Long-range spatial correlations and fluctuation statistics of lightning activity rates in Brazil
H. V. Ribeiro, F. J. Antonio, L. G. A. Alves, E. K. Lenzi, R. S. Mendes
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Long-range spatial correlations and fluctuation statistics of light-ning activity rates in Brazil
H. V. Ribeiro , , , F. J. Antonio , , L. G. A. Alves , , E. K. Lenzi , and R. S. Mendes , Departamento de F´ısica, Universidade Estadual de Maring´a, Maring´a, PR 87020-900, Brazil Departamento de F´ısica, Universidade Tecnol´ogica Federal do Paran´a, Apucarana, PR 86812-460, Brazil National Institute of Science and Technology for Complex Systems, CNPq, Rio de Janeiro, RJ 22290-180, Brazil
PACS – Atmospheric electricity, lightning
PACS – Complex systems
PACS – Patterns
Abstract. - We report on a statistical analysis of the lightning activity rates in all Brazilian cities.We find out that the average of lightning activity rates exhibit a dependence on the latitude ofthe cities, displaying one peak around the Tropic of Capricorn and another one just before theEquator. We verify that the standard deviation of these rates is almost a constant function of thelatitude and that the distribution of the fluctuations surrounding the average tendency is quitewell described by a Gumbel distribution, which thus connects these rates to extreme processes. Wealso investigate the behavior of the lightning activity rates versus the longitude of the cities. Forthis case, the average rates exhibit an approximate plateau for a wide range of longitude values,the standard deviation is an approximate constant function of longitude, and the fluctuations aredescribed by a Laplace distribution. We further characterize the spatial correlation of the lightningactivity rates between pairs of cities, where our results show that the spatial correlation functiondecays very slowly with the distance between the cities and that for intermediate distances thecorrelation exhibits an approximate logarithmic decay. Finally, we propose to model this lastbehavior within the framework of the Edwards-Wilkinson equation.
Introduction. –
The search for better understand-ing of earth-related systems is a stimulating challenge of-ten present in the physicists agenda. Examples of suchinvestigations include study of earthquakes [1–4], geomag-netic activities [5–7], climate [8,9] and weather-related sys-tems [10, 11]. Usually, these investigations examine dataaiming to uncover patterns or laws that rule the spatiotem-poral dynamics of the systems. These empirical-based ap-proaches are also very useful to provide basis for compar-ing and testing the increasing number of models studiedin these fields.In the previous context, a well-known phenomenon isthe lightning strike, which displays an intrinsic complexbehavior on its own. Physically speaking, lightning is amassive electric discharge between electrically charged re-gions in one or more clouds, or between clouds and anearth-bound object (cloud-to-ground lightning). Usuallythis complex process involves more than one current flow (a)
E-mail: [email protected] and lasts about half a second, while each one of these flowscan take only milliseconds to happen [12]. Despite light-nings having short duration, they are well-known by caus-ing severe injuries or even death when striking humans oranimals, and for the damage they usually provoke to build-ings and power networks [13,14]. In this sense, monitoringand probing spatiotemporal patterns of lightning occur-rences may not only prevent electrical discharge hazardbut also provide useful markers for thunderstorms, whichare among the the major causes of weather-related dam-ages and economic losses in tropical and middle latituderegions [15].Around 78% of all lightning events occur in tropicaland middle latitude areas, that is, the majority of theseevents are concentrated between 30 ○ N and 30 ○ S lati-tudes [16, 17]. Thus, understanding the lightning activityin these regions of the world is especially important. Wedevote this article to investigate the spatial dynamics oflightning activity rates in Brazil, the largest tropical coun-p-1 a r X i v : . [ phy s i c s . d a t a - a n ] D ec . V. Ribeiro et al. try in the world, which is hit (on average) by more than 50million lightning strikes each year. Furthermore, due itscontinental dimensions and the existence of reliable data,Brazil can be considered as an ideal place for probing spa-tial patterns of lightning activity. Here we shall study thelightning activity rates in connection with the geographiclocalization of all Brazilian cities aiming to uncover sta-tistical patterns and to model these findings within a re-ductionist approach. In following, we present the data wehave analyzed, the procedures we have employed to ana-lyze the data as well as the empirical findings, and finally,we end with a summary and some concluding remarks. Data presentation. –
We have accessed data cor-responding to the lightning activity rates (only cloud-to-ground lightning discharges) of all Brazilian cities madefreely available by the National Institute for Space Re-search (INPE) [18–21]. These rates were estimated byINPE through data obtained by the sensor LightningImaging Sensor [22] that is aboard the Tropical RainfallMeasuring Mission [23]. The rates were calculated usingdata from the years 1998 to 2013 and are considered a re-liable measure for all Brazilian cities. The data consist ofthe geographic location (latitude φ and longitude λ ) andthe average number of lightnings per km per year ( L ).Figure 1 shows the spatial distribution of the lightningactivity rates throughout the Brazilian territory. Tropic of Capricorn 0.02 0.63 19.66Lightnings by km by year Longitude, λ La t i t ude , φ Fig. 1: Spatial distribution of lightning activity rates of theBrazilian cities. Each point represents the geographic location(latitude φ and longitude λ ) of one of the 5,562 Brazilian citiesand the color code indicates the lightning activity rate in thecity. Data Analysis. –
