# Determining the Probability of Lightning Striking a Facility

R.T. Hasbrouck, P.E.

National Lightning Safety Institute
Louisville CO, Tel. 303-666-8817

One objective of a facility lightning hazard mitigation study is to determine the likelihood of its being struck by lightning. In this article, actual site-specific lightning strike data is used to calculate probability.

Estimating Probability General

The probability of lightning striking a particular object situated on the earth (ground) is found by multiplying the object's lightning-attractive area by the local ground-flash density (lightning strikes to ground per km2 per year). The example below considers a low structure surrounded by 12, tall grounded metal light poles.

Caveats

It must be understood that calculations used for determining strike probability are based upon empirical relationships, generally accepted by the research community as reasonably representing the lightning phenomenon. The method presented here provides a reasonable estimate but should not be considered the "final word." Other, more complicated geometric methods can be used but, considering the capricious nature of lightning, it is unlikely they would provide significantly improved results.

A complete cloud-to-ground lightning event, referred to as a flash, consists of one or more return strokes. Return strokes are high-peak-amplitude (10s-100s of thousands of amperes) current pulses, each lasting for a few hundred microseconds. Analysis of a large quantity of lightning flash data shows the average number of strokes (multiplicity) per negative (the most common type of lightning) flash to be between three and four. Approximately 25% of all negative flashes also exhibit several hundred amperes of continuing current during an interval lasting 100s of milliseconds following at least one return stroke. In a given flash, consecutive return strokes may strike the ground within several meters of each other, or as far apart as 8 km. Analyses indicate that they exhibit a "random walk" having a mean interstroke distance of 1.8 km. The ground-flash density data used in this report is based upon the location of each flash's first detected stroke, regardless of its amplitude or multiplicity. The author is unaware of any strike probability estimates that take into account the area included in a multi-stroke flash and the current-amplitude distribution of strokes in the flash. Finally, note that the statistically less frequent positive lightning flash usually consists of a single stroke having average and maximum peak amplitudes that are significantly higher than for negative lightning. It is accompanied by continuing current and has a total duration as long as oneñtoñtwo seconds.

Cumulative Probability

Lightning Attractive Area

If the surface beneath a storm cloud were perfectly flat, lightning would strike any point on the surface with equal probability. For example, if the ground-flash density were one flash per km2 per year and the area of concern was flat, with an area 0.1 km2, the probability of its being struck would be 0.1 in any given year (a return frequency of 10 years per flash). However, a conductive object that is taller than the surrounding area exhibits a lightning attractive area greater than the ground surface area it occupies. The probability of its being struck is a function of its ground surface area, height, and the striking distance between the tip of the downward-moving stepped leader and the object (Ref. 1). For negative lightning, the stepped leader is a negatively charged channel that travels in discrete jumps from cloud to Earth.

Striking distance, the stepped leader's final jump to the conductive object, varies with the amount of charge carried by the channel. (Note: For the sake of simplicity, striking distance calculations don't take into account upward-moving, positively-charged streamers. These streamers emanate from conductive objects under the influence of the stepped leader's strong electric fieldómuch as hairs rise up toward a statically charged comb held over one's head.) Since the magnitude of this charge also determines return-stroke peak-current amplitude, greater striking distance is associated with a larger amplitude return stroke, i.e., it jumps farther to reach the object (Ref. 2). Thus, for a given ground surface area and object height, the greatest attractive area is associated with the lightning stroke having the largest peak amplitude.

Ground-Flash Density

In the United States, actual cloud-to-ground lightning strike data is detected and archived by the National Lightning Detection Network (NLDN). Global Atmospherics, Inc. (GAIóTucson, AZ) analyzes the data and produces ground-flash density maps for user-specified areas. For our example, the map was based upon 29,207 negative and positive flashesófive years of site-specific dataócovering an area of 1.3 ´ 104 km2. The average overall flash density was 0.45 flashes/km2/yr, ranging from < 0.25 to < 0.5 flashes/km2/yr within a 4-km radius of the facility. The following should be taken into account when considering the GAI data. NLDN detection efficiency (DE)ói.e., the percentage of all lightning flashes that were detected and recordedóhas improved over the five-year period during which our data was acquired (1990ñ95). Initially, DE was reported as 65ñ70%; the currently stated value is 85ñ90% (for Ipk > 5 kA). Assuming a five-year DE average of 75% (considered by GAI to be a reasonable estimate) gives a corrected facility flash-density range of < 0.33 to < 0.67. The median value of 0.5 flashes/km2/yr was used for our probability calculations.

