## 3 Results  
In order to demonstrate the superiority of the proposed method, some simulations were performed in MATLAB on a PC with i7 CPU. To focus on the difference between the two methods, the discretization step along the lightning channel, \(dz^{\prime}\), was adopted as the variable, while the other parameters are listed in Table 2. The main purpose of this paper was to deal with the calculation problem of the horizontal electric field and it is noted that the horizontal electric field has great attenuation when it is more than 1 km away from the lightning return channel, so the lightning striking distance \(r\) is restricted to 0-1 km. According to the geometric formula, the height difference caused by the curvature of the earth is below 8 cm, which is very small compared with the distance of 1 km [26], so the influence of the curvature of the earth is ignored in this paper. Since both of the methods include numerical integration with respect to \(dz^{\prime}\), the value of \(dz^{\prime}\) would inevitably influence the accuracy. Theoretically, the smaller the value of \(dz^{\prime}\), the more accurate the result. However, there must be a trade-off between the adoption of \(dz^{\prime}\) and the efficiency. Firstly, the effect of \(dz^{\prime}\) on the calculation accuracy of the two methods was studied, which is shown in Figure 3. By comparing Figure 3a with b, it can be seen that \(dz^{\prime}\) has a great influence on the horizontal electric field's calculation of the conventional method, while it has basically no influence on that of the proposed method. When \(dz^{\prime}\geq 0.05\) m, the horizontal electric field obtained by the conventional method will deviate from the real value. However, no matter what value of \(dz^{\prime}\) is taken by the proposed method, the final calculation results are consistent. Therefore, the proposed method is more accurate when the \(dz^{\prime}\) is the same. The calculation formulae of the proposed method are composed of two parts: The arithmetic part as shown in Equation (13) and the integral part as shown in Equation (14). In order to better explain the reason why the proposed method is not sensitive to \(dz^{\prime}\), these two parts were analyzed respectively. For the arithmetic part, it is an error-free analytical result completely independent of \(dz^{\prime}\) in the proposed method (analytical calculation) while an inexact numerical result sensitive to \(dz^{\prime}\) in the conventional method (numerical calculation), as shown in Figure 4a,b. This indicates that the proposed method is \begin{table}
\begin{tabular}{c c c c c c c c} \hline \(t\) & \(v\) & \(r\) & \(z\) & \(c\) & \(\alpha\) & \(\mathbf{d}t\) & \(\epsilon_{0}\) \\ \hline
30 \(\mathrm{\SIUnitSymbolMicro s}\) & \(1.3\times 10^{8}\) m/s & 50/250/500 m & 10 m & \(3\times 10^{8}\) m/s & 1700 m & 10 ns & \(8.854\times 10^{-12}\) \\ \hline \end{tabular}
\end{table} 
Table 2: Basic parameters used in the simulation examples. Figure 3: The time-domain evolutions of the horizontal electric field at the observation point under different \(dz^{\prime}\); (**a**) calculated by the conventional method; (**b**) calculated by the proposed method. more accurate in this part than the conventional method. For the integral part, its accuracy would be dependent on \(dz^{\prime}\). However, as can be seen in Figure 4c, the results are coincident together, although \(dz^{\prime}\) takes different values. That is to say, an adoption of \(dz^{\prime}\) = 2 m is enough to achieve the accurate result. This is why the proposed method can guarantee the accuracy with a relatively larger \(dz^{\prime}\).