The paper presents a reconstruction of the current moment waveform of the gigantic jet observed optically last winter in Europe, based on the magnetic field component of the ELF electromagnetic field, recorded by the Hylaty station in Poland. Gigantic jets have only been observed so far on a few occasions, and there is still relatively little known about them. In order to analyze the recorded signal we have developed a new technique, which makes it possible to obtain the actual current moment waveform of the lightning discharges associated with the gigantic jet by eliminating from the waveform the effects of both the impulse response of the receiver and the Earth-ionosphere propagation channel. The proposed method can be also used to analyze other waveform observations, especially in the ELF and VLF frequency bands.
 A strong electromagnetic signal was recorded on 12 December 2009 by the Hylaty ELF station in Poland and several other ELF and VLF observatories worldwide. The Extremely Low Frequency (ELF) range is defined by the ITU (the International Telecommunication Union) as frequencies ranging from 3 to 30 Hz, but in atmospheric physics it often refers to frequencies up to 3 kHz. The Very Low Frequency (VLF) range is defined as frequencies from 3 to 30 kHz. van der Velde et al.  determined that the source of the recorded signal was a gigantic jet (GJ), which is the most spectacular Transient Luminous Event [Lyons et al., 2003]. Transient luminous events (TLEs) are large-scale optical events that occur at stratospheric and mesospheric altitudes and are directly related to the electrical activity in underlying thunderstorms [Pasko, 2010]. Several different types of TLEs above thunderstorms have been documented and classified. These include “elves,” “sprites,” “halos,” “blue jets” and “gigantic jets.” Gigantic jets establish a direct path of electrical contact between thundercloud tops and the lower ionosphere and so far have been observed only on a few occasions [Pasko et al., 2002; Su et al., 2003; van der Velde et al., 2007; Cummer et al. 2009; Kuo et al., 2009].
 Preliminary analysis of the magnetic field component that was recorded by the Hylaty ELF station indicates that a rapidly rising phase of the discharge current was followed by a continuous discharge phase that lasted more then 100 ms. The continuous phase of the discharge process was long enough to be recorded by most ELF recording stations worldwide [van der Velde et al., 2010]. Due to the low-pass specificity of ELF receiver circuitry, recordings of the continuous phase are distorted to various extents, depending on the characteristics of the receivers that were used. This challenged us to find a practical method that would enable reliable rectification of the signal we received. The method has been further developed so that it allows us to reconstruct the actual source current moment waveform from the magnetic field B(t) recorded by any ELF receiver with a known transfer function using the frequency-dependent propagation characteristics of the Earth-ionosphere waveguide on the source-receiver path.
 The usual technique used in similar situations, in which the continuous phase is important, involves deconvolution [Cummer and Inan, 2000; Li et al., 2008; Cummer et al., 2009]. But this operation may fail to provide reliable results on experimentally recorded waveforms because of computational issues. Deconvolution, unlike convolution, is not a straightforward operation due to its nonunique nature, i.e., there are many waveforms that satisfy the forward convolution almost equally well [Cummer and Inan, 2000]. This and other issues related to the deconvolution technique of current moment reconstruction are described in detail by Cummer and Inan . A modified version of the deconvolution method has been proposed in order to calculate the current moment waveform of sprite-associated lightning discharges [Li et al., 2008] and gigantic jet related discharges [Cummer et al., 2009] and it applies regularization parameters that enforce more smoothness on the later part of the current moment waveform. Fullekrug et al.  presented the current moment waveform of a short lightning discharge. It was determined from the inverse Fourier transform of the lightning current spectrum, by taking into consideration that the recorded magnetic field depends on the lightning current via the channel transfer function, but without taking into account the receiver's complex transfer function. The transfer function of the receiver has a strong influence on the recorded waveform and has to be taken into account especially when the continuous phase of the discharge is long, as it is in the case of the gigantic jet analyzed here.
 A new technique that we present in this paper allows us to avoid doing deconvolution calculations and relies on convolution only. This has practical advantages when analyzing experimentally recorded signals. Moreover, it allowed us to remove both the influence of the propagation channel and the distortions caused by the impulse response of the receiver from the signal. Unlike the methods described by Jones , Burke and Jones  and Sentman , which enable the extraction of one- or two-parameter lightning current moments, our technique reconstructs an arbitrary current moment waveform.
2. Magnetic Field Component of the ELF Electromagnetic Field Recorded by the Hylaty Station
 The Hylaty ELF station is located in the Eastern Carpathians in Poland (49.2035°N, 22.5438°E). The great circle distance between the station and the gigantic jet location (41.99°N, 7.61°E [van der Velde et al., 2010]) is 1407 km. The Hylaty station is equipped with a two-axis magnetometer (BN–S, BE–W), that has a 0.1–52 Hz frequency bandwidth and a sampling frequency of fs = 175.957207 (http://www.oa.uj.edu.pl/elf/), designed for Schumann resonance observations [Kułak et al., 2003]. The station is located in the sparsely populated Eastern Carpathians Mountains, far from major power lines. This makes the noise level from the power system very low, usually below 10 pT at 50Hz, and allows us to record the magnetic field component without using a notch filter.
