## 4 Simulation Results
### Retrieval Using Gaussian Basis Functions  
In this set of simulations, we replace the \(160\times 25\) rectangular basis functions in Section 4.2 with the same number of Gaussian basis functions, with the center location of each function unchanged. The width of the Gaussian functions (\(\sigma\)) is set to 1.5 km. To compute the integral in Equation (17), let us divide the sample link \(k\) into a large number (\(S\)) of segments of the same length \(l\). Each segment is so small in relation to the width of the Gaussian function that \(\mu(x,z)\) can be regarded constant for anywhere on a segment. Consequently, Equation (17) becomes  
\[q_{k,n}=\sum\
olimits_{i=1}^{S}l\times\exp(-\frac{\left(x_{k,i}-X_{n}\right)^{ 2}+\left(z_{k,i}-Z_{n}\right)^{2}}{2\sigma^{2}}), \tag{38}\]  
where \(\left(x_{k,i},~{}z_{k,i}\right)\) is the coordinates of the center of the \(i\)-th segment of sample link \(k\). After the differential matrix \(\tilde{\mathbf{Q}}\) is calculated, the same least-squares algorithm is employed to retrieve \(\{\hat{w}_{n}\}_{n=1}^{N}\). However, intuitively, a different PL function \(B(x)\) needs to be adopted because of the different nature of the Gaussian basis function, compared with the rectangular basis function. Multiple simulations suggest that a simple PL function such as the following would generate satisfactory results:  
\[B(x)= \begin{cases}3.0&\text{for }x<0.5\\ 2.1&\text{for }x\geq 0.5\end{cases} . \tag{39}\]  
A contour map of the final retrieved field after the PL function is applied is shown in Figure 5a. It has good agreement with the reference field. The RMSE of this field is 0.207 dB/km, which is smaller than the rectangular case in Figure 3e. Examining the change in specific attenuation along a vertical line shows in detail the performance of different retrieval methods. For instance, Figure 5b is for a vertical line located at \(x=21\) km (shown by a black dashed line in Figure 5a) where the cloud is relatively thin. The green line shows how the specific attenuation changes with height in the reference field. The blue and red lines are for the retrieved fields using the PL function with respectively small and large rectangular basis functions. For the retrieved field with Gaussian basis functions (Figure 5a), the specific attenuation is plotted in black. The same plots are done for heavier clouds (vertical line located at \(x=45\) km, marked by a black dashed line in Figure 5a), shown in Figure 5c. For all three types of basis functions, the retrieved lines are able to closely follow the reference line and the level of error is the lowest for small rectangle basis functions. When the Gaussian basis functions are employed, as they are continuous, continuity is guaranteed in the retrieved field. The peak of the Gaussian retrieved line is generally lower than that of the reference line. This is partly due to the larger value of \(B(x)\) chosen to achieve a lower overall RMSE and partly due to the number of basis functions of the retrieved field being less than the number of grids in the reference field. In addition to the RMSE (Equation (37)), the Pearson correlation coefficient (PCC) is also employed to compare the retrieval results across different methods. PCC is defined by  
\[\text{PCC}=\frac{\sum_{i=1}^{W}\left(R_{i}-\overline{R}\right)\left(R_{i}^{ \prime}-\overline{R^{\prime}}\right)}{\sqrt{\sum_{i=1}^{W}\left(R_{i}- \overline{R^{\prime}}\right)^{2}}\sqrt{\sum_{i=1}^{W}\left(R_{i}^{\prime}- \overline{R^{\prime}}\right)^{2}}}, \tag{40}\]  
where \(\overline{R}\) is the mean of \(R_{i}\) and \(\overline{R^{\prime}}\) is the mean of \(R_{i}^{\prime}\). The scatter plots of the retrieved specific attenuations against the reference values at the comparison points can be found in Figure 6. Panel (a) is for the retrieved field by the averaging method (the result in Figure 3c), and the PCC is 0.986. Panel (b) is for the retrieved field by the PL function (the result in Figure 3d), with the PCC being 0.973.