Validation of single-station lightning location technique

Authors


Abstract

[1] Lightning discharges are powerful impulsive sources of electromagnetic energy over a wide bandwidth allowing passive methods to determine lightning location through the energy released by the lightning discharge. While multistation lightning location techniques provide high location accuracy, in some instances it is impossible to deploy a multistation network, and for this reason, techniques have been developed to allow single-station lightning location. We consider the validation of the “Kharkov” single-station lightning location method proposed by Rafalsky et al. [1995a], through comparing the locations of ELF/VLF observed sferics determined by the Kharkov technique with the positions of coincident lightning flashes recorded by a commercial lightning detection network. Making use of 85 sferics observed at Robertson Army Barracks, near Darwin, Australia, in the period from 20 November 1997 until 7 January 1998, we find that that ∼68% of Kharkov method bearing estimates are accurate to within 4.1° and that ∼68% of the Kharkov method range estimates are accurate to within 73.2 km.

1. Introduction

[2] Lightning discharges are powerful impulsive sources of electromagnetic energy over a wide bandwidth (up to optical), with significant radiated electromagnetic power from a few hertz to several hundred megahertz [Magono, 1980], and the bulk of the energy radiated in the frequency bands <30 kHz. Passive lightning location methods rely upon the energy released by the lightning discharge; acoustically (thunder), optically (lightning), and electromagnetically [Uman, 1987]. Today, commercial lightning location networks are in operation in many regions of the world, using multiple stations to locate the source of lightning discharge electromagnetic radiation pulses. An example is the U.S. National Lightning Detection Network (NLDN), which in 1996 used 106 sensors located over the continental United States to achieve a typical accuracy of 0.5 km [Cummins et al., 1998]. A discussion of multiple-station lightning detection techniques is given by Orville [1995].

[3] While multistation lightning location techniques provide high location accuracy, the facilities involved are necessarily complicated, with relatively high cost. In some instances it is impossible to deploy a multistation network, and for this reason techniques have been developed to allow single-station lightning location (which might be more appropriate for an aircraft or vessel). Single-station lightning location techniques are often a combination of a method of direction finding and means by which the distance of the lightning discharge can be estimated. A review of the various techniques by which estimates of direction and distance can be determined has been given by Rafalsky et al. [1995a].

[4] Recently, a new single-station lightning location technique was put forward to be applied to three-component field measurements (two horizontal magnetic and one vertical electric) of lightning electromagnetic pulses (“sferics”) in the ELF/VLF band [Rafalsky et al., 1995a]. Direction finding is performed using the Poynting vector calculated in the time domain over the full frequency range of the measurements. The distance is estimated from the phase spectrum of the first-order transverse electric (TE) waveguide propagation mode, by comparison with models of propagation in the Earth-ionosphere waveguide. The authors apply their technique, which we term the “Kharkov method,” in honor of the affiliation of most of the authors (Kharkov, Ukraine), to 28 sferics recorded on board a scientific vessel during voyages in the Atlantic and Indian Oceans. Rafalsky et al. [1995a] estimate the accuracy of their single-station lightning location method, the Kharkov method, to be a few degrees for the source bearing and 5% for the distance but noted the need for a confirmation of these estimates, as they did not possess an independent lightning data set with which to make comparisons. A subset of the same work has been published as Rafalsky et al. [1995b].

[5] In this paper we consider the validation of the Kharkov single-station lightning location method, by comparing the locations determined by the application of this method to ELF/VLF observed sferics with positions recorded by a commercial lightning detection network.

2. Lightning Data Sets

[6] A sferic recording system was deployed in late 1997 at Robertson Army Barracks (12°26′26″ S, 130°58′51″ E), henceforth known as ROB, located near the city of Darwin, Northern Territory, Australia. This system operated as part of campaigns to study lightning and red sprites (luminous discharges at high altitudes above thunderstorms), the results of which have been described in several studies [e.g., Dowden et al., 2001]. During the Darwin 1997–1998 summer campaign, lightning location data were purchased from a commercial Australian lightning location network, Kattron. By comparing the Kattron lightning locations with those determined by the Kharkov method applied to sferics recorded at ROB we investigate the accuracy of the single-station technique.

