*2.4. Methods*

Figure 2 represents the flow chart of the proposed methodology applied in the current study. SM2RAIN-CCI is compared and evaluated against the RGs during 2000–2015; while, SM2RAIN-ASCAT is evaluated during 2007–2015. Moreover, due to the difference in time duration of the considered PPs, the performance of each dataset is compared with TMPA during the shared period from 2007 to 2015. To ascertain the impartial inter-comparison of SM2RAIN-ASCAT with SM2RAIN-CCI and TMPA, SM2RAIN-ASCAT has been re-gridded using nearest neighbor algorithm at 0.25◦ spatial resolution. Figure 2 shows that inter-comparison of daily precipitation estimates from SM2RAIN-CCI, SM2RAIN-ASCAT, and TMPA with RGs is conducted to assess the spatial consistency of precipitation datasets over Pakistan. Further, the performance of precipitation datasets is also assessed based on precipitation topography and seasonal time scale using continuous and categorical metrics. Finally, systematic and random error components of SM2RAIN-CCI and SM2RAIN-ASCAT are also assessed utilizing a decomposition method (the final step in Figure 2).

**Figure 2.** A general framework for the proposed methodology.

Three continuous metrics were used to assess the performance of SM2RAIN-CCI and SM2RAIN-ASCAT against the RGs. These metrics were based on inter-comparison of precipitation estimates from each product with precipitation records from RGs on a pixel scale. The Bias (B), unbiased root mean square error (ubRMSE), and Theil's U coefficient (*U*) are considered in this study (Table 2). B is used to estimate the over/under-estimation of precipitation by each product, where positive values indicate overestimation and negative values underestimation. ubRMSE measures the difference between estimated and observed precipitation values. ubRMSE attain only positive values where minimum values indicate better performance. Theil's U predicts the forecasting accuracy of PPs with respect to RGs. The lower bound of Theil's U is zero, indicating the perfect forecast; the value of 1 indicates that PPs forecast the same error as the naïve no-change extrapolation. A value greater than 1 indicates the worst forecasting and has to be rejected [60].

**Table 2.** Statistical metrics used to assess the performance of SM2RAIN-CCI and SM2RAIN-ASCAT. *E* is the estimated precipitation (PPs), *O* is the observed precipitation records from RGs, *n* is the total number of data points, *E* is the average of estimated precipitation, and *O* is the average of observed precipitation.


Moreover, the method developed by AghaKouchak, et al. [61] is used to decompose the total mean square error (MSE) in PPs into random (MSEr) and systematic (MSEs) errors, which are presented as

$$\text{MSE} = \text{MSE}\_{\text{f}} + \text{MSE}\_{\text{s}} \tag{2}$$

$$\text{MSE}\_r = \frac{1}{n} \sum\_{i=1}^{n} \left( E\_i - E\_i^\* \right)^2 \tag{3}$$

$$\text{MSE}\_{\mathbb{S}} = \frac{1}{n} \sum\_{i=1}^{n} \left( E\_i^\* - O\_i \right)^2 \tag{4}$$

$$E\_i^\* = a \times O\_i + b \tag{5}$$

*E*∗ *<sup>i</sup>* (mm/day) in Equations (3)–(5) is calculated on the daily scale by least squared linearly regressing the PPs estimates against the RGs at each pixel. Therefore, *a* and *b* in Equation (5) are offset and scale parameters [62].

In addition, the performance of PPs was also assessed using the modified Kling–Gupta efficiency (KGE) score. KGE combines correlation coefficient (r), bias ratio (β) and variability ratio (γ) [63], which is represented as follows:

$$KGE = 1 - \sqrt{\left(r - 1\right)^2 + \left(\beta - 1\right)^2 + \left(\gamma - 1\right)^2} \tag{6}$$

$$\mathbf{r} = \frac{\sum\_{i=1}^{n} \left( E\_i - \overline{E} \right) \left( O\_i - \overline{O} \right)}{\sqrt{\sum\_{i=1}^{n} \left( E\_i - \overline{E} \right)^2} \sqrt{\sum\_{i=1}^{n} \left( O - \overline{O} \right)^2}} \tag{7}$$

$$
\beta = \frac{\overline{E}}{\overline{O}} \tag{8}
$$

$$\gamma = \frac{(\text{CV})\_{\text{E}}}{(\text{CV})\_{\text{O}}}, \text{ (CV)}\_{\text{E}} = \sqrt{\frac{1}{n} \sum\_{i=1}^{n} \left(\overline{\text{E}\_{i} - \overline{\text{E}}}\right)^{2} / \overline{\text{E}}}, \text{ (CV)}\_{\text{O}} = \sqrt{\frac{1}{n} \sum\_{i=1}^{n} \left(\overline{\text{O}\_{i} - \overline{\text{O}}}\right)^{2} / \overline{\text{O}}}\tag{9}$$

where CV denotes the coefficient of variation. The perfect values of KGE, β and γ are all 1.

The precipitation detection competences of SM2RAIN-CCI and SM2RAIN-ASCAT are examined using four categorical metrics (Table 3), including the probability of detection (POD), false alarm ratio (FAR), critical success index (CSI), and Bias score (BS). POD represents the capability of PPs to detect precipitation events. A threshold of 1 mm is considered to distinguish precipitation from dry days (no precipitation) at any time scale [64]. FAR indicates the fraction of predicted precipitation event that did not occur. POD and FAR ranges between 0–1 with a perfect values of 1 and 0, respectively [62]. CSI is a fraction of precipitation events correctly detected by PPs. The CSI value ranges between 0 to 1, with the perfect value of 1 [65]. BS represents the fraction of all PPs precipitation events that were correctly predicted. The range of BS values is 0 to 1 with perfect score equal to 1.

**Table 3.** Categorical metrics, where *hit* indicates the number of precipitation events correctly detected both by PPs and RGs, *miss* indicates the number of precipitation events not detected by PPs but recorded by RGs, *false\_alarm* is number of precipitation events detected by PPs while no precipitation records are available at RGs, and *N* is the total number of events.