 The corrected azimuthal magnetic field component, i.e., after subtraction of the propagation delay, that was recorded by the Hylaty ELF receiver is shown in Figure 1. Field impulse number 1 which is related to the observed GJ initially rises sharply, its risetime (about 17 ms) being limited only by the receiver's bandwidth. After the field reaches the maximum it decreases slowly, and the changes are sufficiently low to be correctly recorded by the receiver.
 After approximately 80 ms from the impulse number 1, another impulse of the same polarity appeared (impulse number 2, Figure 1), shortly after that impulse number 3 appeared, with opposite polarity. It should be noted that the appearance of the third impulse might have been related to the then still ongoing receiver's reaction to impulse number 2. In order to understand this, the impulse response of the receiver g(t) has to be analyzed. As illustrated in Figure 2, the impulse response has a strong influence on the observed waveform.
 The receiver at the Hylaty station, like many other ELF stations worldwide, is designed for spectral observations and is equipped with a Chebyshev anti-aliasing filter that is optimized for this purpose. Therefore, the observation of impulse waveforms with this type of the receiver is highly distorted, due to ripples in the impulse response caused by a steep roll-off in the receiver's frequency response. This must be taken into account when analyzing recorded magnetic field components and reconstructing the source current moment waveforms.
3. Method of Reconstructing the Current Moment's Waveform
 The spectrum of the recorded magnetic field is related to the spectral density of the source current moment in the following way:
where (f) is the spectral density of the source current moment, (f) is the transfer function of the Earth-ionosphere waveguide and (f) is the transfer function of the receiver. Note: macron signs, i.e., horizontal bars above the letter, denote complex variables.
can be called the transfer function of the inverse channel, that takes into account the frequency-dependent propagation characteristics of the Earth-ionosphere waveguide and the transfer function of the receiver. Furthermore, the channel can be interpreted as a system that has the source current moment waveform as its input signal and the receiver's output magnetic field as its output signal.
 By returning to the time domain with (3) we obtain the impulse response of the inverse channel (t), after which the current moment's waveform of the source can be obtained from equation (2) as:
where * denotes convolution and all the elements in the equation (4) are real functions. Reconstruction of the source current moment waveform by convolution has an important practical advantage over the deconvolution method. It does not lead to computational issues, does not require any additional manual tuning and has a satisfactory accuracy even in case of a relatively important noise contribution to the recorded B(t) signal. The proposed method transforms the recorded field waveform into a source current moment waveform by using the transfer function of the Earth-ionosphere waveguide and by removing the influence of the receiving system, which is particularly strong if a narrowband ELF receiver is used. The time resolution of the obtained s(t) waveform is the same for the recorded B(t) waveform; it thus depends on the receiver bandwidth.
Equation (4) describes a process of filtration of the recorded B(t) signal by a filter with the k(t) impulse response. The exact start and stop moment's in the analysis are not crucial in this procedure, therefore it can be adapted for continuous analysis of longer time intervals of B(t).
4. Current Moment Waveform Reconstruction Based on the Magnetic Field Recorded at the Hylaty Station
 In order to reconstruct the current moment waveform of the gigantic jet registered by the Hylaty ELF station on 12 December 2009, one needs to estimate the transfer function (f) or the impulse response w(t) of the Earth-ionosphere waveguide. Fullekrug et al.  calculate the channel transfer function using the normal mode expansion with the frequency-dependent ionospheric height given by Sentman . Cummer et al.  estimate the propagation channel impulse response with a full wave 2-D cylindrical FDTD simulation [Hu and Cummer, 2006]. In our case the distance from the source is only 1407 km and in this paper we have analyzed the magnetic field component, which, unlike its electric counterpart, has a small amplitude of the resonance component near the source, we used a simplified solution for the direct wave [Kułak et al., 2006]. Therefore, the resonance effects related to the around-the-globe wave, which appeared about 150 ms after the first phase of the discharge, remained in the current moment waveform. These effects can be taken into account using the theory presented by Mushtak and Williams [2002, equation (14)], being a uniform simplification of the propagation parameters of the two-dimensional telegraph equations (TDTE) [Kirillov et al., 1997].
 In the present case, we make use of a fully analytical solution for Maxwell equations for a vertical electric dipole (VED) placed in the Earth-ionosphere waveguide, known as the Bannister formula [Casey, 2002]. Supplementing the Banister formula [Casey, 2002, equation (4.20c)] with the attenuation factor and the receiver's transfer function [Kułak et al., 2010] we obtained the spectrum of the magnetic field waveform in the receiver's output as follows:
where α(f) is the attenuation rate of the waveguide, r is the distance from the dipole, hm(f) is the magnetic height of the Earth-ionosphere waveguide, vph(f) is the phase velocity of the electromagnetic wave in the waveguide, and H1(2) is the Hankel function of the second kind and first order.