2.1. ELF/VLF Sferic Recording System

[7] The sferic recording system deployed at ROB allowed the transient sferic waveforms from up to four ELF/VLF antennas to be detected, recorded, and assigned a precision (<1 μs) GPS-derived time stamp. The system was connected to two ferromagnetic loop antennas (Bx and By components) and a conventional vertical whip antenna (Ez component). The magnetic sensors were solenoids with axes accurately horizontal and aligned true north-south and east-west. Their orientation was also checked by the mean direction of arrival near midday of the 19.8 kHz transmission from a 1 MW VLF transmitter some 2000 km distant. Before deployment, mutual calibration of the magnetic sensors was undertaken as required for their application in direction finding techniques. In addition, a precision frequency synthesizer function generator was used to ensure the equivalence of phase-frequency characteristics for the electric and magnetic field sensors from a few tens of hertz to 20 kHz, and beyond. Sferics were detected through simple threshold comparison: If the absolute value of a sample from the omnidirectional Ez antenna exceeds a constant threshold, then a sferic is assumed to have occurred. Once a sferic was detected, a sferic record, consisting of 1024 waveform samples (34.16 ms of time) from each antenna (with GPS timestamp), is logged, consisting of 128 pretrigger and 896 posttrigger samples. Further information on the hardware and software design of this system is given by Brundell [1999] (available from University of Otago at http://www.physics.otago.ac.nz/archives/archives.html). In the Darwin 1997–1998 summer campaign the ROB sferic recording system operated during local night (1000–2100 UT) from 12 November 1997 up to 7 January 1998.

2.2. Kattron Lightning Location Data

[8] Kattron, an Australia-based company, operates a commercial time of arrival (TOA) lightning location network, using a network of Lightning Positioning and Tracking System (LPATS) TOA receivers. The LPATS receivers were supplied by Global Atmospherics, Inc., and are similar to those used as part of the NLDN [Cummins et al., 1998]. In 1996 the Kattron network was made up of six LPATS receivers, positioned to achieve subkilometer location accuracy and high detection efficiencies over most of the high population density regions in Australia (primarily the southeast of the country). The estimated location accuracy and detection efficiency for the six-LPATS Kattron network are shown in Figures 1 and 2, respectively.

Figure 1.

The estimated location accuracy of the Kattron LPATS TOA lightning location network. The contours have units of kilometers.

Figure 2.

The estimated detection efficiency of the Kattron network.

[9] Lightning location data sets from the Kattron network were purchased for the duration of the Darwin 1997–1998 summer campaign. The data set was prefiltered by Kattron to contain only those lightning strokes occurring north of latitude 27°S. The Darwin region is well outside of the primary coverage area of the Kattron network. Location accuracy is still adequate at approximately 10–18 km RMS (Figure 1) although the detection efficiency is very low at <5% (Figure 2).

3. Application of the Kharkov Method

[10] The “Kharkov” method of single-station lightning location has been described in detail by Rafalsky et al. [1995a]. At the surface of the conducting Earth, there are only three significant electromagnetic components: the vertical electric field and the two orthogonal horizontal magnetic fields. These define a cylindrical polar coordinate system, centered on the lightning discharge and with the z axis in the vertical direction. The vertical electric field (Ez) and the transverse magnetic field (Hϕ) are components of the TEM and TM modes, while the radial magnetic field component (Hr) is solely associated with the TE mode. A single mode (TE1) can then be selected by examining only the frequency components of Hr that lie inside the range from above the cutoff of the first mode to below the cutoff of the second-order mode. Having isolated a single mode, the distance of the lightning discharge from the receiver can be estimated by minimizing the difference between the phase spectrum of this mode and that calculated from

equation image

where Δϕ1 is the phase of the first mode at (angular) frequency ω at a distance r from the lightning discharge, c is the speed of light of an EM wave in vacuum, and S1 is the sine of the modal angle (in general, complex).

[11] In order to make the coordinate transform (rotation) from Cartesian horizontal magnetic fields into the transverse and radial field components (cylindrical coordinates) it is necessary to know the direction of the lightning discharge at the receiver. The essence of the Kharkov method is the initial use of the time domain Poynting vector direction finding (TPDF) technique [Rafalsky et al., 1995a] to determine the bearing of the lightning discharge from the receiver (θ), after which a rotation is applied to Hx and Hy to obtain Hr and Hϕ, from which the range of the flash can be determined by finding the distance for which the modeled and experimental TE1 phase spectrum best match.

