Analytical description of ELF transients produced by cloud to ground lightning discharges

Transcription

1 JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 115,, doi:1.129/29jd1333, 21 Analytical description of ELF transients produced by cloud to ground lightning discharges A. Kułak, 1,2 Z. Nieckarz, 3 and S. Zięba 2 Received 2 August 29; revised 4 February 21; accepted 2 April 21; published 1 October 21. [1] Measurements of extremely low frequency (ELF) transients can be used to obtain model parameters and rates of the strong cloud to ground (CG) lightning discharges that generate them. In this study, we present an analytical description of a CG lightning discharge which takes into account both a return stroke and a continuing current. Using quasi TEM (q TEM) solutions for the Earth ionosphere waveguide for distances within the near zone we show that the ELF spectrum of transients strongly depends on the percentage contribution of the continuing current to the discharge process. It is supposed that during typical observational conditions the only parameter of an ELF transient that can be measured sufficiently accurately is the impulse amplitude. In relation to the observational conditions the consequences of the calibration of ELF receivers on the measured ELF impulse amplitude are discussed, particularly the role of antialias filters. It is demonstrated that measurements of the ELF transient amplitude at a known distance to the source allows the total dipole moment of the discharge to be calculated and allows contributions of the return stroke and continuing current to this process to be estimated. Furthermore, some techniques of the signal spectral filtration that enable independent measurements of the return and continuing phase of the discharge are discussed. Citation: Kułak, A., Z. Nieckarz, and S. Zięba (21), Analytical description of ELF transients produced by cloud to ground lightning discharges, J. Geophys. Res., 115,, doi:1.129/29jd Introduction [2] Electromagnetic fields in the extremely low frequency (ELF) range in the Earth ionosphere waveguide consist of background noise originating from global lightning activity and ELF transients generated by intense lightning discharges. Waveforms of these ELF transients are diverse: most of them have the shape of a single spike followed by a more or less distinguished response impulse caused by the propagation in the Earth ionosphere cavity [Ogawa et al., 1966; Jones, 197a; Burke and Jones, 1995]. It is generally accepted that this type of ELF transient come from intense cloud to ground (CG) lightning discharges, both negative and positive [Rycroft, 1994; Holzworth et al., 25]. [3] Currently the primary techniques employed in the research studies of atmospheric discharges use optical measurements carried out near thunderstorm cells [Malan and Collens, 1937; Schonland, 1956; Olsen et al., 26] and broadband radio emission emitted by the discharge current in the VLF, HF, VHF, and ELF frequency bands [Rakov et al., 21; Pavanello, 27]. At larger distances ( 1 Mm) electromagnetic pulses associated with atmospheric discharges are strongly attenuated in the HF and 1 Department of Electronics, Academy of Mining and Metallurgy, Krakow, Poland. 2 Astronomical Observatory, Jagiellonian University, Krakow, Poland. 3 Institute of Physics, Jagiellonian University, Krakow, Poland. Copyright 21 by the American Geophysical Union /1/29JD1333 VHF bands, but in the VLF and ELF bands they propagate as sferics on distances reaching more than 1 Mm. Results obtained from VLF sferic observations are used in the studies of the propagation properties of the Earth ionosphere waveguide [Jones, 197a; Cummer, 1997; Cummer and Inan, 2] as well as to obtain some parameters describing lightning discharges such as return stroke current, current moment change, or charge moment change. [4] Measurements of ELF fields are mainly used for observations of Schumann resonances (SR), which are used effectively in research studies of global thunderstorm activity [Williams, 1992, Nickolaenko and Rabinowicz, 1995; Price and Melnikov, 24; Sátori et al., 28; Nieckarz et al., 29] and of modeling the global atmospheric electric circuit (GEC) [Rycroft et al., 27]. ELF transients are studied relatively rarely and their observations are treated mainly as an additional source of information about the lightning activity of different thunderstorm centers [Polk, 1969; Huang et al., 1999; Hobara et al., 26; Greenberg and Price, 27]. However, as waveforms of ELF pulses also depend on the discharge process, the ELF transients can be used similarly to study discharges as it takes place in the VLF frequency range. [5] In this paper, we show how to use ELF broadband observations of strong enough CG lightning discharges for a study of their physical parameters. For this purpose, we assume that a discharge process can be described by some analytic formulas which consider both return stroke and continuing currents. We use typical values characterizing these currents for examining the relationship of their spectra 1of1

