## Abstract

We present and analyse the collective statistical results of a long-term field study quantifying the number of lightning strikes to buildings equipped with lightning protection systems. The observations were made over the period 1990–2003 in a region with one of the highest lightning activities in the world. The data sample comprised 86 structures with a mean height and exposure time of 57 metres and 6.9 yr, respectively. These structures were subjected to 384 flashes over the total observation time of 592 yr. The positioning of the protective air terminals on these structures was carried out with an improved electrogeometric model of lightning attachment.

The collective observational data have been compared with the expected number of incident flashes using a variety of analytical relations and statistical tests. The expected strike frequency was determined with three different, contemporary models of lightning attachment to structures. The protection level or interception efficiency estimated from the models is compared with the actual value derived from the long-term field observations involving measurements of captured strikes and evidence of lightning by-pass events.

The results show that the observational data used in this study are well-described by a Poisson distribution. There is a highly significant positive correlation between the observational data and the number of strikes expected from the application of the theoretical models. Finally, the observed and expected values for the mean interception efficiency of the lightning protection systems in the study are shown to be in good agreement.

## 1. Introduction

### (a) Scope

An important step in the provision of an effective lightning protection system (LPS) for structures is the positioning of the air terminals or air terminations. Logically, this step is based on the interception protection provided by each terminal. At present, a number of methods are outlined in national and international standards for the placement of air terminals. These include the ‘Protection Angle’, ‘Mesh’ and ‘Rolling Sphere’ methods, as described by Golde (1977), Lee (1978, 1979) and Szczerbinski (2000).

Over the last two decades, new data on lightning parameters and long sparks have become available. Using these new data, a number of workers have developed more detailed models of the lightning attachment process. These models incorporate the role of upward leaders, dependence of striking distance on structure geometry in addition to the downward leader charge, definition of attractive radius, tortuosity of the downward leader path, etc. Some examples of these new models include the improved electrogeometric model (EGM) (Eriksson 1987*a*,*b*), ‘leader progression model’ (Dellera & Garbagnati 1990; Bernardi *et al*. 1996), ‘leader inception theory’ (Rizk 1994*a*,*b*), ‘critical range of field intensification’ (Petrov & Waters 1995; Petrov *et al*. 2000) and ‘stabilization field’ (Bondiou-Clergerie *et al*. 1999; Lalande *et al*. 2002). The results of these investigations hold great promise for the future in terms of more comprehensive modelling of the lightning attachment process.

On a more practical level, it is rather surprising to find that there have been no systematic, quantitative field studies of the effectiveness of interception protection provided by air terminals. A small number of descriptive studies have been carried out on the topic of lightning damage and the occurrence of interception failures, or ‘by-passes’, of LPSs. For example, Golde (1977) mentions nineteenth century studies of damage caused by lightning that by-passed air terminals and then only refers to systematic studies in Germany reported by Walter (1937). More recently, a photographic survey in Malaysia has added to the database of descriptive information about lightning damage to structures and by-pass events (Hartono *et al*. 2001).

However, a thorough literature search going back to 1880 reveals that no quantitative study has been carried out which compares the ‘interception efficiency’ of a LPS with the level predicted or expected from the lightning attachment model used to position the protective air terminals of the LPS. Here, interception efficiency is simply defined to be the proportion of lightning flashes intercepted by the LPS out of the total number of flashes incident to the structure.

In a recent paper (Petrov & D'Alessandro 2002*a*,*b*), we reported on a long-term field study of lightning strikes to structures in Hong Kong. In that study, no by-pass data were available, so the analysis was restricted to a comparison of the strike data with ‘attractive radius’ models of lightning attachment to structures. In essence, the expected number of strikes were compared with the actual number of incident strikes to the structures in the study.

In the present study, we have repeated this analysis on a new set of data which has been taken from field observations of lightning strikes to structures in Malaysia. The difference in this case is the availability of by-pass data, which has made it possible to analyse and compare the ‘theoretical’ and ‘observed’ interception efficiency of LPSs for the first time.

In the remainder of this section, we provide a brief description of the air terminal placement method and the hardware used to register the strikes. We also present some of the theory relevant to the analyses which follow.

### (b) Air terminal positioning method

The placement of air terminals on structures is commonly performed with the Rolling Sphere Method (Lee 1978, 1979), which is based on the simple EGM for striking distance (Golde 1977). The simple EGM assumes that the ‘striking distance’ is a function of only the prospective peak stroke current and so the RSM is relatively easy to apply in practice. However, the simple EGM does not account for the physical basis of the upward leader inception process, the importance of the structure height and the geometry of the launching point. Hence, an improved EGM must allow for the intensification of the ambient electric field created by the grounded structure, structural features and air terminals. Since the majority of lightning strokes terminate on the corners and nearby edges and sharp features of unprotected structures, it is important to assign a strike probability to prospective strike points on a structure using ‘electric field intensification’ as a fundamental parameter.

