## 5 Discussion
### The Selection of Variogram Model  
In this section, we will discuss the selection of variogram model that is used to reconstruct the rainfall field in this paper. The application of the OK algorithm is usually under an assumption that the distribution of property in observed field is close to Gaussian. According to the data used in Section 5.2, we can see that the rain intensity in our experimental region is an obvious non-Gaussian distribution (shown in Figure 12a) that the probability of light- and moderate rainfall is higher than heavy- and extreme rainfall. Therefore, it is necessary for us to find an approach to ensure the accuracy of reconstruction results. As we can see from Equation (17), the variogram model plays an important role in the OK algorithm because it directly relates to the calculation of \(\lambda\) in Equation (14). In non-Gaussian distributions, the variogram model may be sensitive to the field with high kurtosis or skewness. In Figure 12b,c, as the cumulative density function reaches to 0.9, the values of kurtosis and skewness are approximately equal to 10 and 2.5, respectively. Therefore, we will focus on the other 10% rainfall fields with high kurtosis and skewness to analyze the performances of different  
Figure 11: The statistical relative errors between OEL and CMORPH for different rain intensity ranges. Figure 10: The comparisons between inversed rain intensity of all grid cells derived from OELs network and original values from CMORPH during plum rain season (June, July) of 2016, 2017, 2018 and 2019.  
variogram models. Here, the gaussian-, spherical- and stable models are compared with each other and their expressions are [42]:  
\[\textit{Gaussian}:\ \gamma^{\prime}(h)=c\left(1-\exp\left(-\left(\frac{h}{r} \right)^{2}\right)\right) \tag{29}\]  
\[\textit{Spherical}:\left\{ \begin{array}{cc}\gamma(h)=d\left(\frac{3}{2}\frac{ h}{r}-\frac{1}{2}\left(\frac{h}{r}\right)^{3}\right)&\text{if}\ 0\leq h\leq r,\\ \gamma(h)=c&\text{if}\ h>r.\end{array} \right. \tag{30}\]  
\[\textit{Stable}:\gamma(h)=c\left(1-\exp\left(-\left(\frac{h}{r}\right)^{a} \right)\right) \tag{31}\]  
where \(c\) denotes the sill that represents the maximum value of the fitted variogram, and \(r\) denotes the range distance as \(\gamma\)reaching to \(c\). It is noted that the formula of stable model is like gaussian model but has an unknown exponent \(a\)that needs to be fitted. A good performance of spherical model has been proved by [PERSON] et al. who achieved the reconstruction of rainfall field by using microwave links [17]. In Figure 13, the gray point denotes the measured semi-variance from OELs network, and the values of each rainfall field are scaled into [0, 1] to show the total change trend of semi-variance. The normalization can reduce effects caused by the magnitude of rain intensity. Here, the data of rainfall in May, August and September of 2016, 2017, 2018 and 2019 is added into that used in Section 5.2 because the rainfall in Nanjing is mainly concentrated in these five months. The solid lines in this figure denote the fitted variogram functions and the vertical lines represent the average absolute error between fitted- and measured value in each lag distance. It can be seen that the changes of the semi-variance of these four years are very similar to each other. As for the averages of unnormalized values of these four years, the correlation coefficients among them are higher than 0.996 and the standard errors are lower than 2.8 mm\({}^{2}\)/h\({}^{2}\). The result indicates that the variogram of rainfall field in our experimental region has an almost invariable form. In each subplot, we can observe that all above three variogram models can represent the change trend of semi-variance whose determination coefficients are higher than 0.97 and the average absolute errors are lower than 0.15. It is hard to judge which model is better only according to the consistency with measured data. Therefore, the rainfall fields with high kurtosis and skewness are used to test the three models.