1 Introduction

Lightning-generated whistlers, which are commonly detected by ground very-low-frequency (VLF) stations and satellites, have been studied for decades. These signals are right-hand polarized whistler-mode waves with frequencies between the proton gyrofrequency (fcH+) and electron gyrofrequency (fce). Whistlers are often referred to be generated from the common CG (Cloud-to-Ground) lightning; therefore, we cannot exclude the possibility that these waves are generated by lightning in the upper atmosphere, the transient luminous events (TLEs): including sprites, halos, elves, blue jets, and gigantic jets. Indeed, whistlers have been observed by DEMETER at the same time that sprites were detected at Langmuir in a satellite-ground observation campaign (Lefeuvre et al. 2009).

Sprites are luminous emission events spanning altitudes of 40–90 km above active thunderstorms (Franz et al. 1990; Sentman et al. 1995; Pasko et al. 1997). The halo is a pancake-shaped glow that appears at altitudes of 80–85 km with a 50–70 km diameter centered at the point of CG lightning (Frey et al. 2007). Sprites and halos can be triggered by the short-lived quasi-static electrical fields induced by CG lightning with large charge moment change (CMC). The CMC associated with sprites is ≧ 120 C-km (Hu et al. 2002) and the CMC associated with halos is ≧ 500 C-Km (Williams et al. 2012). Elves are at ~ 90 km and are induced by the electromagnetic pulse (EMP) generated by intensive lightning with high peak current > 60kA (Fukunishi et al. 1996; Inan et al. 1996; Barrington-Leigh et al. 2001; Mende et al. 2005). Gigantic jets (GJs) and blue jets are both upward discharges from cloud top (Pasko et al. 2002; Su et al. 2003; Wescott et al. 1995). Yet GJs can extend to the altitude up to 90 km above the Earth’s surface, near the bottom of the ionosphere.

Thus far, among these TLEs, sprites and GJs are the only two types for which recordings of associated radio signals have been explicitly reported (Cummer et al. 1998, 2009; Su et al. 2003; van der Velde et al. 2007, 2010; Huang et al. 2012). This means that out of all TLEs, they are the most likely to generate whistlers. As mentioned above, observations of whistler-like emissions associated with sprites have been reported (Lefeuvre et al. 2009); however, it is uncertain whether the emissions are attributed to the measured magnetic signatures radiating from the lightning or the later-produced sprites (Cummer et al. 2006; van der Velde et al. 2006). Hence, this study selected GJs as the basis for our investigation into whistlers generated from the electromagnetic waves emitted by the TLE event itself.

The electromagnetic features of GJs were statistically analyzed by Huang et al. (2012) based on data from the NCKU ELF/VLF station and ULF station, as well as ground optical observations made on August 31 of 2010 near the typhoon Lionrock, located approximately 500 km from the above stations. Most of the peak current moment ranged from 60 to 159 kA-km. Based on observed features on optical images, these GJs can be grouped into three forms: tree-like, carrot-like, and an intermediate type called tree-carrot-like. Among these, tree-like GJ is the one with the strongest signals (Huang et al. 2012). The detected signals comprised two distinct parts: initiating lightning and GJ, with the former occurred earlier from 24 to 225 ms with a median value of 57 ms. They also reported the f-t spectrograms of the detected signals. These f-t spectrograms revealed that the initiating lightning and the GJ differ in their power distributions across frequencies. The power of GJs decays rapidly above 3 kHz (Fig. 2 within the above reference), while common lightning usually peaks at 5 to 10 kHz, at distances beyond approximately 50 km (Rakov 2013). Thus, characterizing the whistlers generated by tree-like GJs and determining how they differ from those generated from common lightning could prove useful in the detection of these phenomena.

This study explores the characteristics and detectability of tree-like GJ-generated whistlers, including the accompanied initiating lightning, by deriving their waveforms and f-t spectrograms in the ionosphere. We also investigated the ways in which they differ from whistlers generated by common lightning. We then seek to determine the geographical locations most amenable to their detection. This provides information important to ground-satellite campaigns seeking to observe TLE-generated waves and/or to find them in the associated data, so as to further understand the relationship between TLEs and their associated electromagnetic phenomena as well as their impact on the ionosphere and magnetosphere.

