A Novel Method for Single-Station Lightning Distance Estimation Based on the Physical Time Reversal

Abstract

A single-station lightning location has the obvious advantages of low cost and convenience in lightning monitoring and warning. To address the critical challenge of distance estimation accuracy in this technology, we propose a novel physical time-reversal (PTR) method to utilize the full wave information of both the ground wave and the sky wave in the detected signal. First, we improved the numerical model for accurately calculating the lightning sferics signals in the complex propagation environment of the Earth–ionosphere waveguide using the measured International Reference Ionosphere 2020. Subsequently, the sferics signal with multipath effect is transformed by time reversal and back propagated in the numerical model. Furthermore, a broadening factor reflecting the waveform dispersion in the back propagation is defined as the single-station focusing criterion to determine the optimal lightning propagation distance, considering the multipath effect and the focus of the PTR process. The experimental results demonstrate that the average root mean square error (RMSE) and the mean relative error (MRE) of the PTR method for the lightning distance estimation in the numerical simulation within the range of 100–1200 km are 5.517 km and 1.21%, respectively, and the average RMSE and the MRE for the natural lightning strikes to the Canton Tower from the measured data in the range of 181.643–1152.834 km are 9.251 km and 2.07%, respectively. Moreover, the correlation coefficients of the detection results are all as high as 0.999. These results indicate that the PTR method significantly outperforms the traditional ionospheric reflection method, demonstrating that it is able to perform a more accurate single-station lightning distance estimation by utilizing the compensation mechanism of the multipath effect on the sferics. The implementation of the proposed method has significant application value for improving the accuracy of single-station lightning location.

1. Introduction

As a common natural phenomenon in the atmosphere, lightning can pose a great threat to human society [1]. At present, the most widely used lightning detection means in commercial operations is the ground-based lightning detection system. For locating lightning events in oceans and sparsely populated land areas, long-distance, single-station detection has great applications [2,3,4].

The lightning electromagnetic pulse (LEMP) generated by the large-current process of lightning strikes is mainly in the 3–300 kHz very-low-frequency/low-frequency (VLF/LF) band, and it is usually referred to as “sferics”, which can propagate for long distances in the Earth–ionosphere waveguide (EIWG). Therefore, VLF/LF detection has a great advantage in long-distance lightning remote sensing [5,6].

In order to monitor large-scale lightning strikes, VLF/LF lightning location systems (LLSs) have been established in many areas [7,8,9,10]. To ensure the accuracy of lightning detection, the GPS timing system of each station in an LLS needs to be synchronized with high precision to maintain the accuracy of the arrival time of the LEMP received by each station [8,11]. Meanwhile, each station in the LLS needs to have a good internet communication function so as to share the received LEMP with the data center [4,12]. Moreover, there are considerable construction and maintenance costs for a multi-station LLS. Therefore, single-station lightning location technology has also been investigated in some previous studies [13,14]. While also reducing construction and maintenance costs, the single-station technique is also suitable for shipboard installation and self-supporting use.

The principle of the single-station lightning location technique is the combination of the direction-finding method and the distance-estimation technique. For lightning direction finding covering hundreds to thousands of kilometers, the magnetic-direction-finding (MDF) method is usually adopted, which uses the ratio of the horizontal magnetic field signals of the LEMP received by two orthogonal magnetic loop antennas in the east–west and north–south directions to determine the azimuth of the lightning with an accuracy of about ±1°, which is about 17.4 km at 1000 km [7,15]. The other factor influencing the location accuracy is the distance estimation technique, which is the key challenge with the single-station lightning location technique [14,16].

At present, there have been diverse methods proposed for distance estimation in single-station lightning location. The most direct relationship between a received signal and distance is amplitude attenuation [17]. However, it is almost impossible to accurately determine the lightning’s distance by the amount of amplitude attenuation due to the randomness of the intensity of the natural lightning current. Subsequently, it was proposed to use the wave impedance for near-range locations based on its monotonous relationship with the propagation distance [18]. Chen et al. [19] carried out field experiments on single-station lightning location to verify the performance of the wave impedance method, and the results show that the method performs satisfactorily at a range of 15–60 km with the error and relative error being 0.1–4 km and 6%, respectively, but the location results are poorer at a long distance. Sao et al. [20] also utilized a modified wave-impedance method, which calculates the phase difference between the electric field and magnetic field of the LEMP. Same as the wave-impedance method, the detection range of the method is only at the near distance and limited by the relatively simple wave-propagation model at that time, and the actual detection performance is restricted.

