Determining the striking distance of lightning through its relationship to leader potential



[1] A new method for estimating lightning leader potential in negative stepped leaders from electric field changes during the first return stroke process in cloud-to-ground flashes is applied for determining the striking distance to a ground structure. This method may serve as an alternative to calculating the striking distance from formulas that relate it to the peak return stroke current. Presented are the characteristics of downward leaders (potential, vertical extent, and the length of the final step) and the charges transferred during return strokes; these were obtained from electric field measurements during a thunderstorm in Florida using the line charge model. The final step lengths of the leaders are compared with striking distances calculated using the peak current values of return strokes taken from National Lightning Detection Network data for the same flashes.

1. Introduction

[2] In the literature on lightning protection, the striking distance of downward leaders in cloud-to-ground (CG) flashes to ground structures has been related to the peak first return stroke current, and thus to the severity of the flash, using a variety of empirical formulas. One can obtain direct current measurements of return strokes at selected ground installations (usually tall structures) that are frequently struck by lightning. These current measurements are limited in number, and the current values obtained are strongly affected by the dimensions of the structure, and also by the conditions (climatological and topographical) that influence the nature of local thunderstorms. Because of these factors, the data cannot characterize the entire range of CG flashes, or be used universally in lightning protection practice.

[3] A computer-simulated model of leader progression and its interaction with a tall mast by Mazur et al. [2000] indicates that striking distance is determined mainly by the step length of the downward leader near the ground. From a physical standpoint, one is justified in using the leader potential, rather than peak current of the first return stroke in determining the step lengths of the downward leader, because there is no unique physical relationship between the last step length and the maximum return stroke current. The leader's potential, however, has not been applied previously for this purpose, because it could not be measured. A new approach, called a “line charge model” of the return stroke [Mazur and Ruhnke, 2002], proposes to determine leader potential and other characteristics of the first leader in CG flashes from measurements of electric field changes produced by a return stroke.

[4] In this paper, we (1) evaluate further the line charge model by using a new data set from a multistation slow antenna system, (2) present the characteristics of negative leaders obtained using the model, and (3) compare the final step length values (determined by knowing the leader potential) with striking distances calculated using an empirical formula that relates the striking distance to the peak return stroke current.

2. Concept of the Line Charge Model

[5] We start with a brief description of the line charge model that was introduced in Mazur and Ruhnke [2002]. By assuming that the leader is equivalent to a conducting wire extended within the ambient E field of a thundercloud, and that the total charge of the bipolar, bidirectional leader is zero before touching the ground, the leader potential that was equal to ϕ before touching the ground shifts to zero upon contact with the ground. This shift of potential is equivalent to adding a constant charge-per-unit length, q, along the leader channel during the return stroke process. The charge-per-unit length, q, deposited by the return stroke in a leader channel of capacitance-per-unit length, c, is calculated as:

equation image

[6] The electric field change on the ground, ΔERS, produced by the return stroke traversing the vertical leader channel of length, Z, at distance, D, from a sensor, is:

equation image

[7] The variables q, D, and Z are obtained by solving a system of nequation (2), where n is the number of “slow antenna” sensors at various distances from a lightning strike, each measuring ΔERS values. One needs to know the capacitance-per-unit length c of the leader channel, in order to calculate the leader potential ϕ using equation (1). For a vertical, thin, long conductor perpendicular to the ground (with diameter d and length Z), that approximates a leader channel, the capacitance-per-unit length c is calculated using the formula:

equation image

[8] The relationship between the diameter of the leader channel d and its charges is based upon the assumption that the charges are distributed in a corona sheath (with E field of ≤3 MV m−1) around the current-carrying core of the channel, as shown here:

equation image

[9] The leader potential is calculated using the formula:

equation image

[10] The estimated length of the last step of the negative stepped leader, Lst, is a function of both the leader potential ϕ, in kV, and the constant electric field along the negative streamer zone ahead of the leader tip, which is 750 kV m−1[Bacchiega et al., 1994]:

equation image

[11] We postulate that the last step of the downward stepped leader, calculated using formula (6), is longer than, or equal to, the striking distance to the ground structure (or to an upward leader from the structure). The striking distance would be the actual length of the last step. Thus the length of the last leader step is a good measure for striking distance, on the basis the conclusion that an upward leader from a ground structure does not deviate significantly from the vertical during its interaction with a downward stepped leader [Mazur et al., 2000].