2 Methods

Modeling wave features requires the calculation of waveforms and spectrograms. Whistler-mode wave packets of different frequencies propagate at different group velocities; therefore, the wave component of each frequency is treated separately to acquire the wave packet. Figure 1 presents the signals of a disturbed electric field collected by Huang et al. (2012) during a tree-like event that occurred at 16:11:28.516UTC on August 31 of 2010. The top panel presents signals related to the initiating lightning and the bottom panel deals with signals related to the GJ, showing several low-frequency oscillations in the given time span of 10 ms. On the same observational day there were also other five tree-like GJ events detected nearby the same region. Figure 2 shows their associated electric field signals, including the event shown in the bottom panel of Fig. 1, and they all exhibited similar features. Therefore, this published example event (Fig. 1), possessing the strongest peak current, is chosen to be the representative one for modeling. Fast Fourier Transformation (FFT) is used to obtain the Fourier coefficient of each Fourier frequency from the values shown in Fig. 1 in order to determine each wave component. Figure 3a presents the derived amplitude versus the frequency in a window of Δtw = 10 ms (frequency resolution 100 Hz) from the initiating lightning and GJ as presented in Fig. 1. At frequencies below 4 kHz, the amplitudes of the GJ (blue line) are obviously far larger (i.e., > 0.01 V/m) than those of the initiating lightning (black line). Above 5 kHz, the amplitudes of the GJ decrease dramatically while the amplitudes of the initiating lightning remain the same (~ 0.01 V/m). The amplitudes of the GJ peak at 300 Hz, 900 Hz, and 2.2 kHz. For the sake of comparison, Fig. 3b displays the same parameters derived from electric field signals detected at our ELF/VLF station. These are associated with common lightning observed by ISUAL of Formosat-2 mission, which occurred approximately 400 km from the station. As shown in the figure, the spectrum of common lightning is quite different from that of GJ, with peak amplitude at ~ 5 kHz and still possess certain levels of amplitudes above 10 kHz, and are distributed somewhat randomly. Figure 1 shows the measured electric field versus time E(t), which can also be expressed as a summation of Fourier components as follows:

$$E(t) = \frac{{a{}_{0}}}{2} + \sum\limits_{n = 1}^{N} {\left( {a_{n} \cos (2\pi nf_{0} t) + b_{n} \sin (2\pi nf_{0} t)} \right)}$$
(1)

where the fundamental frequency is determined as f0 = 1/Δtw, n is the number of harmonic frequencies of f0, N is the total number of Fourier components, and an and bn are derived from the real and imaginary parts of the nth Fourier coefficients with Fourier frequency f = nf0. The waveform of the GJ within a given window is denoted as EGJ(t), which is based on Eq. (1) using \(a_{0}\) = 0. The waveform of the initiating lightning is denoted as EIL(t), which is also based on Eq. (1) but uses different coefficients. The nth frequency components are denoted as EGJ,n and EIL,n, respectively.

Fig. 1
figure 1

Electric field signal measured from NCKU ELF/VLF station. Time starts at 16:11:28.516UTC on August 31 of 2010, with a period of 10 ms. (Top) signals associated with the initiating lightning; (Bottom) Signals associated with the tree-like gigantic jet

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Fig. 2
figure 2

Electric field signals associated with the 6 tree-like gigantic jets measured from NCKU ELF/VLF station on August 31 of 2010. Time expressed in hh:mm:ss.sss (UTC) denoted on the plot corresponds to time at 0 on x-axis for each GJ. The blue-colored one is the signal of the example GJ shown in the bottom panel of Fig. 1

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Fig. 3
figure 3

The amplitude (x-axis) versus frequency (y-axis) for observed signals of a initiating lightning (black line) and gigantic jet (blue line) at 21.13° N, 119.08° E (L = 1.085) above the typhoon Lionrock and b common lightning detected at 23.01° N, 124.04° E (L = 1.105) at 15:02:16.562UTC on October 21, 2010. Data is selected from a different day because the weather at the time of the GJ event was overcast, making it difficult to judge whether the flashes at the bottom part of the clouds were from common lightning