3.1. Implementation

[12] The Kharkov method has been implemented in software as outlined above. Three different models of the Earth-ionosphere waveguide have been considered to calculate S1 and hence perform the phase matching using a least squares fit. To allow comparison between the performance of the different waveguide models, their output parameters will be subscripted as follows: a, perfectly conducting parallel plates; b, parallel plates; and c, curved plates. In order to select the TE1 mode, only frequency components of Hr from ∼1850 to 3400 Hz were included in the matching. The first two of these were considered by Rafalsky et al. [1995a]. The height of the ionosphere (upper plate of the waveguide) was examined over a range of values from 84.5 km to 87 km, which are representative of nighttime conditions. A single height was determined for each Earth-ionosphere waveguide model, so as to minimize the mean and standard deviation of the least squares fitted phase matches.

3.1.1. Perfectly Conducting Parallel Plates

[13] This is the simplest of models for the Earth-ionosphere waveguide, where both plates are perfectly conducting (metal) and flat so equation image, where k is the wave number and h is the height of the ionosphere.

3.1.2. Parallel Plates

[14] In this case S1 is found from the solution of the flat-plate waveguide mode equation [Wait, equation 4.1, p. 147, 1996],

equation image

where Rg and Ri are the reflection coefficients from the ground and ionosphere, respectively, and C1 is the cosine of the modal angle (which is related to the sine of the modal angle through the trigonometric identity equation image). Rg is determined using the Fresnel formula for a sharply bounded homogeneous media [Landau and Lifshitz, 1960]. However, for Ri the Fresnel expression is modified as described by Barr [1970] through stratification to describe the ionospheres changing refractive index with height. The ionospheric electron density is described using a Wait ionosphere [Wait and Spies, 1964] where the electron number density (m−3) Ne increases exponentially with altitude z,

equation image

where β is given per kilometer and h′ is a reference height (in kilometers). We take h′ = 86 km and β = 0.5 km−1 as representative of nighttime conditions. The effective collision frequency profile is that suggested by Morfitt and Shellman [1976], where

equation image

The ground is represented by a conductivity of 10−3 S/m and relative permittivity εr of 15 [Morgan, 1968]. As most of the lightning-receiver paths end up being overland, these parameters are reasonable. The solution of (2) for C1, and hence S1, is undertaken using the Nelder-Mead simplex method [Lagarias et al., 1998].

3.1.3. Curved Plates

[15] In this case the same approach for solving S1 is taken as in section 3.1.2, expect that instead of using the flat-plate waveguide mode equation we use a curved Earth waveguide modal equation in a low-frequency approximation [Wait, equation 22, p. 200, 1996],

equation image

where a is radius of the Earth. An equivalent expression has been developed by Budden [equation 9.20, p. 142, 1961], who notes that it is applicable for frequencies <8 kHz. The same ionospheric and ground model is used for this case as was applied in section 3.1.2.

3.2. Case Study

[16] Figure 3 shows an example of a sferic received at ROB by the sferic recording system. This sferic was received there at 1027:24.696 UT on the 5 December 1997. This same event was detected by the Kattron TOA network located at latitude 17.6602°S, longitude 125.5781°E, at 1027:24.695 UT. Kattron therefore places the lightning at a distance of 819 km and bearing of 224.5° east of north from ROB (using the WGS-84 ellipsoid). Figure 4 shows the instantaneous Poynting vector of this sferic, i.e., for each of the 1024 samples of the sferic we have plotted, in polar form, the vector Px(i) = −Ez(i)Hy(i), Py(i) = Ez(i)Hx(i), where i is the sample index (1 to 1024). The Poynting vector is integrated over the duration of the sferic to obtain the time domain Poynting vector from which the bearing to the sferic is estimated. For this sferic, a bearing of 226.9° east of north was obtained. This compares favorably with the bearing value from the Kattron location estimate.

Figure 3.

The Ez, Hx, and Hy components of a sferic received at Robertson Barracks (ROB), Darwin by the sferic recording system at 1027:24.696 UT on day 339 of 1997.

Figure 4.

The received instantaneous Poynting vector at ROB for the sferic shown in Figure 3.