3. Measurements

[12] In our field measurements, we used a GPS-synchronized sensor system consisting of a capacitive antenna (Figure 1), an amplifier with high-pass characteristics and a 0.3 s time constant, and a laptop PC with a 12-bit A/D converter. The converter's time resolution of 100 μs allowed us to identify the beginning and end of the return stroke in the E field change record. A 1 s duration of each record was sufficient to identify CG flashes by the shape of their waveforms. The electronic components (amplifier and laptop PC) of the sensor system were carried inside a passenger car, and powered by a car battery; the antenna was carried in the car's trunk, but unfolded at the observation site, and placed about 25 m away from the car (Figure 2). This mobile arrangement allowed us to use remote sites far from 60 Hz noise sources.

Figure 1.

The capacitive antenna (“slow antenna”) used in the mobile sites.

Figure 2.

One of four mobile sites employed in field observations during the 2001 storm season in Ruskin, Florida.

[13] Our network of sensor stations consisted of four manned mobile sites, and one unmanned stationary site (at the WSR-88D weather radar location in Ruskin, Florida) with guaranteed stable AC power. The sites for the mobile sites were selected to be within 4–7 km from each other, easily accessible from the road, and ideally, with the widest possible open space surrounding it. It was a challenge to find such sites in the densely populated area of Ruskin, Florida, with numerous power lines above the ground. Because some sites could be less than ideal, thus producing additional, local errors in the E field change measurements, we conducted another calibration of pairs of sensors already in place, using selected recordings of distant flashes located (according to NLDN data) at an equal distances from each sensor in a pair. The “anchoring” sensor in each pair was always the one located at a site with ideal conditions. Altogether, there were four pairs, each consisting of the anchoring sensor plus one of the remaining four sensors. The anchoring sensor was considered to be perfectly calibrated, and its value of ΔERS was used to calculate the local correction factors for the other sensors. The calibration of the anchoring sensor was done in comparison with a horizontal wire antenna, calibrated in absolute terms by measuring the capacitance of the coaxial cable from the antenna to the amplifier, the capacitance of the amplifier, the antenna's height above the ground, and by calculating the wire's capacitance.

4. Data Processing and Results

[14] An example of the ΔE record of a negative CG flash is shown in Figure 3, in order to illustrate the way to measure ΔERS. The lower point of the upward change, ΔERS, corresponds to the transition from the leader to the return stroke phase; this point is easily determined by a rather sharp change in the slope of the waveform. The upper point of ΔERS corresponds to the end of the return stroke process, when the slope gradually changes from tilted to horizontal. There is more uncertainty is determining the upper point of the ΔERS, because this change in the slope is not always clearly pronounced, and may appear differently in ΔERS records of different sites. Where the slope of the field change near the upper point does not become horizontal, but turns instead into a very slow rise, the upper point of ΔERS marks the amplitude of the tangential line at the time of the beginning of the field change.

Figure 3.

An example of (a) a slow antenna record of a leader return stroke sequence in a CG flash and (b) points of the record used for measuring the ΔERS. Values in parentheses correspond to time and voltage at the point marked with the arrow. For the given CG flash, the duration of the E field change during the return stroke process is 1.5 ms, and ΔERS = 69.9 V/m (at the input of the amplifier).

[15] The records of E field changes correspond to the voltage on the input of an amplifier after passing the RC circuit. In order to restore the original record of the E field change on the sensor's antenna, we performed a deconvolution of the waveform from the effect of the RC circuit that has a time constant τ (see Appendix A). The convolution effect is noticeable, however, only in waveforms that change slowly, comparable to a time constant value (e.g., E field changes after the return stroke). The fast E field changes (e.g., ΔERS) lasting a few ms are not affected by this convolution effect.

[16] Using the measured E field change during the first return stroke, ΔERS, and applying the line charge model, we calculated the characteristics of negative leaders for 42 negative CG flashes during the thunderstorm on 28 July 2001 in the Tampa Bay area of Florida near Ruskin (Figure 4 and Table 1). We employed the least squares method and the MathCad program to solve the system of equation (2), considering the solution as the final and acceptable one when the magnitude of the fitting error reached its minimum.

Figure 4.

Locations of CG flashes (dots) relative to five slow antenna stations (circles) for the storm on 28 July 2002.