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The next step involves determining the group delay by solving the group velocity Vg at each Fourier frequency. This will be used for the wave packet propagation. Since ducted mode propagation of low-latitudinal whistlers had been suggested to be possible based on analysis of ground observations for waves and conjugate lightning activities (Srivastava et al. 2013; Gokani et al. 2015), the wave angles would be small and here the waves are assumed to propagate along the field line for the assessment purpose. Therefore, the cold plasma dispersion relation for right-hand whistler-mode waves is reduced to n2 = R (Stix 1992), where n is the refractive index that obeys n2 = c2k2/ω2, k is the wave vector, ω is the angular frequency, and c is the speed of light. Thus Vg is \(\partial {\upomega }/\partial k\). To assess the absorption of amplitudes with the propagation of waves, we take into account the component R of the dielectric tensor in a multi-ion plasma with collision terms. R is defined as follows:

$${\text{R}} = 1 - \frac{{\upomega _{{{\text{pe}}}}^{2} }}{{\omega \left( {\omega - \Omega_{e} - i\nu_{e} } \right)}} - \mathop \sum \limits_{\alpha } \frac{{\omega_{p\alpha }^{2} }}{{\omega \left( {\omega + \Omega_{\upalpha } - i\nu_{\alpha } } \right)}}$$
(2)

ωpe is the electron plasma frequency, ωpα is the ion plasma frequency for ion species α, Ωe and Ωα are the gyrofrequencies of electrons and ion species α, respectively, νe is the effective collision frequency for electrons and να is the effective collision frequency for ion species α.

In the ionosphere, collision frequency νe is taken as the summation of electron-neutral collision frequency νen and the electron–ion collision frequency νeα of all ion species (Singh 1966; Aggarwal et al. 1979). The value of νen and νeα can be derived from Eq. (3) (Thrane & Piggott 1966) and Eq. (4) (Nicolet 1953):

$$\nu_{{{\text{en}}}} = \left( {1.11 \times 10^{ - 7} {\text{n}}\left( {{\text{N}}_{2} } \right) \, + 7 \times 10^{ - 8} {\text{n}}\left( {{\text{O}}_{2} } \right)} \right)u$$
(3)
$$\nu_{e\alpha } = \left( {34 + 8.36 \log_{10} \frac{{{\text{T}}^{\frac{3}{2}} }}{{{\text{n}}_{e}^{\frac{1}{2}} }}} \right){\text{n}}_{{\text{e}}} {\text{T}}^{ - 3/2}$$
(4)

In the above equations, n(N2) and n(O2) are the number densities of atmospheric gases N2 and O2, u is the electron energy in eV, ne is the number density of electrons, T is the plasma temperature. The profiles of N2 and O2 were obtained from MSISE-00 Model (Picone et al. 2002). The profiles of other parameters for charged particles were acquired from IRI-2007 (Bilitza and Reinisch 2008).

In the calculation of να, the ion–electron collision frequency ναe is obtained from νeα in Eq. (4) as ναe = (me/mα)1/2νeα (e.g., Hsieh 1968), where mα is the mass of ion species α. The ion-neutral collision frequency ναn is obtained from including both nonresonant and resonant collision frequencies as discussed in Schunk and Nagy (2009) and Ieda (2020). The ion–ion collision frequency ναα are much smaller than ναe. Therefore, the effective ion collision frequency να can be taken as the summation of ναe and ναn. Figure 4a shows the effective collision frequencies derived for each species along the field line of L = 1.085 from the starting location where the GJ occurred (100 km high, 21.13° N, 119.08° E), to its conjugation point (100 km high, 5.83° S and 119.23° E) in the ionosphere. The peak value of νe occurs at latitudes distant from the equator where peak electron densities occur. The effective ion collision frequencies are denoted in different line styles and colors as shown at the rightmost side, their values indeed contribute to the variation of R in Eq. (2). To assess changes in the waveforms as the waves propagate, the initial waveform at the starting location, is written as E0 = EIL(t = 0, Δtw) + EGJ(t = τ, τ + Δtw), where EIL is the waveform of the initiating lightning and EGJ is the waveform of the GJ; t is time and τ is the time GJ signal was detected so that τ = 11Δtw since the initiating lightning and the GJ are 117 ms apart. The above expression denotes that EIL is set in the time window between t = 0 and t = Δtw and EGJ is set in the time window between t = τ and τ + Δtw. Inputs EIL and EGJ for E0 are based on Eq. (1) from Fourier coefficients acquired from the observed signal (Fig. 1). For the calculation implemented in this study, the observed electric fields from our stations are taken as the input values at this starting location where the GJ occurred as aforementioned, and the conjugation point refers to the conjugated site of this location of GJ. Our stations are 500 km away from where this GJ occurred; therefore, the actual amplitudes should be larger.

