Experimental waveforms of natural ELF transient signals (Q-bursts) are compared with computations based on the analytical time domain solution. The computations were performed for the uniform Earth-ionosphere cavity model with a linear frequency dependence of the ELF propagation constant. The vertical electric field was recorded in the frequency band from 1 Hz to 11 kHz by a ball antenna in fair weather conditions at Kochi (geographic coordinates: 33.3°N and 133.4°E) during the period of 2003–2004. Waveforms of discrete pulses were compared, each showing a high similarity. This confirms the validity of the model and the consistency of the source-observer distance found experimentally, directly in the time domain.
 The term Q-burst introduced by Ogawa et al. [1966a, 1966b] is related to discrete natural ELF radio pulses detected worldwide [Ogawa et al., 1967] and lasting for 0.3–1.5 seconds. These transient events, recently called ELF transients, originate from powerful remote strokes of lightning that provide signals well above the continuous Schumann resonance (SR) background by a factor of ten. The precise wideband waveforms were recently published, recorded simultaneously in the wide (1 Hz–11 kHz) and in the narrow (4–30 Hz) frequency bands with a sampling rate of 22 kHz [Ogawa and Komatsu, 2007].
 The spectral technique used traditionally for finding the source–observer distance (SOD) was developed by Jones [1970a, 1970b] and, since that time, has been applied experimentally in many studies [e.g., Jones and Kemp, 1970; Lazebny and Nickolaenko, 1976; Burke and Jones, 1995; Boccippio et al., 1995; Sato et al., 2003]. The technique exploits the characteristic spectral pattern in the pulse spectra conditioned by the interference between direct and antipodal radio waves arriving at an observer. Of course, the waveform of an ELF transient signal varies with the SOD: it is composed of the direct and delayed antipodal pulses arriving at an observer from the parent stroke, and both are followed by the round-the-world waves.
 The time domain solution was constructed as a formal Fourier transform of the zonal harmonic series representation (ZHSR) for the ELF fields in the Earth-ionosphere cavity. The latter is used to describe the global electromagnetic (Schumann) resonance [see, e.g., Nickolaenko and Hayakawa, 2002]. When treating the ELF radio propagation from natural sources of radiation, the current moment uniform in frequency is used (delta pulse in the time domain). As a result of formal Fourier transform of the ZHSR term by term, an infinite series of the time-dependent items is obtained, each corresponding to a particular resonance frequency of the Earth-ionosphere cavity [Nickolaenko et al., 1999, 2004; Nickolaenko and Hayakawa, 2002]. Expansion in the time domain converges everywhere, provided that the time moment is greater than zero (the time when the parent lightning stroke occurs). Recently, the formal time domain solution was introduced with an accelerated convergence [Nickolaenko et al., 2004], which is used in the present study.
 We compare below the model pulses with the samples of Q-bursts presented in the experimental study by Ogawa and Komatsu . Wideband (1 Hz–11 kHz) ELF/VLF records were performed with a sampling frequency of 22 kHz at Kochi, Japan (geographic coordinates: 33.3°N and 133.4°E). A classification was introduced for the signals according to the waveforms: W, V, and V–V types. The sole pulse resembles the letter V when it arrives from a positive lightning stroke at a distance of 5 to 12 Mm (1 Mm = 1000 km), and it is regarded correspondingly. When the SOD grows, the direct and antipodal waves combine into the V–V form, and at the place close to antipodal point (17 Mm or more) they merge into the W pulse.
2. ELF Propagation Model
 Our goal is to compare model waveforms with the experimental vertical electric field records. Only the time domain data are used here. We have to note that the SODs initially used were found experimentally directly from the delays between the direct and antipodal pulses [Ogawa and Komatsu, 2007]. We pick up these distances, compute relevant waveforms by using the time domain algorithm [Nickolaenko et al., 2004], and compare computed and experimental results. Such an approach may be regarded as “independent”, because it compares the experimental and theoretical data with all parameters specified beforehand. The only matching concerns with the visual position of the pulse onset and its amplitude. The job of adjusting the distance estimates is addressed afterward.
 A direct comparison of “pure” waveforms is made for the first time, owing to distortions pertinent to the ordinary SR receiving equipment. The latter usually works in the 4–40 Hz band and inevitably includes the notch filters of industrial frequency and its harmonics. Therefore the authentic waveform could only be computed, not measured. The novel wideband experimental data were collected in an exclusively clean field site, which allowed us to resign the filtering. The 1 Hz–11 kHz bandwidth of the receiver significantly exceeded that of natural radio signals from the remote strokes (4–100 Hz). Thus we can acknowledge that the bandwidth is “infinite” both in theory and in experiment.
