## 4 Simulation Results
### Retrieval Using Rectangular Basis Functions  
In this section, we use \(N\) rectangular basis functions obtained by Equation (27) to retrieve the attenuation field. Here, \(N\) is assumed to be \(320\times 50\) and width \(u\) for each function to be 250 m. In other words, the retrieved attenuation field is defined by a grid exactly the same as that of the reference field. Theoretically, parameter \(\Delta\) for the differential approach in Equation (21) needs to be small so that the assumption that \(C(k)\) stays static can be met. However, simulations show that \(\Delta=1\) will generally make the differential distances (\(q_{k+\Delta,n}-q_{k,n}\)) very close to zero and thus make the algorithm sensitive to noise and estimation errors. After some experiments, parameter \(\Delta\) is set at 3, which seems to produce the best retrieval outcomes. After calculating the differential matrix \(\mathbf{Q}\), a constrained least-squares algorithm using an interior-point method (Isqlin in Matlab, see [26; 27]) is employed. The upper and lower bounds for \(w_{n}\) are set to 5 and 0, respectively. In addition, we assume that there being no cloud in the bottom \(320\times 20\) unit squares is prior knowledge. In all of the following simulations, therefore, the weights for the basis functions below 5 km are considered zero and only the weights for the basis functions above 5 km (\(320\times 30\) basis functions in this case) are retrieved. Figure 2(b) shows the retrieved attenuation field obtained by the least-squares algorithm. Note that the retrieved specific attenuation is greater than the reference value (overestimation) for many of the unit squares. This is because, in the first iteration of the algorithm, by assuming \(F_{n}\) in Equation (13) to be zero, the impact of \(F_{n}\) is included in the path-integrated attenuation \(A_{I}\) and makes it effectively larger. The retrieved attenuation field can be used to update \(F_{n}\) and therefore enable the second iteration of the least-squares. The second retrieved field will be underestimated because \(F_{n}\) is overestimated, as the attenuation field is overestimated in the first iteration. As a result, the retrieved fields will oscillate between iterations, but averaging the attenuation fields of two consecutive iterations will yield satisfactory results for rain field estimation [6]. Figure 2(c) shows the final retrieved field by averaging the third and fourth iterations. It can be seen that, for both heavy cloud areas where there is large vertical development and thin cloud areas, the retrieved field and the reference field have good agreement. Assuming that the ground receiver antenna has high directivity, we can simplify the relation between the sky noise and the path-integrated attenuation by assuming that the sky temperature picked up by the antenna is from one direction only, i.e.,  
\[T_{sky}(k)=t_{m}\bigg{(}1-10^{-\frac{A_{I}(k)}{10}}\bigg{)}. \tag{34}\]  
As a result, from Equations (8) and (34), we can conclude that \(F_{n}\) increases with \(A_{I}\) in a nonlinear manner. The nonlinear relation can be observed in Figure 4, where \(F_{n}\) (red line), \(A_{I}\) (blue dashed line) and \(T_{sky}\) (black line) are plotted over the entire course of the overpass. The left panel is for the receiver located at 40 km and the right panel is for the receiver located at 60 km. The ratio of \(F_{n}/A_{I}\) is large for small \(A_{I}\) (e.g., \(\frac{F_{n}}{A_{I}}\approx 1.8\) when \(A_{I}=0.1\) dB) and is small for large \(A_{I}\) (e.g., \(\frac{F_{n}}{A_{I}}\approx 0.6\) when \(A_{I}=6\) dB). For the iterative approach that generates Figure 3c, the least-squares computing needs to be run four times, and each time it deals with \(400\times 91=36,400\) equations. The relation between \(A_{I}\) and \(F_{n}\) in the above can be utilized to reduce the number of the least-squares computing to one time only, thereby reduce the computing complexity of the retrieval process.