## 3 Application: Models and Hypotheses Considered for Analysis  
The geometry of the problem is shown in Figure 7. The aim is to evaluate lightning electromagnetic fields over a mountainous terrain by making use of the FDTD technique. We assume a pyramidal mountain with a height H\({}_{\text{m}}\), which is varied from 0 (flat ground case) to 1000 m, while its cross-section is assumed to have an area of A = 40,000 m\({}^{2}\). The ground and mountain are assumed to be a perfect electric conductor (PEC) and the ground depth h\({}_{\text{m}}\) = 100 m. The vertical electric field and the radial magnetic field components are calculated for three observation points A, B, and C. These points are, respectively, 100, 400, and 500 m far from the lightning channel base, and all located on the ground surface. In the FDTD calculations, a vertical array of current sources was used for the modeling of the lightning channel [46]. The modified transmission line model with exponential decay (MTLE) [47; 48] was used for modeling the return stroke channel. For subsequent stroke current waveforms, a summation of two Heidler functions was used [49], which is given in Equation (1):  
\[\text{i}(\text{t})=\left[\frac{\text{I}_{01}}{\text{\eta}_{1}}\ \frac{\left(\frac{\text{t}}{\text{T}_{11}} \right)^{\text{n}_{1}}}{1+\left(\frac{\text{t}}{\text{T}_{11}}\right)^{\text{ n}_{1}}}\ \text{e}^{\frac{\text{t}}{\text{T}_{21}}}\ +\frac{\text{I}_{02}}{\text{\eta}_{2}}\ \frac{\left(\frac{\text{t}}{\text{T}_{12}} \right)^{\text{n}2}}{1+\left(\frac{\text{t}}{\text{T}_{12}}\right)^{\text{n} 2}}\ \text{e}^{\frac{\text{t}}{\text{T}_{22}}}}\right] \tag{1}\]  
where I\({}_{01}\) = 10.7 kA, \(\text{T}_{11}\) = 0.25 u\(\text{s}\), \(\text{T}_{21}\) = 2.5 u\(\text{s}\), n\({}_{1}\) = 2, \(\text{\eta}_{1}\) = 0.639, I\({}_{01}\) = 6.5 kA, \(\text{T}_{11}\) = 0.25 u\(\text{s}\), \(\text{T}_{21}\) = 230 u\(\text{s}\), n\({}_{1}\) = 2, and \(\text{\eta}_{1}\) = 0.867, respectively [49]. The current decay constant and the return stroke speed are considered to be \(\lambda\) = 2000 m and v = 1.5 \(\times\) 10\({}^{8}\) m/s, respectively. The time increment was set to 9.5 ns. The dimensions of the computational domain are \(1500\times 1500\times 1500\) m with a spatial mesh grid composed of square cells of \(5\times 5\times 5\) m. In an unbounded space, the electromagnetic response analysis of a structure absorbing boundary conditions in which unwanted reflections are suppressed must be applied to planes to truncate and limit the open space and the working volume, respectively. In order to avoid reflections at the domain boundaries, Liao's condition is adopted as the absorption boundary condition [50]. The horizontal magnetic field (H\({}_{\text{y}}\)) components at the observation points A, B, and C are calculated assuming different heights for the mountain. The results are presented in Figure 8.  
According to Figure 8, the time delay, peak value, waveshape, and amplitudes of the horizontal magnetic fields (H\({}_{\text{y}}\)) are affected by the presence of the mountain. Although the effect of the mountain on the time delay (peak field and onset time) was negligible for point A, its effect on points B and C (right side of the hill) was significant [51]. The results represented in Figure 8 have been validated using canonical structures against the simulation results obtained in [51]. The only difference was that the GPU was used and the dimensions were 1500 \(\times\) 1500 \(\times\) 1500 m in this study, whereas a CPU processor was used in [51]. Figure 9 presents, for validation purposes, the obtained results compared with those obtained using a 2D-FDTD method. Figure 7: The simulation domain of the 3D-FDTD method and geometry for the lightning electromagnetic fields over mountainous terrain: (**a**) three-dimensional perspective, (**b**) side view, (**c**) top view.