2 with pulse amplitudes measured by ELF receivers. Our analysis shows that from the broadband ELF observations of transients it is possible to estimate the duration of the continuing current phase of discharge, to calculate the amount of energy emitted during this phase, and to determine the percentage contribution of the continuing current to the total charge transfer. We also point out that synchronous observations of transients made in the ELF and VLF bands provide additional data for verification of different models describing the physics of intense cloud to ground lightning events. 2. Lightning Discharge as a Source of the Electromagnetic Field Inside the Earth Ionosphere Waveguide [6] Since the first observations of Schumann resonances [Balser and Wagner, 196], it has been generally accepted that electrical cloud to cloud and cloud to ground lightning discharges were the main source of the ELF fields in the Earth ionosphere waveguide [Raemer, 1961; Galejs, 1961; Polk, 1969]. If a single discharge transfers charge q along the lightning conductive channel of length l, the derivative of the charge moment of the discharge s =d(ql)/dt decides about the efficiency of the field generation. So the electromagnetic field associated with the lightning discharge depends on changes of both the channel length l(t) and the current flowing in the channel i = dq/dt [Schonland, 1956; Ogawa, 1995]. Changes of the channel length are important for the electromagnetic field emission at the ELF and VLF bands and also in the ULF frequency band as pointed out by [Surkov et al., 25]. [7] According to Rakov et al. [21] there are three main modes of charge transfer to ground during lightning discharge: the leader/return stroke sequence, the continuing current, and the M component. The return stroke sequence can aggregate several strokes, and the amplitude of the first one is about tens of kiloamperes, and the duration is tens of microseconds. A typical continuing current has the amplitude of 1 2 A and the duration tens of milliseconds. Waveforms of this current can be diverse [Campos et al., 27]. Because of its long duration the continuing current transports about 75% of the total charge transferred during the discharge. The very short duration of the return stroke causes generation of electromagnetic EM fields over a large frequency range up to tens of megahertz. In the ELF range both the return stroke sequence and the continuing current mode are important, but up to 1 Hz or more the continuing current plays a significant role Return Stroke [8] The waveform i r (t) of the current flowing during a return stroke can be described by a superposition of two or more exponential functions [Bruce and Golde, 1941; Jones, 197b; Uman and Krider, 1982; Uman, 1987; Nickolaenko and Hayakawa, 1998, 22]. In our discharge model we choose the simplest waveform of the return stroke current, which was first proposed by Bruce and Golde [1941] and used by Jones [197b] and Cummer [1997]: h i r ðþ¼i t r e t a e t b i : ð1þ The reasonable values of the parameters in this formula [Rakov and Uman, 23, Table 1.1], which we used in our model calculation, are: I r = 25 ka, t a = 1 ms, and t b =5ms. The current described by (1) reaches the maximum value I p = xi r, where x.81 results from the accepted values of t a and t b. The integral of the current i r (t) gives the total charge q r = I r t q transported by the return stroke, where t q = t a t b describes the time duration of the stroke. Using the above numerical values of I r, t a, and t b, we obtain q r 2.4 C and t q 95 ms. Assuming that the return stroke current propagates up the lightning channel with a speed v(t) =v r exp( t/t v )[Schonland, 1956; Ogawa, 1995; Rakov and Uman, 23], where v r =1 8 m/s and t v =33ms, it is possible to calculate the channel length from l r (t) =l [1 exp( t/t v )], where l = v r t n is the