Such an improved EGM was presented by Eriksson (1979, 1987*a*,*b*) for power transmission line modelling. Eriksson's basic model has since been extended to common buildings through computer modelling of electric fields around practical, extended structures and by the application of the concept of ‘competing features’ (D'Alessandro & Gumley 2001). This concept considers all points on a structure as potential strike points that can compete with the air terminals. The capture volumes of the competing points are compared with those of the air terminals. The latter must overlap the former for a pre-specified interception efficiency.

The capture volume of each prospective strike point is determined by the ‘striking distance surface’, which relates to the level of protection afforded, and a leader propagation condition which defines a ‘velocity-derived boundary’. The volume defined by the intersection of the velocity boundary with the striking distance surface is known as the ‘collection volume’ of the point, as shown in figure 1. Hence, the method has become known as the Collection Volume Method (CVM). A more detailed description of the CVM can be found in D'Alessandro & Gumley (2001) and D'Alessandro (2003).

All of the LPS air terminals in this study were placed in optimum locations on the structures using the CVM. By ‘optimum’, we mean that the CVM was used to produce the most efficient positioning of the air terminals for the stated interception efficiency or *protection level*. The chosen interception efficiency is typically between about 80 and 99%. These values are derived from a standard cumulative frequency distribution of peak lightning stroke current, e.g. as given by Berger *et al*. (1975) or Anderson & Eriksson (1980). For example, approximately 91% of all strikes result in a peak current greater than 10 kA. On average, at least this portion of the strikes will be intercepted by the LPS. Up to, but not necessarily all of, the remaining 9% of low-intensity strikes may by-pass the LPS. This concept forms the most fundamental risk-management principle in lightning protection.

### (c) Strike counting method

The number of strikes to the LPS on each structure in this study were obtained from lightning event counters (LEC) placed around the conductor through which the lightning current is taken down to ground (the ‘downconductor cable’). The operation of a LEC unit is based on a current transformer principle, where the primary of the transformer is the lightning downconductor cable and the current in the secondary of the transformer, a specially designed ferrite-core inductor with known parameters, charges a capacitor. The charging process reaches a point where an electro-mechanical counter is incremented, but this only occurs if the current magnitude in the primary is beyond a minimum ‘threshold’ value. The basic principle is illustrated in figure 2.

All of the LEC units were tested and calibrated using an 8/20 μs current impulse. These tests showed that the units would not register an impulse in the downconductor cable if the current magnitude was less than 1.5 kA. The specifications of the electro-mechanical counters state that the maximum count rate is 10 Hz, i.e. strokes separated in time by less than 100 ms would not be registered. However, from a series of laboratory tests on a sample of counters and, more importantly, on a batch of LEC units as a whole, it was found that the counter response time was more short, i.e. 30±2 ms. The relevance of this parameter is discussed further and taken into account in §2*b*. The LEC units were not designed to record additional information such as the magnitude or polarity of the lightning stroke current.

### (d) Attractive radius concept

The concept of ‘attractive radius’ is often used to estimate the potential number of strikes to structures. The attractive radius depends on the parameters of the structure and the lightning discharge. For example, observations show that the number of strikes is a statistical quantity with a strong dependence on the random nature of the return-stroke current amplitude and on the structure height.

The ‘equivalent exposure area’ for a free-standing structure can be obtained from(1.1)where *f*(*i*) is the probability density function for the current amplitude distribution and *R*_{a} is the attractive radius of the structure. The average strike frequency (strikes per annum) to the structure can be calculated using the relationship(1.2)where *N*_{g} is the average ground flash density.

It is important to make a distinction between striking distance, *R*_{s}, and attractive radius, *R*_{a}. The striking distance is defined as the distance between the object to be struck and the tip of the downward-moving leader at the instant that the connecting (upward) leader is initiated from the object (Uman 1987). It depends solely on the charge of the downward leader and hence return-stroke current.

On the other hand, a calculation of attractive radius also takes into account the geometry of the object to be struck, i.e. the structure and air terminal. This is done by including physical considerations of the ionization processes of streamer and leader development (Petrov *et al*. 2000; D'Alessandro & Gumley 2001). In general, the attractive radius has a smaller magnitude than the corresponding striking distance, although the striking distance can be used in place of the attractive radius if only a rough approximation is required.

In this paper, the following relations for attractive radius will be evaluated against the field data.

Petrov & Waters (1995), Petrov *et al*. (2000):(1.3b)

Rizk (1994*a*,*b*):(1.3c)where *h* is the height of the structure in metres and *I*_{p} is the return stroke peak current in kA.

In the analysis presented in the remainder of this paper, equation (1.3*a*) will be used because of its direct relevance to the method used for placing air terminals, i.e. the CVM. However, the models of Petrov *et al*. and, where possible, Rizk, as shown in equations (1.3*b*) and (1.3*c*) shall also be compared with the experimental data.

## 2. Analysis

### (a) Description of the raw data

To meet the main goal of this study, it is clear that the field observations of lightning strikes to structures need to provide, as a minimum, a count of the incident strikes and by-pass events. Also, to achieve statistical validity, sufficient data need to be collected. This requires a high lightning area and a reasonably long period of time for the observations.

