A Lightning Very-High-Frequency Mapping DOA Method Based on L Array and 2D-MUSIC

Abstract

Lightning Very-High-Frequency (VHF) radiation source mapping technology represents a pivotal advancement in the study of lightning discharge processes and their underlying physical mechanisms. This paper introduces a novel methodology for reconstructing lightning discharge channels by employing the Multiple Signal Classification (MUSIC) algorithm to estimate the Direction of Arrival (DOA) of lightning VHF radiation sources, specifically tailored for both non-uniform and uniform L-shaped arrays (2D-MUSIC). The proposed approach integrates the Random Sample Consensus (RANSAC) algorithm with 2D-MUSIC, thereby enhancing the precision and robustness of the reconstruction process. Initially, the array data are subjected to denoising via the Ensemble Empirical Mode Decomposition (EEMD) algorithm. Following this, the covariance matrix of the processed array data is decomposed to isolate the signal subspace, which corresponds to the signal components, and the noise subspace, which is orthogonal to the signal components. By exploiting the orthogonality between these subspaces, the method achieves an accurate estimation of the signal incidence direction, thereby facilitating the precise reconstruction of the lightning channel. To validate the feasibility of this method, comprehensive numerical simulations were conducted, revealing remarkable accuracy with elevation and azimuth angle errors both maintained below 1 degree. Furthermore, VHF non-uniform and uniform L-shaped lightning observation systems were established and deployed to analyze real lightning events occurring in 2021 and 2023. The empirical results demonstrate that the proposed method effectively reconstructs lightning channel structures across diverse L-shaped array configurations. This innovative approach significantly augments the capabilities of various broadband VHF arrays in radiation source imaging and makes a substantial contribution to the study of lightning development processes. The findings of this study underscore the potential of the proposed methodology to advance our understanding of lightning dynamics and enhance the accuracy of lightning channel reconstruction.

1. Introduction

Accurate lightning localization plays a crucial role in monitoring thunderstorm activity and elucidating the mechanisms underlying lightning discharge processes. Lightning mapping methodologies serve as a fundamental tool for advancing research in lightning science and developing effective protection strategies. By leveraging lightning mapping data and analytical results, the reconstruction of lightning channels and the analysis of propagation characteristics in natural lightning events can provide critical insights into the physical mechanisms of lightning discharge. Such insights are invaluable for the design and implementation of lightning protection systems for ground-based infrastructure. The lightning discharge process generates electromagnetic radiation across a wide spectrum [1]. Notably, Very-High-Frequency (VHF) radiation pulses are particularly abundant during lightning events [2], accompanying nearly the entire discharge process rather than being limited to the high-current return stroke phase. The density of VHF signals significantly exceeds that of Very-Low-Frequency (VLF) signals, enabling high-resolution detection and precise localization of both intracloud (IC) and cloud-to-ground (CG) lightning discharges within a defined range. Consequently, the analysis of VHF radiation is of paramount importance for lightning mapping, offering a powerful means to study the spatial and temporal evolution of lightning channels with high accuracy. This capability not only enhances our understanding of lightning dynamics but also contributes to the development of advanced lightning detection and protection systems, ultimately improving the safety and resilience of infrastructure vulnerable to lightning strikes.

