# Incorporating Aleatoric Uncertainties in Lake Ice Mapping Using RADARSAT-2 SAR Images and CNNs
## 5 Results

Table 4: Reported metrics averaged on test dates in Lake Ontario. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Model Predictions**} & \multicolumn{3}{c}{**Ice Chart**} \\ \cline{2-4}  & **Ice** & **Water** & **Marginal Total** \\ \hline Ice & a & b & a + b \\ \hline Water & c & d & c + d \\ \hline Marginal Total & a + c & b + d & a + b + c + d = n \\ \hline proportion correct ice = a /(a + c) & & & \\ proportion correct water = d /(b + d) & & & \\ total proportion correct = (a + d)/n & & & \\ missed ice = c/(c + d) & & & \\ false alarm (missed water) = b/(a + b) & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Contingency table and prediction scores used for model evaluation. \begin{table}
\begin{tabular}{l c c c c c} \hline \hline \multirow{2}{*}{**Model**} & \multicolumn{6}{c}{**Proportion Scores**} \\ \cline{2-5}  & **Correct** & **Correct** & **Total** & **Missed** & **Missed** \\  & **Ice** & **Water** & **Correct** & **Ice** & **Water** \\ \hline Baseline CNN & & & & & \\ (inputs: HH-HV) & 0.991 & 0.468 & 0.967 & 0.293 & 0.025 \\ \hline CNN with bathymetry & & & & & \\ (inputs:HH,HV, bathymetry) & & & & & \\ \hline CNN with bathymetry & & & & & \\ (inputs: HH, HV) & & & & & \\ \hline CNN with aleatoric loss & & & & & \\ and bathymetry & & & & & \\ (inputs: HH, HV, bathymetry) & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Reported metrics averaged on test dates in Lake Erie. Figure 6: Ice maps of Lake Erie for 15 January 2014. **(a)** HH SAR image, the red solid line ellipsoid (region 1) indicates the region with dark ice that is misclassified by water using CNN without incorporating bathymetry, region 2 indicates open water where models misclassify some fraction of it as ice, and the red dashed line circle (region 3) presents ice that is classified as water by models (**b**) Ice concentration chart, (**c–e.g**) CNN predictions with mapped probability of each pixel belonging to water 0 and ice 1. Panel (**c**) is Baseline CNN model, **(d)** CNN with zonal bathymetry, **(e)** CNN with aleatoric uncertainty, and **(g)** CNN with both zonal bathymetry and aleatoric uncertainty. Panels **(f,h)** are aleatoric uncertainties of model predictions corresponding to CNN with aleatoric uncertainty (**f**) and aleatoric uncertainty and bathymetry (**h**). In both Lake Erie and Lake Ontario, CNN is not able to fully differentiate open water from ice under wind-roughened water conditions. In Lake Ontario, this effect is more visible due to the larger region of open water (reduced ice cover) and can be seen in the central parts of the lake for the image considered here (Figure 6c). By adding bathymetry, the number of misclassified ice pixels in the central portion of the lake decreased. Using aleatoric loss and bathymetry further decreases these misclassifications. A notable challenge in all applied models is that the ice next to the northern part of the Lake Ontario is consistently misclassified as water (indicated by the red ellipsoid with pointing arrow in Figure 7a). Ice charts and air temperature profiles before and after this date indicate that this region is covered by thin new ice. This region is dark and lacks visible texture in the SAR image, and it is therefore not surprising that it is classified as water. A similar  
Figure 7: Ice maps of Lake Ontario for 25 March 2014. **(a)** HH SAR image, the red circle indicates the region covered by thin new ice that is misclassified by ice using CNNs, **(b)** Ice concentration chart, **(c–e,g)** CNN predictions with mapped probability of each pixel belonging to water 0 and ice 1. Panel **(c)** is Baseline CNN model, **(d)** CNN with zonal bathymetry, **(e)** CNN with aleatoric uncertainty, and **(g)** CNN with both zonal bathymetry and aleatoric uncertainty. Panels **(f,h)** are aleatoric uncertainties of model predictions corresponding to CNN with aleatoric uncertainty **(f)** and aleatoric uncertainty and bathymetry **(h)**.