## 3 Simulations and Results
### Retrieval of Specific Attenuation
It should also be noted that the noise figure is now already calculated based on \(\hat{a}_{H}(40)\) so the least-squares algorithm can be directly applied to estimate \(\hat{a}_{V}(40)\) without engaging the iterative process of Algorithm 1. Although this would again introduce some error, simulations suggest that the error can be considered negligible. Multiple simulations suggest that the retrieved field is erroneous due to underdetermination in the least-squares algorithm when less than eight SNR measurements from \(40^{\circ}\) (\(\hat{\rho}_{V}(40)\) to \(\hat{\rho}_{V}(51.2)\)) are used. Figure 2(c) shows the retrieval of \(\hat{a}_{V}(40)\) for the same 10 voxels as in Figure 2(b). Here, eight SNR measurements are used but the relative errors for individual voxels are much higher than in the retrieval of \(a_{H}\). For instance, for voxel 5, the relative error of estimated \(\hat{a}_{V}(40)\) is 39%. This indicates that using estimated \(\hat{a}_{V}(40)/\hat{a}_{H}(40)\) of individual voxels to calculate the slope parameter \(\Lambda\) will lead to very large errors. As a result, we propose a new model \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multicolumn{2}{c}{\(\mathbf{a_{H}(k)/}\)} & \multicolumn{4}{c}{\(\mathbf{\Lambda(\text{mm}^{-1})}\)} \\ \cline{2-5} \multicolumn{1}{c}{\(\mathbf{a_{H}(40)}\)} & **2.8** & **3.3** & **3.8** & **4.2** \\ \hline \multirow{3}{*}{\(k\) (\({}^{\circ}\))} & 42.80 & 1.0000 & 0.9996 & 0.9992 & 0.9991 \\ \cline{2-5}  & 54.34 & 1.0000 & 0.9977 & 0.9961 & 0.9955 \\ \cline{2-5}  & 66.68 & 1.0001 & 0.9959 & 0.9933 & 0.9921 \\ \cline{2-5}  & 88.72 & 1.0003 & 0.9943 & 0.9909 & 0.9893 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The specific attenuation for a horizontally polarized link changes while the elevation angle increases from 40 degrees. Nonetheless, the maximum change is only about 1%. for retrieving \(\hat{a}_{V}(40)\) and thus \(\hat{a}_{V}(40)/\hat{a}_{H}(40)\), taking advantage of the assumption that \(\Lambda\) does not change within a layer. Before retrieving \(\hat{a}_{V}(40)\), the entire field is divided into three layers according to \(\Lambda\) and then nine areas as shown in Figure 3d, with each of area being 2 km in width. The same least-squares algorithm is applied to retrieve the specific attenuation for vertical polarization for each area. Figure 3d shows the retrieved \(\hat{a}_{V}(40)\) field in one simulation, in which SNR measurements at eight different elevation angles (\(\beta_{V}(40)\) to \(\beta_{V}(51.2)\)) are \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & \(a_{V}(k)\)/ & \multicolumn{4}{c}{\(\Lambda(\text{mm}^{-1})\)} \\ \cline{3-6}  & \(a_{V}(40)\) & **2.8** & **3.3** & **3.8** & **4.2** \\ \hline \multirow{3}{*}{\(k\)} & 42.80 & 1.0090 & 1.0064 & 1.0052 & 1.0044 \\ \cline{2-6}  & 54.34 & 1.0436 & 1.0344 & 1.0267 & 1.0216 \\ \cline{2-6}  & 66.68 & 1.0773 & 1.0600 & 1.0467 & 1.0381 \\ \cline{2-6}  & 88.72 & 1.1057 & 1.0819 & 1.0639 & 1.0523 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The specific attenuation for a vertically polarized link changes while the elevation angle increases from 40 degrees. The change is much more significant compared to that for a horizontally polarized link. Figure 3: (**a**) The retrieved specific attenuation field for horizontally polarized signals at 40 degrees. (**b**) Retrieval results of horizontal specific attenuations for the ten voxels in the 4 th row from the bottom in 100 simulations: the true specific attenuations are marked by red crosses and the average retrieved specific attenuations are marked by red circles. The blue triangles show the relative error. (**c**) Retrieval results of vertical-specific attenuation for the ten voxels in the 4 th row from the bottom in 100 simulations: the true specific attenuations are marked by red crosses and the average retrieved specific attenuations are marked by red circles. The blue triangles show the relative error.