2, rue d'artois, F-75008 Paris http://www.cigre.org C4-304 Session 2004 CIGRÉ STATISTICS OF LIGHTNING OCCURRENCE AND LIGHTNING CURRENT S PARAMETERS OBTAINED THROUGH LIGHTNING LOCATION SYSTEMS baran@el.poweng.pub.ro Ileana Baran, D. Cristescu, C. Gary 2, A. Voron 3, G. Berger 4 University Politehnica of Bucharest, Bucharest, Romania, 2 Electricité de France, Paris, France 3 Météorage, France 4 Supelec, France. Introduction Lightning data have been collected by the French Lightning Detection and Location Network FLDLN, the so-called Météorage network, over a region situated between 43-45 degrees latitude North and 4-8 degrees longitude East, during the entire year 999. For more clearness the above mentioned region was marked on the map of France reproduced in figure. The area of the observation region can be evaluated to 750 km 2. The events recorded by FLDLN are the return strokes in cloud-to-ground flashes. Each event recorded in the database receives an identification number and it is stored with the following attributes: date (year, month, day), time (hour, second), impact coordinates (longitude and latitude), polarity and crest value of the current. If several successive events are identified as Figure return strokes belonging to the same flash each Position of the observation region on the map of return stroke will receive the same identification France number, which will become the identification number of the flash. It must be noticed that the attributes stored for one event do not allow the identification of upward and downward CG-flashes. The sample of lightning data analysed in the paper consists in 02,985 Cloud-to-Ground (CG) flashes observed during 83 storms over the observation region. The sample can be considered as a large one although it represents only 6.9 % of the total number of CG-flashes recorded by Météorage during the entire year 999 (i.e.,490,80flashes).
Because of the large amount of data, the first step in the data mining process was to group the events, using the attributes stored for each registered return-stroke, in the following categories: Return strokes recorded in 83 storms K TOTAL =,490,80 Outside the observation region,387,86 events Inside the observation region N CG = 02,985 events Single (CG-S) 58,066 events Multiple (CG-M) 44,99 events negative 5927 positive 6,40 negative 43,43 positive 2,004 bipolar 3 502 Multiple CG flashes exhibiting () -all strokes of negative polarity, (2) -all strokes of positive polarity, (3) -flash containing strokes of both polarities (introduced in [2]). 2. Number of CG-flashes in a storm, N CG. The monthly level of the lightning activity during the period of observation can be evaluated using two global indicators, (i)-the number of storms developed over the entire territory during each month, and ii)-the total number of CG-flashes generated during a storm. Both indicators are represented in figure 2. As it could be anticipated, the summer months (June to September) exhibit a much intense lightning activity than the other months of the year. Both indicators manifest a seasonal trend that will be analyzed in a further research covering several years of lightning data. (c) Figure 2 Monthly variation of (a)-the number of storms registered during the 999 year, (b)-the mean value for the total number of cloud-to-ground flashes produced by a storm K TOTAL and for the fraction of flashes recorded in the region of interest, N CG ; (c)-n CG represented alone at an appropriate scale. In order to refine the description of the lightning activity intensity, the total number of CG-flashes observed during a storm, N CG, was treated as a continuous random variate. The empirical cumulative frequency distribution (CFD) and the theoretical cumulative distribution function (CDF) are both represented in figure 3. The fitted distribution is a bi-parametric Weibull with the probability density function (f) and the cumulative distribution function (F) f F CG c c c N CG NCG = exp b b b CG c N CG = exp b ( N ) ( N ) with b-scale parameter (387 CG-flashes) and c-shape parameter (0.55). The shape parameter is smaller than, therefore the probability density function is I-shaped (monotonically decreasing with the increasing of the variate). Consequently the N CG variate does not exhibit a "most probable value". The mean value equals 657 CG-flashes/storm, and with % probability a storm developing in the observed region can produce over 684 CG-flashes. () 2
Figure 3: Empirical (CFD) and theoretical (CDF) cumulative distributions for the number of CG-flashes produced by a single storm (ML standing for Maximum Likelihood) The number of CG-flashes divided by the area of the observation region gives the regional ground flash density (the annual mean regional value for the ground flash density, equals in 999,47 flashes/km 2 /year). Therefore, the results obtained for the variable N CG considered in terms of ground flash density allow analysis of short-term variation in ground flash flashing rates, over areas of immediate storm activity over short term (5 to 30 minutes) or longer periods of observation ( to 3 hours). 