Fig. 4
figure 4

a Calculated effective collision frequencies for electrons and ions versus geographic latitude/altitude. b Calculated Im(kn,j) versus geographic latitude/altitude. The color bar on the rightmost side denotes the frequency of the waves. c Derived AL,n versus geographic latitude/altitude. d Calculated AL,n at the conjugate location vs. wave frequencies below 2 kHz in which the reversal frequency occurs at 700 Hz

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The background geomagnetic field model is inferred from the IGRF-11 model (IAGA working group 2010). The field line was divided at intervals of 10 km in altitude to calculate the attenuation factor for each segment. The attenuation factor for wave amplitude of the nth frequency component in the jth segment is \(e^{{a_{j,n} }}\), where \(a_{j,n} = Im\left( {k_{n,j} } \right)dl_{j}\) with \(k_{n,j}\) is the k for the nth frequency component in the jth segment and dlj is the length of the jth segment. Thus, the path-accumulated attenuation factor for the amplitude of the nth frequency component from the initial segment is \(A_{L,n} = e^{{ \sum \nolimits_{j = 1}^{L} a_{j,n} }}\).

Figure 4b illustrates the variation in \({\text{Im}}\left( {k_{n,j} } \right)\) along the field line. From Eq. (2), we find that this parameter is associated with a combination of the electron term − ωωpe2νe/((ω − Ωe)2 + νe2), denoted as Πe as well as the ion term − ωΣαωpα2να/((ω + Ωα)2 + να2), denoted as Πα. Therefore, higher-frequency components generally show larger negative values for \({\text{Im}}\left( {k_{n,j} } \right)\) due to the presence of -ω in the numerator. The sharp decrease from 19 to 17° N and from 2 to 3° S is due primarily to an increase in Ne in the lower F-region of the related magnetic latitudes so that ωpe2 also increases. The value for AL,n calculated along the field line is presented in Fig. 4c. As expected, higher-frequency components undergo more pronounced damping as they propagate. The total absorption after propagating through the ionosphere derived from AL,n shown in Fig. 4c for 2-kHz component is 29 dB (AL,n ~ 0.035), and for 20-kHz component is 90 dB (AL,n ~ 3.1 × 10–5). These values are consistent with the modeled values of absorption from different approaches in Helliwell (1965) and Graf et al. (2013). However, as shown in Fig. 4d, this trend is valid only when wave frequency f is above 700 Hz. Below this frequency, waves become increasingly damped as their frequencies decrease. This reversal is related to an increase in ion densities as waves propagate to higher altitudes: as ω increases, Πe becomes more negative due to the fact that the value of its denominator decreases because ω − Ωe becomes smaller as ω increases. However, Πα would become less negative as ω increases because the value of its denominator increases as ω increases. This reversal takes place when ωpα2 increases, such that the ratio of Πα to Πe exceeds 10%, at altitudes above 160 km. This indicates that the contribution from ions results in a reversal dependency of wave attenuation versus frequency. This is important for tree-like GJ since the main power of it concentrates on lower frequencies while the power of common lightning spreads to higher frequencies.

Actually, the waves propagate in the form of wave packets in group velocities carrying their energy. Therefore, the wave packets are modeled to analyze their evolutions along the propagation path. Here these packets are built by superposing three sequential Fourier frequency components with the Gaussian distribution function applied. For example, the lowest-frequency packet, denoted as W1, is obtained from \(G\left( t \right)\sum\limits_{n = 1}^{3} {\left( {a_{n} {\text{cos}}(2\pi nf_{0} t) + b_{n} {\text{sin}}\left( {2\pi nf_{0} t} \right)} \right)}\), where \(G\left( t \right)\) is the Gaussian distribution function and the value of n from 1 to 3 corresponds to the Fourier frequency component of 100 Hz, 200 Hz, and 300 Hz. This wave packet then represents the 200-Hz wave packet for 200 Hz is the central frequency. The 300-Hz wave packet, denoted as W2, is accordingly to be \(G\left( t \right)\sum\limits_{n = 2}^{4} {\left( {a_{n} {\text{cos}}(2\pi nf_{0} t) + b_{n} {\text{sin}}\left( {2\pi nf_{0} t} \right)} \right)}\). The rest wave packets for different frequencies are acquired in the same way. The mth wave packet is denoted as Wm. Again, Wm can also be represented by the superposition of its Fourier frequency components as \(\sum \limits_{n = 1}^{N} F_{m,n}\), where Fm,n denotes the nth frequency component for the mth wave packet, like Eq. (1) shows. The original three frequency components still compose the largest portion of Wm. At the Lth segment, the time at which Wm arrives can be obtained according to \(t_{m} = \sum \limits_{j = 1}^{L} dl_{j} /V_{g,m,j}\) where Vg,m,j is the group velocity for Wm in the jth segment. When the wave packet Wm arrives the Lth segment, it would evolve into \(\sum \limits_{n = 1}^{N} A_{L,n} F_{m,n}\), denoted as Wm,L.