We start by investigating the de-pendence of the lightning activity rates L on the latitudes φ of the cities. Figure 2(a) shows a scatter plot of L versus φ where, despite the considerable noise, we can identify some regularities. We observe the existence of systemi-cally large values for the lightning activity rates aroundthe Tropic of Capricorn and also around the Equator. Inorder to check whether these visual patterns are statis-tically meaningful, we have binned the data into 25 win-dows and, for each window, we calculate the average value(square) of the lightning activity rates as well as the 95%bootstrap confidence interval (error bar). As shown infig. 2(a), the average value of L is actually statistically sig-nificantly larger around the Tropic of Capricorn and theEquator. We note yet the existence of maximum valuesjust before these geographic references.We have also studied the dependence of the standarddeviation of the lightning activity rates on the latitude.Figure 2(b) reveals that the standard deviation σ is almosta constant function of the latitude. In particular, we notethat practically all values of σ are within the 95% confi-dence interval (light shaded area). A systematic exceptionoccurs around the latitude −
10, where the standard devi-ations remain outside of the 99% confidence interval for aconstant behavior (dark shaded area).Aiming to further characterize the lightning activityrates, we have evaluated the residuals surrounding the av-erage tendency of L versus φ . In order to do so, we haveinterpolated a spline of third order to the average dataand then we calculate the residuals ξ i = L i − F ( φ i ) , (1)where L i is the lightning activity rate in the city i , φ i islatitude of the city i and F ( φ i ) stands for the value ofthe interpolating function at the latitude φ i . Next, weinvestigate the probability distribution of ξ , as shown infig. 2(c). We have found that the Gumbel distribution[24] P ( ξ ) = β e (− ξ + α )/ β − e (− ξ + α )/ β (2)with the parameters α = − .
24 (location parameter) and β = .
16 (scale parameter) describes quite well our empir-ical distribution. The best fit parameters were obtainedby the maximum likelihood method and the p -value of theCram´er-von Mises test is 0.02, indicating that we cannotreject the Gumbel hypothesis at a confidence level of 99%.In order to overcome possible bias related to the inho-mogeneous spatial distribution of the Brazilian cities, wehave also calculated the distribution of ξ after groupingcities with close latitudes values, considering the same 25windows employed for the average and standard deviationanalyses (figs. 2(a) and (b)). The results show that thedistributions of the residuals by latitude obey the sameGumbel form with parameters α and β very close to thoseobtained for the whole dataset, as shown in fig. 2(d). Itis worth noting that the Gumbel distribution is relatedto the distribution of the maximum of a set of n → ∞ random numbers drawn from a distribution that asymp-totically decays faster than any power law. This resultp-2ong-range spatial correlations and fluctuation statistics of lightning activity rates in Brazil (a) (b)(c) L i gh t n i ng s b y k m b y y ea r , L Latitude, φ T r op i c o f C ap r i c o r n S t anda r d de v i a t i on , σ Latitude, φ P r obab ili t y d i s t r i bu t i on Latitude residuals, ξ −4 −3 −2 −5 0 5 10 Latitude residuals, ξ P r obab ili t y d i s t r i bu t i on Gumbeldistribution −4−20246 −30 −20 −10 0 G u m be l pa r a m e t e r s Latitude, φ values of β values of α (d) * * * * * ** α =-1.24 β =-2.16 Fig. 2: Dependence of the lightning activity rates on the latitude of the cities. (a) Each red dot represents the lightning activityrate versus the latitude of a city. The blue squares are window average values (25 equally spaced windows) and the error bars are95% bootstrap confidence intervals for these means. The continuous line is a spline interpolation of third order to the averagedata. We note that the average value displays a maximum value close to the Tropic of Capricorn (horizontal dashed line) andalso a local maximum just before the Equator (zero latitude). (b) Standard deviation of the lightning activity rates versus thelatitude evaluated over same windows utilized for the average values. The blue squares are the standard deviations and theerror bars are 95% bootstrap confidence intervals. The horizontal dashed line is the average value of standard deviations andthe dark (light) shaded area represents the 95% (99%) bootstrap confidence intervals. We note that except for a few points, thestandard deviations do not display statistically significant deviations from a constant behavior. (c) Probability distribution ofthe residuals of the lightning activity rates around the average tendency described by the spline interpolation. The bins werechosen through the Wand’s procedure and the inset shows the empirical distribution in log-log scale. The continuous line is theGumbel distribution (eq. 2) with location parameter α = − .
