power-law behavior S(f) / f�a, where f is the frequency and S(f) the power spectrum,with the so-called fractal exponent a measuring the strength of the clustering. The clustering behavior of apoint process leads to a power-law (fractal) shape of some of the statistics, used to describe its properties[12], and that allows estimating the fractal exponent a [13], whose numerical value is an index of the presenceof clusterization within the process. If the point process is Poissonian, the event occurrence times are uncor-related; for this memoryless process a � 0. On the other side, a 5 0 is typical of point processes with self-sim-ilar behavior; self-similar meaning that parts of the whole can be made to fit to the whole in some way byscaling. Thus, we can understand that the estimation of the fractal exponent a for a lightning sequence playsan important role in the general characterization of the mechanism underlying the lightning phenomenon.

103 104 105

100

101

102

103

N

I (A)

N

I (A)

N

I (A)

N

I (A)

Lombardia 2002 Positive Negative

103 104 105 106

100

101

102

103

104

Lombardia 2003 Positive Negative

104 105

100

101

102

103

Campania 2002 Positive Negative

103 104 105 106

100

101

102

103

Campania 2003 Positive Negativz

Fig. 2. Distribution of intensity current for the negative and positive lightning.


The analysis of the temporal properties of a lightning sequence implies that this sequence is appropriatelydescribed, in order to apply to it statistical methods able to reveal the presence of clusterization. A discrete-time process can be derived from the stochastic point process in two equivalent ways: (1) using the intereventtime series or (2) forming its relative counting process. In the first representation a discrete-time series isformed by the rule si = ti+1 � ti, where ti indicates the time of event numbered by the index i. In the secondrepresentation, the time axis is divided into equally spaced contiguous counting windows of duration T to pro-duce a sequence of counts {Nk(T)}, where Nk(T) represents the number of events falling into the kth window ofduration T. The duration T of the window is called counting time or timescale. The latter approach considersthe lightning as the events of interest and assumes that there is an objective clock for the timing of the events.The former approach emphasizes the interspike intervals and uses the event number as an index of the time.

Both representations allow us applying several statistical techniques to a lightning sequence, in order toanalyze its time behavior, and, in particular, to recognise time-clustering properties. In the following subsec-tions we will describe five methods: two of them (coefficient of variation and interevent-interval histogram) arerelated to the interevent-interval representation, while the remaining three (Allan factor, Fano factor andcount-based periodogram) are related to the counting process representation. These methods have been exten-sively used to analyze the time-clustering properties of earthquake sequences and have been reviewed in Tel-esca et al. [14].

3.2. Methods

Several statistical measures can be used to characterize the time properties of a lightning point process. Weused five different measures: interevent-interval histogram (IIH), coefficient of variation (CV), Fano factor(FF), Allan factor (AF) and count-based periodogram (PG).


Since statistics of all orders are necessary to define uniquely a stochastic point process, these five measures,being first- and second-order measures, do not provide complete information about the process. Nevertheless,when used in conjunction they are useful for providing a basic characterization of the process on all timescales.We describe them.

3.2.1. Coefficient of variation (CV)

A commonly used measure to evaluate the clustering behavior of a point process is the coefficient of var-iation CV, defined as

CV ¼ rs=hsi; ð1Þ

where hsi is the mean of the interevent times and rs is the standard deviation: a Poissonian process (completelyrandom) has a CV = 1, but a clusterized process is characterized by a CV > 1. This coefficient does not giveinformation about the timescale ranges where the process can be reliably characterized as a clustered process.Nevertheless, a complex phenomenon can be deeply known only if the different timescales governing itsdynamics are well understood.

