Propagation effect of the lightning electric fields along rough sea surface and the effects on ToA-based lightning location systems

Abstract

This paper based on Barrick’s formulations and Wait’s formulations, we analyzed the lightning vertical electric field and it is derivative in the time domain along rough sea surface. Furthermore, we discussed that it effects on Time-of-Arrival (ToA)-based lightning location systems (LLS). We found that the wave-shape and time-delay of the vertical electric field can be significantly affected when propagates along rough the sea surface. The rough sea surface has a few effect on the magnitude of the vertical electric fields. With the observation distance increase, the rise time of the wave-shape becomes long with the increasing of the wind speed. The peaks of the field derivative obviously decrease with the increasing of the wind speed, and the farther observation distance is, the stronger attenuation is. The time-delay of the field derivative increases as the wind speed increases. The observed time-delay resulting from the propagation along rough sea surface might impact the location accuracy of ToA-based LLS.

Keywords

Lightning vertical electric field rough sea surface Qingdao buoy station

1. Introduction

The electromagnetic fields generated by lightning return strokes have sub-microsecond components which play a major role in the interaction of these electromagnetic fields with structures [1, 2], and the hazard of lightning to transmission mainly concentrates on the power supply system and transmission system [3, 4, 5]. The high-frequency components in the electromagnetic fields undergo rapid attenuation when they propagate over finitely conducting ground [6, 7, 8, 9, 10, 11, 12]. Weidman et al. [13, 14] and Willett et al. [15] found that the experimentally obtained spectrum under maritime conditions is rapidly attenuated when the frequencies are higher than about 10 MHz, and they extended that the frequency spectrum is generated by lightning return strokes up to 20 MHz. Ming and Cooray [16] theoretically analyzed the propagation effect of the rough sea surface of the lightning electromagnetic fields by using Wait’s formulations and found that the attenuation of frequencies higher than about 10 MHz significantly. Zhang et al. [17] further analyzed the propagation effects of the roughness of the ocean surface on the lightning vertical electric fields, which based on an improved two-dimension (2D) fractal rough ocean surface model.

For the accuracy of the Wait’s formula, Shoory et al. [18] have examined the accuracy of Wait’s formula for the analysis of the propagation of lightning-radiated electromagnetic fields over a mixed propagation path and found that Wait’s formula can also reproduce the distant field and it is waveform with a good accuracy at far distances of 10–100 km from the lightning channel. Zhang et al. [19] further examined the accuracy of Wait’s formulations, which are at distances of 200–1000 m along a mixed ocean-land path by using FDTD and found that the Wait’s formulations also have an acceptable accuracy at close distances of hundred of meters.

More recently, Li et al. [20, 21, 22] analyzed the propagation effects of irregular terrain on lightning electromagnetic fields by using FDTD and the effects on ToA-based LLS, it shows that the time delays caused by the propagation along an irregular terrain and might impact the accuracy of ToA-based lighting location systems. LLS based on the ToA technique have been used worldwide for the lightning detection [23, 24, 25]. And lightning detection inshore has attracted more and more attention in recent years, especially in the coastal regions of China.

However, all the studies presented a theoretical analysis of the propagation effects on lighting electromagnetic fields over irregular terrain. Therefore, this paper selected a thunderstorm weather process in China, using the sea-wave data for Qingdao buoy station in the northeastern part of China, to analyze the lightning electromagnetic fields along a rough sea surface and that effects on ToA-based LLS.

2. Analysis method and computational models

We have assumed for the ground to be perfectly conducting, the lightning channel is straight and vertical to the ground, and the vertical electric field in the time domain at any point on the ground level can be expressed as follow [26, 27].