To meet these requirements, our study was based in Kuala Lumpur and nearby regional centres of the Klang Valley region of Malaysia. Klang Valley is ideal for such a study because it has one of the highest levels of lightning activity in the world. This region typically receives 15–20 ground flashes per square km per year (Yahaya & Zain 2000). Finally, all of the buildings in the study were equipped with LPSs and LEC units for measuring the number of incident strikes.

Table 1 summarizes the raw field data. A preliminary study on a subset of the present data was presented by D'Alessandro & Darveniza (2001). Four of the sites from the 2001 analysis have been removed from the sample due to a lack of continuing access to the LEC unit readings and three other sites were removed from the sample due to uncertainties regarding the shielding effects of adjacent, taller buildings. The present analysis is also more complete in the sense that the sample size has been expanded by 36% with the addition of 18 new sites (24 LPSs) and the mean observation period has been extended by 42% due to 3.5 yr of additional exposure time.

Of the 86 LPSs in the study, almost two-thirds of these had recorded one or more strikes over the observation period. Retention of the remaining one-third of zero-strike data is important because it is characteristic of the statistical nature of the lightning strike process.

### (b) Flash counting issues

In relation to the total of 817 counts registered by the LEC units, we need to consider the LEC characteristics described in §1*c*, together with the main lightning parameters, in order to arrive at the true number of flashes for the analysis. The relevant lightning parameters can be summarized as follows.

The multiplicity of negative lightning flashes is well known (Golde 1977; Uman 1987), i.e. a single, negative flash is comprised of a series of distinct ‘subsequent strokes’, typically 3 or 4. For the purposes of this paper, it is important that the counter readings represent flashes or first strokes, rather than all of the strokes in the flash.

According to published lightning data (Berger

*et al*. 1975; Uman 1987), the median time interval between subsequent strokes is approximately 33 ms. This median value implies that, on average, the LEC units will not register subsequent strokes as they have a 30 ms response time. However, we must consider the frequency distribution of these interstroke intervals. A median value of 33 ms implies that approximately 50% of subsequent stroke time intervals may exceed this value. Hence, we can expect to count up to half of the subsequent strokes.Typically, up to 10% of all flashes are positive, 90% of which display no multiplicity (Uman 1987).

It is known that approximately 50% of multiple strokes in a flash have multiple termination points, typically two (Rakov & Uman 1990; Ishii

*et al*. 1998). This will reduce the number of subsequent strokes registered at a single point.

These parameters and distributions can be used to arrive at a ‘counter factor’, *α*. Simple hand calculations using median values give counter factors in the range 2.0–2.2. Monte Carlo simulations were also performed (D'Alessandro & Darveniza 2001), using published frequency distributions of stroke multiplicity, interstroke interval and multiple termination points for subsequent strokes (Uman 1987; Rakov & Uman 1990; Ishii *et al*. 1998). From sets of several thousand trial flashes, the counter factor was found to range from 1.5 to a worst-case value of 2.8.

*Hence, a counter factor of α=2.5 was chosen as a conservative value*. The number of events registered by the LEC unit for each LPS must be divided by this factor in order to arrive at the true number of flashes. Unity LEC readings are retained as single-stroke flashes without any correction factor.

In an earlier paper (Petrov & D'Alessandro 2002*a*,*b*), it was explained that, for a ‘tall’ structure (typically greater than 100 m in height), lightning events may be counted that are due purely to upward flashes from the structure. In the present study, this issue is not so much of a problem because the mean structure height is considerably less than the 2002 study of strikes to structures in Hong Kong. Nevertheless, the same procedure was applied to the present data. This can be summarized as follows.

In Petrov & D'Alessandro (2002*a*,*b*), it was shown that the probability *p*_{u} of an upward flash from a structure may be expressed as(2.1)where *N*_{u} and *N*_{d} are the number of upward and downward flashes, respectively. The quantity *N*_{u} is a function of the critical electric field for upward leader inception and a shape parameter which depends most strongly on structure height, and *N*_{d} is computed using equations (1.1) and (1.2). Hence, the quantity *N*_{u} was calculated for each structure and subtracted from the measured number of flashes to each structure in order to arrive at the true number of downward flashes.

Therefore, before carrying out any analysis on the data, firstly we applied the ‘counter factor’ to remove subsequent strokes and count only incident flashes. Secondly, we corrected the data for the upward flashes that become more probable as the structure height increases. Hereafter in the paper, all references to ‘strikes’ and ‘strike statistics’ can be interpreted as meaning individual flashes or first strokes, obtained after appropriate corrections were made to the raw data.

### (c) Calculated protection levels

As noted earlier, the air terminals comprising the LPSs in this study were placed in the optimum locations on the structures using an improved EGM, namely the CVM. A CVM design for a structure produces the most efficient positioning for the stated interception efficiency or protection level. In practice, protection levels in the range 80–99% are desired.