 By comparing (1) and (5) we can identify (f), i.e., the channel transfer function of the Earth-ionosphere waveguide:
(Note: (f) is the response of the system to the analytic signal (t), given for f > 0.) The phase velocity and the attenuation rate can be calculated according to the two-dimensional telegraph equation (TDTE) method [Mushtak and Williams, 2002] as follows:
and e(f) and m(f) are the complex characteristic altitudes of the Earth-ionosphere waveguide. Since the event appeared at 2336 UT, the characteristic altitude should be calculated for a typical nighttime path. Greifinger et al.  used aeronomical data to construct daytime and nighttime conductivity profiles of the lower characteristic layer. The model has the form of double-exponential functions that represent the “knee” -like transition from ion-dominated to electron-dominated conductivity. For the nighttime profile the scale heights are 10.7 and 2.0 km, respectively, and the “knee” frequency is equal to 7.7 Hz. Based on the this model, analytic approximations of the lower complex characteristic altitudes for the 3–100 Hz frequency range were presented by Greifinger et al. [2007, equations (7), (16), (18)]. For a typical nighttime path, they can be written in short form, in kilometers, as follows:
 Since Kirillov et al.  developed their model of the magnetic characteristic altitude for frequencies beyond the Schumann resonance range and since the work on a practical analytical model for the lower ELF band is still in progress (V. Mushtak, personal communication, 2010), in order to calculate the magnetic altitude we made use of full-wave results for the phase velocity and attenuation rate presented by Galejs . Using the complex electric altitude from (8a) and (8b), it is possible to find an analytic formula for the complex magnetic altitude so that the propagation parameters (7a) and (7b) with use of (7c) fit those given by Galejs [1972, Figure 7.12] for the Schumann resonance range. This is how we obtained easy to use analytic formulas for the real and imaginary parts of the complex magnetic altitude. A simple logarithm dependence, similar to the Kirillov et al.  formulas, gives a sufficient approximation to the values obtained by Galejs in the Schumann resonance range. The formulas have the following form:
 In order to reconstruct the current moment waveform s(t) that accompanied the GJ phenomenon the following steps were performed: (1) The B(t) waveform, recorded with sampling frequency of fs = 175.95721 Hz, was resampled to fs = 1000 Hz using standard Matlab function resample. (2) According to the formula (3), the analytic inverse channel transfer function (f) was calculated for the distance r = 1407 km. (3) Impulse response (t) of the inverse channel was calculated using the one-sided inverse fast Fourier transform of (f) and the real form of the k(t) was restored. (4) Following equation (4), the convolution of the resampled B(t) with the impulse response of the inverse channel k(t) was calculated.
 Resulting current moment waveform s(t), restored using the proposed inverse channel method, is shown in Figure 3. From Figure 3 it results that the continuous phase initiated by the impulse number 1 might have spread in time even for several milliseconds and had an exponential profile. The time constant of the current fading was about 70 ms. Note that the fast impulse 1, related to optical appearance of the GJ, was preceded by a small, but steadily increasing, current moment that lasted for about 50 ms. Impulses 1a and 2a, that appeared after about 150 ms from the discharges 1 and 2 consecutively, are related to the around-the-globe wave. Its effects remained in the waveform as consequence of the simplified propagation formulas we have used. For the same reason the time integrated charge moment change for the analyzed event (Figure 4) is a little overestimated. Its final value reaches 15 500 (C km). It has to be noted, that in the case of long lasting waveforms, credibility of the Earth-ionosphere waveguide parameters in the lower-frequency range is essential. This issue concerns all of the currently used methods of the current moment reconstruction. The final value of the charge moment includes also 1000 (C km) from the short impulse number 2.
 The presented inverse channel method for reconstructing source parameters is a general-purpose method and it can be used for any propagation model and receiver's transfer function, especially in the case of wide bandwidth receivers in the ELF and VLF bands. Moreover, other ELF stations worldwide can use the presented method to reconstruct current moment waveforms from past and future records, using the gigantic jet EM signature that was analyzed in this paper as a point of reference. It is worth noting that the presented method makes it possible to remove the oscillations related specifically to the receiver's impulse response from the observed waveforms, as they are frequently misinterpreted as actual external events.
 We would like to thank Ferruccio Zanotti, the observer of the gigantic jet, Anna Odzimek, who prompted the development of the method presented here, as well as the whole scientific team involved in the analysis of the event, especially Jozsef Bór, Steven Cummer, Jingbo Li, and Oscar van der Velde.