final length l 3.3 km. [9] The product of the current i r (t) and the channel length l r (t) gives the current moment of the return stroke s r (t). s r ðþ¼i t r ðþl t r ðþ¼s t r n o e a t e t b þ e t bv e av t ; ð2þ where s r = I r l = s r Am, t av = t a t v /(t a + t v ) 25 ms, and t bv = t b t v /(t b + t v ) 4.3 ms. [1] The current moment waveform was proposed by Jones [197b] and used successfully by Cummer [1997] for calculations of VLF waveforms of sferics generated by atmospheric discharges. The electric dipole moment of the return stroke (the charge moment) is given by p r ¼ s r ðþdt t ¼ s r r ; where the time constant t r = t a t b + t bv t av t r 75 ms characterizes the equivalent time of the discharge. For the model values s r and t r we obtain a typical value p r Cm. [11] The Fourier transform of the current moment (2) gives the amplitude spectrum of the current moment. This spectrum is a decreasing function of frequency having four characteristic frequencies: f a 1.6 khz, f b 32 khz, f av 6.4 khz, and f bv 37 khz, defined as f = 1/(2pt) and related to the time constants t a, t b, t av, and t bv. Below the frequency f a 1.6 khz the spectrum is flat and can be described by s r ð3þ ðf Þ s r ð a b þ bv av Þ ¼ s r r ¼ p r : ð4þ The flat spectrum implies that the return stroke can be considered as an infinitely short pulse as far as ELF fields are considered. As the return stroke is short the conducting channel spreads out to the total length just at the final phase of the discharge process when the current is already small. Therefore the channel length which acts as an efficient transmitting antenna should be smaller than l. We define this effective length l r directly from the definition of the dipole moment p r = q r l r, where q r is the total charge transferred by the return stroke current (1). Comparing p r = q r l r with (3) gives l r ¼ l ð5þ 2of1

3 Figure 1. (left) Current moment spectrum of the model discharge with the unit dipole moment of the return stroke p r = 1 (Cm). (right) The percentage contribution of the return stroke current spectrum to the total current spectrum integrated over a receiver frequency bandwidth from to a frequency f, indicated along the x axis. Plots are computed for g/h = 1, 2, 3, 4 and for the cutoff frequency f c 25 Hz. where h = t r /t q is a coefficient describing the shortening of the channel length. In our model h 75/95.79 and the effective channel length l r 2.6 km Continuing Current [12] The continuing current i c is initiated at the end of the return stroke sequence. This current flows along the final channel length l and equals about 1% of the return stroke peak current I p. We assume that the continuing current is described by a rectangular pulse I c = 15 A of width t c =4 ms carrying charge q c = I c t c. Its current moment is given by s c ðþ¼i t c ðþl t ¼ s c P t c ; ð6þ where s c = I c l and the electric dipole moment of the continuing current is p c = s c t c = q c l. The absolute value of the current moment spectrum amplitude is expressed by s c ðf Þ ¼ s c c jsincð c f Þj ¼ p c jsincð c f Þj; ð7þ where sinc(x) = sin px/px. Significantly large amplitudes of this spectrum are below the first zero of the sinc function which is defined by the frequency f c =1/t c. For an average duration of the continuing current 4 ms, the cutoff frequency is about f c 25 Hz. However, for about 3% 5% of lightning discharges, the duration of the continuing current is longer than 1 ms which yields a lower cutoff frequency (f c < 1 Hz) [Rakov and Uman, 23]. The long duration of the continuing current results in a greater charge transfer to ground during this mode than during the return stroke. We introduce parameter g: ¼ q c q r describing the share of the charge transfer between the continuing and return stroke currents mode. From measurements made in the near zone the value of g is about 2 3[Rakov and Uman, 23]. ð8þ 2.3. Calculation of Current Moment Spectra of the Model Discharge [13] Now using (1), (2), (3), (5), and (6) we can describe the current moment of the entire discharge in the ELF range as follows: st ðþ¼s r ðþþs t c ðþ¼p t r ðþþ t 1 P c t c ¼ p r ðþ t ð9aþ According to (4) and (7) and using directly (9a) we obtain the amplitude current moment spectrum of the two discharge components: f sð f Þ¼p r þ p c sinc f c ¼ p r 1 þ sinc f f c ¼ p r ð f Þ; ð9bþ where (t) ( f ) are the pair of the Fourier transforms describing the waveform of the model discharge in the time and frequency