## 3 Satellite Precipitation Uncertainty Distribution Approximation
### Sampling Uncertainties  
In the same manner, Figure 8 depicts the performance differences between a single Gaussian model and the mixed Gaussians model, but for the sampling uncertainties (over-ocean) scenario. Although both the Gaussian and GMM models were able to reproduce the general observed sampling error distributions (seen in Figure 2), considerable performance differences were observed, especially in capturing the distribution maximum densities.  
Figure 7: 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 the precipitation (\(x\)-axis, in mm day\({}^{-1}\)) for the RE_C\({}_{04}\) 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 and the density peaks, respectively.  
The positive skewness of the sampling error distributions, especially at low-to-moderate precipitation rates (Figure 2), was not properly captured by considering a single Gaussian model. The main impact was observed on the density peak location, which consequently contributed to large positive and negative residuals (>0.2). An alternation between under- and over-estimations, indicating a considerable skewness, was observed for the Gaussian model. Although it tended to decrease as a function of the precipitation accumulation, this feature remained throughout the entire precipitation range. Such distribution had a strong impact at precipitation accumulations of up to 20 mm day\({}^{-1}\), for which the uncertainty distributions presented an overestimation for both the density peak (from 50% to 10%) and the median (between 7% and 15%).  
In short, the GMM model could better approximate the sampling uncertainties, being consistent for the most part of the precipitation range. GMM slightly overestimated the density peak at precipitation and at accumulations of lower than 1 mm day\({}^{-1}\). The outer-formances were clearly evident for precipitation amounts of greater than 20 mm day\({}^{-1}\), where the RE distributions became more symmetrical (less skewed). The GMM model could well represent the sampling uncertainties, providing less residuals and correctly matching the medians and density peaks. The uncertainty underestimations at about +50%, for precipitation of up to 5 mm day\({}^{-1}\), are worth noting. No considerable residuals (>\(\pm\)0.2) were observed over much of the precipitation range.  
Figure 8: Density scatterplot of the (**a**) Gaussian and the (**b**) GMM (with \(M=2\)) models and its respective residuals (**c,d**) in representing the relative error distributions (\(y\)-axis, in %) as a function of the precipitation (\(x\)-axis, in mm day\({}^{-1}\)) for the error sampling (by TAPEER1.5 product), during the JJA of 2014 and over ocean surfaces (between 35\({}^{\circ}\)N/S global zones). Black crosses mean the 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\({}_{\rm P}\)), respectively.