The VHF band can be used to reconstruct high-resolution lightning channel development and is suitable for studying small-scale lightning discharge processes [3]. With the development and progress of electronic technology and high-precision time response techniques, VHF radiation source localization technology has also advanced rapidly. Currently, VHF localization techniques are primarily divided into two categories: Time of Arrival (TOA) localization [4,5,6] and interferometric localization [7,8,9]. In TOA localization, the three-dimensional spatial position of VHF radiation sources is calculated based on the time differences of lightning electromagnetic radiation pulses arriving at different stations within a detection grid. This allows for the reconstruction of the entire lightning discharge process. Depending on the baseline length, TOA techniques are divided into short-baseline TOA localization (with baseline lengths typically ranging from several meters to tens of meters) and long-baseline TOA localization (with baseline lengths of about tens of kilometers). Long-baseline TOA localization is commonly used for monitoring large-scale thunderstorm activities [10]. Taylor [11] first designed a short-baseline VHF radiation source localization system based on time difference methods. The system includes five VHF broadband receiving antennas with an operating frequency range of 20–80 MHz. However, this system requires high consistency among channels and can only distinguish independent pulse events during the lightning process. Krehbiel [12] and Hamlin [13] developed a long-baseline time difference localization system called the Lightning Mapping Array (LMA), with a center frequency of 63 MHz and a bandwidth of 6 MHz. This system was successfully used in the Severe Thunderstorm Electrification and Precipitation Study (STEPS) conducted in the western United States in the summer of 2000, yielding many valuable observational results. For individual lightning flashes, the system can typically detect hundreds to thousands of radiation events, allowing for a detailed depiction of the three-dimensional spatiotemporal evolution of lightning discharges. VHF interferometric localization primarily uses interferometry to measure the elevation and azimuth angles of radiation sources, thereby mapping the two-dimensional development structure of lightning discharges. Warwick et al. [14] first applied a narrowband interferometer to the localization of lightning radiation sources. However, due to interference pattern ambiguity, the final localization results were also highly ambiguous. Hayenga and Warwick [15] improved upon this system, developing the first narrowband interferometer for two-dimensional localization of lightning radiation sources using VHF electromagnetic radiation pulses. Shao et al. [16] first designed a broadband interferometer. Due to the broader spectrum of lightning radiation collected, this system is more susceptible to external noise interference.

In recent years, advanced array signal processing techniques, including Multiple Signal Classification (MUSIC) and Time Reversal, have been increasingly applied to lightning mapping research. These methodologies have demonstrated significant potential in enhancing system noise resistance and improving the accuracy of lightning radiation source localization, thereby advancing our understanding of lightning discharge mechanisms. To augment the detection capabilities of broadband VHF lightning detection systems, Wang et al. [17] proposed a continuous broadband VHF time-reversal localization method based on the time-reversal principle. By constructing a uniform L-shaped array, they validated the method’s advantages in localizing weak radiation sources and resolving multiple sources through comprehensive numerical simulations and case studies. Further refining the performance of broadband VHF lightning detection systems, Wang et al. [18] proposed a broadband VHF array lightning radiation source localization method based on the MUSIC algorithm in spatial spectrum theory. This approach was designed to improve interference resistance, suppress sidelobe effects, and enhance spatial resolution. Utilizing a uniform L-shaped array, the authors conducted detailed spatiotemporal analyses of lightning discharge evolution. The method’s effectiveness and reliability in localizing weak radiation sources and resolving multiple sources were rigorously validated through a combination of data simulations, drone-based experiments, multi-source localization tests, and observations from artificially triggered lightning experiments. These advancements underscore the transformative potential of integrating sophisticated signal processing techniques into lightning mapping systems, offering new opportunities for high-precision lightning research and improved lightning detection capabilities. Chouragade et al. [19] proposed the RV-MUSIC method for lightning VHF imaging and compared it with IMUSIC, finding that their proposed method can effectively reduce computational complexity. Du et al. [20] used ESPRIT algorithm for lightning VHF imaging on L-shaped and circular arrays, and their method verified that DOA method can effectively solve imaging problems.

The performance of lightning mapping systems is significantly influenced by the number of array elements and the array configuration. To address the challenges associated with non-uniform and uniform L-shaped multi-element arrays, this paper proposes a novel lightning VHF radiation source mapping method based on the 2D-MUSIC algorithm. Given the substantial volume of VHF band data typically collected, computational errors can arise during covariance matrix calculations. To mitigate this issue, the Random Sample Consensus (RANSAC) algorithm [21] is first employed to reduce the dimensionality of the original data. The effectiveness of the dimensionality-reduced data is rigorously validated using Probability Density Functions (PDF) and Cumulative Distribution Functions (CDF). Subsequently, the Ensemble Empirical Mode Decomposition (EEMD) technique is applied to reconstruct the original data. The covariance matrix of the reconstructed signal is then computed and decomposed into its eigenvalues and eigenvectors. The collected signals are mapped into the signal subspace and noise subspace, and the direction of signal incidence is estimated by exploiting the orthogonality between these subspaces. Theoretically, since measurement noise is uncorrelated with the lightning signal, this method exhibits strong noise resistance. The feasibility of the proposed approach is demonstrated through numerical simulations, achieving elevation and azimuth angle errors of less than 1 degree. The method is also applied to real lightning observation data to analyze the lightning mapping process. The results show that the method can clearly reconstruct the lightning discharge channel and its development process.