The finalized waveform at the Lth segment is the summation of wave packets Wm,L generated from initiating lightning and the GJ, which is written as follows with IL denotes initiating lightning:

$$E_{L} \left( t \right) = \sum\limits_{m = 1}^{N - 2} {\left[ {W_{IL,m,L} \left( {t_{m} ,t_{m} + {\Delta }t_{w} } \right) + W_{GJ,m,L} \left( {t_{m} + \tau , t_{m} + \tau + {\Delta }t_{w} } \right)} \right]}$$
(5)

Furthermore, in the satellite/ground observations of whistlers, sequential whistlers with similar dispersions were indeed observed. These are usually generated from sequentially-occurring common lightning. Nevertheless, the possibility remains that these observed whistlers are generated from tree-like GJs, wherein their initiating lightning cannot be excluded. Therefore, we also model the features of waveforms from two events of sequentially-occurring common lightning, based on Eq. (1) and the values shown in Fig. 3b, under the same ionospheric parameters, with the same value of τ adopted in Eq. (5), in order to compare them to tree-like GJ-generated waveforms.

3 Modeling results

3.1 Whistlers generated from initiating lightning and tree-like GJ

Figure 5a presents the superposed waveform E0 at the starting location. It was derived from Eq. (5) based on Eq. (1) from the waveform of each Fourier frequency up to 50 kHz. The signal starts at t = 0 ms. The starting signal is associated with the initiating lightning beginning at UTC16:11:28.516. The disturbed signal occurring at 11Δtw later is associated with the GJ. The calculated input waveform at t = 0 ms precisely matches the observed waveform shown in Fig. 1. Initially, the amplitudes are on the order of 1 V/m. Figure 5b presents the corresponding f-t spectrogram of power below 10 kHz due to the rapid decay in the amplitudes of higher-frequency components (Fig. 4c). It can be clearly seen that the power of the GJ at lower frequencies is stronger than that of the initiating lightning. The maximum power is set at dB = 0, shown in red at the rightmost color bar, which corresponds to the maximum amplitude of the GJ at f = 300 Hz (Fig. 3a).

Fig. 5
figure 5

a Calculated waveforms [EL(t) vs. time, Eq. (5)] for initiating lightning and GJ at the starting location, where proton gyrofrequency fcH+ = 631.6 Hz, Ne = 1.976 × 109 m−3, NO+% = 95%, and O2+% = 5%. b Corresponding f-t spectrogram of a

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Figure 6a, b present the calculated waveforms for signals from initiating lightning and GJ, respectively, at the highest altitude of the field line in the ionosphere. Longer durations compared with those in Fig. 5a exist because of collision effects in the media that signals have traveled through. An example blow-up diagram for the 500-Hz wave packet is specifically displayed within Fig. 6b to exhibit the detail look of the waveform within a 10 ms frame. Note that the portions with no wave packets may be filled up if the resolution of Fourier frequency component is increased, but the time delays for higher frequency wave packets which travel faster would be hardly differentiated. Figure 6e, f also present the calculated waveforms for signals from initiating lightning and GJ, respectively but for the location at the conjugate point at the bottom of the ionosphere. The blue dotted line denotes the time when the 50 kHz wave component of the initiating lightning signal arrives. At the highest altitude, this time is t = 55 ms (Fig. 6a); and t = 104 ms for the conjugate point (Fig. 6e). The orange dotted line denotes the time when the 50 kHz wave component for the GJ signal arrives. This time becomes t = 172 ms for the highest altitude (Fig. 6b) and 221 ms for the conjugate point (Fig. 6f).

