As lightning discharges often lead to lightning disasters (Paisios et al., 2007; Ma et al., 2008; Chemartin et al., 2012), the study of lightning physical characteristics has become more and more important (Finke and Hauf, 1996; Christian et al., 2003; Pinto et al., 2003; Soriano et al., 2005; Fleenor et al., 2009). Lightning’s current amplitude of return stroke is one of its important parameters, closely responding to the lightning disaster. The cumulative probability of the lightning current amplitude has been widely used in the lightning protection engineering technology, which represents the distribution of lightning current amplitude (Yue et al., 2013; Zhao et al., 2017). The precise expression is useful for studying lightning activity, protecting the transmission line, and assessing the lightning disaster risk (Metwally t al. 2004; Zhang et al., 2011; He et al., 2017).

At present, more researchers have been studying about lightning current amplitude (Borghetti et al., 2004; Chowdhuri et al., 2005). As early as the 1880’s, Kohlrausch had carried out the work of measuring the lightning current amplitude (Kohlrausch et al., 1888). Pobolansky (1977) proposed a formula for calculating the cumulative probability of lightning current amplitude based on data in Europe, Australia, and the United States (Shim et al., 2002). Thereafter, considerable progress has also been made on proposing different calculation formulas based on data in different regions (Grant et al., 1985; Orville and Huffines, 1999). The approximate expression of the cumulative probability of the lightning current amplitude was first given by Anderson based on the data of Berger, which was later recommended by the IEEE (Anderson et al., 1980). In China, the regulation method is mainly used to calculate the cumulative probability of lightning current amplitude in the lightning protection engineering technology (Du., 1996; Li, R.F et al., 2011). The corresponding logarithmic expression is recommended by GB/T 50064-2014. In the early days, lightning data were lacking, and the parameters of the formula were obtained by inversion using the data recorded by the Xinhang line from 1962 to 1987 (Li et al., 2019). It is still in use today.

Due to the vast territory and environmental differences in China, using the identical parameters in different regions shows great uncertainty. For example, the calculation results indicate that the most prominent manifestation is not wholly accurate, and thus, it perhaps cannot accurately reflect the cumulative probability of lightning current amplitude in the area, which affects its application in engineering. With the widespread application of lightning location systems, most of the systems have been running for more than 10 years. A large amount of lightning data has been accumulated, and so it provides the basis for research on parameters of temporal and spatial differences, such as lightning current amplitude and its cumulative probability (Wang et al., 2016; Zhao et al., 2018; Li et al., 2019). Li, J.Q et al. (2011) found that significant differences lay in the distribution characteristics of the cumulative probability of lightning current amplitude with the polarity in Chongqing, and the cumulative probability formulas recommended by the IEEE had better fitting effects than other formulas. Cheng et al. (2021) found that the cumulative probability curve of lightning current amplitude dropped faster in the interval of 20–50 kA and was consistent with the IEEE recommended formula in Liaoning. These research results provide the distribution characteristics of the lightning current amplitude and cumulative probability that are suitable for local use.

There is a subtropical monsoon climate with frequent lightning disasters in Zhejiang province, which lies in the central-eastern part of China. According to statistics, there were 9,728 recorded lightning disasters from 1998 to 2020, including 357 casualties, 282 injuries, and 307 deaths (Gu et al., 2021). The temporal and spatial characteristics of lightning activities have been mainly studied based on the lightning location system in Zhejiang province (Liu et al., 2009; Cui et al., 2021; Zhang et al., 2021). But the relevant research on lightning current amplitude and cumulative probability has not been carried out. Based on the lightning location system in Zhejiang province from 2009 to 2019, this article studies the characteristics of lightning current amplitude and localizes its parameters in the cumulative probability formula. The accuracy of the calculation results is helpful to the development of the lightning protection engineering technology.

The data are collected by using the lightning location positioning system of the Meteorological Departments in Zhejiang province. The system is uniformly deployed by the China Meteorological Administration, and it has 11 substations in Zhejiang province, which are shown in Figure 1. Also, it has a monitoring accuracy of 300 m and a detection efficiency of over 85%. Based on the principle of magnetic field positioning and time difference positioning, the system can accurately determine the location and time of a lightning occurrence, using satellite positioning systems and geographic information systems. It is mainly used for the detection of cloud-to-ground flashes and provides basic data for the study of the physical characteristics of lightning current.

The cumulative probability formula for lightning current amplitude recommended by the IEEE is as follows: p = 1 1 + ( I / a ) b . (1)

The cumulative probability formula for lightning current amplitude recommended by GB/T 50064–2014 is as follows: lg P = − I c , (2) where I is lightning current amplitude (kA), P is the cumulative probability that the lightning current amplitude is greater than I , a is the median lightning current amplitude, and b is the degree of change in the cumulative probability curve. In Eq. 1, IEEE recommends a = 31, b = 2.6, and I ∈ ( 2,200 ] , and in Eq. 2, GB/T 50064–2014 recommends c = 88.

The parameters recommended by the IEEE are the average results of global lightning current amplitude, and lightning activities vary greatly with time and space, so it is not appropriate to use the average results directly in Zhejiang province, and for Eq. 2, the parameters were obtained by inversion using the data recorded by the Xinhang line from 1962 to 1987. Therefore, the accuracy needs further verification according to the lightning observed data. In order to obtain a more suitable cumulative probability formula of lightning current amplitude and corresponding parameters in Zhejiang province, the least squares criterion is often used, and the Levenberg–Marquardt method is a commonly used method.