$\displaystyle E_{v,\infty}(0,d,t)=\frac{1}{2\pi\varepsilon_{0}}\int_{0}^{L^{% \prime}(t)}\frac{2-3\sin^{2}\theta(z^{\prime})}{R^{3}(z^{\prime})}\int\limits_% {t_{b}(z^{\prime})}^{t}i\left({z^{\prime},t-R(z^{\prime})/c}\right)dtdz^{% \prime}{}+\frac{1}{2\pi\varepsilon_{0}}\int_{0}^{L^{\prime}(t)}\frac{2-3\sin^{% 2}\theta(z^{\prime})}{cR^{2}(z^{\prime})}i\left({z^{\prime},t-R(z^{\prime})/c}% \right)dz^{\prime}{}-\frac{1}{2\pi\varepsilon_{0}}\int_{0}^{L^{\prime}(t)}% \frac{\sin^{2}\theta(z^{\prime})}{c^{2}R(z^{\prime})}\frac{\partial i\left({z^% {\prime},t-R({z}^{\prime})/c}\right)}{\partial t}dz^{\prime},$ (1)

Where $\varepsilon_{0}$ is the dielectric constant of free space, ${z}^{\prime}$ is the return stroke current pulse front which is “seen” by an observer on the ground. $t_{b}(z^{\prime})$ is the time of $z^{\prime}$ , $t_{b}(z^{\prime})=(z^{\prime 2}+d^{2})^{1/2}/c$ , $d$ is the horizontally observation distance from the lightning channel. $c$ is the speed of light. $L^{\prime}(t)$ is the length of the channel which an observer “sees” at time $t$ , $t=L^{\prime}(t)/v+\left[{L^{\prime}(t)^{2}+d^{2}}\right]^{1/2}/c$ , $v$ is the return stroke speed. $i(z^{\prime},t-R(z^{\prime})/c)$ is the return stroke current along the channel. $\theta(z^{\prime})$ is the azimuth between $z^{\prime}$ and observation point. $R(z^{\prime})$ is the distance between $z^{\prime}$ and observation point, the parameters are shown in Fig. 1.

Figure 1.

Schematic diagram used in deriving the expressions for the lightning electromagnetic fields on the ground level generated by lightning return stroke.

2.1 Wait’s formulations for the lightning vertical electric field

According to Wait’s formulations, the vertical electric field generated by the lightning return strokes along sea surface are given as below [16]:

$\displaystyle E_{\nu,\sigma}(0,d,t)=\int\limits_{0}^{t}E_{v,\infty}(0,d,t-\tau% )w(0,d,\tau)d\tau,$ (2)

Where $E_{v,\sigma}(0,d,t)$ and $E_{v,\infty}(0,d,t)$ are the lightning vertical electric field in the time domain on the finitely and perfectly conducting ground level, respectively. $w(0,d,t)$ is the inverse Fourier Transformation of the attenuation function $W(0,d,j\omega)$ in the frequency domain.

The attenuation function $W(0,d,j\omega)$ is written as follow [28, 29, 30], which can be replaced by the value corresponding a dipole located at the lower end of the channel [31]:

$\displaystyle W(0,d,j\omega)=1-j\sqrt{\pi p}\exp(-p)\text{erfc}(j\sqrt{p}),$ (3) $\displaystyle p=-\frac{j\omega d}{2c}\Delta_{\textit{eff}}^{2},$ (4)

Where $j=\sqrt{-1}$ , $\Delta_{\textit{eff}}$ is the effective surface impedance corresponding to the sea surface, “erfc” is the complementary error function.

According to Barrick [32, 33], the effective impedance $\Delta_{\textit{eff}}$ of rough ground surface can be given as:

$\displaystyle\Delta_{\textit{eff}}=\Delta+\Delta^{\prime},$ (5) $\displaystyle\Delta=\frac{k_{0}}{k}\left({1-\frac{k_{0}^{2}}{k^{2}}}\right)^{1% /2},$ (6) $\displaystyle k=k_{0}(\varepsilon_{r}-j60\sigma\lambda_{0})^{1/2},$ (7) $\displaystyle k_{0}=\omega(\mu_{0}\varepsilon_{0})^{1/2},$ (8) $\displaystyle\lambda_{0}=c/(\omega/2\pi),$ (9)

Where $\Delta$ is the normalized surface impedance of the smooth ground with a conductivity $\sigma$ and a relative dielectric constant $\varepsilon_{r}$ , and $\Delta^{\prime}$ is increase in the surface impedance due to the rough ground surface. $\varepsilon_{0}$ and $\mu_{0}$ are the dielectric constant and magnetic permeability of free space respectively.

