## 3 Satellite Precipitation Uncertainty Distribution Approximation
### Constellation Changes-Induced Uncertainties  
Systematic analysis of the accumulated precipitation from 0.5 to 100 mm day\({}^{-1}\) confirmed the case study and revealed the better performance of the mixed model than for the single Gaussian model. The better representation of the observed RE density distributions provided by the GMM model is not related to the experimental configuration  
Figure 5: The GMM goodness-of-fit through the Akaike information criterion (AIC) statistics, as a function of the precipitation (\(x\)-axis, in mm day\({}^{-1}\)). Comparisons between the three error sources considered: (**a**) due to constellation changes induced uncertainties, (**b**) due to sampling uncertainty and (**c**) through a comparison between multiple satellite-based with rain-gauges network. AIC results were normalized by the maximum value of all error sources. (Figures 6 and 7), despite the better performance for the RE_C04 experiment (seen the AIC goodness-of-fit comparisons in Figure 5). In the cases of more intense accumulation, even if gain is not as dramatic when using the mixture, the two Gaussian fit outperformed the single Gaussian approach. Overall, the largest residual concentrations were of up to 10 mm day\({}^{-1}\) and were more prominent in the single Gaussian model. The Gaussian model largely overestimated (underestimated) the positive (negative) RE distributions, especially in the RE range from 25% to 60% (from \(-\)30% to 0%), which presented \(\varepsilon\) of greater (lower) than \(\pm\)0.2. The Gaussian model also overestimated both the maximum peak (mode) and median of the uncertainty density distributions in about 20% of low-to-moderate rain accumulations. In addition, the Gaussian model was not able to represent the uncertainty distributions at extremes precipitation accumulations (i.e., \(\geq\)100 mm day\({}^{-1}\)). In this case, the Gaussian model uncertainty distributions were less biased (density peak centered roughly the zero--less positively skewed), overestimating the density peak of roughly 17% of uncertainties ranging from \(-\)5 to 15%. This occurred differently than for the observed RE distributions (seen in Figure 1a), which had a slightly right-skewed distribution. To offset that, given the different density distribution shape compared to that which was observed, the Gaussian model also led to an underestimation of the negative uncertainties ranging between 15% and 30%. Figure 6: Density scatterplot of the (**a**) Gaussian and the (**b**) GMM (with \(M=2\)) models and their respective residuals (**c,d**) in representing the relative error distributions (\(y\)-axis, in %) as a function of precipitation (\(x\)-axis, in mm day\({}^{-1}\)) for the RE_C09 experiment, during the JJA of 2014 and over land surfaces (between 55\({}^{\circ}\)N/S global zones). Black crosses indicate residual values above/below the \(\pm\)0.2 density values. Solid and dashed black lines in c-d correspond to the differences between the model and the observed median (Q\({}_{2}\)) and the density peaks (D\({}_{P}\)), respectively. On the other hand, the GMM model could better fit the entire precipitation accumulation range (Figure 6b,d). The outperformance of the GMM model is evidenced in multiple aspects (e.g., residual, density peak, median distribution) and indicates that the uncertainties remain, mainly, in representing the maximum density peaks at approximately up to 10 mm day\({}^{-1}\) (in the range of \(-8\)% RE on average). Above the precipitation accumulation of 80 mm day\({}^{-1}\), the GMM model slightly overestimated the density peak (less than 5%) surrounding the 0 and \(-20\)% RE range. The maximum underestimations (lower than \(-0.2\)), were found to be of lower than 1 mm day\({}^{-1}\) (in the \(-16\)% RE range). In general, no considerable (greater than 0.2) RE overestimations were found using the GMM model for both the RE_C\({}_{99}\) and RE_C\({}_{04}\) (Figure 7) experiments.