## 4 Discussion  
We have presented a three-step algorithm that combines BCNNs with Bayesian decision theory, for support in environmental monitoring. In the first step a BCNN is optimize on labeled simulated data. In the next step, the BCNN is used to generate a posterior predictive distribution of class labels. In the final step, the optimal monitoring strategy is calculated based on the posterior predictive distribution and given operational costs. Our case study indicates that MC dropout and BCNN are effective tools for detecting CO\({}_{2}\) seeps in marine waters. The overall predictive power is large, as seen through the ROC curve in Figure 4. While the majority of the time series are either classified as leak or no-leak with little uncertainty, a small proportion will be inconclusive; see Figure 5. As expected, the classification uncertainties increase with the distance to the leak source location. This distance depends on the seepage flux rates and can be modeled from the outcome of the BCNN; see Figure 6. \begin{table}
\begin{tabular}{c c c} \hline \hline  & **Leak** & **No-Leak** \\ \hline Confirm (\(\alpha_{1}\)) & \(\lambda_{11}=\kappa\) & \(\lambda_{12}=1\) \\ Not Confirm (\(\alpha_{2}\)) & \(\lambda_{21}=\gamma\lambda_{11}\) & \(\lambda_{22}=0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Overview of the cost functions applied in this example. The cost function can be interpreted as having no cost to doing the correct decision. Confirming a no-leak has some operational cost, e.g., a ship must mobilize and search/confirm that there is no leakage. Confirm a leakage has a cost of \(\kappa\) times the operational cost. The cost of not confirming a true leakage is gamma times the cost of confirming a leakage. We assume there is no cost with not looking for a leak when there is no leakage. Figure 12: The figure shows the percentage of the total area to be monitored where the optimal decision would be confirming a leak. The cost function is altered by varying the parameter \(\kappa\) and \(\gamma\) in Equation (7) and Table 3. The three different lines represents varying \(\kappa\), and the the x-axis shows varying \(\gamma\) for each scenario. Increased cost difference between the operational cost and the cost of confirming a leak, result in a higher degree of confirmation, faster. (**Left Panel**) The MAP of the predictive posterior distribution. (**Right Panel**) The expectation of the predictive posterior distribution. Using the mode instead of the expectation results in less confirmation/mobilization. Demanding high detection probability reduces the detectable area of a monitoring location, i.e., you will need to measure closer to a leak in order to achieve the desired accuracy; see Figure 10. This motivates the use of optimal decision strategies, that is finding appropriate action that should be initiated based on the prediction, its uncertainty, and costs associated with taking wrong and right decisions. We gave an example of an optimal strategy choice in a binary classification setting in Section 3.7 for different costs, showcasing the algorithm. There is extensive literature, including several deep-learning approaches, on monitoring and forecasting of air quality time series. For example [PERSON] et al. (2015) used a recursive and feed forward neural network and [PERSON] et al. (2016) a RNN to forecast and monitor air quality from time series data. In context of oceanography monitoring, [PERSON] et al. (2017) used hyper-ellipsoidal models for detecting anomalies in a wireless sensor network and validated their approach on data from Great Barrier Reef. None of these approaches take uncertainty into account. [PERSON] (2018) recently published a literature review of machine-learning applications in oceanography. As \(\mathrm{CO_{2}}\) is naturally present in the ocean and is highly variable, it is difficult to attribute measured \(\mathrm{CO_{2}}\) to an unplanned release. [PERSON] et al. (2019) developed pH anomaly criterion for determination of marine \(\mathrm{CO_{2}}\) leaks.