3. Polarity of CG-flashes. The proportion of positive flashes (single and multiple) observed in 59 storms is represented in figure 4.a where, for the P-chart: (i)-the plot points are the proportion of CG-flashes recorded in each storm, (ii)-the centre line is the mean value of the proportion computed with all the observations i.e. p=0,069; (iii)-the upper (UCL) and lower (LCL) confidence limit for each sample (flashes observed in a storm) are sample's size dependent and were computed with the following relations: LCL j = max UCL j = min { p ( p ( / n j, 0} p + ( p ( / n, { } j where, the sample size n j is the number of flashes recorded for the j-storm. The red (round) points correspond to samples for which the observed proportion is outside the accepted band. In relation with the P-chart, the histogram of the actual values of the proportion is also represented in figure 4.a. The overall mean value of the proportion is 0.069 positive CG-flashes (6.9 %), but 85 of the actual values (55.2 %) fall outside the confidence limits computed with (2): for 36 values (23.4 %) p < LCL and for 49 values (3.8 %) p > UCL. It must be noticed that the overall mean value results from 0.6 % single positive CG-flashes and only 2.5 % positive multiple CG-flashes, suggesting that single positive CG-flashes are 4 times more probable than multiple ones. It is obvious that the proportion of positive CG-flashes exhibits a great variability, which cannot be explained only by a seasonal trend. (2) 3
(a) Figure 4: Number of positive CG flashes: (a)-proportion observed in 59 storms (classified in ascending order of N CG ), (b)-scatter plot of single N CG -S and multiple N CG -M positive flashes versus the total number of CG flashes in a storm (storms classified in ascending order of N CG ). (b) The number of positive single and multiple CG-flashes recorded in each of the individual storms is represented in figure 4.b versus the total number of CG-flashes, together with the lines having the slope equal to the mean values of the proportion of single and multiple positive flashes. The general mean value obtained for the proportion of positive flashes (i.e. 6.9 %), can be compared with the corresponding overall mean reported in [] (i.e..4 %), based on 3226 downward CG-flashes observed in different geographical regions, values of the proportion ranging from 0 % to 9. %. 4. Incidence of single stroke CG-flashes. For the representative data summarized in [] the proportion of single flashes (all types) has an overall mean value of 45 % resulting from 6428 observed CG-flashes grouped in 7 samples, each sample coming from a different geographical region. It must be noticed that the reported incidence of single flashes varies from 3 % in Southern Africa to about 76 % in Switzerland. The evident considerable dispersion might arise out of several causes such as the variety of measuring methods adopted, which imply different range sensitivity, or the influence of geographic and climatic conditions. For the data analyzed in the present paper, the proportion of single CG-flashes was computed for each sample observed in 38 storms and represented in figure 5 together with the upper and lower confidence limits given by relation (2). Figure 5: Variability of single CG-flashes proportion in 38 storms (storms classified in the ascending order of N CG ) The resulting overall mean value of the proportion is 0.563 (56.3 %) to be compared with 45 % in []. In the observed storms, the proportion varied between 0 (no single CG-flashes) to 0.98 (storms for which practically all CG-flashes were single). It must be also noticed that the single CG-flash proportion displayed a great variability: in 32 storms (20.8 %) the proportion was smaller than LCL and in 45 storms (29.2 %) greater than UCL. Further research will try to analyze the possible seasonal trend or geographical dependency of the proportion. 4
5. Distribution of the number of strokes per flash. It is already well established that in a CG-flash the first stroke can be followed by several subsequent strokes. As far as practical engineering systems are concerned, an adequate knowledge of the probability of occurrence of a certain number of strokes in a flash is clearly required. The number of strokes per flash (N SF ), can be considered as a random variable of the discrete type (A- the support of the variable being the subset of positive integers). For such a random variable, the induced probability (X = n) of the event (X=n) is called the probability density function (hereafter abbreviated PDF), a real-valued function f (, n N that satisfies the following properties: n= ( n) > 0, f ( n) f = (3) The induced probability (X n) of the event (X n) defines the function F(n) by F ( n) = ( X n) = f ( k) n (4) k= F(n) is a step function called cumulative distribution function (abbreviated CDF). The events (X > n) and (X n) have the induced probability given by ( X > n) = F( n) = f ( k) n k= n ( a) ( X n) = F( n) + f ( n) = f ( k) ( b) k= In a first approach, the random variable N SF was analyzed using a sample that includes all types of CG-flashes. The type (b) induced probability defined in (5) and denoted as CFD*, is represented in figure 6 together with the corresponding distribution, based upon 6428 observations, given in []. (5) Figure 6: Complement of the observed cumulative frequency distribution (CFD*) and of the proposed theoretical cumulative distribution function (CDF*) for the number of strokes per flash (all flashes disregarding the polarity of component strokes). Direct comparison between the two empirical distributions displays some important differences. To begin with, the weighted mean reported in [] is 3.32 strokes/flash and.96 strokes/flash in the analyzed sample. In second place, the probability of having more than 0 strokes/flash is about 5.0 % in [], a value 00 times greater than the value 0.43 % for the analyzed sample. To find more about the distribution of N SF, we tried to identify a theoretical distribution to fit the data. The tested model was the geometric distribution, as it is known that for uniformly distributed random data, the probability that a certain event completes a cycle of length n is given by the geometric distribution characterized by, 5
