Channel Development and Electric Parameter Characteristics of Regular Pulse Bursts in Lightning

Observations of a regular pulse burst (RPB) at the end of a K‐event are analyzed utilizing a simple geometric model and particle swarm optimization (PSO) to estimate the currents and propagation speeds of successive pulses of the RPB. The results show that the current of successive pulses is strongly overlapped and, for typical speeds of continuously propagating K‐events, are unphysically large (88 kA), exceeding the currents of most strokes to ground. By default, the unphysical nature of the result, coupled with very high frequency interferometer observations of an RPB in Florida, shows that the propagation speed of the pulses is significantly faster than expected, namely ∼0.6–1.8 × 108 m/s. This reduces the inferred current from 88 kA down to 6–18 kA, typical of intracloud events. The fast propagation speed of the stepping is explained by successive pulses retracing much of the path of the preceding pulses due to the successive pulses being strongly overlapped.


Introduction
In the observations of electric and magnetic fields during lightning discharges, there is often a regular burst pulse sequence having similar pulse characteristics, with the time interval between pulses also being regular.These pulses are usually superimposed on more complex electromagnetic field waveforms.Krider et al. (1975) studied the typical waveforms of these pulses and summarized their main characteristics.From an analysis of the shape and time interval between pulses, they were considered to be generated by intracloud (IC) dartstepped leaders.The arrays of concentrated intermittent pulses were called regular pulse bursts (RPBs).Rakov et al. (1992Rakov et al. ( , 1996) ) analyzed the characteristics of millisecond-scale pulse waveforms during K-events and M-processes of lightning and then analyzed RPBs related to these processes in cloud and ground lightning in detail.It was found that the RPBs in cloud and ground lightning have similar characteristics.Each pulse cluster is usually composed of dozens of successive pulses, with the pulse widths being only a few microseconds, averaging 6-7 μs, separated in time by 10-15 μs.The amplitude of RPBs during cloud to ground flashes is usually two orders of magnitude smaller than the initial peak of their return strokes.Following up on Rakov et al.'s (1992) study, Davis (1999) carried out a more in-depth investigation of RPBs using five-station • A new, innovative method is developed for estimating the currents and extents of fast electric field change pulses • The method is applied to the study of a regular pulse burst, showing that it consisted of a succession of fast overlapping events • The speed of the pulses is inferred to be significantly higher than that of continuously-developing K-leader Supporting Information: Supporting Information may be found in the online version of this article.dE/dt observations to determine the 3-D location and development of 200-600 μs-duration pulse train segments.From the 3-D observations, Davis estimated the propagation speeds of the bursts during IC flashes to range from 1.2 to 5.6 × 10 6 m/s, with a mean of 2.7 × 10 6 m/s.The stepping lengths between successive pulses ranged from 7 to 30 m, with an average of 14 m.Wang et al. (2009) analyzed the waveform characteristics of 69 RPBs.The typical width of the pulses was 1 μs, and the interval between pulses was 4-8 μs.Regular pulse bursts can be divided into four types (normal, inverted, symmetrical and torsional) according to the time interval and polarity conversion of adjacent pulses in the pulse cluster.Normal RPBs are similar to the IC dartstepped leader in negative ground flashes.Liu (2012) used an interferometer to locate and analyze two RPB processes.The RPBs at the end of a cloud flash had a propagation speed and mode similar to those of K events, and the very high frequency (VHF) radiation signal of the RPBs corresponded well with the low frequency (LF)/very low frequency (VLF) electric field pulse signal.Kolmašová and Santolík (2013) analyzed the pulse width and pulse interval of RPBs using magnetic field detection signals and obtained waveform characteristics similar to those of the above electric field signals.Zhu et al. (2014) used synchronous observations of the electric field, magnetic field and VHF lightning radiation signal to compare the RPBs in the K process.The RPBs coincided with pulses of VHF radiation, with an average pulse interval of 6.9 μs.Wang et al. (2014), using a VHF location technique and broadband electric field change measurements, studied the channel development and waveform characteristics of RPBs using during 15 min of lightning activity.Regular pulse bursts occurred in 83% of the lightning discharges, with most RPBs occurring in the middle and late stages of the discharges.Generally, RPBs were related to K-events, M-processes, or dart-stepped leaders.Recently, with advancements in detection techniques (Hare et al., 2021;Jensen et al., 2021Jensen et al., , 2023;;Shao et al., 2023), researchers have been gradually revealing clearer dynamic developmental images to depict RPBs and their associated characteristics in the K-event, confirming RPBs to be strong discharge events related to the K-leader process (Kolmašová et al., 2023), with the propagation speed of the K-leader typically falling within the range of 10 6 to 2.5 × 10 7 m/s.
While the above studies have substantially enhanced our understanding of RPBs, much remains to be determined concerning their currents and overall development.The present study combines observational data obtained from the Low-Frequency Electric-field Detection Array (LFEDA), developed by the Lightning Research Team of the Chinese Academy of Meteorological Sciences, with the transmission line model (Karunarathne et al., 2014;Nucci et al., 1988;Rakov & Dulzon, 1987;Uman & McLain, 1969) to determine the development and electrical characteristics of a 104 RPB sequence.The study utilizes for the first time a newly-developed inversion process and particle swarm optimization (PSO) for fitting the observed RPB waveforms.

