## 4 Data and Methodology
### Methodology  
#### 4.2.1 Statistical Analysis  
A comprehensive study was performed on the correlation of IMERG, 3B42, and Era-Interim with the station precipitation data over Iran. In the first step, statistical indices such as bias, multiplicative bias (MBias), relative bias (RBias), mean absolute error (MAE), root mean square error (RMSE), and linear correlation coefficient (CC) were determined. The bias is defined as the average difference between _in-situ_ observations and satellite/model precipitation estimates (S/M-PE), and can be either positive or negative. A negative bias indicates underestimation by satellite precipitation while a positive bias indicates overestimation.  
\[Bias=\frac{\sum_{i=1}^{N}\left(P_{S_{i}}-\ P_{O_{i}}\right)}{N}\ \left(mm\right) \tag{1}\]  
The multiplicative bias (MBias) is the ratio of S/M-PE over rain-gauge value such that a perfect estimation would result in an Mbias of unity. Underestimation will lead to values less than unity, and overestimation to values greater than unity.  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Products** & \multicolumn{4}{c}{**Number of Pixels**} \\ \hline Study Area & No. of Synoptic Stations & IMERG & 3B42 & ERA-Interim \\ Guilan (G8) & 12 & 23 & 23 & 20 \\ Bushehr (G4) & 9 & 15 & 18 & 12 \\ Kermanshah (G5) & 12 & 24 & 27 & 18 \\ Tehran (G2) & 10 & 16 & 22 & 19 \\ \hline \end{tabular}
\end{table}
Table 2: Number of stations and pixels in each area.  
Figure 3: Map of meteorological synoptic stations.  
\[MBias=\frac{\sum_{i=1}^{N}P_{S_{i}}}{\sum_{i=1}^{N}P_{O_{i}}} \tag{2}\]  
The relative bias (RBias) describes the systematic bias of satellite-based precipitation and behaves the same as bias.  
\[RBias=\frac{\sum_{i=1}^{N}\left(P_{S_{i}}-\ P_{O_{i}}\right)}{\sum_{i=1}^{N}P_{O _{i}}}\ \times 100 \tag{3}\]  
The mean absolute error (MAE) is used to represent the average magnitude of the error.  
\[MAE=\frac{\sum_{i=1}^{N}\left|P_{S_{i}}-\ P_{O_{i}}\right|}{N}\ (mm) \tag{4}\]  
The root mean square error (RMSE), which gives a greater weight to the larger errors relative to MAE, is used to measure the average error magnitude.  
\[RMSE=\sqrt{\frac{1}{N}\ \sum_{i=1}^{N}\left(P_{S_{i}}-\ P_{O_{i}}\right)^{2}}\ (mm) \tag{5}\]  
The correlation coefficient (CC) is used to assess the agreement between satellite/model-based precipitation and rain-gauge observations. The value of CC is such that \(-1<\text{CC}<+1\). A CC value of \(+1\) indicates a perfect positive fit. If there is no linear correlation or a weak linear correlation, CC is close to zero.  
\[\text{CC}=\frac{\sum_{i=1}^{N}\left(P_{S_{i}}-\ \overline{P}_{S}\right)\ \left(P_{O_{i}}-\ \overline{P}_{O}\right)}{\sqrt{\sum_{i=1}^{N}\left(P_{S_{i}}-\ \overline{P}_{S}\right)^{2}}\sqrt{\sum_{i=1}^{N}\left(P_{O_{i}}-\ \overline{P}_{O}\right)^{2}}} \tag{6}\]  
where \(P_{S_{i}}\) is the value of satellite/model precipitation estimates for the \(i\)th daily event, \(P_{O_{i}}\) is the value of rain-gauge observation for the \(i\)th daily event, \(N\) is the number of observed days, \(\overline{P}_{S}\) is the average value of satellite/model precipitation estimates for \(N\) observed days over each pixel or grid points.