domain, respectively. According to (9b) spectral amplitudes grow at frequencies lower than the cutoff frequency f c when the contribution of the continuing current described by g = q c /q r increases. [14] For example, taking the coefficient of the channel shortening h.8 and g 2.5, g/h 3, from (4) and (9) we see that the spectral amplitudes are four times higher (1 + g/ h 4) at the low end of frequencies with respect to the flat spectrum at higher frequencies. Figure 1 shows the current moment spectra calculated for four different values of g.itis clear that the influence of the return stroke current on the amplitude spectrum increases with wider range of frequencies Df considered (or measured (this is usually determined by the bandwidth of a receiver system) for Df = 24 Hz; the contribution of the return stroke current to the spectrum increases to 9% and 7% for g/h = 1 and 4, respectively). Thus as far as broadband analysis is considered, satisfying condition Df f c, the spectral amplitudes are practically dependent solely on the return stroke current. Closer examination of the four curves in Figure 1 (right) implies 3of1

4 Table 1. Electrodynamic Parameters of the Return Stroke and the Continuing Current of the Model Cloud to Ground Lightning Discharge a Return Stroke Current h i I r =25kA i r (t) =I r e a t e t b t a = 1 ms, t b =5ms t q = t a t b =95ms I p = xi r = 2.3 ka Charge q r = I r t q 2.37 C Channel l r (t) =v r tn [1 exp( t/t v )] v(t) =v r exp( t/t v ) v r =1 8 m/s, t v =33ms l = v r t v 3.3 km l r = l h 2.6 km Current moment s r (t) =i r (t)l r (t) =s r e a t e t b þ e t bv e av t Charge moment p r = R1 s r ðþdt t ¼ s r r Current Charge Channel Current moment Charge moment that the contribution of the return stroke current to the measured amplitude decreases from 76% for g/h = 1 to 44% for g/h = 4 in the case when the bandwidth of the ELF receiver is the Schumann range: Df = 6 Hz (from Hz to 6 Hz). Negligence of the role of continuing currents on the excitation of the Earth ionosphere cavity can cause errors in interpretation of SR spectra. [15] To summarize, we list the main electrodynamic parameters of our model discharge in Table 1. We present the typical values of the parameters used in our calculations. They are selected from values presented in Rakov and Uman [23]. 3. Electromagnetic Fields Generated by Cloud to Ground Lightning Discharges in the Earth Ionosphere Waveguide [16] The discharge of the vertical electric dipole (VED), created by the separation of electric charges inside a thundercloud, to the ground by a cloud to ground lightning generates an electromagnetic field impulse propagating in the Earth ionosphere waveguide. In the ELF range the length of discharge channel l fulfills the condition of a short dipole antenna which is valid for l h and l l, where h is the height of the ionosphere. Observing the field impulses at the lowest end of the ELF range, we practically confine to sources at distances r inside the near zone (r l/2p). The size of the near zone is usually more than tens of megameters. For distances r < h the influence of the ionosphere on the field distribution can be neglected so the solutions of Maxwell equations for the near zone in free space can be used: 2s E z ¼ i 4"r 3 H ¼ s 4r 2 ; t av = t a t v /(t a + t v ) 25 ms t bv = t b t v /(t b + t v ) 4.3 ms s r = I r l Am t r = t a t b + t bv t av 75 ms l r = l h 2.6 km p r = q r l r Cm Continuing Current I c = 15 A, t c =4ms q c = I c t c 6C l = v r t v 3.3 km s c = q c l /t c Am p c = q c l Cm a Model parameters, x.81, h = t r /t q.79, g = q c /q r 2.5. ð1þ where E z is the electric vertical component and H the magnetic azimutal component of the EM field and " is the dielectric permittivity of the medium. At distances r h and frequencies lower than the characteristic frequency of the Earth ionosphere waveguide, determined by the condition l <2h, the solutions of the equations describing the VED fields are given by E z ðkr Þ ¼! s 4h H ð2þ ðkrþ H ðkrþ ¼ i k s 4h H ð2þ 1 ðkrþ; ð11þ where H (2) (x) and H 1 (2) (x) are Hankel functions of the second kind and zero and first order, and m is the medium magnetic permeability [Wait, 197; Burrows, 1978]. [17] These solutions determine amplitudes of the waves running directly from sources. At distances 1 3 Mm the running waves dominate over the resonance response of the Earth ionosphere cavity [Kułak et al., 26]. Inside the near zone h r < l the Hankel functions cannot be reduced