For single-station lightning-distance estimation at long distances, the long-distance propagation characteristics of sferics in the EIWG are often used to establish a relationship with the distance of the radiation source, and the detection range is extended to hundreds of thousands of kilometers. Nagano et al. [13] derived lightning distances based on the time delay difference between the direct ground wave and the multiple-reflected sky wave of sferics full waveforms in the EIWG. The experimental results show that the error of the distance estimation of the method is around 10% in the range of about 200 km for sferics with obvious multi-reflection characteristics. Ramachandran et al. [14] utilized that the oscillations of the sferics field pattern are formed as quasi-periodic waveforms of different frequencies after the effect of dispersion has propagated over long distances, from which the group velocity in the narrow band is used to estimate the propagation distance. Additionally, Koochak et al. [21] showed that by utilizing the time difference in arrival (TDOA) between the VLF and the extreme-low-frequency (ELF) signals in sferics, the propagation distance of lightning signals in the EIWG can be roughly calculated, and the average relative error of the distance estimation can be 6.7% compared with that of the location results of the National Lightning Location Network (NLDN) for the sferics that do not cross the day–night terminator.

In recent work, machine learning methods have also begun to be applied to the research on single-station lightning location [15,16,22]. However, the propagation of sferics generated by lightning in the EIWG is affected by complex propagation environments such as lossy ground, the Earth’s curvature, and ionospheric variations [23,24,25], and the establishment of a reliable machine learning model for the single-station lightning distance estimation requires an extremely large number of a priori samples in a complex and variable environment.

Overall, the problem of low reliability still exists in current methods for single-station lightning distance estimation. On the one hand, the most notable is the inability of the wave-impedance method to be applied to long-distance detection. Moreover, the long-distance methods rely on the determination of the time of arrival (TOA) in sferics waveforms that contain multipath components. However, the determination of TOA becomes less and less accurate as the propagation distance increases due to the effect of propagation dispersion, and a new perspective is needed to investigate the intrinsic relationship between the propagation distance and the measured waveform. On the other hand, only over-idealized sferics propagation computational models have been adopted in previous methods, and there is still a need to further develop the utilization of high-precision propagation models for the complex environments in the EIWG.

Considering the above problem, we propose to use the novel physical time-reversal (PTR) method to implement the single-station lightning distance estimation, which utilizes the full waveforms of both the ground wave and sky wave of the sferics signals. Compared with the traditional time-reversal (TR) method used only for multi-station lightning location systems [26,27,28,29], the PTR method utilizes the auxiliary differential equations finite difference time domain (ADE-FDTD) numerical model to consider the effects of complex propagation environments and multipath effects, thereby realizing the physical propagation process of TR. In our previous work in [25], PTR is applied to the multi-station detection signal, and the focus coordinate of the back propagated multiple signals is used to locate the lightning strike point. For the single-station location problem, the focus of multi-station signals cannot be utilized; thus, we propose to use the focus of the ground wave and the sky wave here.

The innovations in our research in this paper primarily focus on two aspects. On one hand, the ADE-FDTD numerical model applicable to different diurnal periods is developed based on the measured International Reference Ionosphere 2020 (IRI2020) for calculating the sferics signals generated by lightning strikes propagating in the complex propagation environment of the EIWG. The IRI2020 ensures the accuracy of the calculation results by providing accurate ionospheric parameters that match actual characteristics [30]. On the other hand, a novel broadening factor reflecting the waveform dispersion variation in the back propagation is used as a single-station focusing criterion to accurately obtain the lightning propagation distance according to the multipath compensation mechanism of the PTR principle.

The remainder of the paper is organized as follows. Section 2 describes the principles of the ADE-FDTD numerical model used for the sferics calculation and the process of lightning distance estimation with the PTR method, and depicts the experimental setup of the lightning observation experiment at the Canton Tower. Then, the simulation experiments for the sferics calculation and feasibility verification of the PTR method are carried out by ADE-FDTD in Section 3. In Section 4, the actual detection performance of the method is demonstrated by using the measured lightning strikes to the Canton Tower events. Finally, the work is summarized in Section 5.

2. Theory and Methods

When sferics signals generated by lightning propagate in the EIWG, there will be sky waves due to the strong reflection effect of the ionosphere in addition to ground waves propagating along the ground. And such a multipath propagation effect is a prerequisite for realizing the single-station lightning distance estimation. According to the time-reversal principle, signals of multipath effect with different time delays will be superimposed at the original location of the radiation source after they are transformed by time reversal and transmitted from the received station for back propagation, which is known as the multipath effect compensation mechanism by time reversal [26,31]. We utilize this principle to obtain the propagation distance of the radiating source in a single sferics signal. Furthermore, the ADE-FDTD numerical model, considering the effects of the complex propagation environment in the EIWG, is used to accurately calculate the physical process of the time-reversal signals of the sferics in the backward propagation.