$\displaystyle\Delta^{\prime}=\frac{1}{4}\int\limits_{-\infty}^{+\infty}d\gamma% \int\limits_{-\infty}^{+\infty}G(\gamma,\eta)V(\gamma,\eta)d\eta,$ (10) $\displaystyle G(\gamma,\eta)=\frac{\gamma^{2}+b\cdot\Delta\cdot(\gamma^{2}+% \eta^{2}-\omega\gamma/c)}{b+\Delta\cdot(b^{2}+1)}+\frac{\Delta\cdot(\gamma^{2}% -\eta^{2})}{2}+\Delta\cdot\omega\cdot\gamma/c,$ (11) $\displaystyle b=\frac{c}{\omega}\left[{\left({\frac{\omega}{c}}\right)^{2}-% \left({\gamma^{2}+\frac{\omega}{c}}\right)^{2}-\gamma^{2}}\right]^{1/2},$ (12)

Where $V(\gamma,\eta)$ is the sea-wave spectrum of rough sea model, which is very important for calculating the normalized rough (rough normalized) surface impedance. $\gamma$ and $\eta$ are the spatial wave numbers (or spatial frequencies) along (long) $x$ and $y$ directions.

2.2 Mathematical representations of sea-wave model

There are several models available in the literature for the sea-wave spectrum of rough sea model. Here we considered that the two-dimensional (2D) improved fractal sea-wave spectrum, the sea-wave spectrum solves the problem that the omnidirectional sea-wave spectrum of fractal sea-wave model could not satisfy the positive power law when spatial wave numbers were smaller than the fundamental wave number, and the expression is given as in Wang et al. [34]:

$\displaystyle V(\gamma,\eta)=S(\gamma,\eta)D(\gamma,\eta,\phi)$ (13)

Where $V(\gamma,\eta)$ is the sea-wave spectrum, $S(\gamma,\eta)$ is the isotropic spectral density, and $D(\gamma,\eta,\phi)$ is the function associated with the propagation direction of the sea wind.

$\displaystyle S(\gamma,\eta)=\left\{{\begin{array}[]{l}-\frac{\tau^{2}\chi^{2}% }{2\ln a}k_{0}^{2(d-\xi)}(\sqrt{\gamma^{2}+\eta^{2}})^{[-2(d-\xi)-1]},\sqrt{% \gamma^{2}+\eta^{2}}\geqslant k_{0}\\ \\ \frac{\tau^{2}\chi^{2}}{2\ln b}k_{0}^{2(d-3)}(\sqrt{\gamma^{2}+\eta^{2}})^{[-2% (d-3)-1]},\sqrt{\gamma^{2}+\eta^{2}}\geqslant k_{0}\end{array}}\right.$ (14) $\displaystyle D(\gamma,\eta,\phi)=1+\frac{4\pi}{\sqrt{\gamma^{2}+\eta{}^{2}}}% \sum\limits_{l=1}^{\infty}U(\sqrt{\gamma^{2}+\eta^{2}},2l)\cos[2l(\phi-\beta_{% 0})]$ (15)

Where $\phi$ is the angle of the propagation directions between the sea wave and the wind, $\beta_{0}$ is the angle of the propagation directions between the lightning electromagnetic fields and the wind. $U(\sqrt{\gamma^{2}+\eta^{2}},2l)$ is given as:

$\displaystyle U(\sqrt{\gamma^{2}+\eta^{2}},2l)=\frac{1}{4\pi}\int\limits_{-\pi% }^{\pi}\alpha_{0}\text{sec}h^{2}(\alpha_{0}\varphi)\exp(-j2l\varphi)d\varphi$ (16) $\displaystyle\left\{{\begin{array}[]{l}\alpha_{0}=2.61(\sqrt{\gamma^{2}+\eta^{% 2}}/k_{0})^{1.3},0.65\leqslant\sqrt{\gamma^{2}+\eta^{2}}/k_{0}\leqslant 0.95\\ \alpha_{0}=2.61(\sqrt{\gamma^{2}+\eta^{2}}/k_{0})^{-1.3},0.95<\sqrt{\gamma^{2}% +\eta^{2}}/k_{0}<1.6\\ \alpha_{0}=1.24,\textit{other}\\ \end{array}}\right.$ (17)

In this paper, the sea-wave data for Qingdao buoy station on 8 August 2015 is selected to input the sea-wave model.