Fig. 6
figure 6

a Calculated waveforms [EL(t) vs. time, Eq. (5)] of initiating lightning at the highest location of this field line at 535 km high, 7.72° N and 119.4° E, where fcH+ = 469.7 Hz, Ne = 1.072 × 1011 m−3, O+% = 95.6%, N+% = 2.85%, H+% = 1.04%, from the initial values of initiating lightning shown in Fig. 5a. b Calculated waveforms at the highest location based on the initial values of GJ shown in Fig. 5a. The blow-up diagram displays the waveform for the frequency component of 500 Hz. c The superposed waveforms from those shown in a and b. d Corresponding f-t spectrogram of c. The dispersion D is 12 s Hz1/2. e Calculated waveforms [EL(t) vs. time, Eq. (5)] of initiating lightning at the conjugate location of this field line at 100 km high, where fcH+ = 633.6 Hz, Ne = 1.977 × 109 m−3, NO+% = 91%, O2+% = 9%, from the initial values shown in Fig. 5a. f Calculated waveforms at the conjugate location at 100 km high based on the initial values of GJ shown in Fig. 5a. g The superposed waveforms from those shown in e and f. h Corresponding f-t spectrogram of g. The dispersion D is 22 s Hz1/2

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Figure 6c shows the total waveform EL(t) by superposing the waveforms of Fig. 6a, b. The waveform before 172 ms is generated purely from the initiating lightning for this event [part (1), denoted in red]. Part (2) begins after 172 ms whereupon the waveforms are generated from the initiating lightning as well as the GJ. Nevertheless, detailed analysis shows that the initial amplitudes of GJ at higher frequencies are so low (Fig. 6b) that the fluctuations (~ 0.005 V/m) before 240 ms in Fig. 6c are primarily due to the initiating lightning. After t = 240 ms, the 8.6-kHz wave packet from GJ arrives and contributes the dominant amplitude. The amplitude continues increasing with time because lower-frequency wave components (with less damping) continue arriving, with a dip only at t ~ 360 ms corresponding to the decrease in amplitude occurring at ~ 2 kHz (Fig. 3a, blue line). Amplitude peaks up to 0.027 V/m at about 640 ms can be attributed to the incoming 500-Hz wave component. Part (3) in this figure marks the period during which the incoming amplitude begins decreasing due to wave components below 500 Hz. The corresponding f-t spectrogram in Fig. 6d clearly shows the dispersive features of these generated whistlers, with the later one exhibiting far more power at lower frequencies (< 4 kHz) produced from the GJ signal. At this location, the maximum power has decreased to dB = − 15 compared with that at the starting location (Fig. 5b). Note that the wave power show in a grid this spectrogram is the summation of power of all Fourier frequency components of the corresponding wave packet.

At the conjugate point at an elevation of 100 km, the bottom of the ionosphere, the part after 104 ms is entirely attributed to initiating lightning (Fig. 6e), as the part (1) in Fig. 6g. This part lasts until 221 ms, however, the amplitude is so weak that it cannot be seen clearly on the plot. After t = 221 ms, part (2) begins, during which the amplitude generally increases with time. Due to signals from the GJ (Fig. 6f), the amplitude after 450 ms, when the 4.2-kHz wave packet arrives, is dominant. The maximum amplitude at t = 1010 ms can be attributed to the 600-Hz component. This amplitude drops to approximately 0.2% of the original signal on the order of 0.01 V/m. After t = 1010 ms, the amplitude mostly decreases with time. This marks the start of part (3). Theoretically, the lowest-frequency (100 Hz) wave packet EGJ would arrive at 2397 ms, though only 200-Hz wave packet can be modeled here. The corresponding f-t spectrogram is presented in Fig. 6h. Here the maximum power has decreased to dB = − 24.5 and the power dominance of GJ-generated components is clearly shown below 4 kHz.