2.1. ADE-FDTD Numerical Model and Configuration for Calculating Sferics

The configuration of the ADE-FDTD model used for calculating the sferics is shown in Figure 1. The whole model consists of the Earth at a depth of 2 km and the atmosphere containing the D layer (about 60–90 km in altitude) and E layer (about 90–140 km in altitude) of the ionosphere at an altitude of 150 km from the bottom to the top. The convolutional perfectly matched layer (CPML) is applied to truncate the boundary of the model to simulate the propagation to infinity. Moreover, a vertical lightning channel at a height of 7.5 km is set at the center axis, and the return strike speed is set to 0.5c (c denotes the lightspeed in vacuum, which is taken as 2.998 × 10⁸ m/s here). Additionally, lightning VLF/LF observation stations at different distances can be installed on the horizontal ground to receive sferics signals.

The environmental parameters of the numerical model take into account the effects of the ionosphere, lossy ground, and the Earth’s curvature that affect the propagation of sferics signals in the EWIG. The configuration with respect to the setting of the electromagnetic parameters of the lossy ground and the Earth’s curvature correction using the modified refractive index (MRI) method is consistent with the ADE-FDTD model that we have developed in our previous work [25]. The new improvement here is that we use the measured ionospheric parameter model to optimize the ADE-FDTD for calculating the propagation of sferics signals with the multipath effect in EWIG with high precision.

Figure 2 shows the variation with time of the measured electron density profile of the ionosphere above the Canton Tower (23.106°N, 113.323°E) during a day, as given by the latest IRI2020 [32]. In contrast to the traditional exponential ionospheric model based on statistical results that only represent the average conditions of ionospheric parameters, IRI2020 can provide the actual variation characteristics of the ionosphere at any given time [33]. Therefore, to ensure the consistency of the ionospheric parameters with the actual environment and that the model is adaptable to different diurnal periods, we adopt the measured electron density of the ionosphere below 150 km from the IRI2020 in the ADE-FDTD.

2.2. Single-Station Lightning Distance Estimation Based on the PTR Method

The principle of the time-reversal method is that the received multipath signals are subjected to the operations of time domain reversal and back propagation to compensate for the time delays in the original received signals, and that the process enables coherent superposition at the original radiation source position so as to achieve the results of target enhancement and imaging location [26,27]. Specifically, the PTR refers to the physical implementation of back propagation for time-reversal signals within a physical system with a consistent propagation medium along the original path.

Since the propagation of sferics in the EIWG has the characteristic of waveguide propagation, its waveform contains multipath signals with different time delays. After the sferics signal is transformed by time reversal and transmitted from the observation station, the multipath signals in the sferics signal will reach the original position of the radiation source at the same time and superimpose on each other to form the main correlation peak when they propagate along their respective paths in the backward propagation, which is the named focus here. Accordingly, the propagation distance of the lightning signal can be determined.

To visualize the PTR method, the forward propagation process of the sferics signal containing ground wave and the first sky wave in the EIWG, and the backward propagation of the time-reversal signal of sferics for single-station lightning distance estimation are shown in Figure 3, both of which are given by the ADE-FDTD numerical model. To simplify the explanation of the process, only the propagation path of the first sky wave reflected once by the ionosphere is considered here.

When the sferics signal propagates along N paths in the EIWG due to multiple reflections between the bottom of the ionosphere and the Earth, the signal S arriving at the lightning observation station can be expressed as [25]

$$\mathit{S}\left(t\right)={\displaystyle \sum _{i=1}^{N}H}\left(d_i\right){\mathit{S}}_0\left(t-{\displaystyle \frac{d_i}{v}}\right)$$

(1)

where S₀ is the lightning radiation source with the main frequency band in the full band of VLF/LF, v is the propagation velocity of the sferics signal, and H(d_i) is the transfer function in the EIWG for a multipath signal with a propagation path distance of d_i that is the ADE-FDTD model used to calculate the sferics. As shown in Equation (1), the multipath signals in the sferics signal propagated along paths of different distances have different time delays when they reach the observation station.

Then, the sferics signal received by the observation station in Equation (1) is transformed by time reversal to obtain S(−t). Since negative time is not feasible in practical experiments, the time-reversal signal is generally denoted as S(T − t), where T is a time window that is long enough to ensure the sferics signal propagates along the farthest distance path. And this process is expressed as

$$\mathit{S}\left(T-t\right)={\displaystyle \sum _{i=1}^{N}H}\left(d_i\right){\mathit{S}}_0\left(T-t+{\displaystyle \frac{d_i}{v}}\right)$$

(2)