Geometric Distribution ability Function (PDF) Distribution Function (CDF) ( N n) SF Complement to unity = = n ( NSF = n) = p ( n ( NSF n) = ( ( NSF n) + ( NSF = n) n n ( + p ( = where p is the reciprocal of the number of uniformly distributed states. The empirical distribution function was fitted (in the least square approach), with a distribution resulting from the mixture of two geometric distributions represented by the equation: CDF ( N SF n) = CDF = a CDF+ ( a) CDF = p ( p ) n ; CDF 2 = p ( p ) n 2 2 with the mix proportion a = 0.62 and the parameters of the two mixed geometric distributions p =/2.40 for CDF and p 2 =/.24 for CDF2. The complement to unity for both distributions was represented on figure 7 by the curves CDF* and CDF2*. On the same figure the curve CDF* is the complement to unity of the resulting distribution function (CDF in relation (6)). A possible reason for the mixed distribution can be understand by means of empirical distributions represented in figure 7, for flashes having negative or positive polarity (the bipolar flashes being ignored). 2 (6) Figure 7: Complement to unity of the observed cumulative frequency distribution for N SF in positive and negative flashes. As it can be seen, the negative flashes exhibit more strokes per flash than positive flashes (with a probability of about 0.02 % a negative flash can have more than 5 strokes while a positive flash can have more than 4 strokes). Further research will be needed to identify the underlying process responsible for this behavior. 6. Crest value of stroke current The present analyze covers only two groups of data: single strokes and first strokes in multiple negative flashes. Both empirical distributions are represented in figure 7 together with three theoretical representations based on observations on the peak current amplitudes for negative downward flashes to structures less than 60 m height, namely: a)-the interpolating function introduced in [], consisting of two segments originating from the following log-normal distributions: Validity interval Imax < 20 ka Imax > 20 ka Parameter Scale parameter (ka) 6 3 Shape parameter (log 0 units) 0.753.925 6
b)-the distribution derived in [3], which is a mixture of two elementary log-normal distributions with the composition law given by f ( IMAX ) = a f + ( a) f 2 with the mix proportion, a = 0.0, and the shape and scale parameters as follows: Scale parameter (ka) 0.50 36.45 Shape parameter (log 0 units) 0.097 0.90 c)-the distribution adopted by the IEEE, which is not strictly a distribution function, but it is easy to use: ( I MAX ) = + ( I MAX 3) 2. 6 It must be noticed that all quoted distributions are in good agreement with the empirical frequency distribution to which they are fitted. Figure 8: Complement to unity of the observed cumulative frequency distribution for the lightning current crest value, single and first return stroke in negative flashes Each of the cumulative frequency distributions based on the present sample of lightning data has its distinctive features, and both are dissimilar to the accepted distributions. None of the two empirical distributions could be represented by a lognormal approximation or other skew distribution (for example Weibull), but the sample median value and the 95 %-quantiles appear in the table in figure 8, and can be compared one to the other and both with the corresponding values obtained from the different theoretical distributions. Beside the lognormal distribution inferred using the maximum likelihood procedure, an attempt was made to obtain a robust estimation for the cumulative frequency distributions of the first and subsequent return strokes in negative flashes. The robust fit procedure uses an iteratively reweighted least squares algorithm, with the weights at each iteration calculated by applying the bisquare function to the residuals from the previous iteration. This algorithm gives lower weight to points that do not fit well. The results are less sensitive to outliers in the data as compared with ordinary least squares regression. The results are represented in figure 9 for the first return stroke (component) and subsequent return strokes 2 to 4. Both distributions are represented in figure 9 together with the parameters (mean and scale factor, µ and σ) It is obvious that the current's distribution for the first return stroke is not a lognormal one, and a first hypothesis that should be checked is the hypothesis of a mixture of two or more elementary distributions following the model identified in [3]. 7
Figure 9 Observed cumulative frequency distribution for the lightning current crest value, first return stroke and subsequent strokes in negative flashes The distributions inferred for subsequent strokes are closer to log-normal distributions but they exhibit deviations in the upper tail of the distribution. The parameters of the log-normal distributions are given in the table bellow, for each of the 5 strokes identified. Significant differences can be noticed only between the mean value of the first and subsequent strokes distributions. 7. References [] R. B. Anderson, A. J. Eriksson, Lightning parameters for engineering applications. ELECTRA, no.69, pp.65-02, 980. [2] K. Berger, R. B. Anderson, H. Kröninger, Parameters of lightning flashes. ELECTRA, no.4, pp.23-27 [3] Ileana Baran, D. Cristescu, C. Bouquegneau, Lightning peak current amplitude (I PEAK ) and impulse charge (Q IMP ) a new proposal for probability distribution functions Proceedings 0 th International Symposium on High Voltage Engineering, Montreal, 997. 8