## 3 Simulations and Results
### Retrieval of Specific Attenuation  
In the simulations, we use a LEO satellite with a circular Keplerian motion trajectory of an orbital height of 1000 km and an inclination angle of 96\({}^{\circ}\). The trajectory is on the same vertical plane as the rain field. There are 14 receivers evenly placed at the ground level, from \(-3.25\) km to 3.25 km, with any two adjacent receivers being 0.5 km apart. The SNRs for the horizontally and vertically polarized links are generated using exactly the same parametric configurations as in previous studies [19; 20]. It is worth noting that the estimation error of the SNRs is taken into account by using the Cramer-Rao lower bound (CRLB) [45]. This introduces an error of approximately 0.01 dB in the SNR measurements \(\hat{\rho}_{V}(k)\) and \(\hat{\rho}_{H}(k)\). We assume that there are 50 SNR measurements from 50 different elevation angles available, evenly distributed in the overpass of the satellite. We use a differential approach [20] and an iterative process [19] to estimate the specific attenuations of all voxels as well as the noise figure \(F_{n}\). Ideally, we want to retrieve \(\hat{a}_{H}(40)\) and \(\hat{a}_{V}(40)\) only from \(\hat{\rho}_{H}(40)\) and \(\hat{\rho}_{V}(40)\). However, results suggest that links at 40\({}^{\circ}\) only are not sufficient for the least-squares algorithm to solve the unknowns. On the other hand, for horizontally polarized links, the change in attenuation caused by the change in the elevation angle is relatively small within the targeted range of \(\Lambda\). Table 3 lists the \(a_{H}(k)\) to \(a_{V}(40)\) ratio for different elevation angles and different values of \(\Lambda\) calculated  
Figure 2: (**a**) The true specific attenuation field for horizontally polarized signals with 40\({}^{\circ}\) elevation angle. Warmer colors indicate higher specific attenuations. (**b**) The true specific attenuation field for vertically polarized signals with 40\({}^{\circ}\) elevation angle. by the T-Matrix method. It shows that the change in specific attenuation as the elevation angle increases never exceeds approximately 1%. In other words, considering \(a_{H}\) being unchanged for all elevation angles will not introduce a significant error in the retrieval algorithm. Consequently, we just utilize the measurements of \(\hat{\rho}_{H}\) from all elevation angles to estimate \(\hat{a}_{H}\), which will be regarded as \(\hat{a}_{H}(40)\). Same as previous studies [19], we use an iterative process (summarized in Algorithm 1) to update the noise figure for a more accurate estimation of \(\hat{a}_{H}(40)\). Figure 2(a) shows the estimated \(\hat{a}_{H}(40)\) for the entire field in one simulation. It can be seen that the retrieved field shows very close agreement with the true field in Figure 1(a). In 100 independent simulations, we record \(\hat{a}_{H}(40)\) for the 10 voxels in the 4 th row from the bottom (marked in Figure 2(a)) with nonzero true specific attenuations. As shown in Figure 2(b), the true specific attenuations are marked by the red crosses and the average retrieved specific attenuations over 100 simulations are marked by the red circles. The average relative error, which is calculated by averaging over 100 simulations the ratio of the absolute error to the true specific attenuation, is shown by the blue triangles. It can be concluded that retrieval of specific attenuations is relatively more accurate for voxels with high specific attenuations. More specifically, the relative error for the middle four voxels is below 2%. For vertically polarized links, the change in \(a_{V}\) with the elevation angle will be too large to ignore. Table 4 shows the value of \(a_{V}(k)/a_{V}(40)\) for different values of \(\Lambda\). It can be seen that \(a_{V}(54.34)\) already exceeds \(a_{V}(40)\) by over 4% when \(\Lambda\) is 2.8 mm\({}^{-1}\). As a result, only \(\hat{\rho}_{V}(k)\) measured close to \(40^{\circ}\) can be used in the least-squares algorithm without introducing too much error.