3. Simulated results and analysis

3.1 Obtained sea-wave data of Qingdao buoy station

Qingdao buoy station is located on the Changmenyan Island which is 23.3 kilometers from the coast of Qingdao. Figure 2 presents the location map of Changmenyan buoy station. Figure 3 presents the ten-minutes-mean wind speed on 8 August 2015. However, the ten minutes-mean wind speed varies significantly at different times. The maximum wind speed reaches to 11.2 m/s.

Figure 2.

Location map of Changmenyan buoy station.

Figure 3.

Ten minutes-mean wind speed data of Qingdao buoy station on 8 August 2015.

3.2 Propagation effect of lightning electric fields

In order to calculate the electromagnetic fields generated by return strokes. It is necessary to know the spatial and temporal variation of the return stroke current and return stroke velocity. In this paper we have employed the modified transmission line with linear current decay with height (MTLL) [35] to simulate the distribution of the lightning return stroke current along the channel. The expression is given as:

$\displaystyle i\left(z^{\prime},t-z^{\prime}/v\right)=i\left(0,t\right)\left({% 1-z^{\prime}/H}\right),$ (18)

The lightning channel height is assumed to be $H=$ 7.5 km and return stroke speed is $v=1.3\times 10^{8}$ m/s. The return stroke discharging current includes two components, a breakdown current and a corona current. Two components can be calculated using the expression suggested by Heidler [36]:

$\displaystyle i(0,t)=I_{01}\frac{(t/\tau_{11})^{2}}{\left[{(t/\tau_{11})^{2}+1% )}\right]}e^{-t/\tau_{12}}+I_{02}\frac{(t/\tau_{21})^{2}}{\left[{(t/\tau_{21})% ^{2}+1)}\right]}e^{-t/\tau_{22}}$ (19)

Where $I_{0}$ is the amplitude, $\tau_{1}$ is the current rise-time constant, and $\tau_{2}$ is the current decay time constant. Table 1 shows the parameters of the subsequent return strokes [37].

Table 1

Parameters of the subsequent return strokes

$I_{01}$ /kA	$\tau_{11}$ / $\mu$ s	$\tau_{21}$ / $\mu$ s	$I_{02}$ /kA	$\tau_{12}$ / $\mu$ s	$\tau_{22}$ / $\mu$ s
10.7	0.25	2.5	6.5	2	230

Figure 4 presents the channel-based current waveform corresponding to a typical subsequent return stroke.

Figure 4.

The channel-based current waveform corresponding to a typical subsequent return stroke.

Figure 5.

The propagation effect of sea surface on the lightning vertical electric field and its derivative at distances of 1 km (a, b), 10 km (c, d), 30 km (e, f), 100 km (g, h) from the lightning channel.

The direction of the field propagation is assumed to be the same as that of the sea wind, and the field attenuation is maximum in such a case. Figure 5 shows the propagation effect of sea surface on the lightning vertical electric field and it is derivative at distances of 1 km, 10 km, 30 km, 100 km from the lightning channel. Curve 1 corresponds to the perfectly conducting sea, curves 2–5 correspond to the sea surface caused by the wind speed $v=$ 1.4 m/s, 4.7 m/s, 7 m/s, 11.2 m/s respectively. Please notes that the rough sea has no or almost no effect on the size of the vertical electric field. When horizontal observation distance is near the lightning return stroke channel ( $d=$ 1 km), the rise time of the wave-shape caused by rough sea surface that is similar to the perfect conducting sea. Wind speed plays an important role on the lightning electric field when the observation distance increase. With the observation distance increases, the rise time of the wave-shape becomes long with the increasing of the wind speed. In addition, the peak of the field derivative obviously decrease with the increasing of the wind speed, and the farther observation distance is, the stronger attenuation is.