to elementary functions, but inside the wave zone, and for large values of kr 1, it is possible to use an asymptotic approximation [Burrows, 1978]. Solutions (11) do not take into account the dipole altitude and the focusing effect caused by the Earth curvature. In the near zone, which is considered here, the focusing effect is not important as at distances 5 Mm; the correction for this effect is smaller than 1%. [18] Further in this paper, we will analyze only the magnetic component H of the field generated by the model discharge producing the current moment spectrum (9). According to (1), for r < h the amplitude of the spectral density of the magnetic component reproducing the original spectrum of the discharge current moment is given by Hr; ð f Þ ¼ p r 4r 2 ðf Þ: ð12þ For r > h, when (in a simplified way) an exponential attenuation factor exp( ra) is involved to take into account the loss of the field energy inside the waveguide, the amplitude of the spectral density of the magnetic component is given, after (11), by fp r Hr; ð f Þ ¼ 2hðfÞvðfÞ ðf ÞH ð2þ 1 2r f e rðf Þ ; ð13þ v where the height h( f ), the phase velocity n( f ) and the attenuation rate a( f ) are known (see (14)) functions of frequency. At lower frequencies the amplitude spectral density varies irregularly because of the presence of the continuing current in the discharge as well as because of oscillations of the Hankel function H 1 (2) (r, f ), characteristic for these distances. At higher frequencies the density increases approximately linearly with frequency. [19] Next we will apply formula (13) to compute the field amplitude spectrum using the analytic expression by Ishaq and Jones [1977] and Jones [1999] for the propagation parameters of the Earth ionosphere waveguide: the phase velocity n( f ) and the attenuation rate a( f ). c ¼ 1:64 :1759 ln f þ :179 ln f v ð Þ2 ; ¼ 7: f :64 1 : m ð14aþ 4of1

5 Figure 2. (left) Frequency response of six pole Bessel, Butterworth, and Chebyshev antialias filters having the same attenuation of 35 db at 9 Hz (half of the sampling frequency used). The bandwidths measured at the 3 db point are: 27., 42.5, and 51.9 Hz, respectively. (right) Response of these filters to an input broadband impulse. Besides, for the two characteristic altitudes h e ( f ) and h m ( f ), describing how high the ionosphere is penetrated by the vertical electric and magnetic field components of 6 1 Hz waves, we will use expressions from Kirillov et al. [1997]: h e ¼ 6:5 1 4 f þ 373 ln ½mŠ; 1 h m ¼ 9: ln f 1 ½mŠ: 4. Broadband Observations of Field Impulses ð14bþ [2] Broadband recordings of electromagnetic fields in the VLF, HF, and VHF frequency ranges are the foundations of some fundamental techniques investigating atmospheric lightning discharges [Cummer, 2; Rakov and Uman, 23; Rodger et al., 26]. Here we discuss further application of ELF receivers and ELF observations, already applied in the Schumann resonance studies, to the observations of ELF field impulses generated by intense lightning discharges Role of Antialias Filters [21] A typical path of an ELF receiver includes broadband amplifiers, a low pass antialias filter and an analog to digital converter. The input of the receiver is connected to active antennas detecting magnetic or electric components of EM fields. For spectral studies it is usually required that the output characteristics of the antennas and the receiver are flat over the frequency range from a few to tens of hertz. Above the working frequency f h the characteristic amplitudes should decrease rapidly because of the threat of aliasing with signals of frequencies higher than half of the sampling frequency f s /2. The choice of the sampling frequency f s depends on the steepness of the slope of the lowpass filter, and in this case it is usually required that the attenuation at frequency equal to the half of the sampling frequency f s /2 is higher than 3 db. The steeper the filter, the lower the sampling frequency can be and the smaller the stream of data that is produced. [22] In practice, application of a six pole Chebyshev lowpass filter allows the use of a sampling frequency f s =3f h. However, such filters tend to have large group delay variations, which produce significant distortions of the output pulse shapes. Low pass Bessel filters exhibit excellent delay characteristics resulting in an optimal time domain response, but they do not have these flat characteristics and are not useful for spectral measurements. Using