3.2.2. Interevent-interval histogram (IIH)

The interevent-interval histogram (IIH) estimates the probability density function of interevent-intervalmagnitude, and illustrates the relative frequency of occurrence of an interevent-interval of size s. The IIH(s)completely describes the behavior of the point process only if the interevent intervals are independent andidentically distributed random variables, like in Poisson process [12]; in fact, for this memoryless process,the occurrence times are uncorrelated, and the IIH(s) behaves as a decreasing exponential function ke�sk ofthe interevent-interval s, for s P 0, with k the mean rate of the process. For a fractal (with clustering prop-erties) process the IIH(s) generally has a power-law dependence on the interevent-interval s [12], with scalingexponent 1 + a, where a is the fractal exponent.

3.2.3. Fano factor (FF)

The Fano factor (FF) is defined as the variance of the number of events in a specified counting time or time-scale T divided by the mean number of events in that counting time; that is

FFðT Þ ¼ hN2kðT Þi � hN kðT Þi2

hN kðT Þi; ð2Þ

where h. . .i indicate the average value. In order to evaluate the presence of scaling, the timescale T is variedand a relationship FF(T) � T is obtained.

The FF(T) of a fractal point process with 0 < a < 1 varies as a function of counting time T as

FFðT Þ ¼ 1þ TT 0

� �a

: ð3Þ

The monotonic power-law increase is representative of the presence of fluctuations on many timescales [11].The scaling exponent a is the so-called fractal exponent. If a > 0 then the represented phenomenon containsclusters of points over a relatively large set of timescales [13]. If a � 0, the lightning occurrence process is Pois-sonian and the occurrence times are uncorrelated. The crossover timescale T0 is the fractal onset time, andmarks the lower limit for significant scaling behavior in the FF [13], so that for T� T0 the clustering propertybecomes negligible at these timescales. Thus, T0 is estimated as the timescale over which the FF increases as apower-law function of the timescale T. FF assumes values near unity for Poisson processes.

3.2.4. Allan factor (AF)

Another measure, related to the variability of successive counts, useful to detect the event clustering in apoint process, is the Allan factor (AF) [15], defined as the variance of successive counts for a specified countingtime T divided by twice the mean number of events in that counting time

Fig. 3. Interevent time series of the four sequences plotted in Fig. 1.


AFðT Þ ¼ hðNkþ1ðT Þ � N kðT ÞÞ2i2hN kðT Þi

: ð4Þ

This measure reduces the effect of possible nonstationarity of the point process, because it is defined interms of the difference of successive counts. As for the FF, varying the timescale T allows producing a rela-tionship between AF(T) and T, useful to detect scaling behavior in the sequence.

The AF of a fractal point process varies with the counting time T with a power-law form:

AFðT Þ ¼ 1þ TT 1

� �a

ð5Þ

with 0 < a < 3 over a large range of counting times T [16]; T1 is the fractal onset time for the AF and is esti-mated as the crossover timescale between Poissonian and scaling behaviors. It is related to T0 by [12]

T a0T�a

1 ¼ 2� 2a: ð6Þ

As for FF, AF assumes values near unity for Poisson processes.

3.2.5. Count-based periodogram (PG)

The count-based periodogram (PG) [12], that is the periodogram of the sequence of the counts, is anotherstatistical measure that allows estimating the fractal exponent a. This measure performs an estimate of thepower spectral density, which gives information on how the power of the process is concentrated at various

Camp 2002 Camp 2003 Lomb 2002 Lomb 20030

5

10

15

20

25

30

35

40

45

50

CV

area

Total Positive Negative

Fig. 4. The coefficient of variation CV.