[17] Once the arrival angle has been determined, a rotation transformation is performed on the magnetic field vector to obtain the transverse, Hϕ, and radial, Hr, magnetic components, as shown in Figure 5. The phase of Hr is then acquired by fast Fourier transform, from which we determine the distance at which the modeled phase, in this case using perfectly conducting metal plates, best matches that received at ROB. For this sferic, the best match to the experimentally observed phase spectrum (Figure 6) is obtained at a distance of 836 km which again compares favorably with that calculated from Kattron's location estimate for this event.

Figure 5.

The transverse (Hϕ) and radial (Hr) magnetic components for the sferic shown in Figure 3, produced by a rotation transformation on the magnetic field vector.

Figure 6.

Example of the phase spectrum matching by which the range of the sferic shown in Figure 3 is determined. The modeled phase (line) is shown for the distance that best fits the experimentally observed phase (pluses).

4. Results

4.1. Coincident Lightning

[18] In order to make comparisons between the location estimates for lightning discharges by the Kattron network and Kharkov method, sferics received at ROB were selected which occurred within ±10 ms of a lightning event detected by Kattron. For each of these coincident events, the Kattron latitude and longitude was converted into a bearing and range from ROB using the WGS-84 ellipsoid. For the period from 20 November 1997 until 7 January 1998, a total of 85 matching lightning events were found.

4.2. Comparison With Kattron Lightning

4.2.1. Bearing Differences

[19] In Figure 7 we have plotted the bearing of the lightning derived from the Kattron location estimate (θ1) against the bearing obtained from the Kharkov method (θ2). As the bearing is determined from the TPDF technique, the result is independent of h. A strong linear relationship is observed with a correlation coefficient of 0.997. The mean of the bearing difference (Δθ = θ2−θ1) is 3.7°, and the standard deviation (σΔθ) is 4.5°. The 95% confidence interval for the mean of our 85 events is therefore 2.7°–4.7°, and thus there is a statistically significant systematic offset between the bearings measured using the two methods. Examination of average systematic errors in U.S. DF stations have shown deviations of similar size [Orville, 1987]. Lightning flash bearings observed in the ELF/ULF frequency range have been shown to exhibit deviations of ∼10° due to the anisotropic conductivity of the ionosphere [Füllekrug and Sukhorukov, 1999], which may also be contributing to the observed offset.

Figure 7.

The Kharkov method bearing from ROB to the lightning against that determined from the Kattron position estimate. There is a strong linear relation.

4.2.2. Range Differences (Perfectly Conducting)

[20] We now consider the difference between the range to the lightning derived from the Kattron location estimate (r1) and the range obtained from the Kharkov method using perfectly conducting parallel plates (r2) where the ionospheric height, h, is taken to be 87 km. Again, there is a strong linear relationship between these two quantities, with a correlation coefficient of 0.986. The mean of the range difference (Δra = r2r1) is 13.5 km, and the standard deviation equation image is 73.3 km. The 95% confidence interval for the mean of our events is therefore −2.4 km to 29.4 km. This interval includes zero, and thus there is no statistically significant systematic offset between the ranges measured using the two methods. These quantities are shown in Table 1, along with the mean value of the least squares phase matching difference equation image, and the percentage difference between the two methods for discharges determined by Kattron to be >1000 km from ROB. This percentage difference is 4.6%, rather close to the accuracy suggested for the Kharkov method by its developers [Rafalsky et al., 1995a].

Table 1. Parameters Describing the Comparison Between the Range Estimates From Kattron and the Kharkov Method for Different Ionospheric Modelsa
 h, kmμΔr, kmσΔr, kmμΦ, degμΔr > 1 Mm, %
  • a

    The mean (μΔr) and standard deviation (σΔr) of the range difference are shown, along with mean value of the least squares phase matching difference (μΦ), and the percentage difference between the two methods long-range discharges.

Metal (a)87.013.573.39.04.6
Parallel (b)86.011.274.58.54.8
Curved (c)86.027.5473.88.15.9
 85.011.774.512.24.7

4.2.3. Range Differences (Parallel Plates)

[21] The values for mean equation image, and standard deviation equation image of the range difference between the Kattron estimate and Kharkov method using the Wait parallel plates with h = 86.0 km is given in Table 1. As in the parallel plate case above, the linear relationship is strong (correlation coefficient of 0.986), and there is no statistically significant systematic offset between the range measured using the two methods (95% confidence interval for the mean is −5.0 km to 27.4 km). Note that the Wait flat-plate waveguide mode equation produces a slightly better phase match than for the perfectly conducting plate model (8.5° c.f. 9.0°).