The structure of this paper is as follows: In Section 2, the principle of 2D-MUSIC under the L-array model is introduced. In Section 3, the feasibility of the proposed method is verified through simulations. In Section 4, the practicality of the proposed method is validated by mapping real measurement data. Finally, in Section 5, the proposed method is summarized and discussed.

2. The Principle of 2-D MUSIC Under the L-Array Model

2.1. System and Array Structure

The system architecture of this paper is shown in Figure 1. The VHF antenna array consists of seven broadband omnidirectional bicone antennas. The signals received by each antenna are transmitted through low-loss coaxial cables, then pass through a bandpass filter with a frequency range of 27.5 MHz to 60 MHz to eliminate interference from broadcast signals below 30 MHz and above 60 MHz. The filtered signal is then sent to an eight-channel acquisition card and recorded on an industrial computer. Additionally, the system is equipped with a set of low-frequency fast electric field antennas as the trigger source for the signals, with an operating bandwidth of 100 kHz to 2 MHz.

In this paper, an L-shaped array is used, and the array structure is shown in Figure 2. The array is arranged in an L-shape in the plane, with N antennas on both the x-axis and y-axis. The red star in the figure represents the location of the radiation source. Considering the maximum number of channels of the data acquisition card, the number of antennas in the VHF imaging array is set to 7, i.e., N = 4. The adjacent distance between the coordinates on the x-axis and y-axis is denoted as $d=[0,c_1{d}_m,\cdots ,c_{N-1}{d}_m]$, where $c_1,c_2,\cdots ,c_{N-1}$ is a positive integer.

When the array is arranged in a uniform L-shape, the distance between each array element is equal, denoted as $c_1=1,c_2=2,c_3=3,d_m=8$. When the distance is $c_1=1,c_2=4,c_3=6,d_m=4$, the array is non-uniform L-shaped.

2.2. Preprocessing

Due to the large volume of VHF signal data, computational errors may occur when calculating the covariance matrix. To address the issue of large data volume, this paper employs the RANSAC algorithm to process the original data. RANSAC algorithm is an iterative method used to estimate the parameters of a mathematical model from a set of observation data containing outliers, where outliers have no impact on the estimated values. For a set of observation data and a parameterized model used to interpret the observation data, the points that satisfy the model are considered interior points, and vice versa are considered exterior points, which can be regarded as noise. The RANSAC model is used to fit the data by least squares. To improve the localization accuracy, the EEMD technique is then utilized to perform denoising and filtering on the waveforms.

To demonstrate whether the distribution of VHF signal data remains consistent after sampling, the PDF and CDF are employed for verification. The PDF distribution is shown in Figure 3, and the CDF is shown in Figure 4. The PDF and CDF of the original VHF data and the sampled data are consistent, and the result of the Kolmogorov–Smirnov (KS) test is H = 0. This indicates that the RANSAC algorithm can effectively sample VHF data.

2.3. The Principle of the 2-D MUSIC Algorithm

The fundamental principle of the MUSIC algorithm is to perform eigenvalue decomposition on the covariance matrix of the array output data. This process yields a signal subspace corresponding to the signal components and a noise subspace that is orthogonal to the signal components. The orthogonality between these two subspaces is then utilized to estimate the direction of signal incidence.

In DOA estimation algorithms based on the eigenvalue decomposition of the antenna array covariance matrix, the MUSIC algorithm has universal applicability. As long as the arrangement of the antenna array is known, regardless of whether the array elements are evenly spaced, high-resolution estimation results can be obtained.