3.3 Resulting time errors in ToA-based LLS

To minimize the effect of the lossy ground, the several methods have been suggested for the determination of the time of arrival associated with measured waveforms in ToA-based LLS [38, 39]. In this paper, we will consider two methods: (i) the time corresponding to the peak field, and (ii) the time corresponding to 50% of the field peak.

Table 2
Time corresponding to the peak lightning electric field ( $\mu$ s)

Distance from the lightning channel, d (km)	Wind speed, $v$ (m/s)
	0 m/s		1.4 m/s		4.7 m/s		7 m/s		11.2 m/s
1 km	1.48	$\mu$ s	1.48	$\mu$ s	1.48	$\mu$ s	1.48	$\mu$ s	1.48	$\mu$ s
10 km	0.845	$\mu$ s	0.855	$\mu$ s	0.865	$\mu$ s	0.865	$\mu$ s	0.875	$\mu$ s
30 km	0.81	$\mu$ s	0.83	$\mu$ s	0.835	$\mu$ s	0.835	$\mu$ s	0.865	$\mu$ s
100 km	0.8	$\mu$ s	0.825	$\mu$ s	0.83	$\mu$ s	0.84	$\mu$ s	0.875	$\mu$ s

Table 3

Time-delay corresponding to Table 2

Distance from the lightning channel, d (km)	Wind speed, $v$ (m/s)
	1.4 m/s		4.7 m/s		7 m/s		11.2 m/s
1 km	0	$\mu$ s	0	$\mu$ s	0	$\mu$ s	0	$\mu$ s
10 km	0.01	$\mu$ s	0.02	$\mu$ s	0.02	$\mu$ s	0.03	$\mu$ s
30 km	0.02	$\mu$ s	0.025	$\mu$ s	0.025	$\mu$ s	0.055	$\mu$ s
100 km	0.025	$\mu$ s	0.03	$\mu$ s	0.04	$\mu$ s	0.075	$\mu$ s

Table 4

Time corresponding to 50% of the peak lightning electric field

Distance from the lightning channel, d (km)	Wind speed, $v$ (m/s)
	0 m/s		1.4 m/s		4.7 m/s		7 m/s		11.2 m/s
1 km	0.23	$\mu$ s	0.235	$\mu$ s	0.235	$\mu$ s	0.235	$\mu$ s	0.235	$\mu$ s
10 km	0.19	$\mu$ s	0.2	$\mu$ s	0.2	$\mu$ s	0.2	$\mu$ s	0.205	$\mu$ s
30 km	0.19	$\mu$ s	0.205	$\mu$ s	0.205	$\mu$ s	0.205	$\mu$ s	0.215	$\mu$ s
100 km	0.185	$\mu$ s	0.21	$\mu$ s	0.21	$\mu$ s	0.215	$\mu$ s	0.23	$\mu$ s

Table 5

Time-delay corresponding to Table 4

Distance from the lightning channel, d (km)	Wind speed, $v$ (m/s)
	1.4 m/s		4.7 m/s		7 m/s		11.2 m/s
1 km	0	$\mu$ s	0.005	$\mu$ s	0.005	$\mu$ s	0.005	$\mu$ s
10 km	0.01	$\mu$ s	0.01	$\mu$ s	0.01	$\mu$ s	0.015	$\mu$ s
30 km	0.015	$\mu$ s	0.015	$\mu$ s	0.015	$\mu$ s	0.025	$\mu$ s
100 km	0.025	$\mu$ s	0.025	$\mu$ s	0.03	$\mu$ s	0.045	$\mu$ s