Camp 2002 Camp 2003 Lomb 2002 Lomb 20030.00.10.20.30.40.50.60.70.80.91.01.11.21.31.41.51.61.71.81.92.0

αIIH

Area

Total Positive Negative

Fig. 5. The scaling exponent a calculated as the power-law index of the interevent-interval histogram (IIH).


frequency bands. The calculation of the PG by means of a count-based approach implies the division of thetotal observation period into a number N of nonoverlapping windows of length TW. The sequence of thecounts is then Fourier transformed and the PG is obtained by the squares of the coefficients of the Fourierrepresentation of the series of counts. Generally, in order to use the Fast Fourier Transform (FFT) the num-ber N of the windows of duration TW is a power of two. For point processes with scaling properties the PGdecreases as a power-law function of the frequency f over a significant range of frequencies, S(f) / f�a. Ofcourse, for a finite size fractal real process the power spectral density behaves as an inverse power-law functionin a limited range of frequencies, approaching an asymptotic value at high frequencies, at which the behaviorof the process can be considered Poissonian. The numerical value of a is an indicator of the presence of clus-terization in the process [12]. If the point process is Poissonian, the occurrence times are uncorrelated; for thismemoryless process, the PG is approximately flat for any frequency bands and a � 0. On the other side, a 5 0is typical of point processes with scaling behavior. This method introduces a bias at higher frequencies, sincethe fine time resolution information is lost as a result of the minimum count-window size. But, the estimationof the exponent a principally involves lower frequencies where this bias is negligible.

100 101 102 103 104 105 106 107100

101

102

1033-4h

Campania 2002total

αh=0.290+0.005

αL=0.923+0.003

FF

(T)

100

101

102

103

FF

(T)

100

101

102

103

104

FF

(T)

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101

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103

FF

(T)

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FF

(T)

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(T)

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104

FF

(T)

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FF

(T)

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FF

(T)

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FF

(T)

100

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103

FF

(T)

100

101

102

FF

(T)

T (s)

100 101 10 2 103 104 105 106 107

T (s)

100 101 102 103 104 105 106 107

T (s)

100 101 102 103 104 105 106 107

T (s)

100 101 102 103 104 105 106 107

T (s)

100 101 102 103 104 105 106 107

T (s)

100 101 102 103 104 105 106 107

T (s)

100 101 102 103 104 105 106 107

T (s)

100 101 102 103 104 105 106 107

T (s)

100 101 102 103 104 105 106 107

T (s)

100 101 102 103 104 105 106 107

T (s)

100 101 102 103 104 105 106 107

T (s)

3-4 h

Campania 2002negative

αH=0.291+0.005

αL=0.919+0.003

4-5 h

Campania 2002positive

αH=0.205+0.008

αL=0.778+0.005

3-4 h

Campania 2003total

αH=0.255+0.007

αL=0.910+0.004

3-4 h


αH=0.265+0.007

αL=0.916+0.002

4 h


αH=0.210+0.007

αL=0.768+0.003

2-3 h

Lombardia 2002total

αH=0.424+0.006

αL=0.927+0.002

2-3 h

Lombardia 2002negative

αH=0.422+0.006

αL=0.924+0.002

4 h

Lombardia 2002positive

αH=0.378+0.009

αL=0.805+0.002

3-4 h

Lombardia 2003total

αH=0.379+0.005

αL=0.925+0.002

3-4 h


αH=0.378+0.006

αL=0.919+0.002

3-4 h


αH=0.324+0.006

αL=0.839+0.003

Fig. 6. Log–log plots of the FF curves for all the lightning sequences (total, negative and positive) in the two areas investigated.


4. Data analysis

Now we analyze the time-scaling properties of the distribution of the lightning sequence occurred in Camp-ania (southern Italy) and Lombardia (northern Italy) in 2002 and 2003.

Fig. 3 shows the interevent times of the four sequences.Fig. 4 shows the coefficient of variation CV calculated for each lightning sequence. The coefficient of var-

iation has also been calculated for the only positive and only negative flashes. The value of CV ranges betweenapproximately 9.1 and 38.6. The high value of the coefficient of variation suggests that the sequences arestrongly clusterized. The positive lightning have lower coefficient of variation than negative flashes.

