## 3 Retrieval Method  
With \(A_{F}(k)\) and \(\hat{\rho}(k)\) known, the retrieval process mainly involves handling the two unknown terms \(C(k)\) and \(F_{n}(k)\) as well as solving the cloud attenuation field. Consider a vertical rectangular attenuation field of interest. As the overpass time is very short for the LEO satellite (e.g., 289 s for a satellite at 1000 km of orbital height and a minimum elevation angle of 40 degrees for ground receivers), the attenuation field is regarded as static for the purpose of retrieval. It is described by a function of two variables \(R(x,z)\) representing the specific attenuation in dB/km at coordinates \((x,z)\) km, where the \(x\)-axis represents the horizontal direction along the ground track of the satellite and the \(z\)-axis is the vertical direction starting from sea level. The relation between the path-integrated attenuation and the attenuation field function is obtained by [19]  
\[A_{I}(k)=\int_{L_{k}}R(x,z)dL, \tag{14}\]  
where \(L_{k}\) denotes the signal line path for the \(k\)-th sample (see Figure 2). It is assumed that \(R(x,z)\) can be approximated by a linear combination of \(N\) basis functions \(\{b_{n}(x,z)\}_{n=1}^{N}\):  
\[R(x,z)=\sum\
olimits_{n=1}^{N}w_{n}b_{n}(x,z), \tag{15}\]  
where \(w_{n}\) is the weight for \(b_{n}\). Equation (14) then becomes  
\[A_{I}(k)=\sum\
olimits_{n=1}^{N}w_{n}\int_{L_{k}}b_{n}(x,z)dL. \tag{16}\]We define  
\[q_{k,n}=\int_{L_{k}}b_{n}(x,z)dL, \tag{17}\]  
and \(q_{k,n}\) represents the line integral of the \(n\)-th basis function along the \(k\)-th sample path. Once the basis functions have been determined, \(q_{k,n}\) can be obtained solely from the geometrical conditions of the satellite and the ground receiver at the time that the signal is sampled. As a result, \(q_{k,n}\) can be calculated offline and stored away for later use. If the path-integrated attenuation \(A_{I}\) can be measured, the retrieval task then becomes a matter of solving the series weights \(\{w_{n}\}_{n=1}^{N}\). However, the path-integrated attenuation is only measured indirectly through the SNR estimation. We rewrite Equation (13) as  
\[C(k)-V(k)-F_{n}(k)=\sum\
olimits_{n=1}^{N}w_{n}q_{k,n}, \tag{18}\]  
where \(V(k)\) is the sum of two known terms:  
\[V(k)=\hat{\rho}(k)+A_{F}(k). \tag{19}\]  
Note that \(A_{F}(k)\) can be calculated by Equation (12) but \(C(k)\) and \(F_{n}(k)\) are unknown. For a similar problem in [6], \(C(k)\) is regarded as a constant and solved together with \(\{w_{n}\}_{n=1}^{N}\) using a least-squares method. However, in reality, parameter \(C(k)\) could change during one overpass [15; 20], so forcing it to be a constant could introduce considerable errors. An improved approach [15] to eliminate \(C(k)\) before solving \(\{w_{n}\}_{n=1}^{N}\) was introduced as follows. Considering another sample link \((k+\Delta)\) which is very close in time to sample link \(k\), Equation (18) becomes  
\[C(k+\Delta)-V(k+\Delta)-F_{n}(k+\Delta)=\sum\
olimits_{n=1}^{N}w_{n}q_{k+ \Delta,n}. \tag{20}\]  
The difference between \(C(k+\Delta)\) and \(C(k)\) is ignored because the two links are very close in time, Equations (18) and (20) yield  
\[V(k)-V(k+\Delta)+F_{n}(k)-F_{n}(k+\Delta)=\sum\
olimits_{n=1}^{N}w_{n}(q_{k+ \Delta,n}-q_{k,n}), \tag{21}\]  
Without prior knowledge of the path-integrated attenuation, noise figure \(F_{n}(k)\) is completely unknown. One viable approach is to initially regard \(F_{n}(k)\) as zero for the first least-squares computing. According to Equation (8), this means that \(T_{sky}\) is zero, which is physically incorrect [21]. However, it has been shown that the retrieved \(\{w_{n}\}_{n=1}^{N}\) can be used to update \(F_{n}(k)\) and in turn correct \(\{w_{n}\}_{n=1}^{N}\) in an iterative manner [6]. In this paper, we adopt this iterative approach, as well as explore an  
Figure 2: Calculation of the path-integrated attenuation, where the locations of the ground receiver and the satellite are marked by points A and B, respectively.