Assuming there are K far-field signal sources incident on a certain spatial planar array, the array consists of M elements, with the positions of the elements being $\left(x_i,y_i\right)\left(i=1,2,3,\cdots ,M\right)$. VHF signals with different azimuth angles $\theta$ and elevation angles $\phi$ are incident from the far-field onto a non-uniform linear array containing M elements. Lightning VHF signal is a typical broadband signal. In order to apply array signal processing algorithm, we first decompose the VHF signal into several narrowband components through Discrete Fourier Transform (DFT) and then apply array signal processing algorithm to each narrowband component, respectively. The incident signal source is $S(t)$. The far-field signals received by the array elements are transformed as shown in Equation (1):

$$\left[\begin{array}{c}{X}_1(t)\\ X_2(t)\\ ⋮\\ X_M(t)\end{array}\right]=[a({\theta }_1,{\phi }_1),a({\theta }_2,{\phi }_2),\cdots ,a({\theta }_K,{\phi }_K)]\left[\begin{array}{c}{S}_1(t)\\ S_2(t)\\ ⋮\\ S_K(t)\end{array}\right]+\left[\begin{array}{c}{N}_1(t)\\ N_2(t)\\ ⋮\\ N_M(t)\end{array}\right]$$

(1)

Among them, $a({\theta }_K,{\phi }_K)$ is the direction vector of the Kth signal source, which can be represented by Equation (2):

$$a({\theta }_K,{\phi }_K)=\left[\begin{array}{c}1\\ \exp (-j2\pi {\displaystyle \frac{x_2\sin {\theta }_K\cos {\phi }_K+y_2\sin {\theta }_K\sin {\phi }_K}{\lambda }})\\ ⋮\\ \exp (-j2\pi {\displaystyle \frac{x_M\sin {\theta }_K\cos {\phi }_K+y_M\sin {\theta }_K\sin {\phi }_K}{\lambda }})\end{array}\right]$$

(2)

In the formula, $\lambda$ represents the wavelength.

According to the definition of the matrix, it can simplify Equation (1) to the form shown in Equation (3):

$$X(t)=A(\theta ,\phi )S(t)+N(t)$$

(3)

In the formula, $N(t)$ represents the noise signal vector.

The known VHF signal data consists of a finite number of snapshots taken over a finite time range. Within this finite time range, it is assumed that the direction of the spatial radiation source signal does not change, and although the envelope of the spatial radiation source signal varies with time, it is considered to be a stationary random process whose statistical properties do not change over time. Since the incident signals and noise are mutually independent, the cross-covariance matrix between the array elements can be obtained, as represented by Equation (4):

$$R_X=E\left[X{X}^{\mathrm{H}}\right]=E\left\{\left[A(\theta ,\phi )S(t)+N(t)\right]{\left[A(\theta ,\phi )S(t)+N(t)\right]}^{\mathrm{H}}\right\}$$

(4)

where $\mathrm{H}$ denotes the Hermitian transpose.

Given the universal applicability of the MUSIC algorithm, a non-uniform linear array can also achieve high-resolution estimation results. Since the incident signals and noise are uncorrelated, the array covariance matrix $R_X$ can be decomposed into two orthogonal subspaces: the signal subspace and the noise subspace. The covariance matrix is subjected to eigenvalue decomposition, as represented by Equation (5):

$$R_X=U_S{\displaystyle {\sum }_S{U}_S^{\mathrm{H}}}+U_N{\displaystyle {\sum }_N{U}_N^{\mathrm{H}}}$$

(5)

where $U_S$ represents the matrix of eigenvectors corresponding to the incident signal, which are associated with the larger eigenvalues. $U_N$ is the matrix of eigenvectors corresponding to the noise, which are associated with the smaller eigenvalues (tending towards zero). ${\displaystyle {\sum }_S}$ is a diagonal matrix composed of the larger eigenvalues, and ${\displaystyle {\sum }_N}$ is a diagonal matrix made up of the smaller eigenvalues (approaching zero).

