## 2 Materials and Methods
### Methodology
#### 2.3.3 Satellite Precipitation Data Fusion  
The AHP method is a powerful and flexible decision theory, and it is the most commonly used subjective method to determine the weights of criteria/structures; it can also help researchers find the most suitable solution among numerous combination options [80; 99; 100]. Its main steps are as follows:  
1. Construct a hierarchical structure, determine the judgment matrix according to the quality evaluation indicators and prior knowledge, and assign different values to represent the difference in the importance of the indicators. 2. Calculate the eigenvalues and eigenvectors of the judgment matrix: \[\lambda_{\mathrm{max}}=\frac{1}{n}\sum_{i=1}^{n}\frac{(\mathrm{X}\omega)_{i}}{ \omega_{i}}\] (14) where \(\lambda_{\mathrm{max}}\) indicates the largest eigen value of the matrix, \(\omega\) indicates the weight vector, \(\omega_{i}\) indicates the weight of the i-th evaluation index, and n indicates the number of rating indicators. 3. Consistency verification:\[\text{CR}= \frac{\text{CI}}{\text{RI}} \tag{15}\]  
where CR is the Consistency Ratio, and RI represents the average stochastic consistency index for the judgment matrix. The CI is the consistency index of the judgment matrix, which can be calculated using Equation (16):  
\[\text{CI}= \frac{\lambda_{\text{max}}-\text{n}}{\text{n}\,-\,1} \tag{16}\]  
where \(\lambda_{\text{max}}\) is the largest eigen value of the matrix, and n is the number of indicators. If CR < 0.1, it means that the result has passed verification; otherwise, the judgment is inconsistent and the weight needs to be re-estimated. AHP is widely employed in multidisciplinary fields, but its subjectivity is a disadvantage that cannot be ignored, which brings uncertainty to the results [101]. The EW method is currently used as the most common objective method for determining standard/structural weights [102]. EW does not consider information other than the data, and the determination of the weight of each indicator is based on the information of the data itself and the relationship between the indicators [103]. It reflects the amount of information provided by the indicator by calculating the information entropy of an indicator, so as to determine the role of the indicator in the comprehensive evaluation [104]. It can be implemented according to the following steps:  
1. Standardize evaluation indicators. Positive impact indicators: \[\text{Y}_{\text{ij}}\,=\,\frac{\text{X}_{\text{ij}}-\text{min}(\text{X}_{ \text{ij}})}{\text{max}(\text{X}_{\text{ij}})-\text{min}(\text{X}_{\text{ij}})}\] (17) Negative impact indicators: \[\text{Y}_{\text{ij}}=\frac{\text{max}(\text{X}_{\text{ij}})-\text{X}_{ \text{ij}}}{\text{max}(\text{X}_{\text{ij}})-\text{min}(\text{X}_{\text{ij}})}\] (18) where \(\text{X}_{\text{ij}}\) is the value of the j-th evaluation indicator of the i-th evaluation object; and \(\text{Y}_{\text{ij}}\) is the standardization of the standardization. \(\text{max}(\text{X}_{\text{ij}})\) and \(\text{min}(\text{X}_{\text{ij}})\) represent the maximum and minimum values of \(\text{X}_{\text{ij}}\), respectively. 2. Calculate indicator proportion: \[\text{P}_{\text{ij}}\,=\,\frac{\text{Y}_{\text{ij}}}{\sum_{i=1}^{\text{n}} \text{Y}_{\text{ij}}}\,\text{i}=1,2\ldots\text{n;}=1,2\ldots\text{m}\] (19) where \(\text{P}_{\text{ij}}\) is the proportion of the indicator value of the j-th indicator of the first i-th evaluation object. 3. Calculate the entropy value and information entropy redundancy corresponding to each index as follows: \[\text{E}_{\text{j}}=-\frac{1}{\text{ln}(\text{n})}\sum_{i=1}^{\text{n}}\text {P}_{\text{ij}}\text{ln}(\text{P}_{\text{ij}})\] (20) \[\text{D}_{\text{j}}=1-\,\text{E}_{\text{j}}\] (21) where \(\text{E}_{\text{j}}\) is the entropy value of the j-th indicator, n is the value of the evaluation index number, and \(\text{D}_{\text{j}}\) is the information entropy redundancy of the j-th indicator. 4.