Next, S(T − t) will be transmitted as the excitation source from the observation station for backward propagation. Since the propagation medium of the process is the same as that of forward propagation, the signal S_TR arriving at the original lightning radiation source according to Equation (1) can be obtained as

$$\begin{array}{cc}{\mathit{S}}_{TR}\left(t\right)& ={\displaystyle \sum _{i=1}^{N}H}\left(d_i\right)\mathit{S}\left(T-t-{\displaystyle \frac{d_i}{v}}\right)\hfill \\ & ={\displaystyle \sum _{i=1}^N{\displaystyle \sum _{i=1}^{N}H\left(d_i\right)}H}\left(d_i\right){\mathit{S}}_0\left(T-t+{\displaystyle \frac{d_i}{v}}-{\displaystyle \frac{d_i}{v}}\right)+ {\displaystyle \sum _{i=1}^N{\displaystyle \sum _{k\ne i}H\left(d_i\right)}H}\left(d_k\right){\mathit{S}}_0\left(T-t+{\displaystyle \frac{d_i}{v}}-{\displaystyle \frac{d_k}{v}}\right)\hfill \\ & ={\displaystyle \sum _{i=1}^N{H}^2}\left(d_i\right){\mathit{S}}_0\left(T-t\right)+{\displaystyle \sum _{i=1}^N{\displaystyle \sum _{k\ne i}H\left(d_i\right)}H}\left(d_k\right){\mathit{S}}_0\left(T-t+{\displaystyle \frac{d_i}{v}}-{\displaystyle \frac{d_k}{v}}\right)\hfill \end{array}$$

(3)

The second term in Equation (3) indicates that the multipath signals in the time-reversal signal of sferics are unable to coherently superpose after propagating along paths with different distances from their previous ones, and their amplitudes are smaller compared to the focused signal amplitude due to attenuation in the complex propagation environment of the EIWG. Meanwhile, the first term in Equation (3) is the main correlation peak formed by the multipath signals in the time-reversal signal of sferics after propagating along their respective original paths, while arriving at the radiation source location at the same time and coherently superimposed. The main reason is that the physical process of time reversal causes multipath signals in the sferics to propagate backward with the same total delay T as the forward propagation. Thus, the propagation distance of the lightning radiation source can be further obtained.

The multiple sky waves in the actual sferics signal are inconspicuous and difficult to identify accurately due to the effects of noise and attenuation from the lossy long-distance propagation of multiple reflections. We only utilize the two-path signals, including direct ground wave along the ground and the obvious first sky wave reflected only by the ionosphere in the sferics signal for single-station lightning distance estimation with the PTR method.

2.3. Experimental Setup and Data

Advanced Direction-time Lightning Detection (ADTD) is a lightning observation system capable of implementing wide-range lightning detection constructed by the Institute of Electrical Engineering, Chinese Academy of Sciences (CAS). Its sensors use low-level triggering mode to record the VLF/LF signals of lightning strikes with a bandwidth of 3–400 kHz and are equipped with GPS-synchronized timing modules. Affected by environmental noise and the lightning radiation source intensity, the furthest effective detection distance of ADTD stations is around 1000 km.

In order to verify the actual performance of single-station lightning distance estimation, we selected eight ADTD lightning observation stations near the Canton Tower (23.106°N, 113.323°E) to validate the detection results of the lightning strikes to the Canton Tower events in April 2024. The horizontal distribution of the experimental site and the ADTD stations for detecting lightning strikes to the tall tower events is shown in Figure 4. The red cross in the figure represents the location of the Canton Tower with a height of 600 m, which is the tallest tower building in China, and its spire is prone to lightning strikes on the tower, creating a good opportunity for the observation of natural lightning. The eight ADTD observation stations in the vicinity of 181.642–1152.834 km from the Canton Tower detect and record the occurrence of lightning strikes to the Canton Tower and obtain the sferics signal data. Then, the recorded sferics signals are used to perform the single-station distance estimation by the PTR method, respectively, and the actual performance at different distances for the single-station lightning distance estimation can be confirmed.

3. Numerical Results and Simulation Analysis

To verify the feasibility of single-station lightning distance estimation by utilizing the PTR method, a simulation experiment was carried out according to the experimental setup. First, the accuracy of the ADE-FDTD model in calculating the propagated sferics in the EIWG is examined by comparing it with measured signals from the ADTD lightning observation stations, and the multipath effect of the sferics signal containing both ground wave and sky wave is analyzed. Subsequently, the process of the PTR method for single-station lightning distance estimation is validated using the simulated sferics signals.