Butterworth filters could be a compromise, but this leads to a reduction of the flat part of the receiver characteristics instead. Figure 2 shows a comparison of impulse responses of receivers with different antialias filters but having the same 35 db attenuation at 9 Hz. Because of the relatively large bandwidth provided by the Chebyshev filters, these are the filters most frequently used for spectral measurements Distortion of Impulse Shape [23] The spectral analysis of short duration impulses is usually burdened with a large error caused by random noise present inside a time window used for such analysis. In general, broadening of the time window does not improve the signal to noise ratio because of the nonstationary character of the background noise and insufficient averaging process. [24] Time analysis has more advantages for the study of such signals. Signal parameters such as the receiving time, the signal amplitude, and its waveform can be determined in VLF frequency range. However, in the ELF range this analysis is limited as the bandwidth of receivers is much narrower than the spectral width of measured impulses. For this reason the time determination error is large, and the information about the original waveform is almost totally lost. In addition, 5of1

6 observational practice shows that noises of different origin change the impulse shape, and this makes the analysis of the impulse waveform even more difficult. In many cases the interference noise is composed of various small amplitude short impulses shifted randomly in time. In these circumstances the only measurable parameter of the investigated impulse is its amplitude. Here we do not consider questions related to analysis of signals spread in time such as SR transients (Q bursts) or those produced in the atmosphere by sprites where their waveforms can be obtained. [25] If the spectrum of a field impulse is flat in the ELF range, as in the case of fields generated by a discharge return stroke, it is possible to obtain information about the impulse through the measurement of the time of the incoming signal peak and its amplitude at the output of a receiver. However, for impulses emitted by the continuing current phase of the discharge, which have a rippled spectrum inside the receiver bandwidth, the output amplitude is a complicated function of the impulse s spectral properties, and any reconstruction of the parameters of such signal is ambiguous Measurement of Impulse Amplitude [26] On the assumption that only amplitudes of strong atmospherics can be measured, we discuss the problem of how the output impulse amplitude depends on the original signal parameters, the transmittance function of the Earthionosphere waveguide, and the receiver path parameters. This will also determine the number of measurements within the waveguide required to reconstruct this parameter properly. [27] First we consider a receiver response to a short field impulse H(r, t) produced by our model discharge producing the current moment (9) and observed at a distance r from the source. The Fourier transform H(r, f ) of the impulse received by an antenna is given by (12) and (13). If we assume that the receiver response g(t), produced by the entire signal transmission path consisting of reception antennas, amplifiers, antialias filters, and converters, is known, the output signal a(t) of the receiver for the input impulse H(r, t) is given by the convolution g(t)*h(r, t). According to the Rayleigh theorem the energy of the output signal is " ðþ¼ r jar; ð tþj 2 dt ¼ jar; ð f Þj 2 df ¼ jhr; ð f ÞgðfÞj 2 df : ð15þ Input impulses of broadband spectra cause the impulses observed at the receiver output to take the shape of narrow triangles of height H pulse and width Dt, which is comparable to the reciprocal of the equivalent energetic receiver bandwidth Df [Kraus, 1966, p. 244]: R 1 2 jgðfþjdf Df ¼ R 1 : ð16þ jgðfþj 2 df For example, the equivalent energy bandwidths of the Bessel, Butterworth, and Chebyshev antialias filters shown in Figure 2 are 56., 61.6, and 66.1 Hz, respectively. Assuming that the width Dt = c/df, where c is a coefficient 1, the energy of the triangle pulses can be expressed by " ðþ r 1 2 H pulse 2 Dt ¼ 2Df H pulse 2 ; ð17þ and by comparing (15) and (17) we obtain vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Z u2df 1 H pulse t jhr; ð f ÞgðfÞj 2 df : ð18þ In fact, c depends on the receiver frequency response. For a given receiver this parameter can be