Return-Stroke Peak-Current Amplitude

Over a number of decades, researchers have measured and recorded a variety of lightning parameters, with much of the data resulting from strikes to tall instrumented steel towers. Along with current rate of rise and total charge transfer, peak return-stroke current is considered to be one of lightning's most significant threat parameters. For the generally accepted frequency distribution of peak currents for negative lightning, the 1st-percentile value, 200 kA (i.e., 99% of all lightning is of lower amplitude), is generally considered to constitute a severe negative stroke.

Although the NLDN detection efficiency is less than 100%, GAI reports that it is low-peak-current (i.e., < 5 kA) events that are missed. Thus, had all flashes been detected, the distribution of peak-current amplitudes would be expected to show a somewhat lower average value. Facility Lightning Attractive AreaóSince the 12, 32-m-tall perimeter light poles for our example appeared to be likely lightning strike pointsóat least for large-amplitude flashesóthey were used in calculating the facility's lightning-attractive area. For the sake of simplicity, structure height was not included in our equation. It is reasonable to expect that some low-amplitude strokes can be expected to bypass the poles and attach to the structure.

As previously discussed, attractive area must take into account the peak amplitude of return-stroke current. Thus, an attractive area must be calculated for each current amplitude. The following method for dealing with the distribution of return-stroke currents is attributed to the late J. Stahmann of Boeing/Kennedy Space Center (Ref. 3). Stahmann assigned return-stroke peak currents from a large body of available data to decilesói.e., 10% of the total number of flashes being considered were placed into each of ten bins. The mean peak current per decile was then calculated.

Facility Strike Probability

Stahmann's mean peak-current per decile values were used to find the per-decile attractive area. The effect of the tall light poles on attractive area (Aa) can be seen in Table 1. Although the surface area encompassed by the poles is 45 km2, the attractive area for a 6-kA stroke is 77 km2, and 171 km2 for a 112-kA stroke. The product of the attractive area times the ground-flash density provided per-decile probability, the sum of which gave a cumulative probability. The reciprocal of cumulative probability is the mean return period (average strike frequency). For our example it was determined that some point of the facility will be struck by lightningóof some amplitudeóapproximately once every 17 years.

 Decile # Ipk (kA) Ds (m) r (m) Aa (m2) Po Pc R (yr/fl) 1 6 33 33 76.764 3.8E-03 2 13 53 48 93.489 4.7E-03 3 18 65 56 101.496 5.1E-03 4 23 76 62 108.624 5.4E-03 5 28 88 68 115.399 5.8E-03 6 35 101 74 122.391 6.1E-03 7 45 118 81 130.658 6.5E-03 8 57 138 89 140.196 7.0E-03 9 77 168 99 153.061 7.6E-03 10 112 215 113 171.380 8.6E-03 6E-02 17

Area enclosed by light poles: l = 312 m, w = 144 m (l xw = 44,928)m2)
h = height of poles above ground level = 32 m
Ipk = average peak return-stroke current per decile ó kA
Ds = lightning striking distance = 10 x Ipk0.65
r = radius of light pole's attractive area = (2 x Ds x h ñ h2)0.5
Aa = attractive area/decile = (l + 2r) x(w + 2r) ñ 10 x[(4 ñ p)/4] x r2
Fg = ground flash density = 0.5 fl/km2/yr {using GAI flash density analysis}
Po = strike probability/decile = Aa x(0.1 x Fg) x10ñ6
Pc = cumulative probability = SPo
R = mean return period (i.e., average strike frequency) = 1/Pc

Conclusion

Reasonable strike probability estimates can be made using site-specific, ground-flash density values that are based upon actual lightning data. Strike estimates are interesting and although their results provide an indication of lightning strike return frequency, they should not be considered as absolute. Perhaps their most useful function is to permit determination of the relative effects of changes made to a facility. Examples of such changes are: increased attractive areaóeither by extending the facility's surface dimensions and/or height (adding a vent stack or tower); placing an identical facility in a location having a significantly different ground-flash density.

REFERENCES

• 1. Golde, R.H., "Protection of Structures Against Lightning," Proceedings of the Institute of Electrical Engineers, Vol. 115, No. 10, pp. 1523ñ1529, 1968.
• 2. Golde, R.H., "The Lightning Conductor," in Golde, Lightning, Vol. 2, p. 560, Academic Press, London, 1977 (the striking distance equation attributed to E.R. Love).
• 3. Stahmann, J.R., "Launch Pad Lightning Protection Enhancement by Induced Streamers," Boeing Aerospace Operations, Kennedy Space Center, FL.

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