## 3 Satellite Precipitation Uncertainty Distribution Approximation
### The GMM Model Distributions and Performances  
Figure 4 shows an example of the observed uncertainty distributions compared with the GMM and a simple Gaussian fit model for two daily precipitation categories/bins, i.e., at low 1.04-1.25 mm day\({}^{-1}\) (Figure 4a) and large 69.39-83.3 mm day\({}^{-1}\) (Figure 4b) accumulations, through the C\({}_{99}\) and the C\({}_{\text{REF}}\) experiments. As mentioned, with increasing rainfall intensity, the right-skewed uncertainty distribution shape began to acquire a more symmetrical distribution shape with the median centered at zero (less unbiased). For instance, in the low accumulation case, the RE distribution presented a negative median of about \(-9.8\%\), while the median for precipitation intensity was \(-2.7\%\). At low daily precipitation amounts (Figure 4a), the outperformance of the GMM model was clearly observed in representing both the RE under- and over-estimations compared to the Gaussian model. However, the models' representations of the RE distributions at low to moderate daily precipitations were more problematic due its larger sample and the heterogeneity of observations that led to the highest densities (peak numbers and locations). This consequently resulted in larger residuals, especially at the RE underestimations where the maximum densities were found, as the main factor responsible for the precipitation underestimations at this precipitation classes. In contrast, the uncertainty overestimations (positive RE) were well fitted by the GMM model. Yet, despite still being positively skewed, the Gaussian model density peak was more shifted around zero, while the GMM and observed density peaks were approximately \(-25\%\). At large precipitation accumulations (Figure 4b), both the Gaussian model and the GMM model could reproduce the unimodal RE distributions, resulting in significantly small residual magnitudes, especially at the density peak, and with only slight differences between each other. Figure 4: An example of over-land global uncertainty distributions (\(x\)-axis, in %) during the JJA of 2014 for the daily precipitation bins (**a**) 1.04–1.25 mm day\({}^{-1}\) and (**b**) 69.39–83.3 mm day\({}^{-1}\), resulted from the comparison between the C\({}_{99}\) and the C\({}_{\text{REF}}\) experiments. The GMM fit with M = 2 (blue line) is contrasted with a simple Gaussian fit (red line) to represent the observed distribution (black line). Note that GMM “modes” (M#1 and M#2) are in grey filled area dashed curves. The perpendicular dashed lines indicate the Bayesian boundaries between the two GMM modes (in blue) and its correspondent mean values (in grey). Systematic performance, by comparing the GMM and Gaussian models, and as a function of the precipitation accumulation in addition to the contrast between the three types of uncertainty, was better illustrated in the AIC goodness-of-fit statistics (Figure 5). Overall, the GMM skills varied according to precipitation intensity, which was linked to the nature of the precipitation uncertainty source. This also included the constellation changing assumption that presented the largest sensitivity to the AIC results and stability between multiple satellite precipitation products. The superior goodness-of-fit statistics were found for the most part for large precipitation accumulations (substantially increasing greater than 10 mm day\({}^{-1}\)) regardless of the satellite error source. The maximum AIC results were observed in the range from 5 to 10 mm day\({}^{-1}\), except for the sampling error (Figure 5b), which also presented high and constant AIC values at precipitation accumulations of 0.5-10 mm day\({}^{-1}\). The more different the constellation configuration, the more the model is prone to overfitting (higher AICs) the uncertainty representation (Figure 5a). Figure 5c shows that the GMM proved to be well suited to represent the uncertainties of satellite products when compared to rain gauges, resulting in lower AIC values overall.