24 and scale parameter β = .
16, which were obtained via maximumlikelihood fit. The p -value of Cram´er-von Mises test is 0.02. (d) Values of α and β of the Gumbel distribution when groupingthe residuals by using the same 25 windows employed for the calculation the averages and standard deviations. These valueswere obtained via maximum likelihood fit considering only the windows with more than 250 points within and the asterisksindicate the windows where the Gumbel hypothesis cannot be rejected at a confidence level of 99%. The error bars are 95%bootstrap confidence intervals and the horizontal lines represent the α and β obtained for the whole dataset. thus suggests that the deviations surrounding the aver-age tendency can be also understood as an extreme pro-cess. While it is not trivial to connect these results withthe mechanisms underlying the physics of lightning pro-cess [25], we can consider a cloud-to-ground lightning as anextreme event of the several processes that occur in a thun-dercloud (in addition to large values of the electric fieldsnecessary for the lightning initiation). Actually, cloud-to-ground lightnings are about 25% of the global lightningactivity, all remaining activity does not involve the groundand it is mainly composed by intra-cloud discharges andby (on a much smaller fraction) cloud-to-cloud and cloud-to-air discharges. Therefore, when analyzing only cloud-to-ground lightnings, we are selecting a group of extremeevents and thus approaching somehow the mathematical conditions of the extreme value theory where Gumbel dis-tributions naturally emerge.We now focus on the dependence of the lightning ac-tivity rates L over the longitude λ of the cities. We haveproceeded as previously, that is, we start with a scatterplot of L versus λ , we bin the data into 25 windows andfor each one we evaluate the average of L , and we also in-terpolate a spline of third order to the average data, as it isshown in fig. 3(a). For this case, we do not observe a cleardependence of the average lightning activity rates over thelongitudes within the interval − ≲ λ ≲ −
43. In this inter-val, apart from a peak around the latitude − L displaysan approximate constant behavior. However, we do notea systematic decrease in the average values of L for forlongitudes higher than ≈ −
43. This fact is also visible inp-3. V. Ribeiro et al. S t anda r d de v i a t i on , σ Longitude, λ L i gh t n i ng s b y k m b y y ea r , L Longitude, λ (a) (b)(c) Longitude residuals, ξ P r obab ili t y d i s t r i bu t i on −4 −3 −2 −1 −10 −5 0 5 10 P r obab ili t y d i s t r i bu t i on Longitude residuals, ξ Laplacedistribution −2−101234 −70 −60 −50 −40 −30
Lap l a c e pa r a m e t e r s Longitude, λ values of γ values of µ (d) * * * * * * ** γ =1.40 µ =-0.12 Fig. 3: Dependence of the lightning activity rates on the longitude of the cities. (a) Each green dot represents the lightningactivity rate versus the longitude of a city. The orange squares are window average values (25 equally spaced windows) and theerror bars are 95% bootstrap confidence intervals for these means . The continuous line is a spline interpolation of third order tothe average data. We note that the profile of the average values is almost a plateau in the interval − ≲ λ ≲ −
43 and also thatthe lightning activity rates are substantially smaller for longitudes larger than ≈ −
40. (b) Standard deviation of the lightningactivity rates versus the longitude evaluated over same windows utilized for the average values. The orange square are thestandard deviations and the error bars are 95% bootstrap confidence intervals. The horizontal dashed line is the average valueof standard deviations and the dark (light) shaded area represents the 95% (99%) bootstrap confidence intervals. Similarly to thelatitude case, the standard deviations do not display statistically significant deviations from a constant behavior. (c) Probabilitydistribution of residuals of the lightning activity rates around the average tendency described by the spline interpolation. Thebins were chosen through the Wand’s procedure and the inset shows the empirical distribution in log-log scale. In contrast withthe latitude case, the continuous line is a Laplace distribution (eq. 3) with mean µ = − .