Tables 2 and 4 present the obtained values for the time corresponding to the peak field and 50% of the field peak respectively, which caused by the different wind speed on this weather process. Tables 3 and 5 present Time-delay corresponding to Tables 2 and 4 respectively. The notable part is that when the observation distance below 1 km, the wind speed rarely effect on the time corresponding to the peak field and 50% of the peak field on the lightning electric field. The time-delay of the electric field increases as the wind speed increases. When the distance up to 100 km, the wind speed has a significant effect on the time corresponding to the peak field. For the 11.2-m/s wind speed, the time corresponding to the peak field is 0.875 $\mu$ s, with the time-delay is 0.075 $\mu$ s, and the time corresponding to the 50% of peak field is 0.23 $\mu$ s, with the time-delay is 0.045 $\mu$ s.

3.4 Resulting location errors in ToA-based LLS

In order to illustrate the location accuracy of ToA-based LLS considering rough sea surface model, we select two lightning flash occurred on 8 August 2015. The results showed as Fig. 6a and b. The symbol “ $\bigstar$ ” represents the stations of the LLS around Qingdao. The four stations are Jimo Station (120.469E, 36.392N), Rizhao Station (119.539E, 35.432N), Rongcheng Station (122.493E, 37.17N) and Longkou Station (120.341E, 37.64N). The symbol “ $\bullet$ ” represents the lightning location given by LLS. The symbol “ $+$ ” represents the modified lightning location considering rough sea surface model. Table 6 gives the inputs parameters and results. Note that the rough sea surface has a significant effect on lightning location accuracy when the wind speed reach about 11 km/s, and the location errors caused by rough sea surface are 3771 m and 4090 m respectively.

Table 6
The inputs parameters and results corresponding to Fig. 6

Lightning location	Wind speed,	Time-delay for the four stations ( $\mu$ s)				Lightning location	Location
given by LLS	$v$ (km/s)	Jimo	Longkou	Rizhao	Rongcheng	considering rough	error (m)
		station	station	station	station	sea surface model
120.654E, 36.0536N	11.5	0.018	0.086	0.05	0.196	120.708E, 36.0347N	3771
120.3E, 35.9N	10.8	0.018	0.056	0.044	0.181	120.3467E, 35.8729N	4090

Figure 6.

The effect of rough sea surface on Lightning location measured by ToA-based LLS.

4. Conclusion and discussion

This paper analyzed the lightning vertical electric field and it is derivative in the time domain along a rough sea surface using Barrick’s formulations and Wait’s formulations. The sea-wave data for Qingdao buoy station on 8 August 2015 are selected and it inputted in the sea-wave model.

It was shown that the wave-shape and time-delay of the vertical electric fields can be significantly affected when the propagation along a rough sea surface. The rough sea surface has rarely effect on the magnitude of the vertical electric fields. With the observation distance increase, the rise time of the wave-shape become long with the increasing of the wind speed. The peaks of the field derivative obviously decrease with the increasing of the wind speed, and the farther observation distance is, the stronger attenuation is. As the wind speed increases, the time delay of the field derivative increases. The observed time-delay caused by the propagation along a rough sea surface, which might impact the location accuracy of ToA based LLS, and the location error can reach to several kilometers.

Footnotes

Acknowledgments

This paper was financially supported by the 2016 annual youth research project of Qingdao Meteorological Bureau (Grant No. 2016qdqxq3) and Lightning warning technology in severe convective (storm) weather near Shandong Peninsula (Grant No.QYXM201710). The authors express their sincere thanks to Prof. Qilin Zhang for his helpful advice. Special thanks go to SWMWP for providing sea-wave data.

References

Rubinstein

M.A.

Tzeng

Uman

M.A.

et al., An experimental test of a theory of lightning-induced voltages on an overhead wire, IEEE Trans. Electromagn. Comp. 31 (1989), 376–383.

Master

M.J.

and Uman

M.A.

, Lighting induced voltages on power lines: theory, IEEE Trans. Power. Appl. Syst. 103(9) (1984), 2505–2517.

X.C.

et al., Research on energy coordination between multi-level surge protective devices with the transmission line theory, International Journal of Applied Electromagnetics and Mechanics 48 (2015), 345–356.