10-8 10-7 10-6 10-5 10-410-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Campania 2002total

αHF

=1.51+0.08

αLF

=0.19+0.06

Frequency (Hz)

Pow

er

10-8 10-7 10-6 10-5 10-4

10-6

10-5

10-4

10-3

10-2

10-1

100


αHF

=1.51+0.07

αLF

=0.24+0.06

Frequency (Hz)

Pow

er

1E-8 1E-7 1E-6 1E-5 1E-4

10-8

10-7

10-6

10-5

10-4

10-3 Campania 2002positive

αHF

=1.51+0.09

αLF

=0.30+0.05

Frequency (Hz)

Pow

er

10-8 10-7 10-6 10-5 10-410-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Campania 2003total

αHF

=1.73+0.06

αLF

=0.24+0.06

Frequency (Hz)

Pow

er

10-8 10-7 10-6 10-5 10-410-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100


αHF

=1.62+0.07

αLF

=0.17+0.06

Frequency (Hz)

Pow

er

10-8 10-7 10-6 10-5 10-410-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1


αHF

=1.16+0.07

αLF

=0.23+0.07

Frequency (Hz)

Pow

er

Fig. 7. Log–log plots of the PG curves for all the lightning sequences (total, negative and positive) in the two areas investigated.


Fig. 5 shows the scaling exponent a calculated as the power-law index of the interevent-interval histogram(IIH). The value of the scaling exponent ranges between 0.55 and 1.92. The positive lightning show a lower

10-8 10-7 10-6 10-5 10-410-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Lombardia 2002total

αHF

=1.45+0.06

αLF

=0.29+0.08

Frequency (Hz)

Pow

er

10-8 10-7 10-6 10-5 10-410-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100


αHF

=1.37+0.06

αLF

=0.28+0.08

Frequency (Hz)

Pow

er

1E-9 1E-8 1E-7 1E-6 1E-5 1E-410-10

10-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1


αLF

=0.37+0.09

αHF

=1.68+0.06

Frequency (Hz)

Pow

er

1E-8 1E-7 1E-6 1E-5 1E-410-7

10-6

10-5

10-4

10-3

10-2

10-1

100

Lombardia 2003total

αHF

=1.50+0.06

αLF

=0.22+0.07

Frequency (Hz)

Pow

er

1E-8 1E-7 1E-6 1E-5 1E-410-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1

100


αHF

=1.40+0.07

αLF

=0.22+0.07

Frequency (Hz)

Pow

er

10-8 10-7 10-6 10-5 10-410-9

10-8

10-7

10-6

10-5

10-4

10-3

10-2

10-1


αHF

=1.38+0.07

αLF

=0.24+0.07

Frequency (Hz)

Pow

er

Fig. 7 (continued)


degree of clusterization in all the sequences, indicating a weaker strength of clusterization than that of negativelightning.

4-5 h

α=0.23+0.02

α=1.657+0.007

Campania2002

AF

(τ)

τ (s)

5 h

αH=0.25+0.02

αL=1.621+0.008

Campania2002negative

6-7 h

αH=o.20+0.02

αL=1.15+0.01

Campania2002positive

4-5 h

Campania2003

α=0.22+0.02

α=1.67+0.01

αH=0.23+0.02

αL=1.58+0.01

5-6 h

Campania2003negative

6-7 h

αH=0.26+0.01

αL=1.12+0.01

Campania2003positive

5-6 h

α=0.21+0.03

α=1.707+0.009

Lombardia 2002 4-6 h

400 s

αH=0.22+0.03

αL=1.664+0.009

Lombardia2002negative

4 h

700 s

αH=0.27+0.02

αL=1.18+0.01

Lombardia2002positive

4-5 h

α=0.32+0.02

α=1.662+0.008

Lombardia2003 4 h

400 s

αH=0.35+0.01

αL=1.607+0.009

Lombardia2003negative

αH=0.20+0.02

αL=1.32+0.01

5-6 h

700 s

Lombardia2003positive

100 101 102 103 104 105 106 107

τ (s)100 101 10 2 103 104 105 106 107

τ (s)100 101 10 2 103 104 105 106 107

τ (s)100 101 10 2 103 104 105 106 107

τ (s)100 101 10 2 103 104 105 106 107

τ (s)100 101 10 2 103 104 105 106 107

τ (s)100 101 10 2 103 104 105 106 107

τ (s)100 101 10 2 103 104 105 106 107

τ (s)100 101 102 103 104 105 106 107

τ (s)100 101 102 103 104 105 106 107

τ (s)100 101 102 103 104 105 106 107

τ (s)100 101 102 103 104 105 106 107

100

101

102

103

AF

(τ)

