## 2 Materials and Methods
### Methodology
#### 2.3.1 Geographically Weighted Regression Models  
Traditional regression models, such as the ordinary least square method, assume that the target variable and auxiliary variables are stationary over the entire region. Precipitation is the result of the interaction of various factors, and there exists typical spatial non-stationarity. The traditional global regression model could not reflect the spatial characteristics of precipitation. The GWR model is a regional regression method that can be used to construct non-stationary correlations between target and auxiliary variables [56]. The GWR model assumes that the regression coefficient varies with geographic location, and estimates the regression coefficient for each location and establishes a regression equation to explore spatial heterogeneity [59; 92; 93]. Its mathematical expression is as follows:  
\[y_{i}=\beta_{0}(u_{i},v_{i})+\sum_{p=1}^{j}\beta_{p}(u_{i},v_{i})\ x_{ip}+ \varepsilon(u_{i},v_{i})\ \ \ i=1,2\ldots n \tag{1}\]  
where \(y_{i}\) is the precipitation at position \(i\) as a dependent variable; \(x_{ip}\) is the observed value at position \(i\) of the \(p\)th auxiliary variable; \((u_{i},v_{i})\) is the geographic coordinate at position \(I\); \(\beta_{0}(u_{i},v_{i})\) is the intercept at position \(I\); \(\beta_{p}(u_{i},v_{i})\) is the regression coefficient at position \(i\), which is a function of geographic location; \(p\) is the number of auxiliary variables; and \(\varepsilon(u_{i},v_{i})\) is the independent normal distribution with zero mean errors. The regression coefficient at position \(i\) can be estimated by solving the following matrix equation:  
\[\beta(u_{i},v_{i})=\left(X^{T}(W(u_{i},v_{i}))X\right)^{-1}X^{T}W(u_{i},v_{i})y \tag{2}\]  
where \(\beta(u_{i},v_{i})\) denotes the estimated local coefficient at position \(I\); \(X\) and \(y\) are the auxiliary variables (cloud optical and physical properties) and dependent variable (precipitation); superscript \(T\) represents matrix transpose; and \(W(u_{i},v_{i})\) is the weight coefficient matrix, which mainly depends on the selection of the spatial kernel function [94]. It is vital to note in the GWR model that there is a linear dependence of the column vectors in \(X\), i.e., a collinearity problem in terms of regression, which can cause errors in the solution of Equation (2) [1]. There are two main types of spatial kernel functions, namely the Gaussian function and Bi-square function. 1. Gaussian function  
The Gaussian function uses a continuous monotonically decreasing function to describe the relationship between \(W_{ij}\) and \(d_{\bar{ij}}\). The mathematical relationship is expressed as follows:  
\[W_{ij}=\exp\big{(}-\big{(}d_{\bar{ij}}/b\big{)}^{2}\big{)} \tag{3}\]  
where \(W_{ij}\) is the observation weight at position \(j\) used to estimate the coefficient at position \(i\); \(d_{ij}\) is the distance between position \(i\) and position \(j\); and \(b\) is the bandwidth, i.e., a non-negative attenuation parameter describing the relationship between the weight and the distance function [95], which could be confirmed by the Corrected Akaike Information Criterion (AICc) or cross-validation (CV) method. 2. Bi-square function  
The Bi-square function may help to reduce the computational cost compared with the Gaussian function [86; 90], and its expression is as follows:  
\[W_{ij}=\left\{ \begin{array}{ll}\big{(}1-\big{(}d_{ij}/b\big{)}^{2}\big{)}^{ 2}&d_{ij}\ \leq\ b\\ 0&d_{ij}>\ b\end{array} \right. \tag{4}\]The validation methods of adaptive bandwidth b mainly include AICc and CV methods. The AICc method is widely employed in the GWR model, which could be expressed as follows:  
\[\text{AICc}=2\text{Nlog}_{\text{e}}\bigg{(}\sqrt{\frac{\text{SSE}}{\text{N}- \text{tr}\big{(}\text{S}\big{)}}}\bigg{)}+\text{Nlog}_{\text{e}}\big{(}2\pi \big{)}+\text{N}\big{(}\frac{\text{N}+\text{tr}\big{(}\text{S}\big{)}}{\text{N} -2-\text{tr}\big{(}\text{S}\big{)}}\big{)} \tag{5}\]  
where SSE represents the residual sum of squares, and tr(S) represents the trace of the hat matrix.