## 3 Simulations and Results
### Simulations of Another Rain Event  
A second rain event is utilized to further verify the performance of the proposed method. Shown in Figure 6a, it is an idealized convective storm generated by WRF using the same resolution settings as before. The vertical attenuation field measures 8 km (width)  
Figure 4: Using different numbers of SNR measurements to retrieve the specific attenuation for vertically polarized link at 40 degrees. The circles mark the average retrieved specific attenuations for area V across 100 independent simulations and the error bars show the standard deviation.  
Figure 5: (**a**) The retrieved specific attenuation ratio when the slope parameter for area \(V\) takes different values. Each blue circle represents the mean for 100 simulations, and each error bar shows the standard deviation. The black line is the linear regression result over the blue circles, which is also given by the equation. (**b**) The retrieved slope parameter using the linear relationship. The blue line shows the average for 100 simulations and the red bars show the standard deviation at certain values. (**c**) Scatter plot of the retrieved intercept parameter for voxel A. (**d**) Scatter plot of the retrieved intercept parameter for voxel B.  
by 5 km (height). The highest specific attenuation (for horizontal polarization at 40\({}^{\circ}\)) in all voxels is larger than the frontal rain field in Figure 2a. The DSD profile for each voxel is generated following the exact same process. The entire field is redivided into \(4\times 3\) (marked by white solid lines in Figure 6a) areas, two of which (marked by white dashed lines in Figure 6a) are used to retrieve the DSD parameters. Figure 6b shows the retrieved \(\hat{a}_{V}(40)/\hat{a}_{H}(40)\) when \(\Lambda_{2}\) takes different values, in which the blue circles mark the average retrieved ratio across 100 simulations and the error bars show the standard deviation. Again, using a linear least-squares regression fit of the blue circles, a linear relationship is identified and shown by the black line and the equation in Figure 6b. Figure 6c shows the retrieved \(\Lambda_{2}\) using the identified linear relationship, in which the blue line is the average retrieved \(\Lambda_{2}\) and the black line is a reference line with a gradient of 1. The red bars show the standard deviation across 100 simulations when the true \(\Lambda_{2}\) is at the same selected values as in Figure 5b. Figure 6d is the scatter plot of the retrieved \(N_{0}\) for voxel A in Figure 6a.  
The results above suggest that the performance of retrieval is consistent for both the synthetic convective storm and the simulated frontal rainfall. The proposed methods display great potential in retrieving the slope and intercept parameters of DSD. Although only the retrieval of vertical DSD fields is discussed in this paper, it can be inferred that 3-D retrieval can be achieved through the aggregation of a series of retrieved vertical fields.  
Figure 6: (**a**) Second rain event: the true specific attenuation field for horizontally polarized signals with 40\({}^{\circ}\) elevation angle. (**b**) The retrieved specific attenuation ratio for different slope parameter values. Each blue circle represents the mean for 100 simulations, and each error bar shows the standard deviation. The black line is the linear regression result over the blue circles, which is also given by the equation. (**c**) The retrieved slope parameter using the linear relationship. The blue line shows the average for 100 simulations and the red bars show the standard deviation at certain values. (**d**) Scatter plot of the retrieved intercept parameter for voxel A.