100

101

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103

AF

(τ)

100

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AF

(τ)

100

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AF

(τ)

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AF

(τ)

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102103

AF

(τ)

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AF

(τ)

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102103

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AF

(τ)

100

101

102

AF

(τ)

100

101

102103

AF

(τ)

100

101

102

AF

(τ)

100

101

102103

Fig. 8. Log–log plots of the AF curves for all the lightning sequences (total, negative and positive) in the two areas investigated.


Fig. 6 shows the FF curves calculated for all the lightning sequences. The log–log plots clearly show twoscaling regions, with a crossover timescales at about 2–4 h for the total and negative lightning, while atapproximately 4–5 h for the positive lightning. The values of the scaling exponent ranges around 0.9 for inter-mediate timescales for the total and negative lightning, while is about 0.8 for the positive sequences. It shouldbe noted that the values of the scaling exponents in the intermediate timescale range could be affected by asaturation effect, typical of FF method. For large timescales the exponent varies between 0.2 and 0.4, withhigher value for Lombardia area and total as well as negative sequences.

Fig. 7 shows the results of the PG analysis. Each PG curve is characterized by the same properties of the FFcurves, that is two scaling regions and different values of the scaling exponents in the low frequency (large


timescales) and high frequency ranges (small timescales). The values of the scaling exponents are consistentwith those obtained by the FF analysis, but only for the low frequency ranges. The scaling exponents calcu-lated in the high frequency range between 1.1 and 1.7, well above those calculated with the FF method. Itshould be noted that the PG can be sensitive to nonstationarities in the data, and the values of the scalingexponents calculated by means of the PG method can be affected by this effect.

Fig. 8 shows the results of AF analysis. This method allows the detection of scaling behavior in point pro-cesses characterized by scaling exponents larger than 1 and affected by nonstationarities, since it is defined interms of the first difference between two successive counts. Two scaling regions, linear in log–log scales, arealso visible in the AF plots for each sequence. The values of the scaling exponent are approximately consistentwith those obtained by using the PG method and FF method (only in the large timescale range). The crossovertimescale is quite larger than that calculated by using the FF method.

5. Conclusions

A sequence of lightning can be seen as a stochastic point process, where each event is mainly characterizedby impact location, occurrence time and current intensity. The analysis performed in the present study aims toinvestigate the property of time-clustering in the lightning sequence recorded in two areas in Italy, Campania(southern Italy) and Lombardia (northern Italy). Using different representations of the lightning process(interevent times and series of counts) and several statistical measures we detected time-scaling behavior inthe analyzed sequences, and this suggests the presence of time-clusterization of the lightning events.

The quantification of the time-clusterization of the lightning events is given by the numerical value of thescaling exponent estimated by using the proposed methods. High values of the scaling exponent a denotestrong time-clusterization of the events.

The analyzed sequences have shown two different scaling regions, at intermediate and large timescales, withdifferent values for the scaling exponent. The crossover between the two scaling regions is around 4–5 h, whichis consistent with the mean duration of a thunderstorm. Therefore the intermediate timescale region describesthe clustering behavior of the lightning inside a single thunderstorm, while the large timescale region describesthe clustering behavior of lightning among successive thunderstorms.

Acknowledgement

We acknowledge the CESI-SIRF (Milan) for providing lightning data.

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Time Scaling Analysis of Lightning in Italy

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