12 and scale parameter γ = .
40, whichwere obtained via maximum likelihood fit. The p -value of Cram´er-von Mises test is 0.02. (d) Values of µ and γ of the Laplacedistribution when grouping the residuals by using the same 25 windows employed for the calculation the averages and standarddeviations. These values were obtained via maximum likelihood fit considering only the windows with more than 250 pointswithin and the asterisks indicate the windows where the Laplace hypothesis cannot be rejected at a confidence level of 99%.The error bars are 95% bootstrap confidence intervals and the horizontal lines represent the µ and γ obtained for the wholedataset. fig. 1, where the blue region points out the lower lightingrates in the northeast of Brazil. Analogously to the pre-vious case, we have also studied the standard deviations σ within windows in function of the longitude (fig. 3(b)).Similarly, we have found an almost constant behavior andthat most of the values of σ are inside the 99% confidenceintervals of the average of the standard deviation over thelongitude (though systematic deviations are observed forlongitudes larger than − L versus latitude and L versus longitude occurs for the dis-tribution of the residuals ξ around the average tendency.While the Gumbel distribution is a quite good descriptionfor the dependence of L on the latitude, it cannot describe the dependence of L on the longitude. In fact, the tent-shaped distribution shown in fig. 3(c) is in well agreementwith the Laplace distribution P ( ξ ) = γ e −∣ ξ − µ ∣/ γ , (3)where the parameters µ = − .
12 (mean) and γ = .
40 (scalefactor) were obtained via maximum likelihood fit and the p -value of Cram´er-von Mises test is 0.02, pointing out thatthe Laplace hypothesis cannot be rejected at a confidencelevel of 99%. As in the latitude case, we also calculatedthe distribution of ξ after grouping cities with close longi-tudes values. Intriguingly, the Laplace form also describesthe residuals distributions by longitude with parameters µ p-4ong-range spatial correlations and fluctuation statistics of lightning activity rates in Braziland γ very close to those obtained for the whole dataset, asshown in fig. 3(d). In fact, since the spatial lightning dis-tribution is closely linked to climate phenomena on Earth(such as the general circulation of the atmosphere betweenthe equator and the middle latitude [26]) and therefore itseems unrelated to longitudes, we could expect normaldistributions emerging after accounting the bias relatedto the inhomogeneous spatial distribution of the Braziliancities. However, the lightning rates are also influenced bythe continental and oceanic forms of the Earth, which in-troduce a non-trivial dependence of the lightning rates onlongitudes [27].Another intriguing question is whether there is long-range spatial memory in the lightning activity rates L . Toinvestigate this hypothesis, we evaluate the spatial cor-relation function of the lightning activity rates betweenpairs of cities that are r kilometers distant. Specifically,we have computed C ( r ) = ⟨[ L i − m ( r )][ L j − m ( r )]⟩∣ r i,j = r s ( r ) , (4)where m ( r ) is the mean value and s ( r ) is the standarddeviation of the lightning activity rates of cities separatedby r kilometers, L i is the lightning activity rate in the city i , and ⟨ . . . ⟩∣ r i,j = r stands for the average value over citieswhere the distance r i,j that separates them are equal to r . Due to the discrete nature of our spatial data, we haveactually considered logarithmically spaced intervals of r for evaluating eq. 4. Figure 4 shows the spatial correla-tion C ( r ) in a lin-log scale where it is remarkable a veryslow decay of the correlation function. In particular, wehave identified three regimes: i ) an initial plateau in theinterval 10 km ≲ r ≲
40 km where C ( r ) ≈ ii ) an approxi-mate logarithmic decay in the interval 60 km ≲ r ≲
600 km(shaded area) where C ( r ) ≈ −C ln ( r /R) ; and iii ) a faster-than-logarithm decay for distances larger than ≈
600 km.We thus conclude that the lightning activity rates arelong-range correlated in space, since the correlation func-tion C ( r ) decays slower than any power law function fordistances smaller than ≈
600 km. Another interestingpoint is concerning the relationships between these long-range correlations and the non-Gaussian residual distri-butions previously presented. In several contexts, long-range correlations emerge in coupled-like manner withnon-Gaussian distributions (see Ref. [28] for a specific ex-ample) and here the same happens. Furthermore, if weshuffle the lightning rates among the cities, besides de-stroying the spatial correlations this process drasticallychanges the profile of the residual distributions. Specifi-cally, both residual distributions for latitude and longitudebecome equal and they pass simply to reflect the spatialdistribution of the Brazilian cities.The approximate logarithmic decay is very intriguingand it can be modeled on the theoretical canvas of the C o rr e l a t i on , C ( r ) Distance, r (km) Logarithmic decay
Fig. 4: Slow decay of the spatial correlations in lightning ac-tivity rates. Dependence of the spatial correlation function C ( r ) on the distance r (circles). We note that for smalldistances ( r ≲
40 km) the correlation remains almost con-stant. We next observe an approximate logarithmic decay,where C ( r ) ≈ −C ln ( r /R) describes the correlation functionfor 60 km ≲ r ≲
600 km (shaded area) with
C = .