3.1. Accuracy Test of the ADE-FDTD

The actual sferics signals of lightning strikes to the Canton Tower events received from the eight ADTD stations were used to compare with the results of the ADE-FDTD model calculations. Furthermore, the normalized root mean square error (NRMSE) and the correlation coefficient (CC) are used to evaluate the error and correlation between the simulated and measured sferics signals, respectively, so as to quantitatively assess the actual performance of the ADE-FDTD model for calculating the sferics signals. The calculation formulas for NRMSE and CC are given as

$$\mathrm{N}\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n{\left(\frac{{\mathit{S}}_{\mathit{sim}}- {\mathit{S}}_{\mathit{mea}}}{\mathit{max}\left({\mathit{S}}_{\mathit{mea}}\right)-\min \left({\mathit{S}}_{\mathit{mea}}\right)}\right)}^2}}$$

(4)

and

$$\mathrm{C}\mathrm{C}={\displaystyle \frac{cov\left({\mathit{S}}_{\mathrm{sim}}, {\mathit{S}}_{\mathrm{mea}}\right)}{{\sigma }_{\mathrm{sim}}\times {\sigma }_{\mathrm{mea}}}}$$

(5)

respectively, where cov(S_sim, S_mea) is the covariance between the simulated sferics signal S_sim calculated by ADE-FDTD and the measured sferics signal S_mea, and σ_sim and σ_mea are the standard deviations of S_sim and S_mea, respectively.

Figure 5a–h show the comparison between the measured sferics signals of a natural lightning strike to the Canton Tower from eight ADTD stations around the tower and simulated sferics signals calculated by the ADE-FDTD model. From the comparison, it can be seen that, except for the obvious noise contained in the measured sferics signals, the characteristics of ground waves and first sky waves appearing due to the multipath effect are recognized in both simulated sferics signals and measured signals.

This indicates that the ADE-FDTD can better calculate sferics signals with the characteristics of multipath effect, and there is a good agreement between simulated sferics signals and the measured results. Additionally, it can be seen that as the distance of the observation station increases, the leading edge of the first sky wave gradually approaches the falling edge of the ground wave, which causes the time delay difference between the ground wave and the sky wave to become smaller progressively. This is consistent with the variation characteristics of the multipath effect in practice.

The quantitative results for evaluating the accuracy of the ADE-FDTD in calculating sferics signals are given in

Table 1. Comparison between the sferics signals calculated by the ADE-FDTD and the measured signals received from the eight ADTD observation stations.

Observation Stations	Time-Domain Waveforms
	NRMSE *	CC
S1	0.077	0.879
S2	0.064	0.928
S3	0.112	0.850
S4	0.116	0.903
S5	0.167	0.857
S6	0.101	0.948
S7	0.107	0.948
S8	0.133	0.921

* When comparing the measured sferics signal with the simulated sferics signal, the length of the signal time window is consistent with that shown in Figure 5, and the signal sampling rate is 12 MHz.

. The NRMSE between the simulated and the measured results is lower than 0.167, and the CC is higher than 0.850. These results show that the ADE-FDTD model is able to provide high-accuracy numerical results for calculating the sferics signals of lightning strikes propagating in the EIWG.

3.2. Analysis of the Sferics Signal Propagating in the EIWG

According to the configuration of the ADE-FDTD model in Section 2.1, the forward propagation of the sferics signal in the EIWG can be calculated using the ADE-FDTD after considering the effects of the ionosphere, lossy ground, and the Earth’s curvature, which are the main factors influencing the propagation of sferics signals in the EIWG. Regarding the ionosphere settings, the ADE-FDTD model uses the ionosphere parameters of IRI2020 at 19:00 on 20 April 2024, when the lightning strike to the Canton Tower, to analyze the forward propagation.

The spatial distribution of the electric field of the sferics signals propagating in the EIWG to distances of 200, 500, 800, and 1100 km is shown in Figure 6a–d, respectively, which essentially represents the spatial distribution of the energy of the radiation field of the sferics signals. It is clearly visible that the propagation characteristics of the sferics signal in the EIWG, due to the reflections from the ionosphere and the Earth, and that the sferics signals contain both direct waves and ionospheric reflected waves as a result of the multipath effect. Additionally, the amplitude of the electric field spatial distribution of lightning sferics signals is significantly attenuated with increasing propagation distance due to the lossy environment in EIWG. Moreover, as the propagation distance increases, the polarity of the direct wave remains unchanged. However, the polarity of the first ionospheric reflected wave changes from the same as that of the direct wave to the opposite of it, and the leading edge of the first ionospheric reflected wave gradually approaches the direct wave.

Furthermore, it is found that the multiple ionospheric reflected waves of sferics signals are poorly characterized and difficult to distinguish accurately except for the first ionospheric reflected wave, because of the attenuation effect of the complex propagation environment in the EIWG, as well as the influence of long-distance lossy propagation by multiple reflections from the bottom of the ionosphere and the Earth.

To further analyze the multipath effect, the electric field waveforms of the sferics signals propagating in the EIWG received by ground-based lightning detectors at propagation distances of 200, 500, 800, and 1100 km are illustrated in Figure 7a–d, respectively. It can be seen that the same characteristic of the first sky wave appears in the waveform of the sferics signal owing to the reflection of the ionosphere, and the polarity change in the ground waves and the first sky waves is in agreement with that in Figure 6. Essentially, the ground wave in the sferics signal received by the lightning detector is the sferics signal wavefront of the direct wave, while the sky wave is the reflection of the sferics signal wavefront by the bottom of the ionosphere.