3.2 Whistlers generated from two sequentially-occurring common CG lightning

Figure 7a plots the input waveforms of two common lightning based on Eq. (1). The same time gap as the initiating lightning and GJ (t = 11Δtw at the starting location) is used. The strength of the signal is near 20 V/m (Fig. 7a). Because the initial amplitudes below 10 kHz are generally large (Fig. 3b), most of the f-t spectrogram is red (Fig. 7b). The waveform and the f-t spectrogram at the highest location of the field line are presented in Fig. 8a, b. Parts (1), (2), and (3) denoted in Fig. 8a, c are defined in similar manner to those in Fig. 6c, g. Part (1) marks the period during which the signals are only generated by the 1st lightning event; part (2) marks the period starting from the reception of signals from the 2nd lightning event until the maximum amplitude is recorded; part (3) marks the period in which the amplitude mostly decreases with time. In part (1) of this figure, there are no relatively-small fluctuations generated by the initiating lightning as those shown in Fig. 6c. Overall, the amplitudes in part (2) are increasing; however, they are somewhat randomly distributed over time, compared with those occurring before 240 ms as shown in Fig. 6c. This is because the initial amplitudes of common lightning peak at several frequencies over a broader frequency range (Fig. 3b). The amplitude reaches a maximum at 295 ms when the 1.5-kHz wave packet of the 1st lightning event and the 4-kHz wave packet of the 2nd lightning event arrive together, and then decreases as lower-frequency components arrive [part (3)]. Figure 8c presents the waveform calculated at the conjugation location. Parts (1) to (3) are defined in a manner similar to those in Fig. 8a. Because higher frequency components are associated with more pronounced absorption, signals in part (1) are very small to be distinguished on the plot. The earlier occurrence of the maximum amplitude in Fig. 8c, resulting in shorter time span of part (2), may be a useful indicator to differentiate from the part (2) of Fig. 6g. The peak in amplitude at 770 ms can be primarily attributed to the arrival of the 800 Hz wave packet of the 1st lightning event and the 1.1-kHz wave packet of the 2nd lightning event. These components possess larger amplitudes compared with those at lower frequencies, especially for 800 Hz (Fig. 3b), and are less affected by damping because they are close to the reversal frequency (Fig. 4d). However, the spectrum of the GJ possesses a peak amplitude at a lower frequency of 300 Hz (Fig. 3a); therefore, the maximum amplitude occurs later [end of part (2) in Fig. 6g]. Consequently, part (3), in which the amplitude decreases with time, should generally account for a larger portion for common-lightning-generated whistlers, compared with that of GJ-generated whistlers. On the f-t spectrogram there are two sequential whistlers (Fig. 8b, d), and the distributions of power over frequencies for them would be similar, different from those generated from initiating lightning and GJ (Fig. 6d, h).

Fig. 7
figure 7

a Calculated waveform [EL(t) vs. time] based on two sequentially-occurring events of common lightning based on the amplitude spectrum in Fig. 3b, at the same starting location as that shown in Fig. 5. b Corresponding f-t spectrogram of a

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Fig. 8
figure 8

a Calculated waveforms at the highest location of this field line based on the initial values given in Fig. 7a. b Corresponding f-t spectrogram of a. The dispersion D is 12 s Hz1/2. c Calculated waveforms at the conjugate location of this field line based on the initial values given in Fig. 7a. d Corresponding f-t spectrogram of c. The dispersion D is 22 s Hz1/2

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3.3 Conjugate locations of tree-like GJ events observed by ISUAL

In this subsection we discuss locations likely to allow the detection of tree-like GJ-generated whistlers. These should be directly related to the conjugate locations of the observed tree-like GJ events. The conjugate locations of the events observed by ISUAL for the period from 2005 to 2016 are analyzed using geomagnetic field model IGRF-11 (IAGA working group 2010). Figure 9 presents the results in which the blue dots represent the locations where tree-like GJ events were observed. The conjugate locations (red dots) occur primarily in the western Pacific Ocean, southeastern Pacific Ocean, and northern Africa region. Therefore, regions covering these dots can be the places to detect this type of whistlers.

Fig. 9
figure 9

Geographical distributions of tree-like GJ events observed by ISUAL (blue dots, year 2005 to 2016). A total number of 28 locations is shown. Conjugate points of these locations are denoted by red dots, traced from the field model IGRF-11. The green line represents the geomagnetic equator

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4 Discussion and conclusions