The Low-Frequency Electric-Field Detection Array Lightning Location System
Low-Frequency Electric-field Detection Array is a synchronous electric field signal detection and recording system composed of nine fast antennas (Kitagawa & Brook, 1960).The bandwidth of the fast antennas is 160 Hz-600 kHz, with a decay time constant of 1 ms.Synchronization between different substations is realized by a GPS clock with a temporal accuracy of 50 ns.The data is sampled at a 10 MHz rate with 12-bit resolution.The electric field waveforms used for the inversion calculation are taken from the CHJ and SGC stations, which were at ∼12 and 20 km plan distance from the flash being studied.Fan et al. (2018Fan et al. ( , 2021) ) and Fan, Zhang, et al. (2020) introduced empirical mode decomposition (EMD) (Huang & Wu, 2008;Huang et al., 1998Huang et al., , 1999;;Rilling et al., 2003) to the analysis of LF/VLF electric field signals of lightning.Low frequency filtering and high-frequency noise reduction is applied to the electric field waveforms in 1 ms segments to determine the 3-dimensional structure and development of flashes.Figure 1a shows results obtained for an IC flash that produced a 104-pulse RPB, whose detailed development is shown in Figure 1b.The RPB occurred during an extensive late-stage K-event of the flash (red sources in Figure 1a), as the event developed upward in the storm.

Pulse Extraction During Regular Pulse Bursts
Regular pulse bursts usually occur during K-events, M-processes, or dart-stepped leaders, with their continuous pulsing superimposed on the relatively slow electrostatic field waveform.For the flash of this study, EMD was used to improve the location performance of LFEDA (Shi et al., 2017) by removing background noise, allowing the multi-station electric field waveforms to be accurately matched and located (Fan et al., 2018), and to separate the pulses from the electrostatic background (Figure 2b) while preserving the characteristic features, amplitudes and timing of the impulsive activity (Figures 2c and 2d).

