## 3 Simulations and Results
### Synthetic DSD Field  
A vertical DSD field is generated synthetically for our simulations. The field is considered to remain static during the overpass time of the LEO satellite (approximately 289 s in our simulations). It is 6 km in length and 4 km in height, and divided into \(12\times 8=96\) voxels, with each voxels measuring 500 m by 500 m. The DSD profile for each voxel is then generated as follows. Firstly, the specific attenuation for each voxel for horizontally polarized 30 GHz signals with 40\({}^{\circ}\) elevation angle (\(a_{H}(40)\)) is generated according to a localized frontal rainfall event [19] simulated by the Weather Research and Forecasting (WRF) model. This event is a subset of a larger convection-permitting simulation covering the southern part of the Great Barrier Reef and adjacent coastline. As shown in Figure 2a, the entire rain field is contained within the 96 voxels. The specific attenuation varies from voxel to voxel. For instance, voxel A has the highest specific attenuation (\(a_{H}(40)=3.23\) dB/km) and some voxels on the edge of the field have zero dB/km specific attenuations. Secondly, the shape parameter \(\mu\) for all voxels is assumed to be 1, which is in line with the WDM6 Scheme of WRF. Thirdly, the field is divided into \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{\(Q_{H}(D)\)} & \multicolumn{4}{c}{Elevation Angle (\({}^{\circ}\))} \\ \cline{2-6} \multicolumn{1}{c}{(\(\times 10^{-8}\) m\({}^{2}\))} & **40** & **55** & **70** & **90** \\ \hline \multirow{6}{*}{\(D\) (mm)} & 0.5 & 1.1337 & 1.1336 & 1.1335 & 1.1335 \\ \cline{2-6}  & 1.0 & 23.634 & 23.499 & 23.389 & 23.327 \\ \cline{1-1} \cline{2-6}  & 2.0 & 509.52 & 507.46 & 505.76 & 504.82 \\ \cline{1-1} \cline{2-6}  & 4.0 & 4019.0 & 4139.8 & 4243.7 & 4302.9 \\ \cline{1-1} \cline{2-6}  & 6.0 & 9064.2 & 9318.2 & 9487.6 & 9562.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The extinction cross section for horizontal polarization calculated by the T-Matrix method for various raindrop sizes and elevation angles. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multicolumn{2}{c}{\(Q_{V}(D)/Q_{H}(D)\)} & \multicolumn{4}{c}{Elevation Angle (\({}^{\circ}\))} \\ \cline{2-6}  & **40** & **55** & **70** & **90** \\ \hline \multirow{6}{*}{\(D\) (mm)} & 0.5 & 0.9989 & 0.9994 & 0.9998 & 1 \\ \cline{2-6}  & 1.0 & 0.9700 & 0.9831 & 0.9939 & 1 \\ \cline{1-1} \cline{2-6}  & 2.0 & 0.9045 & 0.9461 & 0.9807 & 1 \\ \cline{1-1} \cline{2-6}  & 4.0 & 0.8585 & 0.9220 & 0.9727 & 1 \\ \cline{1-1} \cline{2-6}  & 6.0 & 0.8366 & 0.9191 & 0.9746 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The extinction cross section ratio of vertical to horizontal polarizations calculated by the T-Matrix method for various raindrop sizes and elevation angles. three layers (middle layer 2 km in height, top and bottom layers 1 km in height) and the slope parameters (\(\Lambda_{1}\), \(\Lambda_{2}\) and \(\Lambda_{3}\)) are assigned for each layer (see Figure 2a). This is based on the assumption that for a particular rain event, the slope parameter is more spatially homogeneous than the intercept parameter [40]. According to the probability density function of \(\Lambda\) in recent studies [41], \(\Lambda_{1}\) and \(\Lambda_{3}\) are set to be 2.9 mm\({}^{-1}\) and 4.1 mm\({}^{-1}\) for the bottom and top layer, respectively. Initially assigned to 3.5 mm\({}^{-1}\), \(\Lambda_{2}\) for the middle layer is able to take different values in the following simulations so that we can test the performance of the DSD retrieval. Finally, the intercept parameter \(N_{0}\) for each voxel is calculated from \(a_{H}(40)\) and \(\Lambda\) using Equations (1) and (3). The canting angles of raindrops follow a Gaussian distribution with a mean of 0\({}^{\circ}\) and a standard deviation of 2\({}^{\circ}\)[42; 43]. In the calculation a MATLAB implementation of the T-Matrix method [44] is applied and the integral is computed numerically from D\({}_{\text{min}}\) = 0.01 mm to D\({}_{\text{max}}\) = 7.0 mm in 0.01 mm increments.