As the propagation distance increases, the peak of the ground wave exhibits rapid attenuation, while the peak of the first sky wave becomes more and more significant as compared to the ground wave. Moreover, the leading edge of the first sky wave gradually approaches the falling edge of the ground wave until they are superimposed, resulting in the time interval between them gradually becoming smaller.

3.3. Simulation Configuration for Single-Station Lightning Distance Estimation by PTR Method

In order to verify the performance of single-station lightning distance estimation by the PTR method, we adopt the ADE-FDTD model to calculate the whole physical propagation process of the PTR method.

Figure 8 shows the simulated sferics signals received from the eight ADTD observation stations. Further, the sferics signals are transformed by time reversal and then transmitted from their respective observation stations at their respective distances for back propagation, where the whole back propagation process of the PTR is calculated by the ADE-FDTD, and the model configuration is the same as that of the forward propagation.

When the time-reversal signal of sferics propagates backward to the original location of the lightning radiation source, the sky wave and the ground wave propagating along their respective original paths will focus and merge into one peak. To further estimate the focusing point corresponding to the propagation distance, a judgment criterion needs to be established. Theoretically, the reflected wave of the sky wave will gradually approach the ground wave and ultimately focus with it in the PTR process [25]. Therefore, we define the integral of the normalized zero-crossing portion of the same-polarity signal centered on the ground wave as the broadening factor to reflect the cumulative degree of waveform broadening during the back propagation process of the entire time reversal, thereby quantitatively describing the variation in multipath compensation. A schematic of the broadening factor is shown in Figure 9.

For the specific calculation of the broadening factor, K_BF is written as

$$K_{BF}={\displaystyle {\int }_{t_1}^{t_2}norm\left(S_{center}^{TR}\left(t\right)\right)}dt$$

(6)

where $S_{center}^{TR}$ is the same-polarity signal centered on the ground wave in the time-reversal signal, and t₁ and t₂ are the zero-crossing times at the left and right ends of its peak, respectively, and the purpose of normalization is to compensate for the propagation attenuation of the ground wave.

3.4. Analysis of Single-Station Lightning Distance Estimation by PTR Method

The back propagation in the ADE-FDTD model of the time-reversal signals of sferics from the farthest observation station S8 is first utilized for the case analysis. Figure 10 shows the variation in broadening factor with propagation distance during the back propagation of the time-reversal signals from observation station S8, demonstrating four distinct evolutionary phases of the broadening factor.

In the first phase, the broadening factor exhibits a gradual increase, corresponding to waveform broadening induced by dispersion effects in the propagation environment on the ground wave. This is followed by an abrupt, rapid rise in the second phase, indicating that reflected sky waves progressively approach the ground wave, further enhancing its broadening. The third phase features a decline, where the fusion of reflected sky wave with the ground wave strengthens the peak amplitude and narrows the waveform, ultimately driving the broadening factor to its unique minimum value at full focusing. Finally, the fourth phase shows a slow subsequent increase, signifying continued broadening of the focused ground wave due to both the end of focusing and the dispersion in the complex propagation environment. The minimum point of the broadening factor marks the focusing of the multipath effect, enabling the single-station lightning distance estimation based on the corresponding propagation distance.

To further illustrate the compensation process for the sferics signals with multipath effect by the PTR method, the spatial distribution of electric field of the time-reversal signals of the sferics from the observation station S8 during back propagation from their own location to the original location of the lightning radiation source in the ADE-FDTD model corresponding to the different propagation distances d₁, d₂, d₃, and d₄ in Figure 10 are shown in Figure 11.

It can be seen that in the case of considering only the direct path and the ionospheric reflection path, the time-reversal signal of sferics will appear as new reflected waves during back propagation in addition to the direct waves of the original sky wave and the original ground wave. That is, a new reflected wave of the sky wave appears along the ionospheric reflection path, and a new reflected wave of the ground wave appears along the ionospheric reflection path as well.

As the propagation distance increases, the distance between the ground wave of the original path and the sky wave of the original path that reaches the ground after reflection by the ionosphere, which we are concerned about, gradually becomes smaller. When the propagation distance is up to d₄ in Figure 11d, the sky wave of the original path and the ground wave of the original path reach the propagation distance of the lightning radiation source at the same time, at which point it can be seen that the paths of the two are overlapping at the ground.

The waveform variations in the electric field during back propagation of the time-reversal signals of the sferics from the corresponding observation station S8 in Figure 11 are given in Figure 12. A similar focusing process can be seen in that the sky wave of the original path gradually approaches the tail of the ground wave of the original path as the propagation distance increases. When the propagation distance is up to d₄ in Figure 12d, the sky wave of the original path and the peak of the ground wave of the original path have overlapped together, indicating that both of them arrive at the location of the lightning radiation source at the same time and have been focused. Meanwhile, the time delay difference between the ground wave of the original path and the reflected wave of the ground wave obviously decreases with the increase in the propagation distance, which is, in fact, consistent with the variation rule of the time delay difference between the ground wave and the sky wave of sferics in forward propagation.