Three-Dimensional Transmission Line Model and Channel Current Inversion Method
Shao (2016) successfully derived a general formulation of the Jefimenko equations that expresses the electric and magnetic fields entirely in terms of currents, rather than charges and currents.The resulting formulation is similar to Uman et al. (1975) equations, in that it has electrostatic, inductive and radiative components, but applies to arbitrarily-oriented current elements rather than vertical currents.The resulting expression for electric fields is as follows: where the outer integral is over the extent of the current vector in Figure 3a, and the propagation velocity v is contained in the volume current density J = ρv.The new formulation allows transmission line models (Karunarathne et al., 2014;Nucci et al., 1988;Rakov & Dulzon, 1987;Uman & McLain, 1969) to be extended to arbitrarily-oriented three-dimensional problems, as in this paper.
Figure 3a shows the three-dimensional transmission line model used for fitting the pulse waveforms and determining their currents and speeds.Fits were obtained for the electric field waveforms at two LFEDA stations (Figures 3b and 3c), ∼12 and 20 km distance from the pulse burst, and were successfully completed for 63 of the 104 pulses.For each pulse, the process begins by fitting a straight line through its source at (x, y, z) that extends through a sufficient number of adjacent located sources to be representative of the local 3-D breakdown orientation and development.The fitted straight line is then truncated such that its (x 1 , y 1 , z 1 ) and (x 2 , y 2 , z 2 ) end points are equidistant from the source in question, with the resulting vector constituting the transmission line.A current is then propagated along the transmission line from its lower to upper end, using Equation 1 to determine E(t).The asymmetric Gauss function (Gurevich & Zybin, 2005;Karunarathne et al., 2014Karunarathne et al., , 2016;;Nag & Rakov, 2010a, 2010b, 2016;Smith et al., 1999;Watson & Marshall, 2007) is used to simulate the current, which is attenuated using the Kumaraswamy distribution (K-distribution) (Jones, 2009;Karunarathne et al., 2014;Kumaraswamy, 1980).The current is expressed by where k = (t 2 t 1 )/t 1 , and α is the morphological control factor of the Gauss function.This piecewise function ensures that the current is zero at t = 0 and t = t 2 .The resulting transmission line is referred to as the MTLK line.
Because the located sources are closely spaced, there is substantial overlap of the MLTK lines for successive sources.That the breakdown of successive pulses during RPBs is substantially overlapped is confirmed by VHF interferometer observations of an RPB obtained in Florida, as shown in Figure S1b.
The inversion process of fitting the pulse waveforms to obtain the unknown source parameters is accomplished utilizing PSO (Bansal et al., 2011;Eberhart & Shi, 2001;Fan, Yao, et al., 2020;Kennedy & Eberhart, 1995;Shaw & Srivastava, 2007;Song et al., 2012).Eight parameters or unknowns (termed "particles") are needed for fitting the waveforms, namely A, α, t 1 , t 2 , v, a , b, x 2 ].The first four are the current parameters from Equation 2; v is the magnitude of the currents' propagation velocity (whose components are determined from the orientation vector), and a and b are the parameters of the K-attenuation (Karunarathne et al., 2014).The final parameter, x 2 , defines the extent of the transmission line and its current.(Because the orientation of the line and its mid-point are known, only a single end-point variable is needed to define the line's length).Particle Swarm Optimization determines the best fit by starting with random guesses of the full set of variables (within specified limits).1,000 such guesses, or particles, were used in the present study, with the best fit of each iteration being selected as the center of the next 1,000 particle iteration, etc., until a stopping criteria is met.Even though the data points for successive pulses substantially overlap, the random nature of the PSO processing causes the results for successive pulses to otherwise be independent of each other.Finally, because waveforms from two stations were used, the criterion for evaluating the quality of the solutions was as follows: where M 1 and M 2 are the peak values of the two stations, giving increased weight to the closest station, E fit 1 and E obs 1 are the fitted and measured electric field values of the first station, and n = 70 was the number of time points for the electric field comparisons.

