2.3.2. Error Analysis of IMERG Products

To assess the performance of satellite-based precipitation products against measured precipitation by rain gauges, widely applied evaluation indices, including the mean absolute error (MAE), Pearson correlation coefficient (CC) [14], and relative bias (rBIAS) [36], were used. In this regard, MAE (Equation (1)) indicates the error distribution and mean magnitude of errors without considering direction. MAE has the same unit as the precipitation data (i.e., mm day−1). These criteria were calculated for each grid covering the attributed rain gauges. Daily and monthly products were separately analyzed, mainly due to the smooth nature of monthly data and superior performance relative to daily products. The CC (Equation (2)) shows the agreement between the precipitation estimated by the satellite and rain gauge measurements. CC is a dimensionless number, which varies between −1 and 1, with CC equal to zero when there is no correlation. The rBIAS (Equation (3)) represents the size and direction of the difference between the two datasets. Positive and negative rBIAS is an overall overestimation and underestimation of the satellite relative rain gauge measurements, respectively [7]. MAE and rBIAS close to 0 and CC close to 1 display the best performance of the IMERG products relative to the rain gauge measurements in this study:

$$\text{MAE} = \frac{\sum\_{i=1}^{n} |S\_i - O\_i|}{n} \,\text{}\tag{1}$$

$$\text{CC} = \frac{\sum\_{i=1}^{n} \left( \mathbb{S}\_{i} - \overline{\text{S}} \right) \left( O\_{i\prime} - \overline{\text{O}} \right)}{\sqrt{\sum\_{i=1}^{n} \left( \mathbb{S}\_{i\prime} - \overline{\text{S}} \right) \sum\_{i=1}^{n} \left( O\_{i\prime} - \overline{\text{O}} \right)}},\tag{2}$$

$$\text{trBIAS} = \frac{\sum\_{i=1}^{n} (S\_i - O\_i)}{\sum\_{i=1}^{n} O\_i} \text{ or } \frac{\overline{S} - \overline{O}}{\overline{O}}, \tag{3}$$

where *Oi* and *Si* are the observed rain gauge and satellite-based precipitation data, respectively, *O* and *S* are the rainfall averages for pixel i associated to the rain gauge, and n is the total number of satellite-gauge data pairs, which are being compared.

In addition, to investigate how often a significant over/under-estimation by the satellite takes place regardless of the overall magnitude and direction of the errors, we introduced two new indices, named over and under, based on introducing a preliminary index of equal, which stands for an insignificant error. This corresponds to an error smaller than 10% as compared to measurements. However, errors smaller than 0.25 mm day−<sup>1</sup> were considered insignificant as well. The over, under, and equal indices are presented as percentages.

Further, to quantify the precipitation detection ability of the satellite-based precipitation estimates against the ground-based observations, two indices, including the probability of detection (POD) and false alarm ratio (FAR) were calculated (Equations (4) and (5)). The POD expresses the ratio of the correct precipitation detection of the satellite and FAR measures the proportion of no-rain events that are recorded as rain by the satellite. The closest values to 100% and 0% display the best satellite performance for POD and FAR, respectively [14]. The following equations define the POD and FAR:

$$\text{POD} = \frac{\mathbf{n}\_{11}}{\mathbf{n}\_{11} + \mathbf{n}\_{10}} \times 100,\tag{4}$$

$$\text{FAR} = \frac{\mathbf{n}\_{01}}{\mathbf{n}\_{11} + \mathbf{n}\_{01}} \times 100,\tag{5}$$

where n11 is the number of rainfall events that are observed by the rain gauge and detected by the satellite, n10 is the number of rainfall events that are observed by the rain gauge but not detected by the satellite, and n01 is the number of rainfall events that are detected by the satellite but not observed by the rain gauge.