 When computing either the waveforms or the spectra of transient ELF radio signals, one has to postulate a frequency dependence of the complex propagation constant ν(f) [Nickolaenko and Hayakawa, 2002]. We apply the dependence derived from experimental SR records at two distant (longitudinally separated) points, Kharkov and Lake Baikal. Details of simultaneous measurements may be found in the study of Bliokh et al. [1977, 1980] or that of Nickolaenko and Hayakawa . Here we mention that three kinds of SR spectra were obtained experimentally: the individual power spectra at each observatory, the complex cross-spectra, and the amplitude spectra of Q-bursts detected at two sites simultaneously. Averaged experimental data are listed in Table 1 and plotted in Figure 1. One may observe that wave attenuation found from the power spectra was the highest, and that from the Q-bursts was the smallest. Then, an intermediate value of ν(f) was inferred from the cross-spectra.
 Three linear functions approximate the experimental data:
Here f is the signal frequency in Hz, and ν(f) is the dimensionless propagation parameter. According to the mode theory, the electromagnetic fields in the frequency domain are proportional to the Legendre function Pν[cos(π − θ)] or to the associated Legendre function Pν1[cos(π − θ)] of the angular SOD θ [Nickolaenko and Hayakawa, 2002]. The real part of dimensionless propagation constant ν(f) is connected with the phase velocity of ELF radio wave, while its imaginary part accounts for energy losses in the Earth-ionosphere cavity. The global or SR occurs at frequencies satisfying the following condition:
where n = 1, 2, 3, etc., is the resonance mode number. The infinite set of relevant complex frequencies found from equation (4) is regarded as the resonance or the eigen-values. The time domain solution is the sum of attenuating in time terms, each corresponding to the particular resonance frequency [Nickolaenko and Hayakawa, 2002].
 The real parts of propagation constants (1)–(3) coincide. All of them correspond to SR peaks at frequencies of 8, 14, 20 Hz etc. separated by a 6 Hz interval. The distinction among the three models lies in the attenuation factors. We choose the intermediate propagation constant (2) for computations. The time domain solution corresponds to the uniform Earth-ionosphere cavity [Nickolaenko et al., 2004].
 A flat infinite frequency response of the receiver was assumed. We also used the constant or “white” source current moment of a lightning discharge Ids(f) = 108A*m, which corresponds to the source delta function in the time domain. The above current moment corresponds to an average lightning stroke having the peak current of 25 kA and the length of 4 km [see Rakov and Uman, 2003]. We are interested here in a qualitative comparison of experimental and computations data, and particular source amplitude was not important at present stage. The stroke polarization was positive (+ cloud-to-ground stroke) except a single case. A visual comparison of pulse amplitudes measured and computed indicates that the above current moment underestimates the observed value by a factor of up to five, which is in agreement with Q-bursts observations [c.f., Ogawa et al., 1966a, 1966b, 1967; Jones, 1970a, 1970b; Jones and Kemp, 1970; Lazebny and Nickolaenko, 1976; Burke and Jones, 1995; Boccippio et al., 1995; Sato et al., 2003].
3. Comparison of Pulsed Waveforms
 We compare in Figures 2–5 the time domain waveforms measured and computed. The experimental data were taken from the study of Ogawa and Komatsu  and we discuss them in the same order as given in their article. High similarity of the patterns is apparent, which confirms the validity of the time domain technique for establishing the SOD for all types of Q-bursts.
Figure 2 shows the model pulsed waveform (line with dots) superimposed on the experimental record shown by the black line. This waveform was attributed to the W-type, and the source distance of 17.4 Mm was established experimentally from mutual delays of individual pulses. Elements of computed and the experimental curves coincide: the direct, antipodal, and the sub-pulse, as well as the round-the-world wave. This coincidence supports the distance experimentally found by Ogawa and Komatsu .
Figures 3a and 3b compare experimental and model data for 18.3 and 18.0 Mm source distances, and again the qualitative similarity is obvious from the plots. When looking at Figure 3, an optimist will note a similar tendency in the distance variations of the intermediate peak (sub-peak) of the W-waveform, which however is not very distinct in the model. The comparison is also done in Figures 4 and 5, here distances are 19.2, 9.7, and 9.8 Mm. The correspondence of the model to the records is obvious in all cases, for all types of the Q-bursts. We must note that appropriate interference of direct and antipodal waves is clearly seen in all the figures and the round-the-world waves are also positioned at the right place.
 However, a closer inspection reveals deviations of model pulses from the records. Such departures might be conditioned by the following reasons: (1) the SOD estimate should be refined by fitting the range in model computations. (2) A more sophisticated ν(f) function could be applied. (3) The source moment depending on the frequency could be taken into account.
4. Discussion and Conclusion
 This investigation was motivated by a possibility to relate directly the computed and recorded waveforms. Previously, such a comparison was impossible owing to particular features of ELF experiments. The model pulsed signals are formed exclusively by the zero-mode radio propagation in the Earth-ionosphere cavity. The field source is a delta pulse, so that it radiates all the frequencies. At the SODs of about 1 Mm, the radiated pulse occupies the band up to 1 kHz, which is conditioned by the wave attenuation increasing with frequency. As the source distance increases, the pulsed energy appears at lower ELF, and it lies below 100 Hz when the distance exceeds 5 Mm. Further concentration of pulsed energy at lower frequencies becomes insignificant with an increasing propagation distance [Nickolaenko et al., 1999, 2004; Nickolaenko and Hayakawa, 2002].