The study counts the observed data of the lightning location system in Zhejiang province from 2009 to 2019 and carries out quality control. According to IEEE regulations, the application range of the current is defined as 2–200 kA. Therefore, records with lightning current amplitudes of less than 2 kA and greater than 200 kA are excluded. The lightning number is counted according to the rule, which is that the current amplitude starts at 2 kA, then 5 kA, and increases by 5 kA until 200 kA. According to the results, the distribution characteristics of the lightning current amplitude are analyzed. According to the definition of the cumulative probability of lightning current amplitude, the percentages of positive flashes, negative flashes, and total flashes are calculated, respectively, based on the above intervals. Finally, the statistical points of the cumulative probability distribution of positive flashes, negative flashes, and total flashes are obtained.

In order to get a more suitable cumulative probability formula of the lightning current amplitude and corresponding parameters in Zhejiang province, the Levenberg–Marquardt method is used to fit the measured data with the formulas recommended by IEEE and GB/T 50064–2014, respectively, and the correlation coefficients of different polarities are obtained. The Levenberg–Marquardt method is a commonly used method that solves the non-linear least squares problem. In order to strengthen the robustness of the algorithm, it adds a damping factor on the basis of the Gauss-Newton method. The following is an example:

The vector p is defined to consist of fitted parameters whose number is M , while the vector X and Y consists of horizontal and vertical coordinates of measured scatter points whose number is N , respectively. There is a corresponding residual between the measured value and the fitted function value at each observation point. The residual functions, which are N form vector, are shown in Eq. 3 and Eq. 4. r j ( p ) = Y j − y ( p , X j ) , (3) r ( p ) = [ r 1 ( p ) , r 2 ( p ) , … , r N ( p ) ] Τ , (4) where y ( p , X j ) is the value substituted p and X j , r j ( p ) is the fitted residual function at the scatter point j , and r ( p ) is the residual function vector.

The objective function F ( p ) is defined as follows: F ( p ) = 1 2 r Τ ( p ) r ( p ) = 1 2 ∑ j = 1 N ( Y j − y ( p , X j ) ) 2 . (5) Here r T ( p ) is the transposition of r ( p ) . It can be seen that F ( p ) is exactly 1/2 of the sum of squares of the fitting residual R . Under the principle of least squares, the vector p ∗ with the smallest value F ( p ) is the optimal solution, and R is also the smallest.

The partial derivatives of the residual vector r ( p ) with respect to each variable in p form the N × M order matrix. It is expressed as J = [ ∂ y 1 ∂ x 1 ∂ y 1 ∂ x 2 … ∂ y 1 ∂ x M ∂ y 2 ∂ x 1 ∂ y 2 ∂ x 2 … ∂ y 2 ∂ x M … … … … ∂ y N ∂ x 1 ∂ y N ∂ x 2 … ∂ y N ∂ x M ] N × M . (6)

In this study, based on Eq. 1, p = [ a , b ] Τ and M = 2. Based on Eq. 2, p = [ c ] Τ and M = 1.

On the basis of the initial value p 0 , a suitable increment Δ p is found to make the objective function F ( p ) change in the direction of the decreasing value, and p = p + Δ p is iterated until it approaches p ∗ ; then, F ( p ) reaches the minimum value. Δ p is obtained by Eq. 7 and Eq. 8. S ( p ) = J T ( p ) J ( p ) + μ   diag ( J T ( p ) J ( p ) ) , (7) S ( p ) Δ p = − J T ( p ) r ( p ) . (8) where μ is the damping factor, diag ( J T ( p ) J ( p ) ) is the diagonal matrix consisting of main diagonal elements J T ( p ) J ( p ) , Δ p is the iterative increment of p , and S ( p ) is the intermediate variable.

In each iteration, the Levenberg–Marquardt method can have various indicators to measure the quality of this iteration, and the value of μ is adjusted accordingly. In this study, the method is to decide whether Δ p is acceptable or not by comparing the sizes of F ( p + Δ p ) and F ( p ) .

In order to verify the effect of fitting, the median current method is applied to test the results by the Levenberg–Marquardt method.

In this work, a total of 8,634,767 flashes are used to analyze the characteristics of lightning current amplitude and cumulative probability, based on the lightning data from 2009 to 2019. In addition, the parameters in the formulas recommended by the IEEE are corrected, which can more objectively reflect the cumulative probability distribution of lightning current amplitude in Zhejiang province.

It is found that the number of negative flashes is far more than that of positive flashes in Zhejiang province. But the average current amplitude of positive flashes is significantly higher than that of negative flashes. Lightning current amplitude is mainly concentrated in the range of 2–100 kA, accounting for 94.3%. In the range of 160–200 kA, the number becomes smaller, accounting for only 0.94%. The seasonal characteristics are significant, and the monthly distribution of the positive lightning current amplitude has a unimodal distribution, while that of negative flashes is relatively concentrated. Positive flashes are frequent in spring, but summer is the season of negative flashes. The cumulative probability distribution of the negative lightning current amplitude is closer to that of total flashes and is significantly different from that of positive flashes. The variation law of total flashes is mainly affected by negative flashes. Based on the Levenberg–Marquardt method, the measured data are fitted with the formulas recommended by the IEEE and GB/T 50064-2014, respectively, and the parameters of different polarities are obtained. By analysis, the cumulative probability distribution formula recommended by IEEE is more suitable for Zhejiang province, which is basically consistent with the results in other regions in China.

In Figures 6B and C, the changing trend of the two curves basically coincides. But there are two intersections between the two curves in Figure 4A, and when the value is higher than 100 kA, the fit of the two curves becomes lower. In other words, for describing the distribution characteristics of the cumulative probability of lightning current amplitude in Zhejiang province, the formulas applying to negative flashes and total flashes are more realistic than those for positive flashes. With more lightning observation data, a subsection refinement of the cumulative probability of the lightning current amplitude can be carried out in the following research, in order to obtain a more accurate cumulative probability formula.