19 and
R = .
32. For r ≳
600 km, we note a faster-than-logarithmdecay.
Edwards-Wilkinson equation [29] ∂∂t L ( r , t ) = D ∇ L ( r , t ) + η ( r , t ) , (5)where L ( r , t ) represents the lighting rate in a point local-ized by the vector r , D is a constant, ∇ is the Laplacian,and η ( r , t ) is an uncorrelated noise (in space and time)with zero mean and finite variance. Because this equationis mainly used to describe stochastic kinetics of a growinginterface, it may appear as an ad hoc hypothesis for mod-eling our data. However, the Edwards-Wilkinson equationcontains two important phenomenological ingredients re-lated to the spatial distribution of the lightning activity:diffusion and randomness. The former (Laplacian term)can be related to the diffusive aspects of the thunderstormsand the second (noise term) reflects the natural complex-ity of the lightning phenomenon (as well as other sourcesof randomness). Nonetheless, modeling the spatial distri-bution of lighting rates only with this equation representsa quite crude approximation where several important pro-cesses are not considered. For instance, we can not expectthis simple model to describe the residual distributions.On the other hand, the autocorrelation function relatedto eq. 5 can be obtained in d dimensions by taking Fouriertransforms in space and time, evaluating the correlationin this double-transformed Fourier space, and next return-ing to the usual space via inverse Fourier transforms (seeRefs. [29–31] for more details). For equal-times and in theequilibrium ( t ≫ R / D , see below), the correlation is writ-ten as C ( r ) ∼ r − d when d ≠
2, and for the two-dimensionalcase (that is, for our case) it is C ( r ) ≈ −C ln ( r R ) for r min < r < R , (6)where C > r min is a short-distance cut-offp-5. V. Ribeiro et al. and R is the length of the system. We thus observe thatthe logarithm decay is intimately connected with the two-dimensional nature of the problem. Furthermore, it isequally intriguing that similar logarithm decays were re-ported in very different context related to turnout ratesin elections [32, 33] and spatial spreading of obesity andother diseases [34], which can be somehow related to rem-nant effects of a universal mechanism underlying thesetwo-dimensional processes. Summary. –
We characterized the lightning activityrates over all the Brazilian cities by taking their geographiclocation into account. We firstly studied the dependenceof the lightning activity rates on the latitude of the cities,where we observed two peaks in the average value of theserates: one just before the Tropic of Capricorn and an-other one just before the Equator. We also investigatedthe fluctuations surrounding the average tendency, wherewe found that the standard deviation is almost a constantfunction of the latitude and that the distribution of thesefluctuations (residuals) is in quite good agreement witha Gumbel distribution. The dependence of the lightningactivity rates on the longitude of the cities was also charac-terized, where we reported that the average value exhibitsan approximate plateau for a wide range of longitude val-ues. For the longitude case, the standard deviation of theresiduals also presents an almost constant behavior; how-ever, the distribution of the residuals was well described bya Laplace distribution. We further investigated the spatialcorrelations of the lightning activity rates between a pairof cities. Our results show that the correlation functiondecays very slowly with the distance between the citiesand that for intermediate distances (60 km ≲ r ≲
600 km)the correlation exhibits an approximate logarithm decay.Finally, we proposed to model the logarithmic decay viathe Edwards-Wilkinson equation, where we observed thatthis behavior is closely related to two-dimensional natureof the system and also that the same behavior was re-ported in a very different context, which somehow pointsto a universal mechanism. ∗ ∗ ∗
We thank Capes, CNPq and Funda¸c˜ao Arauc´ariafor financial support. H.V.R. is especially grateful toCapes/Funda¸c˜ao Arauc´aria for financial support undergrant number 113/2013.
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