determined experimentally through placing an antenna inside the field of unit pulses of spectrum H exp ( f ) = 1 and recording the impulse response g exp (t) and the amplitude H exp at the output. From (15), (16), and (18) it can be obtained that 2 R1 2 g exp ðþ t 2 dt exp ¼ : ð19þ H 2 exp Similarly, it is possible to calculate the parameter c teor if we know a simulated frequency response of a receiver system g teor ( f ) and respective amplitude H teor of the impulse response. For the frequency responses of the three receivers considered here the computed values of c teor are 1.7, 1.5, and 1.2, respectively. [28] From (18), after substitution of (12) or (13), we obtain a linear dependence of the observed impulse amplitude and the electric dipole moment of the return stroke (the charge moment): H pulse ¼ Cp r ; ð2þ where C = C(r, Df, c, f c, h, g, h, v) is a function of the distance r, receiver s parameters Df and c, the continuing current parameters f c, h, and g, and besides, in the case of long distances r within the Earth ionosphere waveguide, parameters h( f ) and v( f ). However, to determine the distance r to the discharge, an independent method, or additional measurement, is required as from the magnetic components H x and H y we can only determine the direction from which the impulse is coming. The accuracy of this formula depends on the specific propagation path from the source to an observer. This issue will be discussed further in the next paper, in which we will present results of simultaneous observations of field impulses in the ELF and VLF frequency range. [29] The formula (2) can also be applied in the reverse direction for the determination of the discharge electric dipole moment, if only the coefficient C is known with sufficient accuracy. From (12), (15), and (18) we obtain that in the simplest case of the near zone C is given by sffiffiffiffiffiffiffiffiffiffiffiffi C 1 2DfW 4 r 2 ; where W ¼ j ðf ÞgðfÞj 2 df ¼ 1 þ sinc ð f =f 2 cþ gðfþ df : ð21þ 6of1

7 Table 2. Predicted Values of the Electric Dipole Moment p r = H pulse /C of the Return Stroke for Different Values of the Continuing Current Parameter g/h g/h I c (A) H pulse C p r = H pulse /C The algorithm outlined above is based on the approximation (given by (17)) that assumes that the shape of the output impulse of the receiver does not differ much from a triangle. Thus, if the input signal contains any spectral components deforming the triangular shape of the output impulse, the coefficient C does not take this into account fully. The influence of the continuing current parameter g/h on the prediction of the electric dipole moment of the return stroke p r = H pulse /C, which is observed by a 6 Hz bandwidth receiver equipped with a low pass antialias six pole Chebyshev filter of Df = 66.1 Hz and c = 1.8, is illustrated in Table 2. The third column gives the impulse height measured at the receiver output for the input signal with the electric dipole moment p r = 1. The fourth column gives the values of the coefficient C computed according to (21) assuming Figure 3. Amplitudes of output signals generated by the model impulse (13) recorded by an ELF receiver equipped with the six pole Chebyshev low pass filter of bandwidth 5, 1, 15, 3, 6, 9, 18, and 24 Hz. (left) The responses due to a return stroke (g/h = ). (right) The receivers responses to the model impulse with f c = 25 Hz and g/h = 3. Inserts in Figure 3 (top) show the response of a receiver with a 6 24 Hz band pass filter. 7of1

8 Figure 4. The amplitudes of the output signals generated (right) by the model impulse (12) and (left) by the return stroke only, observed at ELF receivers equipped with filters having different frequency configurations: low pass filter bandwidth 6 Hz, low pass filter with bandwidth 3 Hz, and band pass filter 3 6 Hz. 1/4pr 2 = 1 for simplicity. The value in the second row and rows below underestimates the dipole moment, because the impulses are spread over longer time at the receiver output because of the continuing current. [3] The dependence of the coefficient C on unknown parameters of the continuing discharge f c, h, and g is an obstacle in practical usage of the function (2) in data reduction. However, Figure 1 shows that the contribution to the spectrum of the continuing current part of the discharge is significant only at lower frequency range, so the effect can be reduced by using a receiver of sufficiently large passband. Figure 3 shows simulated responses