## 2 The Model Backgrounds  
A small current element is considered in a horizontal lightning channel located in the air above a conductive ground at height \(z_{p}\) as indicated in Figure 1.  
In the frequency domain, the phasor at frequency \(\omega\) of the vector potential must satisfy the equations,  
\[ \begin{array}{l}\overset{\rightarrow}{\
abla}^{2}\overset{\rightarrow}{A}+ \frac{\omega^{2}}{c_{0}^{2}}\overset{\rightarrow}{A}=-\mu_{0}\delta\overset{ \rightarrow}{I}\left(z>0\right)\\ \overset{\rightarrow}{\
abla}^{2}\overset{\rightarrow}{A}+\frac{\omega^{2}}{ c_{1}^{2}}\overset{\rightarrow}{A}=0\left(z<0\right)\end{array}  \tag{1}\]  
where \(c_{0}^{2}=\left(\varepsilon_{0}\mu_{0}\right)^{-1}\) is the speed of light in a vacuum and \(c_{1}^{2}=\left(\mu_{1}(\varepsilon_{1}+\frac{\sigma_{1}}{\mu\omega})\right)^{ -1}\). The current element can be expressed as \(\delta\overset{\rightarrow}{I}=\delta I\cdot\overset{\rightarrow}{u}_{x}\) located at the position \((x_{p},y_{p},z_{p})\). The electrical parameters in the air carry the subscript \(0\) with \(\sigma_{0}=0\), while for the conductive medium, the subscript is \(1\). The vector potential must also satisfy the boundary conditions at the air-ground interface, which are expressed in terms of the generated electric and  
Figure 1: Geometry of a horizontal lightning channel over a perfect conductive ground. magnetic fields. For a horizontal waveguide along the \(x\)-axis, at a point with coordinates \((x,y,z)\) located in the air, the vector potential has the following components. \[ \begin{array}{l}A_{x}(x,y,z,\omega)=\frac{\mu_{0}\delta I}{4\pi}\int\limits_{0 }^{\infty}(\frac{\lambda}{\lambda_{0}}e^{-\lambda_{0}|z-z_{p}|}+f(\lambda)e^{ -\lambda_{0}(z-z_{p})})J_{0}(\lambda r)d\lambda\\ A_{y}(x,y,z,\omega)=0\\ A_{z}(x,y,z,\omega)=\frac{\mu_{0}\delta I}{4\pi}\frac{x-x_{p}}{r}\int\limits_{0 }^{\infty}g(\lambda)e^{-\lambda_{0}(z-z_{p})}J_{1}(\lambda r)d\lambda\end{array}  \tag{2}\]  
where \(r=\sqrt{\left(x-x_{p}\right)^{2}+\left(y-y_{p}\right)^{2}}\) and the functions \(f(\lambda)\) and \(g(\lambda)\) must be determined from the aforementioned boundary conditions. Once the boundary conditions are imposed, the expressions for these functions are the following,  
\[ \begin{array}{l}f(\lambda)=\frac{\lambda}{\lambda_{0}}\cdot\frac{\lambda_{ 0}-\lambda_{1}}{\lambda_{0}+\lambda_{1}}\cdot e^{-2\lambda_{0}z_{p}}\\ g(\lambda)=\frac{\gamma_{0}^{2}-\gamma_{1}^{2}}{\gamma_{0}^{2}\lambda_{1}+ \gamma_{1}^{2}\lambda_{0}}\cdot\frac{2\lambda^{2}}{\lambda_{0}+\lambda_{1}} \cdot e^{-2\lambda_{0}z_{p}}\end{array}  \tag{3}\]  
In the Equation (3), \(\gamma_{i}^{2}=-\frac{\omega^{2}}{c_{i}^{2}}\left(i=0,1\right)\) and \(\lambda_{i}=\sqrt{\lambda^{2}+\gamma_{i}^{2}}\). By substituting Equation (3) into (2) and using the Lipschitz identity [10]  
\[\frac{e^{-\beta|\overset{\rightarrow}{r}\overset{\rightarrow}{r}\overset{ \rightarrow}{p}|}}{\left|\overset{\rightarrow}{r}-\overset{\rightarrow}{r} _{p}\right|}=\int\limits_{0}^{\infty}\frac{\lambda}{\sqrt{\lambda^{2}+\beta^{ 2}}}e^{-\sqrt{\lambda^{2}+\beta^{2}}|z-z_{p}|}J_{0}(\lambda r)d\lambda \tag{4}\]  
results  
\[\begin{array}{l}A_{x}(x,y,z,\omega)=\frac{\mu_{0}\delta I}{4\pi}\cdot\frac{e ^{-\beta_{0}R}}{R}+\frac{\mu_{0}\delta I}{4\pi}\cdot\frac{e^{-\beta_{0}R^{ \primeFor the calculation of the fields, the well-known expressions \(\overset{\rightarrow}{B}=\overset{\rightarrow}{\
abla}\times\overset{ \rightarrow}{A}\) for the magnetic field and \(\overset{\rightarrow}{E}=-\overset{\rightarrow}{\
abla}\phi-\frac{\overset{ \rightarrow}{aA}}{\partial t}\) for the electric field will be used, in which the scalar potential will be evaluated, according to the Lorenz gauge, as  
\[\phi(R,t)=-c_{0}^{2}\int\limits_{0}^{t}\overset{\rightarrow}{\
abla}A \cdot d\tau \tag{9}\]  
The electric field at point \((x,y,z)\) created by the current element of the actual horizontal channel at position \((x_{p},y_{p},z_{p})\), has the following Cartesian components. \(a=1\), \(b=-4\cdot 10^{5}\) s\({}^{-1}\), \(c=-1\), \(d=-6\cdot 10^{6}\) s\({}^{-1}\).