## 2 Materials and Methods
### Precipitation Simulation  
The rainfall simulation was carried out using the parameters of conventional water. Nevertheless, different water parameters could be used, such as those shown in Table 4. For this research, we used the specific values \(\epsilon_{r}=80\), \(\mu_{r}=1\), and \(\sigma=10\) mS/m corresponding to the representative rainfall of lake areas in the continental interior [57]. The geometry of the droplets is considered to be that of oblate ellipsoids [58], and a fully vertical drop trajectory is considered. For the droplet terminal drop velocities and sizes, we use limit values reported in the literature [59; 60], which can be seen in Table 5. The terminal velocity of droplets is used as the simulation is intended to be at ground level. Drop sizes and velocities at values shown in Table 5 were generated pseudo-randomly using the Ziggurat method [61], which can be consulted in Appendix A. \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Material** & \(\epsilon_{r}\) **(1)** & \(\sigma\) **(mS/m)** & \(\mu_{r}\) **(1)** \\ \hline Air & 1 & 0 & 1.0003 \\ Distilled water & & 0.01 & \\ Fresh water & 80–81 & 0.1–10 & 1 \\ Salt water (and sea water) & 81–88 & 4000 & \\ \hline Polar snow & 1.4–3 & - & 1 \\ Polar ice & 3.3–1.5 & 0.02–0.003 & \\ \hline Warm ice & 3.2 & \(5.10^{-4}\)–\(8.10^{-6}\) & \\ \hline Pure ice & 3.2 & & \\ \hline Freshwater frozen lake & 4 & & \\ \hline Sea ice & 2.5–8 & & \\ \hline Permafrost & 1–8 & 1–0.1 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Electromagnetic properties of water in the various forms in which it is found on Earth, as well as of conventional air. \begin{table}
\begin{tabular}{l c c c} \hline \hline \multirow{2}{*}{**Characteristic**} & \multicolumn{2}{c}{**5 GHz**} & \multicolumn{2}{c}{**25 GHz**} \\ \cline{2-4}  & **Free Space** & **Rainfall** & **Free Space** & **Rainfall** \\ \hline \(\Delta x=\Delta y\) (m) & 0.0030 & 5.9958491600000004 \(\times\)\(10^{-4}\) \\ \(\Delta t\) (s) \({}^{\dagger}\) & 7.07106781186547 \(\times\)\(10^{-12}\) & 1.4000714267493641 \(\times\)\(10^{-12}\) \\ Nodes (1) & & 3,339,336 & 83,461,716 \\ Iterations (1) & 33,480 & 134,100 & 133,920 & 334,800 \\ \hline \hline \end{tabular} \({}^{\dagger}\) This time step is calculated through the CFL criterion, for air (\(\epsilon_{r}=1\) and \(\mu_{r}=1\)), which yield the smallest \(\Delta t\) possible, ensuring quality of solution for other media.  
\end{table} 
Table 3: Parameters of the four numerical experiments. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Drop Properties** & **5 GHz** & **25 GHz** \\ \hline
**Semi-minor axis (mm)** & \(2\Delta x_{\text{5 GHz}}\)–\(3\Delta x_{\text{5 GHz}}\) & \(5\Delta x_{\text{25 GHz}}\)–\(10\Delta x_{\text{25 GHz}}\) \\
**Semi-major axis (mm)** & \(4\Delta x_{\text{5 GHz}}\)–\(8\Delta x_{\text{5 GHz}}\) & \(8\Delta x_{\text{25 GHz}}\)–\(20\Delta x_{\text{25 GHz}}\) \\
**Eccentricity (1)** & 0.7071–0.7906 & 0.6124–0.8660 \\
**Terminal velocity (m/s)** & 7.61–9.90 & 8.72–9.40 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Physical parameters of the water droplets for the experiments shown in Table 3[59]. To obtain realistic precipitation, the pseudo-random droplet generation process is coupled to the propagation velocities of the electromagnetic waves so that the droplet layers will only be generated every time \(t=M\Delta t\), where \(M\) is a specific number of iterations for each propagation case. The numerical FDTD algorithm coupled to the Ziggurat-based rain simulation can be observed in the flowchart at Figure 3.  
The code was implemented in FORTRAN 2018 standard [62], and the results were visualised manually, generating ppm graphics files. OpenACC pragmas were added to the source code to port the program to the GPU because OpenACC has demonstrated a better performance than OpenMP for a bi-dimensional FDTD numerical scheme [53; 63; 64]. Nevertheless, OpenMP was used to generate the pseudo-random numbers in a parallel way, through the mentioned Ziggurat algorithm (Appendix A).