4.2.4. Range Differences (Curved Plates)

[22] The values for mean equation image, and standard deviation equation image of the range difference using the Wait curved plate approximation with h = 86.0 km is given in Table 1. Once again, the linear relationship is strong (again a correlation coefficient of 0.986), but although there is a very slight improvement in the phase match than for the previous two models equation image, the 95% confidence interval for the mean lies from 27.5 km to 59.6 km, and so this ionospheric model makes a statistically significant, range over-estimate. A slightly lower ionospheric height (h = 85 km) produces ranges for which there is not a statistically significant difference in the determined ranges, as shown in last row of Table 1, but leads to a worse mean least squares phase fit (12.2 compared to 8.1). As such, the “better” ionospheric height can only be determined with reference to the Kattron lightning data, and therefore is not valid for the determination of the accuracy of the Kharkov method. It appears that the low-frequency approximation required for the Wait curved plates model may not hold well enough for its application in the Kharkov method.

4.3. Accuracy of Kharkov Method

[23] The position of the lightning discharges as estimated by the Kattron (pluses) and Kharkov data sets (circle) are shown in Figure 8. The Wait flat-plate waveguide mode equation is used to model the phase spectrum, as this produced the best answers in section 4.2 (the other waveguide models produce visually similar figures). Vectors are shown, joining the two location estimates for each event. For each of the 85 lightning strokes, which will have occurred at a given range (r) and bearing (θ) from ROB, we have two estimates of the range (r1 and r2) and bearing (θ1 and θ2) from the Kattron and Kharkov data sets. Now Δr = (r2r) − (r1r) and Δθ = (θ2 − θ) − (θ1 − θ) are each a measure of the difference in the actual error of the two methods. As the error in the range and bearing from ROB deduced from the Kattron location will not be correlated with the error in the range and bearing deduced from the Kharkov method we can assume that the errors in the two approaches (Kattron and Kharkov) are independent. Thus we can say that

equation image

and that

equation image

that is, the variance of the differences between the two methods is equal to the sum of the variances of the error in each method. From our experimental data set, we have an estimate of σΔr and σΔθ. In order to estimate the standard deviations equation image and equation image of the errors in the range and bearing of the Kharkov method we need to estimate the standard deviations of the errors in the transformed Kattron range and bearing.

Figure 8.

The estimated positions of the 85 lightning discharges in the study. For each event, both the Kattron (pluses) and Kharkov (circles) location estimates are shown, joined by a vector. The position of Robertson Army Barracks (ROB) is shown by a triangle.

[24] Figure 9 shows the error ellipses for the Kattron network, at the location of 2 of the coincident 85 events. Event A occurred at a location of 13.7617°S, 130.8213°E and event B was located at −13.8486°S, 131.8584°E. The error ellipses are computed using a Monte Carlo style simulation based on the assumption the TOA errors are Gaussian with a standard deviation of 2 μs [Cummins et al., 1998]. While the two error ellipses are very similar, when they are transformed into a range and bearing from ROB we find that for event A the expected bearing error is 4.5° and the expected range error is 13.4 km, while for event B we get 0.5° and 17.6 km. The expected range and bearing errors were calculated at the location of each of the 85 events and we take the root mean square of these range and bearing error estimates, to obtain our best estimate of σequation image= 14.1 km and equation image = 1.83° for our data set of 85 events.

Figure 9.

The expected bearing error from ROB of lightning A will be much larger than for B even though the error ellipses at the two different locations are virtually identical. Similarly, the expected range error from ROB for B will be larger than for A.

[25] Using these estimates and equations (6) and (7), we can estimate for the accuracy of the range and bearing obtained using the Kharkov single-station lightning location technique with the Wait parallel plate waveguide as equation image = 73 km and equation image = 4.1°, or that ∼68% of Kharkov method range estimates will be accurate within 73 km, and that ∼68% of bearing estimates will be accurate to within 4.1°. These results are very similar to those which are obtained using the much simpler perfectly conducting plate waveguide model where ∼68% of Kharkov method range estimates will be accurate within 72 km, with an unchanged value for the accuracy of the of bearing estimates. It appears that for our application of the Kharkov technique, the accuracy is not significantly improved by the application of a significantly more complicated waveguide model, requiring greater than four orders of magnitude more computational time. Note that our estimates of Kharkov method accuracy applies only to the conditions under which this analysis was performed, i.e., sferics propagating during summer nighttime over low latitude paths of <2000 km length.

