# Incorporating Aleatoric Uncertainties in Lake Ice Mapping Using RADARSAT-2 SAR Images and CNNs
## 5 Results  
Using typical scores utilized in forecast verification of binary predictands, results are evaluated to infer each model's performance. Four models are compared: (1) CNN baseline model; (2) CNN with zonal bathymetry; (3) CNN with aleatoric loss; and (4) CNN with aleatoric loss and zonal bathymetry. All models use both the HH and HV SAR images as input data. Contingency tables (Table 2) for Lake Erie and Lake Ontario for total number of 8,644,181 and 2,885,246 pixel counts are shown in Tables 3 and 4, respectively. Figure 6 presents results for 15 January 2014, in Lake Erie and Figure 7 shows results of 25 March 2014, in Lake Ontario. Although only one prediction map is shown for each lake, we have observed the same patterns on different dates, and our interpretations are transferable to those dates. Scores in Tables 3 and 4 are quite close with slight improvements when using bathymetry, and bathymetry and aleatoric uncertainty. However, visual comparisons show substantial improvements in specific regions. For instance, visual comparisons show improvements in mapping ice in the western basin of Lake Erie (Figure 6d,g) when bathymetry is added with/without aleatoric uncertainty, respectively. In Lake Erie, when bathymetry information is included, scores from scenes with a large water coverage confirmed that the ability of the model to predict open water in the central portion of the lake is significantly reduced leading to a high number of water points misclassified as ice (shown with a solid red line (labelled as region 2)). Dropped performance in central regions is the expense of using large scale zonal bathymetry whereas same values covers different backscatter signatures. The bathymetry and aleatoric uncertainty together provide a good estimate of water points and ice points, with larger aleatoric uncertainties near the southern shore of Lake Erie. This can also be seen in Figure 6, comparing Figure 6c with Figure 6g in the west basin. Results also show that aleatoric uncertainty is high around the ice edges. We also observe that aleatoric uncertainty is generally correlated with small  
Figure 5: (**a**) Schematic architecture of a CNN using HH and HV backscatter patches for a custom loss function with two output units (**b**) Architecture of a CNN using HH and HV backscatter patches for a custom loss function with two output units and auxiliary dataset of bathymetry before fully connected layer  
scale image features with less than 2 km by 2 km area. Notice also that there is a patch of smooth dark ice in the western basin shown with a red solid line ellipsoid in Figure 6a (region 1) that is incorrectly classified as water in CNN baseline (Figure 6c). When aleatoric uncertainty is included (Figure 6e), this region is still misclassified, but it is also associated with high uncertainty. When both bathymetry and aleatoric uncertainty are included, the classification is improved and the uncertainty in this region is reduced. At the same time, when both bathymetry and aleatoric uncertainty are included, the patch of smooth ice is identified appropriately as an area associated with high uncertainty. We also observed that models were not successful in identifying ice shown with a red dashed line circle in Figure 6a (region 3). Lack of visible texture in the SAR image is the main reason for classifying ice as water. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multicolumn{6}{c}{**Proportion Scores**} \\ \cline{2-7}  & **Correct** & **Correct** & **Total** & **Missed** & **Missed** \\  & **Ice** & **Water** & **Correct** & **Ice** & **Water** \\ \hline Baseline CNN & & & & & \\ (inputs: HH, HV) & 0.965 & 0.763 & 0.819 & 0.017 & 0.390 \\ \hline CNN with bathymetry & & & & & \\ (inputs:HH, HV, bathymetry) & & & & & \\ \hline CNN with aleatoric loss & & & & & \\ (inputs: HH, HV) & & & & & \\ \hline CNN with aleatoric loss & & & & & \\ and bathymetry & & & & & \\ (inputs: HH, HV, bathymetry) & & & & & \\ \hline \hline \end{tabular}
\end{table}