X.C.

et al., Propagation and suppression method of lightning wave in coaxial line, International Journal of Applied Electromagnetics and Mechanics 49 (2015), 315–325.

Z.Q.

et al., Simulation calculation and analysis of lightning induced over voltage on power distribution lines, High Voltage Engineering 39(2) (2013), 415–422.

Rubinstein

, An approximate formula for the calculation of the horizontal electric field from lightning at close, intermediate, and long range, IEEE Trans. Electromagn. Compat. 38 (1996), 531–535.

Cooray

, Propagation effects due to finitely conducting ground on lightning-generated magnetic fields evaluated using Sommerfeld’s integrals, IEEE Trans. Electromagn. Compat. 51 (2009), 526–531.

Cooray

Fernando

Sorensen

et al., Propagation of lightning generated transient electromagnetic fields over finitely conducting ground, J. Atmospheric Sol.-Terrestrial Phys. 62 (2000), 583–600.

Delfino

Procopio

and Rossi

, Lightning return stroke current radiation in presence of a conducting ground: 1. Theory and numerical evaluation of the electromagnetic fields, J. Geophysical Res.: Atmospheres 113 (2008), 79–88.

10.

Delfino

Procopio

Rossi

Rachidi

et al., Lightning return stroke current radiation in presence of a conducting ground: 2. Validity assessment of simplified approaches, J. Geophysical Res.: Atmospheres 113 (2008), D05111.

11.

Cooray

, On the accuracy of several approximate theories used in quantifying the propagation effects on lightning generated electromagnetic fields, IEEE Trans. Antennas Propag. 56 (2008), 1960–1967.

12.

Shoory

Moini

and Rachidi

, Validity of simplified approaches for the evaluation of lightning electromagnetic fields above a horizontally stratified ground, IEEE Trans. Electromagn. Comp. 52 (2010), 657–663.

13.

Weidman

C.D.

and Krider

E.P.

, Lightning amplitude spectrum in the interval from 100 kHz to 20 MHz, Geophys. Res. Lett. 7 (1980), 955–958.

14.

Weidman

C.D.

Krider

E.P.

and Uman

M.A.

, Submicrosecond spectrum of the return stroke waveforms, Geophys. Res. Lett. 8 (1981), 931–934.

15.

Willett

J.C.

Bailey

J.C.

Leteinturier

et al., Lightning electromagnetic radiation field spectrum in the interval from 0.2 to 20 MHz, J. Geophys. Res. 95 (1990), 20367–20387.

16.

Ming

and Cooray

, Propagation effects caused by a rough ocean surface on the electromagnetic fields generated by lightning returns strokes, Radio Sci. 29 (1994), 73–85.

17.

Zhang

Yang

and Li

, Propagation effects of a fractal rough ocean surface on the vertical electric field generated by lightning return strokes, Journal of Electrostatics 70 (2012), 54–59.

18.

Shoory

Mimouni

Rachidi

and Rubinstein

, On the accuracy of approximate techniques for the evaluation of lightning electromagnetic fields along a mixed propagation path, Radio Sci. 46 (2016), 1–8.

19.

Zhang

Fan

Zhang

et al., Examination of the Cooray-Rubinstein (C-R) formula for a mixed propagation path by using FDTD, J. Geophys. Res. 117 (2012), D15309.

20.

Rachidi

Rubinstein

et al., Propagation effects on lightning magnetic fields over hilly and mountainous terrain, in: IEEE Int. Symp. Electromagn. Compat., Dresden, Germany, 2015, pp. 1436–1440.

21.

Azadifar

Rachidi

et al., Analysis of lightning electromagnetic field propagation in mountainous terrain and its effects on ToA-based lightning location systems, J. Geophys. Res. Atmos. 121 (2016), 895–911.

22.

Azadifar

Rachidi

et al., On lightning electromagnetic field propagation along an irregular terrain, IEEE Transactions on Electromagnetic Compatibility 58 (2016), 161–171.

23.

Cummins

K.L.

