*3.3. Evaluation of Statistical Distribution of IMERG Products*

Figure 5 displays the Q-Q plots for each IMERG daily product and different seasons, including winter (December-February), spring (March-May), summer (June-August), and fall (September-November). It should be noted that these plots are not showing data from the tails of the distributions (close to the 100th percentile). For clarity, the Q-Q plots contain rainfall data at least until the 95.5th percentile. As a result, only rainfall lower than 37 mm day−<sup>1</sup> (typical high rainfall amount for Iran) was taken into consideration, allowing a comparison between all Q-Q plots. In addition, the exact location of some typical quantiles is depicted by an arrow and a percentile value to better visualize the data distribution.

As an example, Figure 5a shows that the most obvious departures for the IMERG-Early product in comparison to rain gauge observations in winter start above the 90th percentile of data (i.e., rainfall >3 mm day−1) in the form of an underestimation that gradually increased until the 97.5th percentile (~14 mm day<sup>−</sup>1). This underestimation remained rather constant until the 99.5th percentile (~36 mm day−1) by an amount of about 2.5 to 3 mm day−<sup>1</sup> while there was a slight overestimation for the more frequent rainfall events (lower than the 90th percentile), with rainfall amounts less than 1 mm day<sup>−</sup>1. Figure 5b shows that the IMERG-Late, to a lesser extent, overestimated low rainfall, but larger underestimations are seen for higher rainfall compared to the IMERG-Early product. However, as shown in Figure 5c, the higher accuracy of the IMERG-Final product leads to a significant reduction of departures from the 45-degree line that is observed for the IMERG-Early and -Late products in the form of underestimation. This result shows an acceptable fit for rainfall located approximately between the 95th and 99th percentile (~6–24 mm day−1) while the overestimation extends up to the 95th percentile (i.e., rainfall <7 mm day−1). That could be the reason for the overestimated average winter rainfall for the IMERG-Final product observed in Figure 3.

In Figure 5d, the scatter of points is linearly located to, and slightly above, the 45-degree line for rainfall less than the 99.5th percentile. This can be seen as a shift in the location of the statistical distribution of the IMERG-Early products in comparison to the distribution of rain gauge data. In other words, the IMERG-Early overestimated daily rainfall for values less than 29 mm in spring by an amount of about 1 to 1.5 mm. As seen in Figure 5e, the overestimation was reduced for the IMERG-Late product such that the points between the 97.5th and the 99.5th percentiles nearly overlaid the 45-degree line. On the other hand, as seen in Figure 5f, while the IMERG-Final product resulted in a better fit for the more frequent daily rainfalls in spring (i.e., less than 11 mm), major departures for higher values in the form of underestimation appear close to the 97.5th percentile. Therefore, a better fit for frequent rainfall by IMERG-Final suggested by Figure 5f seems to be the reason for the best estimation of average spring rainfall depicted in Figure 3. This product cannot be used as a firm reference dataset for extreme rainfall studies. Instead, a corrected IMERG-Early or -Late product, by removing the shift value for every data point, will be a better choice for the evaluation of extreme rainfall events in spring, which recently caused severe flood events in the country [37].

**Figure 5.** Q-Q plots of daily rainfall (IMERG and rain gauge) during 2014–2017: (**a**) IMERG-Early for Winter, (**b**) IMERG-Late for Winter, (**c**) IMERG-Final for Winter, (**d**) IMERG-Early for Spring, (**e**) IMERG-Late for Spring, (**f**) IMERG-Final for Spring, (**g**) IMERG-Early for Summer, (**h**) IMERG-Late for Summer, (**i**) IMERG-Final for Summer, (**j**) IMERG-Early for Fall, (**k**) IMERG-Late for Fall, and (**l**) IMERG-Final for Fall. The arrows and percentile values in red depict the location of quantiles.

The worst distributional fit between IMERG daily products and rain gauge observations is observed for summer (Figure 5g–i). Among all daily products, the best fit with measurements in summer for the data lower than the 99th percentile (~2 mm day<sup>−</sup>1) is observed for the IMERG-Final (Figure 5i). However, the tail behavior for summer (>10 mm day<sup>−</sup>1) for the IMERG-Final indicates a large discrepancy relative to the other products (above 99.5th percentile). Finally, for the fall season (Figure 5j–l), all products indicate an overestimation of daily rainfall less than the 95th percentile and an underestimation of daily rainfall above the 95th percentile. However, the magnitude of overestimation is slightly reduced for IMERG-Late and IMERG-Final relative to IMERG-Early. While the IMERG-Final is the best product for fall daily rainfall lower than the 97.5th percentile, the underestimation deteriorates for extreme rainfall (obviously, above the 99th percentile) (Figure 5l). As a common result, while the bias-adjusted GPM IMERG products, which is the case for IMERG-Final datasets, resulted in better match with the gauge measurements for more frequent rainfall events (lower amounts of rainfall), uncorrected datasets of IMERG-Early and –Late products were shown to be more trusted related to the extreme events (heavy rainfall and flooding), especially in spring, summer, and fall seasons, as the bias-corrected data from the IMERG-Final product deteriorated underestimations observed for extreme rainfalls.

Figure 6a–d show Q-Q plots for the IMERG-Monthly product relative to rain gauge measurements for different seasons. The increasing underestimation from the 45-degree line for data above the 95th percentile for all seasons indicates that the IMERG-Monthly product might not be the best choice for the study of extreme monthly rainfall. It is noteworthy that the amount of extreme rainfall varies by season. For example, the average rainfall for the 99.5th percentile based on rain gauge measurements was about 4.3 and 13.5 mm day−<sup>1</sup> for summer and fall, respectively. Based on the IMERG-Monthly product, however, the equivalent values for these months were 2 and 7.8 mm day−1. This implies an underestimation as big as 69 mm for summer and 171 mm for the fall months (considering a 30-day month) that can obviously bias extreme monthly rainfall study based on the IMERG-Monthly product across the country. For more frequent rainfall (below the 95th percentile), especially in spring (Figure 6b) and summer (Figure 6c), the distribution of monthly data from the IMERG-Monthly product adequately fits the measurements, with an exception of overestimation for data lower than the 75th percentile, especially for winter (Figure 6a), fall (Figure 6d), and spring (Figure 6b).

**Figure 6.** Q-Q plots of average monthly rainfall (IMERG and rain gauge data) during the 2014–2017 period for (**a**) winter, (**b**) spring, (**c**) summer, and (**d**) fall seasons. The arrows and percentile values in red depict the location of quantiles.