In an ideal scenario, the signal subspace and the noise subspace are mutually orthogonal, as shown in Equation (6):

$$a^{\mathrm{H}}\left(\theta ,\phi \right)U_N=0$$

(6)

Based on the orthogonal relationship between the noise eigenvectors and the signal vectors, the spatial spectrum function of the array is obtained, as represented by Equation (7):

$$P_{MUSIC}(\theta ,\phi )={\displaystyle \frac{1}{a^{\mathrm{H}}(\theta ,\phi )U_N{U}_N^{\mathrm{H}}a(\theta ,\phi )}}$$

(7)

3. Simulation Analysis

To verify the feasibility of the algorithm, this paper simulates the lightning VHF signal as a Gaussian pulse modulated signal, referring to the actual broadband VHF signal waveform characteristics, represented by Equation (8):

$$s(t)=A_0\sin (2\pi f_{0}t)e^{-4\pi {({\displaystyle \frac{t-{\tau }_1}{{\tau }_2}})}^2}$$

(8)

where $f_0$ represents the center frequency; ${\tau }_1$ and ${\tau }_2$ are delay factors, used to adjust the signal bandwidth; $A_0$ is the signal amplitude, which is set to 1 during the simulation process.

During the numerical simulation process, the bandwidth settings for the VHF antenna actual signal collection are set to $f_0=40\mathrm{MHZ}$ and ${\tau }_1={\tau }_2=80\mathrm{ns}$. The time-domain waveform of the signal is depicted in Figure 5, and the noise signal model is a Gaussian white noise.

The VHF signal frequency band ranges from 30 MHz to 300 MHz. In this paper, the main frequency component of the simulated pulse signal is between 20 MHz and 60 MHz, the sampling rate is set to 312.5 MSPS (Million Samples Per Second), and the sampling length is 512 sampling points.

Feasibility Verification

Assuming the radiation source S is positioned at an azimuth angle of 120 degrees and an elevation angle of 50 degrees incident to the antenna array. To assess the accuracy of the positioning results, Figure 5 is used as the input signal. When the positioning result is within 1 degree of the actual position of the radiation source, it is considered that the algorithm has correctly identified the location. The simulation positioning result is shown in Figure 6. From Figure 6, it can be seen that the peak point is consistent with the pre-specified direction of the radiation source, indicating the feasibility of the algorithm presented in this paper.

In order to verify the performance of EMD and EEMD, 100 ms data were selected and EMD and EEMD were added to the imaging using 2D-MUSIC algorithm, respectively. We found that EEMD located more radiation sources than EMD. The number of radiation sources located by EEMD is 33,174, and the number of radiation sources located by EMD is 24,297. The specific positioning results are shown in Figure 7.

4. Mapping Results

4.1. Non-Uniform L-Shaped Array Mapping Results

Based on the 2D-MUSIC algorithm, this subsection processes observational data from a natural lightning event. The system continuously collected data for an effective duration of approximately 500 ms. The lightning event occurred on 20 August 2021, in Hefei, Anhui Province (event number 20210820-0006). The raw data collected during this observation are shown in Figure 8.

After obtaining the raw VHF signal data from the observation, data windowing and high-energy window screening are required. The windowing process divides the VHF signal into several segments of continuous, appropriately short signal windows. In this paper, the window length is set to 512 sampling points, with an overlap of 75% between adjacent windows. Subsequently, to eliminate invalid windows that contain only noise and to reduce computational load, high-energy window screening is performed on the signals in each window. The threshold screening method is adopted in this paper. The experiment in this paper was conducted on an AMD Ryzen Threadripper 2990WX 32-Core Processor 3.00 GHz CPU and 128 G memory computer using an unoptimized MATLAB2022b.

To demonstrate the mapping performance of the proposed algorithm, we conducted a comparative analysis with interferometry, the traditional 2D MUSIC algorithm, and the EEMD method. EMD has weaker handling capabilities for noise and mode aliasing issues, making it susceptible to noise signals. EEMD introduces random perturbations based on EMD, decomposing the original waveform by adding different noises, thereby improving the noise suppression capability and stability of EMD decomposition. Channel mapping was performed using real lightning signals collected by a non-uniform L-shaped array. Figure 9 presents the azimuth estimation results for the event 20210820-0006, where the horizontal axis represents azimuth (ranging from 0° to 360°), and the vertical axis denotes elevation (ranging from 0° to 90°). The color gradient in the figure indicates temporal progression.