Murphy

M.J.

Bardo

E.A.

et al., A combined TOA/MDF technology upgrade of the US National Lightning Detection Network, Journal of Geophysical Research: Atmospheres (1984–2012) 103 (1998), 9035–9044.

24.

Cummins

K.L.

Murphy

M.J.

and Tuel

J.V.

, Lightning detection methods and meteorological applications, in: IV International Symposium on Military Meteorology, Malbork, Poland, September, 2000, pp. 25–28.

25.

Lewis

Harvey

and Rasmussen

, Hyperbolic direction finding with sferics of transatlantic origin, Journal of Geophysical Research 65 (1960), 1879–1905.

26.

Thottappillil

Rakov

V.A.

and Uman

M.A.

, Distribution of charge along the lightning channel: Relation to remote electric and magnetic fields and to return-stroke models, J. Geophys. Res. 102 (1997), 6987–7006.

27.

Thottappillil

and Rakov

V.A.

, On different approaches to calculating lightning electric fields, J. Geophys. Res. 106 (2001), 14191–14205.

28.

Wait

J.R.

, Transient fields of a vertical dipole over a homogeneous curved ground, Canadian Journal of Physics 34 (1956), 27–35.

29.

Wait

J.R.

, Recent analytical investigations of electromagnetic ground wave propagation over inhomogeneous earth models, Proceedings of the IEEE 62 (1974), 1061–1072.

30.

Wait

J.R.

, The ancient and modern history of EM ground-wave propagation, IEEE Antennas and Propagation Magazine 40 (1998), 7–24.

31.

Cooray

and De la Rosa

, Shapes and amplitudes of the initial peaks of lightning-induced voltage in power lines over finitely conducting Earth: Theory and comparison with experiment, IEEE Trans. Antennas Propag. 34 (1986), 88–92.

32.

Barrick

D.E.

, Theory of HF and VHF propagation across the rough sea, 1, the effective surface impedance for a slightly rough highly conducting medium at grazing incidence, Radio Sci. 6 (1971), 517–526.

33.

Barrick

D.E.

, Theory of HF and VHF propagation across the rough sea, 2, application to HF and VHF propagation above the sea, Radio Sci. 6 (1971), 527–533.

34.

Wang

Guo

and Wu

, Application of an improved 2D fractal model for electromagnetic scattering from the sea surface, Acta Phys. Sin. 55 (2006), 5191–5199 (in Chinese).

35.

Rakov

V.A.

and Dulzon

A.A.

, A modified transmission line model for lightning return stroke field calculations, in: Proceedings of the 9th International Symposium on Electromagnetic Compatibility, Switzerland: Zurich, 1991, pp. 229–235.

36.

Heilder

, Traveling current source model for LEMP calculation, in: Proceedings of the 6th International Zurich Symposium on Electromagnetic Compatibility, Switzerland: Zurich, 1985, pp. 157–162.

37.

Caligaris

Delfino

and Procopio

, Cooray – Rubinstein formula for the evaluation of lightning radial electric fields: Derivation and implementation in the time domain, IEEE Transactions on Electromagnetic Compatibility 50 (2008), 194–197.

38.

Cooray

, Effects of propagation on the return stroke radiation fields, Radio Science 22 (1987), 757–768.

39.

Honma

Suzuki

Miyake

et al., Propagation effect on field waveforms in relation to time-of-arrival technique in lightning location, Journal of Geophysical Research: Atmospheres (1984–2012) 103 (1998), 14141–14145.

Propagation effect of the lightning electric fields along rough sea surface and the effects on ToA-based lightning location systems

Abstract

Keywords

1. Introduction

2. Analysis method and computational models

3.1 Obtained sea-wave data of Qingdao buoy station

Table 2 Time corresponding to the peak lightning electric field ( μ s)

Table 6 The inputs parameters and results corresponding to Fig. 6

Footnotes

Acknowledgments

References

Table 2
Time corresponding to the peak lightning electric field ( $\mu$ s)

Table 6
The inputs parameters and results corresponding to Fig. 6