Table 1. Calculated Characteristics of Leader and Return Stroke From the Line Charge Model
Time, hhmmssq, mC/mQ, Cϕ, MVZ (LCM), mZ (LPCM), mLst, mdQ, CMax Current, ARst, m

[17] A validation of the results of the line charge model may be obtained only for such physically measurable parameters as distance D, and the vertical length of the channel Z, by using independent observations with contemporary lightning mapping techniques. Unfortunately, such validation with a lightning mapping system was not possible during the 2001 storm season, because the system was unavailable. However, we used the National Lightning Detection Network (NLDN) data for locations of CG flashes, to compare them with the locations of the same flashes that had been obtained from calculations using the line charge model. In most cases, the distance between the locations from the NLDN data and those from the model was less than 2 km. The discrepancies may be explained by the deviation of the actual return stroke channel from straight vertical; a fully or partially tilted channel leads to increasing ΔERS values at the sensors in the direction of the tilt, and to decreasing ΔERS at the sensors in the direction opposite to the tilt, thus producing an error in the calculated flash location using the line charge model and, consequently, in other parameters of the leader.

[18] The NLDN system implements time of arrival and direction-finding techniques to determine the locations of ground attachment points of the return stroke channels, which correspond to the locations of the lower, mostly vertical part of the channel. Therefore we decided to use the NLDN data on flash locations to calculate q and Z, by solving the system of equation (2). Flashes for which the significant fitting error indicated that either the location from NLDN was in error, or the lightning channel had characteristics leading to a substantial deviation (more than 2 km) in location from the NLDN data were excluded from consideration.

[19] Some values of Z in Table 1 may exceed the vertical dimensions of the cloud, which may indicate some inadequacy in the line charge model for those flashes. Aware of this, we explored the possibility that this inadequacy could be the result of branching of the intracloud portion of the leader. The return stroke in some flashes propagates partly though a vertical channel between the cloud base and the ground, and partly through the upper in-cloud part of the bidirectional leader that is composed of branches of the positive leader (in negative CG flashes). This possibility is not taken into account in the line charge model, which assumes the leader channel to be vertical. We attempt to address this issue below.

[20] The branching of the intracloud portion of the leader may be equated to a point charge, ΔQ, added at the upper end of the vertical return stroke channel (see Figure 5). Then, for the line with a uniform charge q plus a point charge ΔQ, the expression for ΔERS will change to:

equation image
Figure 5.

The concept of the line plus point charge model. (a) Visual representation of the leader/return stroke channel, made of a nearly linear and vertical part of the channel to the ground and a branched part inside the cloud. (b) The model approximation with the line charge for the channel part below the cloud base and an additional point charge for the branched part inside the cloud.

[21] By assuming for the modified line charge model (called “line plus point charge model”) the same channel location and the value ofq as for the line charge model, we are able to determine variables Z, and ΔQ by solving the system of equation (7), and then deriving variables φ, Q, and Lst.

[22] The results of applying the line plus point charge model to our data (see Table 1) show that the application of this model makes a difference in only 18 cases out of a total of 42, showing charge ΔQ greater than 0.1C, and also a smaller Z than those calculated using the line charge model. For example, in the flash at 223026, Z decreased from 18,050 m to 12,840 m. In 24 flashes, the line charge model provides a good fit for the data obtained, because the line plus point charge model produces identical results.

[23] Table 1 also contains values of the maximum return stroke current from NLDN data, IRS, for all flashes, and of the striking distance, Rst, calculated using the formula from Golde [1977, p. 560],:

equation image

[24] The final step length shows a much greater degree of dispersion and a wider range of values than the striking distances (see Figure 6). The length of the final step of the negative stepped leader, determined by using the leader potential, has an average value of 100.3 m, while the average value of striking distance calculated using the theoretical formula (8) and peak current values for CG flashes from the NLDN data is 46.8 m.

Figure 6.

Striking distance determined from formula (8) versus step length of the leader, determined with the line charge model.

[25] We also tested the postulate by Golde [1977, p.555], that states that charges transferred by the return stroke are proportional to the peak return stroke current. If this assumption were correct, then the return stroke velocity would be approximately the same for all CG flashes. We found, however, a very weak correlation between charge transfer and peak return stroke current (see Figure 7).

Figure 7.

Charge transferred during the return stroke, obtained with the line charge model, versus the peak return stroke current, from the NLDN data.

5. Discussion and Conclusions

[26] The new approach for determining leader potential and other characteristics of the first leader in CG flashes (including the last step length), from measurements of the electric field changes produced by a return stroke (the line charge model), involves remote measurements of lightning flashes. Thus this approach is less intrusive than direct current measurements, which are presently used to evaluate striking distances. This method is easy to implement for obtaining the statistically significant data necessary for evaluation of the range of leader potential values, and also of the range of the last step lengths of the leader in a variety of climatological conditions and geographic locations.