The proposed algorithm demonstrates superior performance in capturing the fundamental structure of lightning compared to other methods. The mapping results reveal excellent continuity of the lightning channel, with the discharge initiating at approximately 250° azimuth and 72° elevation, and terminating at around 185° azimuth and 35° elevation. Through comparative analysis, we have determined that the integration of RANSAC and EEMD effectively eliminates noise interference, thereby enabling clearer and more accurate reconstruction of the lightning channel. This discovery establishes a substantial theoretical framework for advancing studies on lightning discharge phenomena.

4.2. Uniform L-Shaped Array Mapping Results

This section presents the analysis of a real lightning signal captured by a uniform L-shaped array for channel mapping. The lightning event under investigation occurred in Hefei, Anhui Province on 21 July 2023 (event number: 20230721-0005). The original observational data for this event are presented in Figure 10.

To validate the imaging performance of the proposed method on uniform L-shaped arrays, we conducted comparative experiments involving three approaches: interferometry, the traditional 2D-MUSIC algorithm, and our proposed fusion EEMD method. The positioning results, as illustrated in Figure 11, demonstrate the effectiveness of our method. It is important to note that this study focuses on proposing a novel positioning algorithm rather than analyzing the lightning discharge process itself. Based on the positioning results, we observed that this lightning event comprised multiple strikes. Our proposed algorithm successfully reconstructs both the multi-strike process and the fundamental structure of the lightning channel, showcasing its superior capability in accurately mapping complex lightning events.

In conclusion, the proposed method in this study successfully reconstructs the development process of lightning discharge channels across two distinct arrays, thereby establishing a robust framework for subsequent investigations into array configuration optimization and lightning signal direction estimation.

5. Discussion and Conclusions

5.1. Discussion

In this study, we identified several issues that warrant further investigation. First, since the acquisition system is installed on the roof of a building, the complex outdoor environment introduces significant noise into the collected VHF signals. Although the EEMD method was employed for noise reduction in this work, the positioning results still exhibit some “false points”. Future research should focus on developing more advanced noise suppression techniques to address this limitation. Second, due to spatial constraints, the distance between arrays, particularly in the uniform L-shaped configuration, is limited. To mitigate this issue, future efforts should consider deploying the acquisition system in more open areas, which would enhance signal quality and improve positioning accuracy.

This paper only performs two-dimensional positioning of the radiation source, involving azimuth and elevation angles, but not the height of the antenna at the time of lightning occurrence. In later stages, we will conduct three-dimensional lightning positioning by building multiple sites. Additionally, drones equipped with VHF simulators can be used to verify the positioning accuracy of the algorithm in future stages.

5.2. Conclusions

This paper presents a novel approach for DOA estimation of lightning VHF radiation sources by integrating EEMD noise reduction with the 2D-MUSIC algorithm and the RANSAC method. For VHF data acquired from both non-uniform and uniform L-shaped arrays, the proposed method first applies RANSAC to reduce data dimensionality, followed by covariance matrix decomposition to extract signal subspaces corresponding to the signal components and noise subspaces orthogonal to these components. The orthogonality between these subspaces is then exploited to estimate the direction of signal incidence. The following key conclusions are drawn:

(1)

Effectiveness of RANSAC Sampling: After applying RANSAC, the PDF and CDF of the sampled VHF data align closely with those of the original data. The KS test yields H = 0, confirming that the RANSAC algorithm effectively samples VHF data without compromising its statistical properties.

(2)

Feasibility of the Algorithm: The feasibility of the proposed algorithm is validated through numerical simulations. The simulation results demonstrate that the peak point of the localization output aligns precisely with the pre-defined direction of the radiation source, confirming the algorithm’s accuracy and reliability.

(3)

Practical Application and Imaging Performance: The effectiveness of the 2D-MUSIC algorithm is further verified using real-world observation data. The results indicate that the proposed method can accurately reconstruct the structure of lightning discharge channels, providing clear and detailed imaging of lightning development processes.

In summary, the integration of EEMD, 2D-MUSIC, and RANSAC offers a robust and accurate framework for lightning VHF radiation source localization and channel reconstruction. This method not only enhances the precision of lightning imaging but also contributes to a deeper understanding of lightning discharge mechanisms, with potential applications in lightning protection and atmospheric electricity research.