 Available experimental records were always acquired with the receiving equipment that spoiled the genuine time dependence owing to the narrow band receivers (typically 4–40 Hz) containing special notch filters. Measurements at an exceptionally quiet site were managed in practically “infinite” band 1 Hz–10 kHz with the time resolution thus suggesting a verification of model data by experiment. We found and demonstrated a clear qualitative correspondence between the model and the experimentally observed pulses, which is encouraging. On the one hand, it implies that the time domain technique of establishing the source distance works rather effectively. On the other hand, the close reproduction of measurements by the model computations confirms the soundness of the model time domain solution. In this context, a quantitative, more exact comparison and fitting must be an objective of future investigations.
 Let us turn to the property noticed in the measurements [Ogawa and Komatsu, 2007]: the majority of Q-bursts arrive at an observer from great, antipodal distances. Such a feature is readily explained by the distance dependence of the peak amplitude of Q-bursts, which we depict in Figure 6 [Nickolaenko et al., 1999, 2004; Nickolaenko and Hayakawa, 2002]. The SOD is shown along the abscissa in Mm, and the amplitude of initial pulse excursion is plotted on the ordinate in decibels relative to 1 mV/m. The dependence is easily understood if we invoke the geometrical focusing (convergence) of the pulsed wave on a sphere at the source antipode.
 The circular wave front starts to converge after passing the “equatorial distance” of 10 Mm. At the same time, the purely geometrical magnification is combined with the energy loss in the Earth-ionosphere cavity. The attenuation factor increases with frequency, therefore the electromagnetic energy dissipates predominantly in the upper ELF band. Thus an originally “white” Q-burst becomes “pink” with distance. We observe that in the time domain its initial half-wave broadens, as the pulse progresses with distance [Nickolaenko et al., 1999, 2004; Nickolaenko and Hayakawa, 2002].
 Owing to energy losses, the focusing cannot completely restore the initial pulse amplitude. The pulse grows toward the source antipode, but it remains finite at the antipode itself. Figure 6 quantitatively presents this process. One may observe that the amplitude of a Q-burst reaches its minimum around 15.5-Mm distance, and afterward the focusing overcomes the losses of the cavity. The pulse amplitude at 20 Mm exceeds the minimum value by ∼10 dB or by a factor of 3, which explains the observational fact that many Q-bursts arrive from remote strokes.
 The comparisons of model and experiment showed a close similarity, however, slight deviations remain. Departures may indicate some inaccuracy in the SOD, and/or on a necessity to apply a more sophisticated model. We turn to the distance fitting. Relevant plots are collected in Figure 7, where we indicate alterations in the source distance being about −1 Mm. Modified distances improved matching of the waveforms, and the reciprocity of model to the experiment has increased (compare Figure 7 with Figures 2–5). On the other hand, the changes found are close to the accuracy (±1 Mm) of the SOD establishing in the frequency domain [Boccippio et al., 1995]. Therefore we admit that the global ELF radiolocation of powerful lightning strokes is characterized by ±1 Mm standard error, regardless any particular technique used: frequency or the time domain.
 A few words should be mentioned about future investigations. It is clear that waveforms in Figure 5 demand further adjustment of the model. Positioning of the peaks is already perfect, but the model pulses look much narrower than experimental curves. Clearly, modifications must be made in the spectrum of the causative stroke. In the present study, we adjusted only the experimentally found source distances and did not fit the source model in any way. In particular, we used the “white” source everywhere, while for the pulses shown in Figure 5 the “red” source should be applied. Such a source with elevated low frequency radiation generates wider pulses, which would fit the experiment. Relevant adjustment of the source spectrum is easily made in the frequency domain [see Burke and Jones, 1996]. Such a task will be addressed in future, as it is beyond the objectives of our present “time domain” study.
 In principle, modification might be also required of the propagation constant model. Say, a transform might be examined from the linear ν(f) dependence used here to a more complicated function of frequency. Incorporation also could be considered of the ionosphere angular non-uniformity or its anisotropy. Fortunately, a necessity is redundant of such complicated modifications: the data comparison demonstrated that the resemblance already achieved is so high that the residual deviations fall into the interval of measurement accuracy. They could be attributed to the impact electromagnetic interference or to the random nature of natural ELF radio signals, etc. We conclude that the major features observed experimentally are correctly described by the available time domain theory of ELF radio propagation in an isotropic and uniform Earth-ionosphere cavity.
 We would like to thank NiCT for its financial support (R and D promotion scheme finding international joint research).