Rather than relying on the assumed or desired protection level which had been documented at the time the LPS was installed, the analysis of interception efficiencies commenced with detailed CVM calculations for each LPS. In this way, a reliable estimate of the protection level for each LPS at each site was established. The required calculations made some assumptions, based on published parameters and the recommended implementation of the method (D'Alessandro & Gumley 2001). For example, the critical radius concept was used for the determination of leader inception of competing features (*R*_{c}=0.28 m) and air terminals (*R*_{c}=0.38 m). The installation height of all air terminals was taken to be 4 m, a typical height for CVM-placed air terminals. If only one air terminal was deemed to be required for a particular structure, it was assumed to be roughly centrally placed on that structure. Finally, the structures were assumed to be rectangular with no nearby shielding structures or features.

### (d) Occurrence of by-passes

The final task in the analysis was the identification of the number of by-passes on each building. The identification of by-passes was done by visual inspection of the upper periphery and top surface of each building.

All of the buildings in this study were selected so that they were of an extended nature, i.e. of sufficient width so that a by-pass is reasonably probable and measurable. Also, all the structures were of masonry construction. The latter point is important because of the method of identification of by-passes, i.e. by visual inspection, where a by-pass event delivering kiloampere current levels can leave evidence behind. This evidence is typically in the form of minor dislodgement or cracking of some masonry (Wilson 2003). The damage point may be in the size range of millimetres to centimetres or, in the case of very large lightning strikes, much more extensive and obvious.

In the present study, we assumed that one such point corresponds to a unique by-pass event. This is the most conservative scenario, as discussed further in §4. A total of 48 by-passes were identified on this basis. All of the by-pass observations were checked against the photographic survey and documentation of Hartono *et al*. (2001). At the time of completing the data collection, the two sets of observations were in complete agreement for the cases where the two datasets overlapped.

Of the 86 buildings in the study, 27 had received a by-pass. In the majority of these cases, there was only one by-pass. However, one of the buildings accounts for almost one quarter of the 48 by-passes, which raises the question as to whether 11 unique by-passes caused the 11 damage points or a smaller number of by-passes created multiple damage points. This aspect is also discussed further in §4.

### (e) Probability-weighted average attractive area

In order to compute the expected number of strikes to a free-standing structure, the equivalent exposure area, *A*_{eq}, of the structure is required. This can be obtained from equation (1.1). If the current amplitude is log-normally distributed, the probability density function is given by(2.2)where *Ī* is the median current and *σ* is the standard deviation of the natural logarithm of the current amplitudes. The applicability of the log-normal distribution has been well documented (Anderson & Eriksson 1980), though there may be some deviation in the low-amplitude range. Note that the *mean* current, 〈*i*〉, for this distribution is larger than the *median* current. For the values *Ī*=31 kA and *σ*=0.7368 (Anderson & Eriksson 1980), we find that 〈*i*〉≈40 kA.

The attractive radius *R*_{a} depends on the structure height *h* and prospective peak stroke current *I*_{p}. Substituting equations (1.3*a*) and (1.3*b*) from Eriksson (1987*a*,*b*) and Petrov *et al*. (2000), respectively, into equation (1.1) and using the values *Ī*=31 kA and *σ*=0.7368, we find that(2.3a)(2.3b)and from equation (1.3*c*) of Rizk (1994*a*,*b*),(2.3c)Hence, the expected number of downward lightning strikes *per year* to a structure of height *h* can be expressed as(2.4a)

(2.4b)

(2.4c)

The probability *p*(*i*_{min}<*i*<*i*_{max}), where(2.5)is of practical interest. Calculations show that *p*=0.999 for lightning currents in the range 1<*i*<300 kA. Hence, the scatter in the number of strikes due to the spread of lightning currents can be calculated for different structure heights. The spread of values is defined as .

### (f) Summary of the final dataset

Table 2 presents a final, condensed dataset after the relevant corrections, as described above, were carried out on the raw field data. Table 2 also presents the estimated protection levels obtained from the CVM calculations for each LPS and the observed interception efficiency, for ease of comparison. The data have been binned and summarized according to structure height. It can be seen that, for the 86 LPSs in the sample, there were 336 captured strikes and 48 by-passes, i.e. 384 incident flashes, over a total observation period of 592 yr.

In the remainder of the paper, these data will be compared with the theoretical number of incident flashes according to the models of Eriksson, Petrov *et al*. and Rizk using a variety of analytical relations and statistical tests. Also, the interception efficiency, as estimated from the method used for positioning the air terminals comprising each LPS, will be compared with the actual value derived from the long-term field observations.

## 3. Results

### (a) Average parameters

Different average parameters may be obtained from the data. For example, it is possible to obtain an indirect estimate of the ground flash density, *N*_{g}. It follows from the data that 86 structures with an average height *h*=57 m were subjected to *N*_{d}=384 flashes. The total time of observation was *T*=592 yr. The actual number of strikes per annum, is therefore into an area . The area *A*_{eff} corresponds to the attractive area of a single air terminal with an effective height *h*_{eff}, which is struck 384 times over a period of 592 yr.