[27] The line charge model provides a good fit for the E field changes produced by return strokes for most flashes in the data set analyzed in this paper. It is suggested that the line plus point charge model accounts for the possibility of branching in the upper part of the bidirectional leader channel.

[28] There is no certainty that the results obtained either with the line charge model or the line plus point charge model will fully, and with the highest degree of accuracy, describe the features of each and every CG flash. Although the majority of CG flashes have a vertical or almost vertical return stroke channel to the ground, some flashes will have either significantly tilted or highly branched channels to the ground. In cases like these, the results from the model will differ from the features of the real flash. These cases, and also cases with a mistaken location provided by NLDN, would produce a substantial fitting error in the solution. These cases should be screened out in the analysis.

[29] We have shown that applying the line plus point charge model reduces, in a few cases, the values of q, Z and φ. However, in the majority of cases, this model did not produce a significant difference in comparison with the results obtained with the line charge model. Both models represent features of the leader and returns stroke, which could not be determined previously by using the conventional point charge model for CG flashes (see Appendix B). A conclusive test of validity for both line charge and line plus point charge models can be performed when the results of the calculations of variables Z, D (x, y), and possibly Lst are compared with and confirmed by other independent measurements (e.g., with a lightning radiation mapping system and video observations).

[30] The core of our argument for calculating the last step length (striking distance), using leader potential rather than return stroke current, is that the former is the dominant factor in determining the negative leader's step length. As a consequence of the leader process, return stroke current has only a statistical, rather than analytical, relationship to leader potential. Admittedly, the question of the degree of correlation between leader potential and the peak current of the first return stroke is not finally resolved. Their correlation may be more accurately estimated from a comparison of the actual current measurements at a tall structure (rather than the current values deduced from NLDN data, as in our study), with the leader potential values obtained through remote measurements of ΔERS, using the line charge or line plus point charge models.

Appendix A:: Deconvolution of the E Field Waveforms

[31] If E(t) is the voltage at the sensor's antenna, and U(t) is the voltage at the amplifier after passing the RC circuit, then the Laplace transfer of both voltages is related, according to formula (9):

equation image

[32] Here s/(s + a) is a transfer function of the RC circuit, and a is the inverse time constant, τ, of the RC circuit.

[33] From formula (9), it follows that:

equation image

[34] In the time domain, formula (10) will evolve into:

equation image

[35] Figure 8 shows the waveform of ΔE for an eight-stroke CG flash as recorded by our slow antenna system with τ = 0.3 s (curve 1), and in its original form at the antenna input, after deconvolution (curve 2).

Figure 8.

Effect of deconvolution (curve 2) on the ΔE waveform recorded with a slow antenna system that has a time constant of 0.3 s (curve 1).

Appendix B:: Comparison of the Line Charge and the Point Charge Models

[36] The conventional method of determining the charges transferred during a return stroke utilizes the point charge model [e.g., Krehbiel et al., 1979]. Being derived from the “source charge” leader model (which assumes that the return stroke neutralizes the charges carried by the uniformly charged negative leader, so that the former leader channel has a zero charge after the return stroke), the point charge model calculates the value of a fictitious source charge, Qpc, left over in a cloud after the return stroke, and its position (D and Zpc). On the contrary, the line charge model calculates the total charge Q on the former leader channel after the return stroke. Variables Qpc and Zpc in the point charge model are fundamentally different from Q and Z in the line charge model, although the return stroke process is assumed to be the same in both models (depositing the uniform positive charge along the leader channel). The importance of assuming a realistic charge distribution for the leader/return stroke channel is illustrated by the difference between the E field changes as shown in Figure 9, produced by the two models.

Figure 9.

The dependence of an E field change on the distance to the lightning channel, approximated as a point charge Qpc = 8 C at a height of Zpc = 6 km (curve 1), and a line charge Q = 10 C along the vertical channel between the ground and Z = 10 km altitude (curve 2), respectively.

[37] In this example the point charge Qpc and the line charge Q produce similar field changes at distances longer than 6 km, so the ΔE measurements at grater distances will fit well both the point charge and the line charge models. However, in this case, the obtained charge Q differs by 20% from Qpc, which is a significant error. Thus, even for long distances from the sensor, it is imperative to assume the physically correct charge distribution in the leader/return stroke channel, if the charges are deduced from electric field measurements. At distances closer than 6 km, ΔE measurements will fit only the line charge model, if there is a line charge distribution in the channel, and only the point charge model, if there is a point charge producing the electric field.


[38] This research was supported in part by the WSR-88D Operational Support Facility of the National Weather Service. The authors are grateful to Marijo Hennagin-Mazur for her invaluable help in editing the paper.