## 4 Simulation Results  
For X-band SAR working at the oblique angle of \(30^{\circ}\), \(\sigma_{0}\) is approximately \(-7\) dB [29]. Set \(z_{t}=13\) km and \(z_{0}=4.5\) km. From Equations (12), (16) and (18), we can first obtain the normalized radar cross section (NRCS) \(\sigma_{SAR}\) of the double-layer precipitation model. With Formulas (6) or (7), we can obtain the horizontal precipitation variation of the model. Taking a rectangular rain-cloud cell, a trapezoidal rain-cloud cell, a triangular rain-cloud cell and two separate rectangular rain-cloud cells under the rain rate of \(10\) mm/h as Examples 1-4, the precipitation variation simulation result diagrams, given precipitation variation diagrams, comparison diagrams and absolute error diagrams are shown in Figures 2-5. Taking a triangular rain-cloud cell under the rain rate of \(30\) mm/h and \(50\) mm/h as Examples 5-6, the precipitation variation simulation result diagrams, given precipitation variation diagrams, comparison diagrams and relative error diagrams are shown in Figures 6 and 7. The attributes of the six examples, including the maximum rain rate \(R(x)\), the width of the rain-cloud cells \(w\), the shape parameters \(d\) and the shapes of rain-clouds are listed in Table 2. The simulation results of the attributes are listed in Table 3, where \(\hat{R}(x)\) is the simulated result of the maximum rain rate, \(\widetilde{R}(x)\) is the relative error of the maximum rain rate, \(\hat{w}\) is the simulated result of the width of the rain cloud and \(\widetilde{w}\) is the relative error of the width of the rain cloud. Figure 2: Rectangular distribution diagrams for \(R(x)=10\) mm/h. (**a**) Given precipitation distribution; (**b**) Simulated precipitation distribution; (**c**) Given precipitation distribution and simulated precipitation distribution; (**d**) Relative error between given and simulated result. Figure 4: Triangular distribution diagrams for \(R(x)=10\) mm/h. (**a**) Given precipitation distribution; (**b**) Simulated precipitation distribution; (**c**) Given precipitation distribution and simulated precipitation distribution; (**d**) Relative error between given and simulated result. Figure 3: Trapezoidal distribution diagrams for \(R(x)=10\) mm/h. (**a**) Given precipitation distribution; (**b**) Simulated precipitation distribution; (**c**) Given precipitation distribution and simulated precipitation distribution; (**d**) Relative error between given and simulated result. Figure 5: Rectangular bimodal distribution diagrams for \(R(x)=10\) mm/h. (**a**) Given precipitation distribution; (**b**) Simulated precipitation distribution; (**c**) Given precipitation distribution and simulated precipitation distribution; (**d**) Relative error between given and simulated results. Figure 6: Triangular distribution diagrams for \(R(x)=30\) mm/h. (**a**) Given precipitation distribution; (**b**) Simulated precipitation distribution; (**c**) Given precipitation distribution and simulated precipitation distribution; (**d**) Relative error between given and simulated results. According to Figures 2-7, it is obvious that large errors occur at the edges of the precipitation area. The relative errors of the rest areas of the simulation results will be listed for each example. **Example 1**. : _Due to the non-linearity, fluctuations occur at each junction, which is \(\hat{x}_{R}-n\cdot(z_{t}-z_{0})\)\(\tan\theta\) or \(\hat{x}_{R}-n\cdot z_{t}\tan\theta\), where \(n\) is a natural number. Except the right edge, \(\hat{R}(x)\) is in the range of \([9.9,10.07]\)\(mm/h\) in the precipitation area, and the maximum relative error is 1%._ \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Maximum R(x) (mm/h)** & \(d\) (km) & \(w\) (km) & **Shape of Rain-Cloud(s)** \\ \hline
1 & 10 & 0 & 10 & Rectangular \\
2 & 10 & 3 & 10 & Trapezoidal \\
3 & 10 & 5 & 10 & Triangular \\
4 & 10 & 2.5 & 2.5 & Rectangular bimodal \\
5 & 30 & 5 & 10 & Triangular \\
6 & 50 & 5 & 10 & Triangular \\ \hline \hline \end{tabular}
\end{table}