Using *N*_{g}=15–20 km^{−2} yr^{−1} gives *A*_{eff}=0.03–0.04 km^{2}. These estimates can be compared with the theoretical attractive areas given by equations (2.3*a–c*). The theoretical attractive areas are 0.082, 0.046 and 0.102 km^{2}, respectively. Hence, the attractive radius model of Petrov *et al*. (2000) is in closest agreement with the averaged data. If the ground flash density in the region of study in Malaysia is closer to 10 km^{−2} yr^{−1}, then the model of Eriksson (1987*a*,*b*) would be in closer agreement with the data.

It follows from the field data presented in table 2 that the number of strikes to the area occupied by the buildings in the study is substantially larger than the number of strikes that would be incident on the same area of the ground in the absence of the buildings. Therefore, the concept of a ‘coefficient of attraction’, *k*_{a}, may be introduced for the buildings. This coefficient is simply defined as the ratio of the actual strikes to the buildings per year, to the number of strikes per year *n*_{g} into the area *S*_{Σ} occupied by the buildings. The total area of the building roof surfaces in the present study is *S*_{Σ}=0.419 km^{2}. Taking *N*_{g}=15 km^{−2} yr^{−1}, the number of strikes per year into this area is equal to *n*_{g}=*N*_{g}*S*_{Σ}=6.29 yr^{−1}. The actual number of strikes per year is . Thus, the coefficient of attraction .

Finally, the total attractive area is given by and the mean attractive area, . Hence, the mean attractive radius is . The model equations (1.3*a*,*b*) can now be used to make a comparison with this result. For Malaysia, the measured median current is *ca* 33 kA (Yahaya & Zain 2000). Taking this median current and a mean structure of height 57 m, *R*_{a}=126 and 99 m from the Eriksson (1987*a*,*b*), Petrov & Waters (1995) and Petrov *et al*. (2000) models, respectively. Hence, both models are in reasonably good agreement with the averaged data.

### (b) Comparison of actual and expected strikes

In simple terms, a basic measure such as the ‘yield’ of a LPS can be defined, namely the ratio of *actual* to *expected* strikes. The total number of actual strikes is already known, namely *N*_{d}=384. One approach to obtain the total number of expected strikes, *N*_{e}, to the building LPSs is to calculate the attractive area using equations (2.3*a–c*). In this case, *N*_{e}=Σ*n*_{st}[*s*], where *n*_{st}[*s*]=*N*_{g}*A*_{eq}[*s*]*T*[*s*] is the number of strikes to the structure *s* and *T*[*s*] is the period of observation for that structure. A second approach is to use equations (1.3*a–c*) to obtain the expected strikes *N*_{e_I}, using the known, median value for the peak stroke current in Malaysia, i.e. *ca* 33 kA, as noted above. The total number of expected strikes was computed using both approaches and two values of flash density, namely 15 and 20 km^{−2} yr^{−1}. The results for both approaches are displayed in table 3.

In the first case, the value for *N*_{e} derived from the Petrov *et al*. (2000) model is in best agreement with the actual number of strikes for a flash density *N*_{g}=15 km^{−2} yr^{−1}. The models of Eriksson (1987*a*,*b*) and Rizk (1994*a*,*b*) both predict a larger number of strikes. In the second case, for the same flash density, the value *N*_{e_I} from Eriksson's (1987*a*,*b*) model is in best agreement with the data. The Petrov *et al*. (2000) model is in very good agreement with the data when a flash density of 20 km^{−2} yr^{−1} is used. The difference between the results can, in part, be attributed to the different leader inception criteria that are employed by the models, namely, the ‘critical radius concept’ (Eriksson 1987*a*,*b*), ‘critical range of field intensification’ (Petrov & Waters 1995) and ‘leader inception theory’ (Rizk 1994*a*,*b*).

All three theoretical models predict an increase in the number of strikes with increasing structure height. Table 4 shows the observed dependence of the number of strikes on the structure height. The structure heights have been binned into three groups such that each of the groups has a similar observation period of approximately 200 yr.

### (c) Lightning strike frequency: Poisson distribution

We can apply a range of statistical tests to the lightning strike data in order to assess whether they conform to a well-known theoretical distribution. Further details regarding the tests applied hereafter can be found in Press *et al*. (1989), Neave (1979), Bury (1999), Conover (1998) and Frank & Althoen (1994).

The probability *p*(*k*) that a given area will be struck by lightning exactly 0, 1, 2, …, *k* times over a given period of time is usually expressed by the Poisson relation,(3.1)where *μ* is the average number of strikes to the given area *A*_{eq}. The average number of strikes may be estimated as the selective mean,(3.2)where the number of protected structures *n*_{k} which were struck *x*_{k} times per year is presented in table 5.

