## 5 Results and Discussion
### Daily Precipitation  
Figure 8a-i show the daily precipitation distributions for the chosen gauge stations for 2015-2017, respectively. Plots for each gauge should have 365 points in theory (one for each day of the year). However, some plots have fewer points than this because of the disproportionate temporal precipitation patterns in Mexico. Additionally, due to most precipitation falling in the summertime, there are many low values of precipitation throughout the rest of year, which is not easily visible on the graphs.  
For gauge 1084 (Figure 8a,d,g), it can be observed that the values of CC (around 0.86) are high, which indicates that the precipitation data of GPM IMERG and rain gauges were in relatively high agreement compared to the other two single gauge stations. However, the CC values of gauge 25074 (Figure 8c,f,i) are relatively low at around 0.65. This can be explained by the fact that the precipitation intensity is high in that region, which can cause the measurement errors of surrounding rain gauges and result in a low agreement between GPM and gauge data. A similar study [52] was conducted in the Yangtze River basin at a daily scale, presenting a low CC value (0.44) between IMERG and rain gauge data. Furthermore, both of the values of RMSE and MAE for three gauges are acceptable in Table 5, which are mean value of 8.2 mm and 1.12, respectively. In terms of the values of BIAS, most of the values are positive, which indicate that the GPM precipitation product overestimated the precipitation, especially for gauge 7038. Nevertheless, the results show that the GPM precipitation measurements underestimated the precipitation at the national and annual scales. The reason may because the precipitation intensity was relatively high on that day and the wind flow may have impacted the measurement of precipitation by rain gauges, a phenomenon revealed by the literature review. Thus, the precipitation exceeded the capability of rain gauge measurement, which resulted in the instrument error. For the probabilistic statistical indices (POD, FAR and CSI), the values varied by about 0.75-0.89, 0.21-0.31 and 0.68-0.73, respectively. Based on the results, it was found that the performance of the GPM IMERG precipitation product is decent at the daily scale. Moreover, in order to compare the precipitation data of GPM and rain gauges more clearly, Figure 10 was plotted to show the mean daily precipitation between GPM and rain gauges during the period of 2015-2017. Note that the red dash line indicates a year break. From the plot, it is clearly seen that the GPM and rain gauge profiles are in good agreement. The value of BIAS is \(-4.68\%\), so the GPM precipitation product shows a slight underestimation, and the values of RMSE and MAE are good at 2.36 mm and 0.94, respectively. The data quality of the GPM precipitation product is therefore high at the daily scale.  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**Gauge ID** & **Elevation (m)** & **Time** & **CC** & **RMSE (mm)** & **MAE** & **BIAS (\%)** & **POD** & **FAR** & **CSI** \\ \hline \multirow{4}{*}{1084} & \multirow{4}{*}{2032} & 2015 & 0.82 & 7.27 & 0.76 & \(-13.3\) & 0.89 & 0.21 & 0.74 \\  & & 2016 & 0.80 & 4.55 & 0.97 & 26.4 & 0.87 & 0.23 & 0.71 \\  & & 2017 & 0.94 & 4.82 & 0.78 & 4.9 & 0.91 & 0.20 & 0.77 \\  & & Average & 0.86 & 5.55 & 0.84 & 6.0 & 0.89 & 0.21 & 0.74 \\ \hline \multirow{4}{*}{7038} & \multirow{4}{*}{226} & 2015 & 0.79 & 10.10 & 1.21 & 21.5 & 0.78 & 0.28 & 0.71 \\  & & 2016 & 0.57 & 11.18 & 1.21 & 10.7 & 0.85 & 0.24 & 0.76 \\  & & 2017 & 0.84 & 8.94 & 0.98 & 6.8 & 0.81 & 0.26 & 0.72 \\  & & Average & 0.73 & 10.07 &