This study modeled waveforms of whistlers generated from an observed representative tree-like GJ and its initiating lightning that occurring above Typhoon Lionrock in 2010 (Fig. 6c, g). Our results provide a reference for future studies seeking to ascertain whether a detected whistler-like waveform can be attributed to a tree-like GJ event rather than sequential lightning (Sect. 3.2). The features of the waveforms are illustrated by dividing the time period into the three parts in red in Figs. 6c, g and 8a, c. Part (1) counts only the contribution from the 1st lightning (i.e., the initiating lightning in Fig. 6a, e, and the 1st common lightning in Fig. 8a, c). At the highest location, the amplitudes shown from signals dominated by the initiating lightning (before 240 ms) remain fluctuated with small amplitudes on the order of 0.001 V/m (Fig. 6c), while those from common lightning features relatively larger amplitudes on the order of 0.01 V/m [part (1), Fig. 8a]. Part (2) starts with the signal of the electric-field signal of the 2nd lightning (i.e., as related to the GJ-, and the 2nd event of common lightning). Overall, the amplitudes increase with time but are more randomly distributed for whistlers generated from common lightning (Figs. 6c and 8a). Part (3) begins when the amplitudes decrease following the arrival of lower-frequency waves. At the conjugate location around the bottom of the ionosphere, higher-frequency waves above 10 kHz are almost completely attenuated, there is little difference between the waveforms generated by the tree-like GJ and common lightning [Part (1) in Figs. 6g and 8c]. The best way to differentiate between these events would be to compare the time at which the maximum amplitude occurs and the duration of part (3). The whistlers generated by two sequential lightning would reach their maximum amplitude earlier and the period associated with decrease in amplitude [part (3)] would be longer (Fig. 8c).

The f-t spectrograms of emissions from the tree-like GJ event are presented in Fig. 6d, h. Comparing the power of whistlers versus their frequency over time is an effective method of differentiating tree-like GJ-generated whistlers from lightning-generated ones. Tree-like GJ-generated whistlers have distinctly greater power at a few hundred Hz, especially near the reversal frequency [start of Part (3) of Fig. 6g, h]. It should be noted that by taking collisional damping effects in the ionosphere, we can obtain a reversal frequency at 700 Hz (i.e., for this case only). Waves of a higher frequency undergo more damping with an increase in frequency, while waves of a lower frequency undergo more damping as the frequency decreases (Fig. 4d). This reversal can be attributed to the presence of ions and has never been reported before. Nonetheless this is an important factor due to the fact that the power of tree-like GJ is still very high below 1 kHz; however, the power of common lightning is not as strong.

The above results dealing with waveforms and f-t spectrograms provide crucial supplementary information for ground-satellite campaigns seeking to verify whether observed whistlers are associated with GJs observed from the ground. For example, propagation links was required in order to connect the sprites observed from ground stations and whistlers measured by satellites (Lefeuvre et al. 2009). Therefore, a list of what characteristics of what to look for in the waveforms and f-t spectrograms of TLE-generated emissions will be highly valuable for identifying similar emissions in observational data.

Also for the purpose of observational study, we investigated the detectability and possible locations of these signals. At the highest altitude of the field line, we found that the calculated electric fields drop to the order of 0.01 V/m (Fig. 6g). This value can be detected based on the instrument output quantity identified in contemporary satellite missions such as DEMETER (Berthelier et al. 2006). As to the detectability for the order of 1 mV/m at the conjugate location in the ionosphere shown in Fig. 6g, the instruments on satellite missions are able to measure it; however our ground NCKU ELF/VLF station are unable to perform measurements at this level even if no further attenuation when waves propagate to the ground (Huang et al. 2012). Nevertheless, it is essential to note that the input electric fields in this calculation are adopted from values measured 500 km away from the site at which the tree-like GJ occurred. According to a previous study on the attenuation of lightning signal strength as a function of distance (Orville 1991), the rate of amplitude decay versus distance D (in km) is approximately D−1.13. Considering that the top of the GJ is approximately 10 km from the bottom of the ionosphere (100 km high), the actual amplitudes of the signals entering the ionosphere could be as large as 50 times the input values used in this model. Consequently, the order shown in Fig. 6g must be 50 times larger in order for it to be detectable from our ground ELF/VLF station. The geographical locations from which the tree-like GJ-generated whistlers may be observed were derived from the conjugate locations of tree-like GJs observed by ISUAL for the period from 2005 to 2016. These locations are clustered in the western Pacific Ocean, southeastern Pacific Ocean, and northern Africa region (Fig. 9).

An important point raised by this study had to do with the impact of tree-like GJ-generated whistlers. Interactions between these whistlers and electrons may provide a clue due to the fact that the energy range of electrons capable of interacting with waves depends upon the frequency of the wave (e.g., Kennel 1966). Since the peak power of tree-like GJ-generated whistlers occurs at frequencies different from those associated with whistlers generated by common lightning, the interactions with electron can occur at different energies. This paper serves as a reference for related observational and theoretical studies on TLE-generated whistlers.