Transmission Characteristics of Regular Pulse Bursts
From the detailed development of the RPB shown in Figure 1b, in approximately 1 ms the lightning channel developed upward from ∼9.0 to 11.5 km and approximately 1.8 km horizontally, corresponding to a vertical speed of 2.5 × 10 6 m/s and a horizontal speed of 1.8 × 10 6 m/s. Figure 4a shows the piecewise linear current channels obtained from inversions of 63 of the 104 electric field pulses, each of which is represented by a directional arrow between the estimated starting and terminating points of the pulses.Due to the short time interval between successive pulses (∼10 μs), the results for successive pulses are substantially overlapped, forming a relatively complete channel.A histogram of the pulse extents is shown in Figure 4b, and range from ∼43 to 633 m, with an average length of 202 m, but less than 250 m for most pulses.The sum of the apparent extents is 3-4 times the actual extent of the RPBs overall development (∼12 km vs. 3-4 km actual).The difference between the apparent and actual extents is an artifact of the inversions being strongly overlapped, as seen in Figure S1.
Concerning the propagation speeds of the pulse currents, the inversions range between 0.9 and 1.5 × 10 7 m/s, with an average of 1.2 × 10 7 m/s (Figure 4c).At first glance, the indicated speeds are in good agreement with the observations of dart leaders to ground, which were determined by Wang et al. (1999Wang et al. ( , 2016) ) to have a stepping speed of about 10 7 m/s, but the agreement is qualified in the Discussion section.The speeds are also at the upper end of continuously-propagating downward dart leaders, which are typically between ∼5 × 10 6 m/s and 1 × 10 7 m/s (Berger, 1967;Hubert & Mouget, 1981;Idone & Orville, 1982).

Current Characteristics of Regular Pulse Bursts
Figure 5 shows the current waveforms and their statistics obtained from the inversions.The indicated peak currents varied from ∼20 to ∼160 kA.The peak current of most pulses ranged from ∼40 to ∼120 kA, with an average of 88 kA.The indicated rise time of the current pulses is 1.5-2.8μs, with an average of 1.9 μs; the decay time is 1.2-4.2μs, with an average of 2.3 μs; and the duration of the initial current is 2.9-6.3 μs.Generally, the morphological characteristics of the pulse currents are quasi-symmetrical, with the decay time (t 2 t 1 ) being slightly greater than the rise time (t 1 ).
From the inversion results, the current begins to decay after its onset, decreasing exponentially with distance along the channel (Figure 5b).For the K-distribution (Karunarathne et al., 2014), the attenuation rate increases for larger values of the K b-parameter.Assuming that the total length of the channel is L, as b increases to a certain value, the current attenuates to zero well before reaching L. This effect on channel length needs to be taken into account in determine the distribution of channel lengths.In this paper, the effective channel length is determined from when the current decreases to 5% of its initial value, and the attenuation results are normalized to 100 m, as shown in Figure 5b.From the graph, attenuation along the channel is exponential-like, with the attenuation rates for most of the pulse currents being relatively similar.