The above characteristics of the spatial distribution and waveforms of the electric field confirm that the ADE-FDTD numerical model can accurately reconstruct the physical process of the PTR method in back propagation, and further verify that the multipath effect of sferics can be compensated in this process.

The results of the simulated sferics signals from the eight ADTD observation stations in the range of 181.643–1152.834 km for single-station lightning distance estimation using the PTR method are given in Figure 13 and compared with the traditional ionospheric reflection method [13]. For the purpose of reducing the random error caused by using only a fixed equivalent reflectance height in the traditional ionospheric reflectance method, we have calculated the best result of the method with the equivalent reflection height in the range of 70–100 km under a nighttime ionospheric environment as a reference for comparison.

It can be concluded that the traditional ionospheric reflection method exhibits the average root mean square error (RMSE) of 53.440 km, the relative error (RE) below 14.88%, the mean relative error (MRE) of 7.07% and the CC of 0.992. By comparison, the PTR method achieves an average RMSE of 6.040 km, an RE below 1.64%, an MRE as low as 1.04%, and a CC as high as 0.999.

Figure 14 further shows the results of the simulated sferics signals calculated by ADE-FDTD in the range of 100–1200 km for single-station lightning distance estimation using the PTR method and the traditional ionospheric reflection method. It can be seen that the traditional ionospheric reflection method achieves an average RMSE of 49.320 km, an MRE of 6.90%, and a CC of 0.991. In contrast, the PTR method has an average RMSE of 5.517 km, an MRE is as low as 1.21%, and a CC is as high as 0.999.

The simulation results demonstrate that the PTR method significantly outperforms the traditional ionospheric reflection method, indicating that it is able to perform a more accurate single-station lightning distance estimation by utilizing the compensation mechanism of the multipath effect on the sferics.

4. Analysis of Distance Estimation for Natural Lightning Strikes to the Tall Tower

In order to examine the PTR method for the multipath effect of natural lightning strikes, we carried out the analysis by using the lightning strikes to the Canton Tower that happened during 19:13–20:03 on 20 April 2024.

The waveforms of the measured sferics signals for the six lightning strikes to the tall tower events named RST191330, RST191441, RST194401, RST194822, RST195806, and RST200341, respectively, are shown in Figure 15. In practice, the number of signals for each lightning strike received by the ADTD observation stations is different since the strength of the lightning radiation source and the influence of the environmental noise of the propagation process. The weaker RST191330, RST194401, and RST194822 were detected by only three ADTD stations at close range. The RST195806 with the strongest intensity was received by up to seven ADTD observation stations.

In the detection experiments of the six lightning strikes to the Canton Tower, the ADTD observation stations received a total of 24 sferics signals that can be used for single-station lightning distance estimation analysis by the PTR method. As in the same procedure in Section 3.3, these sferics signals are transformed first by time reversal and then transmitted for back propagation from their respective observation stations, where the whole back propagation process of the PTR is also calculated by the ADE-FDTD. Furthermore, the back propagation of the time-reversal signals from the RST195806 at the farthest observation station S8 is first used to perform the case analysis.

The variation in the broadening factor with the propagation distance during the back propagation of the time-reversal signals of the measured sferics at observation station S8 is illustrated in Figure 16. It can be observed that the broadening factor of the time-reversal signals of the measured sferics follows the same four-phase variation pattern as that of the simulated signals, due to the consistency of the ADE-FDTD numerical model. Similarly, the decline in the third phase of the broadening factor represents the propagation process in which the reflected wave of the sky wave gradually converges with the ground wave, enhancing its peak amplitude. The broadening factor reaches its minimum when complete focusing is achieved between the two waves. Consequently, the unique minimum point of the broadening factor that indicates the focusing of the multipath effect enables the single-station lightning distance estimation for measured sferics signals based on the corresponding propagation distance.

To further illustrate the compensation performance for the multipath effect in the measured sferics signals by the PTR, the spatial distribution of electric field of the time-reversal signals of sferics received from RST195806 at the observation station S8 during back propagation from their original locations to the Canton Tower location in the ADE-FDTD model corresponding to the different propagation distances d₁, d₂, d₃, and d₄ in Figure 16 are shown in Figure 17.

Similarly to the results of the simulation experiment, the time-reversal signal of the measured sferics in the back propagation, except for the direct waves of the sky wave and the ground wave, the new reflected waves of both the sky wave and the ground wave will appear as they propagate along the ionospheric reflection path. As the propagation distance increases, the ground wave of the original path and the sky wave of the original path both reach the location of the lightning radiation source at the same time when the propagation distance is up to d₄ in Figure 17d, and it is also found that the paths of the two are overlapping at the ground.