of several receivers with the passband from 5 up to 24 Hz to an excitation generated by a discharge described by (13), which occurred in the near zone, and it is evident that the observed influence of the continuing current on the output impulse amplitude decreases with increasing passband. If a receiver bandwidth is 24 Hz, the continuing discharge mode increases the output signal by only 7.5%. Another method of reduction of the participation of the continuing current in the signal is possible by an application of a band pass filter with the lower band pass frequency f l sufficiently higher than f c. A receiver of 6 24 Hz band pass reduces the contribution of the continuing current in the output signal to only 1%. [31] Most ELF or SR observations are made using receivers with bandwidths not wider than 6 Hz, i.e., covering the first few Schumann resonances. Measurements of the amplitudes of impulses related to lightning discharges still can be made in such case, in a configuration of the receiver with a band pass filter. Examples of simulated recordings of impulses by such receivers equipped with low pass or band pass filters are shown in Figure 4. The contribution of the continuing discharge to the output signal of such a receiver in configuration with a 6 Hz bandwidth low pass filter does not exceed 35%. Narrowing the bandwidth down to 3 Hz increases the amplitude three times. In a receiver configuration with a 3 6 Hz bandwidth band pass filter the contribution is less than 7%. [32] Similar dependences can be noticed during observations of impulses which propagate over long distances inside the Earth ionosphere waveguide. As the spectrum of the input magnetic field impulse (13) depends on the distance to the source, the calibration coefficient C, which results from (13), (15), and (18), is influenced by both the continuing discharge mode and the Earth ionosphere waveguide parameters: sffiffiffiffiffiffiffiffiffiffiffiffiffiffi Df W C ; where W ¼ ðf ÞH ð2þ 1 ðr; f Þe r gðf Þf 2 df : hðf Þvðf Þ ð22þ [33] The independent observations of impulses generated by atmospheric discharges in the ELF and VLF frequency range provide different data for discharge models. According to the formula (2), the ELF field observations allow estimates of the electric dipole moment p r, while the signal amplitude received in VLF range is proportional to the current moment of the return stroke s r [Jones, 197b]. Using the independent measurements of various discharge parameters, we can test for some relation among them, which allows for more precise description of the model discharge. This problem will be considered in our next paper in which we will discuss simultaneous ELF and VLF observations of large number (about 7,) of impulses. 5. Summary [34] ELF impulses generated by cloud to ground lightning discharges can be studied to obtain information about the discharge parameters and thunderstorm activity, similar to what can be obtained from observations in the VLF range. Only the impulses emerging from the background level of Schumann resonances coming from isolated CG discharges located up to a few megameters from an ELF receiver can be analyzed. [35] Continuing current of a CG discharge generating an ELF transient exerts a relatively large influence on the transient s amplitude spectrum. The simple model of such a discharge discussed above, together with the performed calculations, shows to which extent, at frequencies lower than 8of1

9 tens of hertz, the amplitude spectrum is dominated by the main components generated by this current. This creates a possibility of measurement of the contribution of the continuing current to the total discharge using ELF observations of discharge produced ELF transients. We show that the separation of the contributions from the return stroke and the continuing current can be done if the observations of transients are made using broadband receivers (bandwidth of a few hundreds of hertz) with signal filtration. [36] In addition, a parameter which can be relatively accurately measured is the magnitude of the impulse above the average level of the background. However, this impulse amplitude is a function of many physical variables, such as the dipole moment of discharge, parameters of the continuing current, parameters of the Earth ionosphere waveguide, source observer distance and the bandwidth of the receiver, all of which can be calculated or measured. 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