## 3 Reconstruction of Rainfall Field by OELs Network
### Performance of Rainfall Field Reconstruction  
We perform the reconstruction of rainfall field based on the OK and inverse distance weighting (IDW) interpolation, respectively. The corresponding results are compared with CMORPH rainfall fields to assess the performance of the OELs network. The comparisons of four rainfall maps are shown in Figure 4 where the first line is the synthetic field from CMORPH, the second and the third lines are the reconstructed fields based on the IDW and OK, respectively. Although we can roughly observe the peak and distribution of rain intensity from reconstructed fields by IDW, the results based on OK show a better agreement with synthetic field and the crucial characteristics of rainfall field are also described accurately. In addition, Table 2 provides evaluation metrics of the reconstructed  
Figure 3: The simulation process of rainfall field reconstruction.  
fields. Here, _CC_ indicates the spatial correlation of two rainfall fields and \(S\) denotes the entropy used to represent the heterogeneity of rain intensity and its expression is given by:  
\[\left\{\begin{array}{l}S=\frac{-1}{\ln(mm)}\sum\limits_{i=1}^{n}\sum\limits_{j= 1}^{m}\left(\frac{R_{ij}}{F}\right)\ln(\frac{R_{ij}}{F})\\ F=\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m}\left(R_{ij}\right)\end{array}\right. \tag{22}\]  
where \(n\) and \(m\) are the number of grid cell in horizontal and vertical directions.  
In terms of the four given examples, the results derived from OK algorithm are more accurate whose _RMSE_ is lower than 1 mm/h. It also can be seen from the values of _CC_ that the reconstructed- and synthetic field are strongly correlated with each other whose value is higher than 0.89. Although the entropy \(S\) of IDW and OK rainfall fields are both close to synthetic field, showing similar distributions of rain intensity, the results of OK are still better than IDW. Based on above simulations, we achieve the reconstruction of rainfall field by OELs network. The results also demonstrate that the OK interpolation method used in our work is better at addressing this issue than IDW which is adopted in other relevant  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & \multicolumn{2}{c}{_CC (\(\emptyset\))_} & \multicolumn{2}{c}{_RMSE (mm/h)_} & \multicolumn{2}{c}{_S (\(\emptyset\))_} \\
**Date** & **IDW** & **OK** & **IDW** & **OK** & **CMORPH 1** & **IDW** & **OK** \\ \hline May 25 & 0.899 & 0.979 & 0.482 & 0.190 & 1.000 & 1.000 & 1.000 \\ June 29 & 0.912 & 0.979 & 0.659 & 0.281 & 0.881 & 0.958 & 0.909 \\ July 11 & 0.926 & 0.974 & 0.361 & 0.163 & 0.999 & 1.000 & 0.999 \\ August 10 & 0.962 & 0.990 & 1.529 & 0.665 & 0.984 & 0.991 & 0.985 \\ \hline \hline \end{tabular}  
* CMOROH denotes the synthetic rainfall field derived from Climate Prediction Center morphing method.  
\end{table}
Table 2: The evaluation metrics of reconstructed rainfall fields from OELs network based on inverse distance weighting (IDW) and ordinary kriging (OK) interpolation methods.  
Figure 4: The comparisons of reconstructed- and synthetic rainfall field: 17:00, 25 May 2019; 8:00, 29 June 2019; 23:00, 11 July 2019 and 1:00, 10 August 2019  
studies. Although the simulation only achieves the reconstruction of rainfall field in the Jiangning district whose area is approximately 1225 km\({}^{2}\), the results indicate the feasibility and great potentials of widely distributed OELs for national or even global rainfall monitoring. On the other hand, the terrain of Jiangning is mainly plain and the effects of topography need to be considered in practice. To prevent the surface multipath fading and distortion, OEL is generally sited where significant specular and diffuse surface reflections rarely occur and many techniques have been proposed to address this issue [50]. As for the region where we cannot employ a lot of OELs, the interpolation algorithm used in this article shows obvious advantages.