# Quantifying Uncertainties in OC-SMART Ocean Color Retrievals: A Bayesian Inversion Algorithm
## 3 Methodology for Quantifying Uncertainties
### Experimental Setup
As explained in Section 3.4, the measurement error covariance matrix term, \(\mathbf{S_{e}}\) is effectively neutralized by the Jacobian leading to the _a priori_ term dominating to such an extent that if the measurement error term were considered to be zero the calculation of uncertainty would not be significantly changed, as seen in Figure 10B. In fact this scenario would be in line with the approach used by the C2 RCC [25] and SWARM [26] algorithms to attribute uncertainties to IOPs. Without the measurement term, the posterior error covariance matrix can in essence be simplified to a weighted average percent difference between estimated \(R_{\text{rs}}\) retrievals and training simulation data. With this situation in mind now consider our definition of the _a priori_ in Equation (14). We have defined it to be a combination of the retrieval parameters (\(R_{\text{rs}}\) estimates in this case) and average percent errors (APEs), as determined in Section 3.3. Equation (14) implies that the largest uncertainties should occur in bands where both the \(R_{\text{rs}}\) and the APEs are large. The APEs associated with each \(R_{\text{rs}}\) wavelength in Figure 3 are higher for blue bands than for red bands with green bands exhibiting the lowest APEs, meaning that even if the \(R_{\text{rs}}\) estimates were equal for blue and green bands we should expect larger uncertainty estimates in the blue bands. This finding is consistent with the findings reported by others [22; 47] based on an optimal estimation approach. Equation (14) also explains why we are seeing small uncertainties in red bands, because the \(R_{\text{rs}}\) estimates are almost always lower in those bands than in the blue and green bands. Figure 10 shows the \(R_{\text{rs}}\) and uncertainty estimates for the three locations identified in Figure 4. Figure 10A shows the calculated uncertainties estimates while Figure 10B shows the same uncertainty calculation if the measurement error term was set to zero. As stated above, setting the error measurement term to zero hardly makes an impact on the uncertainties. Figure 10C shows the uncertainty calculation if all the APE values were set to 10%, which would imply a poorer match-up of the trained MLNN to the synthetic data. On the other hand, lower APEs would have implied a better match-up and would lead to lower uncertainty estimates. This finding emphasizes the importance of a well-trained neural network. The _a priori_ plays a very significant role in determining the uncertainty estimate and is largely tied to the value of \(R_{\text{rs}}\) at each pixel. This situation leads to a nearly constant relative uncertainty which closely mirrors the APE value for each wavelength. Had the measurement error term played a larger role, we would have expected to see more variability in the relative uncertainty. Figure 10: \(R_{\text{rs}}\) uncertainties for the three locations identified in Figure 4 for several wavelengths. Panel (**A**) shows the actual calculated uncertainty. Panel (**B**) shows the same uncertainty calculation but with the measurement term considered to be zero for all values. Panel (**C**) shows the uncertainty calculation if the APEs of all wavelengths were set to 10%. When looking at the Gulf of Mexico in Figure 6 we can see a similar shape of the \(R_{\rm rs}\) estimate curves for ST4 and ST5, as was seen in Figure 4, for the ST3 location in the open ocean. The ST5 \(R_{\rm rs}\) curve, however, has an elevated green band profile comparable to, though not as dramatic, as that of ST1 in the Chesapeake Bay. The uncertainties for ST5 demonstrate the importance of the APE values. Even though we see higher \(R_{\rm rs}\) estimates in green bands than blue, the blue bands still have larger uncertainties because of their higher APEs. OC-SMART is based on simulated datasets for optically deep water and while it can be applied to coastal regions where shallow waters are present, the waters must be optically deep for OC-SMART to return accurate simulated radiances.