Summary and Discussion
In this paper we analyze impulsive electric field changes of a RPB sequence observed by the LFEDA sferic array to estimate the currents and extents of the individual pulses.The basic analysis approach is essentially the same as in other lightning studies, in that it utilizes a parameterized current waveform and propagation parameters to fit observed impulsive electric field waveforms, and some means for locating the events.It extends previous types of analyses by utilizing the generalized version of Uman et al.'s equations (Shao, 2016) for arbitrary current orientations, and introduces the use EMD techniques for cleaning the waveforms and PSO for determining the parameters of the waveforms.
Although the parent flash of the RPB occurred outside the 9-station LFEDA network, its temporal and spatial development was sufficiently well-determined to allow 63 of the 104 pulses to be analyzed.The waveforms from two LFEDA stations were utilized for the analysis, which was simplified by their electric field change ΔE being dominated by the radiation component, and show that the pulses had a range of currents and extents.As a result of the analysis procedure, which went through the located sources one by one, the inferred extents of the successive sources substantially overlapped each other.In addition, it is important to note that, for events whose extents are small compared to their distance from the observation station, the radiation component depends only on the derivative of the current moment M i , which is ambiguously related to the current and its velocity or extent (e.g., Cummer, 2003;da Silva & Pasko, 2015).In particular, for the simplified case of the current I being constant over a distance Δx, ΔE is proportional to d (M i )/dt = (I⋅Δx)/Δt = I⋅v.As discussed by da Silva and Pasko, this prevents either the current or the velocity/extent from being unambiguously determined.While regular pulses will have a range of moment values, the ambiguity between the current I and propagation speed v is partially or largely responsible for the range of peak currents and propagation speeds in the present paper.
Although there are various parameterized models and techniques for estimating the currents and propagation speeds of electric field change observations, PSO provides a valuable method for optimizing the parameters.For the present study, the inferred extents of the RPB pulses range from ∼43 to 633 m, with an average of 202 m.The peak current has a range of ∼20-160 kA, with an average of 88 kA-unphysically high values, stronger than the currents of most return strokes to ground.From the above discussion, and for a given ΔE and current moment, the inferred value of I is inversely dependent on the propagation speed v, namely, I is proportional to ΔE/v.Thus, the unphysically large value of the pulse current indicates that the propagation speed of the current was faster than 1.2 × 10 7 m/s average value of the PSO analysis.That the analysis did not determine a faster speed is due to its value having been restricted to be between 5 × 10 6 and 1.5 × 10 7 m/s.This range of speed values corresponded to observational results available at the time of the PSO analysis (conducted in 2018, and pandemic delayed in their analysis and publication), and needing to be used to make the 8-parameter search computationally feasible.
Nevertheless, the study confirms the usefulness of the analysis procedure in that, by default, it shows that the speeds of RPBs must exceed 1.5 × 10 7 m/s for their currents to be physically realistic.For example, speeds of 6 × 10 7 and 1.8 × 10 8 m/s (with the latter value being the speed used by the NLDN for determining the peak current of return stroke and cloud events), the estimated peak current value is reduced from 88 kA to 17.6 and 5.9 kA, respectively, well within the range of NLDN-reported cloud events (Nag et al., 2014).Also, the speed of a step would be expected to exceed that of the incoming, continuously-developing K-leader, whose maximum speed is typically ∼2-3 × 10 7 m/s (e.g., Jensen et al., 2023).
Finally, that the propagation speed of successive pulses is extremely fast is seen in VHF interferometer observations (Fan, Krehbiel, Stranley, Rison, et al., 2023;Fan, Krehbiel, Stanley, Zhang, et al., 2023) of a regular pulse sequence in Figure S1, for which the successive pulses have speeds of ∼10 8 m/s and are strongly overlapped.Due to the speeds being comparable to the nominal NLDN speed, NLDN-detected pulse bursts would approximately reflect the actual current values.More significantly, the fast speed of the stepping is explained by successive pulses quickly retracing much of the path of the previous pulses, before the conductivity of the preceding pulse or pulses is reduced.

Figure 1 .
Figure 1.Three-dimensional observations of the cloud flash (a) and its regular pulse bursts (b), showing that the K-event occurred in the final stage of the flash (arrows).(a) Altitude versus time plots and the low frequency/very low frequency electric field measured at the CHJ station, which served as the coordinate origin, (b) East-West vertical projection, (c) Source count versus altitude and the total number of located sources (104 during 1 ms for the regular pulse burst), (d and e) Plan view and North-South vertical projection of the sources (from Fan et al. (2018)).

Figure 2 .
Figure 2. Decomposition, reconstruction, and extraction of the regular pulses from the background electrostatic field change using empirical mode decomposition (b and c) (from the CHJ station), and for filtering out background noise (d).

Figure 3 .
Figure 3. (a) Schematic diagram of the three-dimensional transmission line model.(b and c) Example comparison of the measured and fitted electric field waveforms for a given pulse from the two Low-Frequency Electric-field Detection Array stations.

Figure 4 .
Figure 4. (a) Upward development of the regular pulse bursts in Figure 2a.(b and c) Statistics of the channel lengths and propagation speeds of successive pulses during the regular pulse burst.

Figure 5 .
Figure 5. Results for the channel currents, (a) current waveforms (curves) and peak current statistics (histogram), (b) current attenuation normalized to a 100 m channel, and (c) current decay time (t 2 t 1 ); (d) statistics of the current rise time t 1 ; and (e) current duration t 2 .