## 4 Data Processing  
In order to detect and evaluate the quality and performance of GPM data, the rain gauge data was applied as a reference for comparison. Several conventional statistical performance metrics were selected, which were the Correlation Coefficient (CC) in Equation (1), the Mean Absolute Error (MAE) in Equation (5) [46], the Root Mean Square Error (RMSE) in Equation (4) [47,48] and the Relative Bias (BIAS) in Equation (6). Firstly, the CC described the agreement between satellite estimates and rain gauge observations, and secondly, the MAE, RMSE and BIAS was applied to describe the error and bias of satellite estimates compared with rain gauge observations [49]. The relevant equations are as follows [50,51]:  
Pearson Correlation Coefficient (CC) was calculated as Equation (1):  
\[\text{CC}=\frac{\frac{1}{\hbar}\sum_{i=1}^{n}\left(P_{\text{S}}^{(i)}-\overline {P_{\text{S}}}\right)(P_{\text{G}}^{(i)}-\overline{P_{\text{G}}})}{\sigma_{ \text{S}}\sigma_{\text{G}}} \tag{1}\]  
Figure 5: Map showing the distribution of mean annual precipitation in mm measured by GPM (2015–2017).  
Figure 6: Location of the selected rain gauge stations in Mexico.  
\[\sigma_{S} =\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(P_{S}^{(i)}-\overline{P_{S}} \right)} \tag{2}\] \[\sigma_{G} =\sqrt{\frac{1}{n}\sum_{i=1}^{n}\left(P_{G}^{(i)}-\overline{P_{G}} \right)} \tag{3}\]  
The Root Mean Square Error (RMSE) was obtained by Equation (4) as:  
\[\text{RMSE}=\sqrt{\frac{\sum_{i=1}^{n}\left(P_{S}^{(i)}-P_{G}^{(i)}\right)^{2} }{n}} \tag{4}\]  
The Mean Absolute Error (MAE) was obtained by Equation (5) as:  
\[\text{MAE}=\frac{\sum_{i=1}^{n}\left|P_{S}^{(i)}-P_{G}^{(i)}\right|}{n} \tag{5}\]  
The GPM product overestimates precipitation amounts if MAE value higher than 1, otherwise, the satellite product underestimates the precipitation.  
The Relative Bias (BIAS) was obtained by Equation (6) as:  
\[\text{BIAS}=\frac{\sum_{i=1}^{n}\left(P_{S}^{(i)}-P_{G}^{(i)}\right)}{\sum_{i= 1}^{n}\left(P_{G}^{(i)}\right)}\times 100\% \tag{6}\]  
where \(P_{G}\) and \(P_{S}\) represent the actual precipitation value provided by rain gauges and precipitation estimation provided by the satellite product, respectively. \(n\) is regarded as the number of observations. \(\sigma_{S}\) and \(\sigma_{G}\) are standard deviations of GPM satellite precipitation and standard deviations of rain gauge precipitation, respectively.  
In addition, in terms of probabilistic statistical indices, the probability of detection (POD), the critical success index (CSI) and the false alarm ratio (FAR) was applied to describe the eventuality of GPM satellite precipitation estimates. POD provides the fraction of precipitation events that the satellite distinguishes among all the actual precipitation events, and FAR provides the fraction of unreal events among all the events GPM satellites detect [49]. The CSI was essentially obtained by a function of POD and FAR. These equations were as follows [6; 52]:  
The Probability of Detection (POD) as Equation (7) as:  
\[\text{POD}=\frac{n_{11}}{n_{11}+n_{01}} \tag{7}\]  
The False Alarm Ration (FAR):  
\[\text{FAR}=\frac{n_{10}}{n_{11}+n_{10}} \tag{8}\]  
The Critical Success Index (CSI):  
\[\text{CSI}=\frac{n_{11}}{n_{11}+n_{01}+n_{10}} \tag{9}\]  
where \(n_{11}\) represents precipitation observed by the rain gauge and GPM satellite simultaneously. \(n_{10}\) is precipitation observed by the GPM satellite but not observed by the rain gauge, and \(n_{01}\) is converse to \(n_{10}\). The perfect values of the evaluation metrics are presented in Table 3 below [38; 49]. Moreover, since the expression for precipitation intensity in this study is the collecting of precipitation over a consistent time period, the unit of precipitation is'mm' instead of'mm/h'.