5. Discussion

[26] The Darwin region is located at a distance of approximately 3000 km from the Kattron TOA network, which is based in southeast Australia. To date, there appears to be no published data on the accuracy of TOA lightning location at such large distances from the locating network, as studies have concentrated on characterizing the location accuracy within the network boundaries. The estimated location accuracy of NLDN observations (inside the network) was found to agree well with experimental confirmations of the location accuracy by modeling the DF and TOA errors as Gaussian with standard deviations of up to 1.5° and 2 μs, respectively [Cummins et al., 1998]. For this reason we made use of a 2 μs TOA error to estimate the location accuracy of Kattron's LPATS TOA lightning location network in the Darwin region. Investigations have been made on the effects of propagation on the rise times of lightning radiation fields for paths <500 km over finitely conducting ground [Cooray, 1987]. It was shown that in extreme cases (seawater and low conductivity ground) that the TOA measurements could be in error by as much as 20 μs. By recording both the trigger threshold crossing time and peak time of the received lightning radiation field the LPATS TOA system makes an estimate of the received signal rise time, allowing a correction to be made for these propagation induced timing errors [Casper, 1991]. One can speculate, however, that there will be difficulties in measuring the rise times of low-amplitude signals that just exceed the trigger threshold, as is likely to be the case for the Kattron network observations of lightning from the Darwin region. Additional difficulties are likely when active thunderstorms are located close to the Kattron sensors, leading to a degradation in the signal to noise for Darwin region events. It is therefore quite possible that the TOA errors may be larger than we have estimated, and hence the location accuracy of the Kattron TOA lightning location network may be lower than when have estimated for activity at great distances from the network. As such the Kharkov method may be somewhat better than the estimate given in section 4.3.

[27] Another possible source of lightning location data by which comparisons could be made with the Kharkov method and Kattron network locations are from Earth orbiting satellites. The Lightning Imaging Sensor (LIS) aboard the Tropical Rainfall Measuring Mission satellite detects lightning optically by observing the neutral oxygen line at 777.4 nm [Orville, 1995]. Comparisons of LIS, NLDN, and a VHF radar lightning observations found that the LIS tends to detect cloud to ground (CG) discharges toward the end of the discharge process, probably the late stage in-cloud components of the CG discharge, occurring in the upper parts of the thundercloud. Differences of ∼1 s were possible between the NLDN and LIS detections [Thomas et al., 2000]. While the LIS observations show lightning occurring in the same general location (i.e., within the same thunderstorm), it is not clear that any of our Kattron or Kharkov strokes are coincident with LIS detections.

6. Conclusions

[28] In this paper we have considered the validation of the Kharkov single-station lightning location method proposed by Rafalsky et al. [1995a], through comparing the locations of ELF/VLF observed sferics determined by the Kharkov technique with the positions of coincident lightning flashes recorded by a commercial lightning detection network. While noting the requirement for validation, it was estimated that the accuracy of their single-station lightning location method should be a few degrees for the source bearing and 5% for the distance from the observing station [Rafalsky et al., 1995a]. Making use of 85 sferics observed at Robertson Army Barracks, near Darwin, Australia, in the period from 20 November 1997 until 7 January 1998, we find that ∼68% of Kharkov method bearing estimates are accurate to within 4.1°. Three different models of the Earth-ionosphere waveguide were considered to perform phase matching between the calculated and observed phase spectrum, and hence determine the range of the lightning discharge from the observing station. Of these, the Wait parallel-plate waveguide produced the most accurate positions, although the difference between this and the much simpler perfectly conducting parallel plates was very small, even though the Wait parallel-plate model requires more than an four orders of magnitude more computation time. It is found that ∼68% of the Kharkov single-station lightning location method range estimates are accurate within 73.2 km. The accuracy estimate originally proposed for the Kharkov method is reasonably accurate, at least for lightning located >1000 km from the observing site.

Acknowledgments

[29] This work was supported by the New Zealand Marsden Research Fund contract LFE801 and the University of Otago. The authors would like to thank Neil Thomson of the University of Otago for helpful discussions and Kattron of Chittaway Bay, Australia, for providing the lightning data used in this study.

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