## 4 Evaluation Methods
Accordingly, in this study, a smaller threshold value between 20% of average daily precipitation and 1 mm/day was set to determine precipitation/no-precipitation days [35]. This choice will ensure more corresponding observations for GPPs' capability assessment in regions with lighter precipitations. This dynamic threshold could lead to a spatially non-uniform evaluation; however, since the primary objective of this paper is to compare the performance and capability of different GPPs, its influence on the results is negligible [35]. \[\text{POD}=\frac{\text{Hit}}{\text{Hit}+\text{Miss}} \tag{5}\]  
\[\text{FAR}=\frac{\text{False}}{\text{Hit}+\text{False}} \tag{6}\]  
\[\text{CSI}=\frac{\text{Hit}}{\text{Hit}+\text{Miss}+\text{False}} \tag{7}\]  
The first step to computing the above-mentioned categorical metrics is calculating four elements of Hit, False, Miss, and Correct Negative. Table 1 shows the contingency table between the target (GPP) and reference (synoptic station) precipitations, allowing the calculation of the four elements. For instance, a corresponding observation is identified as Hit if both sources (i.e., GPP and synoptic station) have recorded a rain event. Similarly, if two sources have recorded a no-rain occurrence, the corresponding observation is considered as Correct Negative. In conditions of different responses (i.e., capturing non-identical precipitation/no-precipitation events) from two sources, the observations will be categorized as Miss or False. The point-to-pixel (i.e., reference precipitations as points and target precipitations as pixels) approach was implemented to validate and compare the performance of GPPs using in situ data [83, 84, 85]. In this regard, the target precipitation values were extracted at each synoptic station location and then compared with reference precipitation values of the corresponding synoptic station. Different time scales were considered for comprehensive evaluation and comparison of GPPs. First, daily comparisons were performed between target and reference precipitation values for the entire two decades (i.e., 2000-2020) of observations. In this regard, first, the target precipitation values with sub-daily temporal resolutions were accumulated to generate daily target precipitation values. The comparisons were implemented on daily (i.e., considering all corresponding precipitation values at each synoptic station), monthly (i.e., considering all corresponding precipitation values at each synoptic station at each month), and yearly (i.e., considering all corresponding precipitation values at each synoptic station at each year) time scales. Taylor Diagram [86], focusing on precipitation intensity assessment using continuous metrics, and the Performance Diagram [87], focusing on precipitation occurrence evaluation using categorical metrics, were implemented to illustrate the monthly precision characteristics of GPPs. Later, monthly \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Grided Precipitation Product (GPP)**} & \multicolumn{2}{c}{**Synoptic Station**} \\ \cline{2-3}  & **Yes** & **No** \\ \hline Yes & Hit & False \\ No & Miss & Correct Negative \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The contingency table representing the agreement between Gridded Precipitation Products (GPPs) and the in situ observations from the synoptic stations. and yearly average and accumulated values were computed for both target and reference precipitations to apply monthly and yearly comparisons. The last two time scales allow for better comparison and validation of GPPs, especially targeting the needs of disciplines that employ monthly or yearly precipitation values instead of daily precipitation values. Daily observations were also used to validate the selected GPPs at each synoptic station, considering their geographical coordinates. This analysis was conducted to explore the capability of GPPs in a spatial manner.