We shall attempt to verify the hypothesis that the data follow a Poisson distribution. The Poisson distribution is described by one parameter only, since the mean value *m*_{x} is equal to the dispersion *D*_{x}. Using the data in the present study, we find that the mean value is , and the dispersion is . These values are in close agreement and so it appears that the Poisson distribution hypothesis may be accepted. Comparing actual and theoretical data using Pearson's criterion (see table 5), we find that *Χ*^{2}<*Χ*_{cr}^{2}(0.05;3)=7.8. Hence, the Poisson distribution hypothesis can be accepted.

Kolmogorov's criterion provides a simple way of verifying the correspondence of the actual data to a theoretical distribution. To apply this criterion, it is necessary to calculate the difference between the actual and theoretical distribution functions, namely,(3.3)The value *X* has the distribution *F*^{*}(*x*) if the probability(3.4)is high. The parameter *λ* for the Kolmogorov distribution is given by .

Table 6 shows the computed values for the difference between the actual data and the assumed Poisson distribution (*D*_{P}). In our case, *λ*=0.11 and *p*(*λ*)≈1.000, so the Poisson distribution hypothesis may be accepted. This result is consistent with other data comparisons, e.g. Miyazaki *et al*. (1986).

On this basis, Poisson statistics may be used to predict the number of strikes to a structure of a given height. This is illustrated in figure 3 for a ground flash density of 15 km^{−2} yr^{−1} and the mean observation of the present study. For example, a structure of height 60 m is likely to be struck about five times during a 7 yr period in a region with this level of lightning activity.

### (d) Correlation

The data clearly show that the number of captured strikes increases with structure height. Therefore, there is an implicit correlation between the raw data and model predictions. However, formal calculations of correlation coefficients are useful for quantifying and assigning a level of significance to the relationship between the actual and expected strikes. We computed these coefficients using Pearson's linear, *r*_{P}, Spearman's rank, *r*_{S}, and Kendall's Tau, *r*_{K}, methods.

The Pearson linear correlation coefficients for different height ranges are shown in table 7. A positive correlation is significant for all height ranges with a confidence of 95% or better. Table 8 presents the results of the rank correlation analysis for different height ranges. In general, these correlation coefficients are also significant. However, unlike the linear correlation, the rank methods resulted in a relatively low correlation coefficient when only tall structures were considered. A scan of the data for the 15 structures of height greater than 80 m reveals that there are three LPSs for which no strikes have been registered over a mean period of 9 yr and another five LPSs that have registered only 1 or 2 strikes. Clearly, the small number of data points for this range, plus the fact that half of the sites have received an unexpectedly low number of strikes has affected the rank correlation probabilities.

### (e) Interception efficiency

In general, a valid statistical analysis requires greater than 20 data points. Therefore, to work with strike data from an individual site or structure would require decades of field observations. Even sub-groupings such as those used in table 2 show that analysing only small sections of the data is not appropriate. Hence, to make a comparison of the observed interception efficiency with the desired or estimated value, we need to use the entire dataset.

From the data presented in table 2, it can be seen that the total number of captured strikes was 338 and the number of by-passes was 48 over the observation period. Using these values, the observed interception efficiency *E*_{i} is determined simply from(3.5)where *N*_{c} is the number of captured strikes and *N*_{b} is the number of by-passes (*N*_{c}+*N*_{b} is the total strike incidence). Hence, *E*_{i}=87.5%. This result can also be expressed as a ‘by-pass probability’, namely (100−*E*_{i})=12.5%.

From table 2, the mean value calculated for the interception efficiency or protection level across the whole sample, using the CVM, is 80%. Conversely, we can say the maximum number of ‘allowable’ by-passes is about 20% of the incident strikes. As noted in §1*b*, the contemporary understanding of interception efficiencies and by-pass probabilities is that anywhere *up to* 20% can be expected as the proportion of by-passes. Hence, it can be seen that the field data and the calculations are in good agreement in terms of interception efficiency and by-pass probability.

The protection level may also be estimated from the theoretical models via the integral in equation (2.5). In terms of the interception efficiency of any LPS, there is a current *i*_{min} below which a stroke may not be intercepted. Assuming this minimum current to be the one that results in a total capture area of the air terminals that only just overlaps the roof surface of the building, then(3.6)where *w* is the width and *l* is the length of the building. Then, substituting *I*_{p}=*i*_{min} into equation (1.3*a*) for the Eriksson and (1.3*b*) for the Petrov *et al*. models, rearranging to obtain *i*_{min}, and substituting this value into equation (2.2), the probability can be computed.

Using the median current of 33 kA for Malaysia (Yahaya & Zain 2000), we obtain a probability of 0.14 and 0.17 from the Eriksson and Petrov *et al*. models, respectively. In other words, the estimated protection level from these theoretical models are 86 and 83%, respectively, both in excellent agreement with the observed protection level of *ca* 87%.