It can also be observed from Figure 18, which are the waveform variations in the electric field during back propagation of the time-reversal signals of the sferics corresponding to Figure 17, that the sky wave of original path and the ground wave of original path arrive at the location of the lightning radiation source simultaneously and are focused when the propagation distance is up to d₄ in Figure 18d. Meanwhile, the time delay difference between the ground wave of the original path and the reflected wave of the ground wave also shows the variation rule of decreasing with the increase in the propagation distance.

The above characteristics of the spatial distribution and waveforms of the electric field in back propagation of the time-reversal signals of the measured sferics are also demonstrated that the numerical computation of ADE-FDTD is able to accurately reconstruct the physical processes of the PTR and compensate for the multipath effect of the measured sferics signals in back propagation.

To validate the actual performance of the PTR method at different distances, the results of the single-station lightning distance estimation using the PTR method for all measured sferics signals received from eight ADTD stations at different distances ranging from 181.643 km to 1152.834 km are given in Figure 19. Consistent with the simulation analysis, the actual performance of the PTR method is compared with the best result calculated by the traditional ionospheric reflection method.

As a result, the traditional ionospheric reflection method has an average RMSE of 35.967 km, an MRE of 8.29%, and a CC of 0.996 for single-station distance estimation of sferics signals from natural lightning strikes. This accuracy is comparable to the 10% error reported by Nagano et al. [13] in single-station distance estimation for measured sferics signals at scales around 200 km using the ionospheric reflection method. In contrast, the PTR method attains significantly higher accuracy, with the average RMSE of 9.251 km, the MRE of 2.07%, and the CC as high as 0.999. The experimental results confirm that the PTR method can also achieve accurate single-station lightning distance estimation for natural lightning strikes by utilizing the compensation mechanism of the multipath effect on sferics.

5. Conclusions

The sferics signals generated by lightning strikes exhibit the multipath effect when propagating in the EIWG. We have innovatively adopted the PTR method for the compensation mechanism of the multipath effect to achieve accurate single-station lightning distance estimation. First, we have improved the numerical model for accurately calculating the lightning sferics signals in the complex propagation environment of the EIWG using the IRI2020. Subsequently, the sferics signal with multipath effect is transformed by time reversal and back propagated in the ADE-FDTD model. After that, the broadening factor reflecting the waveform dispersion variation in the back propagation is used as the single-station focusing criterion to accurately obtain the lightning propagation distance according to the multipath effect compensation mechanism of the PTR principle, thus realizing the single-station lightning distance estimation. Furthermore, the performance of the PTR method is validated by the sferics signals of the lightning strikes to the Canton Tower events detected by ADTD observation stations, and the results are compared with those of the traditional ionospheric reflection method. The major conclusions are as follows.

• The ADE-FDTD numerical model that takes into account the effects of the complex environment of the EWIG for calculating the sferics of lightning strikes can be applicable to different diurnal periods after adopting the measured ionospheric parameters of IRI2020. The comparison with the measured sferics signals demonstrates that the ADE-FDTD is able to provide high-precision numerical results for the calculation of sferics signals. Moreover, the physical process of the PTR method in the back propagation can be accurately reconstructed by ADE-FDTD, which is the key numerical way to realize the single-station lightning distance estimation.
• Essentially, the multipath effect of the sferics signal serves as a prerequisite for being able to utilize the PTR method for single-station lightning distance estimation. The proposed spreading factor utilizes the full waveform information to identify the focus in the single-station location while using the PTR method. This innovation directly correlates the focusing point with distance and enhances the criterion accuracy.
• The detection experiment of the lightning strikes to the Canton Tower events can provide precise reference results for the examination of the PTR method, as the exact locations of the lightning strikes are already known. Both simulations and experimental results demonstrate that the PTR method performs effectively for multiple lightning strike cases from the eight stations at varying distances, highlighting the universality of the method.

The major application background of the PTR method is to solve the challenge of distance estimation in single-station lightning location, and it is also expected to be applied to remote sensing studies involving single-station location that have signals with multipath effect in other fields. Compared to previous ideal simplified models and machine learning approaches, this method demonstrates significant improvements in theoretical rigor and detection accuracy. It is important to note that the method is applicable to signals with the obvious multipath effect in most of the sferics, but it is not suitable for those sferics with insignificant multipath effect in practice (e.g., the sky wave of sferics within tens of kilometers is not prominent). Additionally, the accuracy of the ADE-FDTD numerical model depends on precise ionospheric parameters. Moving forward, we will continue to explore the feasibility of applying the PTR method to a limited number of signals lacking multipath characteristics by further analyzing their time-frequency information influenced by propagation distance.