## 4. Discussion

### (a) Ground flash density and its scatter

We have seen above that, to some extent, the comparison of models with field data relies on the value of ground flash density used. For the region of Malaysia under study, we have used the published value of 15–20 km^{−2} yr^{−1} (Yahaya & Zain 2000). It is also important to consider the variability in this parameter. In Petrov & D'Alessandro (2002*a*,*b*), it was shown that a Poisson distribution adequately described the ground flash density and, on this basis, the uncertainties in the estimation of ground flash density were determined after taking into account factors such as year-to-year variability and the observation time span.

The variability in the ground flash density due to the probabilistic nature of lightning occurrence is determined by the dispersion,(4.1)where and . Here, the probabilities *p*_{i} are expressed by the Poisson equation (3.1). Equation (4.1) shows that the average value is equal to the dispersion, and so(4.2)It is also important to consider the scatter of attractive area due to the spread of lightning currents. As for the case of ground flash density, this may be defined as(4.3)where *A*_{eq}, *R*_{a} and *f*(*i*) are given by equations (1.1), (1.3*a*,*b*,*c*) and (2.2), respectively. It is now possible to determine the scatter of lightning incidence, using equation (1.2). The scatter in the number of strikes is given by(4.4)

However, it is known that the values of a random parameter such as *N*_{d} only occasionally overstep the limits of the ‘3*σ* rule’ (van Belle 2002), namely 〈*N*_{d}〉±3δ*N*_{d}, where δ*N*_{d}=√〈*N*_{d}〉. It can be assumed that all of the values fall within this range. So, for the present study, *N*_{d}=384±60, i.e. between 324 and 444 strikes. In table 3, it can be seen that the expected number of strikes according to the model of Petrov *et al*. (2000) falls into this range. In the case of Eriksson (1987*a*,*b*), it is only marginally outside this range.

If we consider the data on a structure-by-structure basis, we also find that the observed and expected number of strikes are in good agreement with the statistical bounds. Figure 4 shows the comparison for ground flash densities of 15 and 20 km^{−2} yr^{−1}. Additional scatter of lightning incidence is observed in one case (for *N*_{g}=15 km^{−2} yr^{−1}). Isolated cases such as this one can be attributed to factors such as ground surface relief, the effect of surrounding structures and measurement uncertainties.

Hence, the local scatter of strikes for a given structure height may be explained by the probabilistic nature of the lightning stroke current and the number of strikes to a given area. On a larger scale, a process such as cloud-to-ground lightning strikes is also going to exhibit significant statistical variability. This variability can be temporal, such as the variations seen from one season to the next. It may also be spatial in nature, which accounts for common observations such as two structures or power transmission lines, in relatively close proximity, receiving a completely different number of strikes over a given period of time.

### (b) Interpretation of captured strikes and by-passes

The area of Malaysia where this field study was conducted has one of the highest frequencies of lightning flashes to ground in the world. This fact makes the region an ideal ‘live laboratory’ for studies of lightning attachment to structures. However, if the goal is to conduct a detailed analysis on the results for one particular structure, then more years of data collection are required. The variability seen in a stochastic process such as lightning strike incidence means that a statistically valid number of strikes to that structure must be reached. For example, a study involving a single structure may require an exposure period of 30 yr, whilst a study on a sample size of 100 structures might only require a few years of exposure.

So, for the purposes of the present study, all of the strike data were combined to allow the analyses to be made with statistical validity. The inherent statistical assumption in this approach is that all the data from the buildings belong to the same ‘population’ and that no buildings have features which differentiate them significantly from the rest of the buildings. On this basis, the uncertainty in the data for ‘captured strikes’ would be dominated by the statistical fluctuations quantified in §4*a*. The uncertainty in the value of 2.5 used to determine the number of flashes per count for the LEC units was discussed in D'Alessandro & Darveniza (2001). It was estimated to be no more than 25%, i.e. considerably less than the *N*_{d} statistical fluctuations expected in the strike process itself.

The matter of by-pass damage points in the case of non-intercepted lightning flashes is more difficult to assess. In §3*e*, each damage point was assumed to be a unique by-pass event. But, it is clearly possible that a single non-intercepted flash with multiple strokes may cause more than one damage point if wind or some other process caused the arc channel and the termination point to move between strokes. This is a well-known phenomenon, e.g. see Rakov & Uman (1990) and Ishii *et al*. (1998). Therefore, there is uncertainty in the actual number of by-pass events. It may be less than the 48 events counted and used in the analysis. Therefore, it is important to note that we have taken the most conservative or ‘worst-case’ value for the observed by-pass probability and, hence, interception efficiency.

Noting the above issues, the expected by-pass probabilities which may be estimated from the Petrov *et al*. model (17%), Eriksson model (14%) and CVM design (less than 20%) are in good agreement with the observed value of 13%. Hence, it can be stated conservatively that the mean interception efficiency or protection level of the LPSs in this study, namely 87%, lies in the range of Levels III–IV according to the IEC standard on lightning protection (IEC 1990).

### (c) Influence of positive flashes

Summer thunderstorms typically produce 1–15% of cloud-to-ground flashes which are positive (Uman 1987). In winter storms, positive discharges can range from 10% to more than 50% of all cloud-to-ground flashes (Goto & Narita 1995). In general, the number of positive flashes also increases with latitude and altitude above sea level (Uman 1987). Theoretical models are usually based on the condition for opposite (or upward) leader development from the structure. These models result in a high sensitivity of attractive radius to the structure height. For positive lightning flashes to ground, the upward leader inception criterion is different and this results in a lower sensitivity of attractive radius to the structure height (Petrov & Waters 1999). For example, we find that the striking distance for positive lightning is given by *R*_{s+}=1.08 *I*_{p}^{2/3} ln(*h*/15+10) for *h*<200 m and *R*_{s+}=0.103[(*h*+30)*I*_{p}]^{2/3} for *h*>200 m. On the other hand, for negative lightning we find that *R*_{s−}=0.8[(*h*+15)*I*_{p}]^{2/3}. As an example, for a structure of height 57 m and a strike delivering a peak current of 30 kA, the striking distance is 27 and 134 m for a positive and negative flash, respectively.

Negative downward lightning ultimately becomes oriented to the highest object because of the relatively easy development of a positive upward leader from the top of the object. This occurs in an electric field of *ca* 500 kV m^{−1} over a critical distance. On the other hand, positive downward lightning does not ‘feel’ the height of the grounded object, since inception of a negative upward leader requires a field of at least 1 MV m^{−1}. Hence, a positive downward leader must approach much more closely to a grounded structure and is often seen to ‘choose’ objects randomly, regardless of their height, shape, etc.

Therefore, it is interesting to consider the influence of positive flashes in this study. Once again, if *N*_{e}[*s*]=*N*_{g}*A*_{eq}[*s*] is the expected number of strikes per year, and we apportion the flash density according to the percentage of positive and negative strikes, i.e.,(4.5)then . The parameter *η* is the resulting fraction of positive polarity discharges, which was found to be in the range 0.05–0.3. However, calculations of the correlation coefficient for different *η* show that *η* is less than 0.1 for the data in the present study because the correlation rapidly decreases for *η*>0.1.

## 5. Conclusions

In 1990, the data collection for an unprecedented, long-term, field study was commenced, with the aim of quantifying the number of lightning strikes to buildings equipped with LPSs. In this paper, we have presented an analysis of the observed lightning strike incidence to a large and varied sample of structures in Malaysia during the period 1990–2003. The positioning of the protective air terminals on these structures was carried out with an improved electrogeometric model of lightning attachment derived from the field and analytical studies of Eriksson (1987*a*,*b*).

The data sample was comprised of 86 structures with a mean height of 57 m and mean exposure time of 6.9 yr. These structures were subjected to 384 flashes over a total observation time of 592 yr.

These field data have been compared with the theoretical number of incident flashes according to the lightning attachment models of Eriksson (1987*a*,*b*), Rizk (1994*a*,*b*), Petrov & Waters (1995) and Petrov *et al*. (2000) using a variety of analytical relations and statistical tests. Also, the calculated interception efficiency was compared with the actual value derived from the long-term field observations involving measurements of captured strikes and lightning by-pass events.

From the theory, analyses and results presented in the previous sections of the paper, the following conclusions can be drawn.

Statistical tests show that the observational data used in this study are well-described by a Poisson distribution and, hence, Poisson statistics.

In general, the variation of strike incidence to any given structure does not overstep the limits of the range determined by the ‘3

*σ*rule’, i.e. 〈*N*_{d}〉±3√〈*N*_{d}〉.With regard to the number or frequency of lightning strikes to the structures in the sample, there is a highly significant positive correlation between the observational data and the number expected from the application of contemporary models of lightning attachment to structures.

Calculations involving mean values suggest that Eriksson's (1987

*a*,*b*) model is in better agreement with the observational data if the ground flash density is 15 km^{−2}yr^{−1}. The Petrov & Waters (1995) model is in better agreement if the ground flash density is 20 km^{−2}yr^{−1}.The expected by-pass probabilities which may be estimated from the models of Petrov

*et al*. (17%), Eriksson (14%) and CVM design calculations (less than 20%) agree with the observed value of 13%, within the uncertainties. Hence, the expected and observed LPS interception efficiencies are in agreement.The mean interception efficiency or protection level of the LPSs in this study, namely 87%, lies in the range of Levels III–IV according to the IEC standard on lightning protection.

This study has shown that it is possible to dimension the interception efficiency of a LPS using real field data. The method described in the paper circumvents the problems associated with laboratory testing, such as scale-size issues and the inability to properly replicate the electric field wavefronts observed in nature.

## Acknowledgments

N.I.P. wishes to thank ERICO Inc. for financial support of this study. The authors are indebted to Prof. Mat Darveniza, whose contribution to the research via earlier work helped lay the rigorous foundation for certain aspects of this paper. We also thank the three anonymous referees, who provided useful comments that helped improve the clarity of the paper.

## Footnotes

- Received March 11, 2005.
- Accepted November 25, 2005.

- © 2006 The Royal Society