# Quantifying Uncertainties in OC-SMART Ocean Color Retrievals: A Bayesian Inversion Algorithm
## 3 Methodology for Quantifying Uncertainties
### Experimental Setup  
Figure 1 shows a flowchart of the processing chain described in Sections 3.1-3.3. For an example case, such as that described in Section 4.1, MODIS L1B radiances and sun-satellite geometry data are used as inputs to OC-SMART which will then produce L2 retrievals of \(R_{\text{rs}}\) used to form the _a priori_ matrix \(\mathbf{S_{a}}\), and Rayleigh-corrected radiances used to form the measurement error matrix \(\mathbf{S_{e}}\). These matrices, along with the Jacobian, are used to form the posterior error covariance matrix \(\mathbf{\hat{S}}(\mathbf{x})\) [see Equation (7)], and by taking the diagonal of this matrix we arrive at an estimate of corresponding \(R_{\text{rs}}\) uncertainties. This experimental setup can be viewed as a framework that can easily be adapted to other sensors, such as those listed in Section 2.4, by substituting those sensors TOA radiance and geometry data in place of MODIS data. Figure 4: **(Left panel)** Locations of interest off the east coast of the United States on 9 March 2016 (MODIS). ST1 (Red) is located in the optically complex Chesapeake bay region, ST2 (Blue) is located in near coastal waters, and ST3 (Black) is located in the open ocean. **(Right panel)**\(R_{\rm rs}\) with uncertainties for three locations of interest and for several wavelengths. Figure 5: Remote sensing reflectance data from MODIS sensor (**Left**), corresponding standard uncertainties (**Center**), and relative uncertainty (**Right**) for several wavelengths off the east coast of the United States on 9 March 2016. Relative uncertainty tends to be consistent for all pixels in a wavelength. Figure 6: **(Left panel)** Locations of interest in the Gulf of Mexico on 4 March 2020 (MODIS). ST4 (Red) is located in the central gulf region while ST5 (Blue) is located in the coastal waters of southwest Florida. (**Right panel**) \(R_{\rm rs}\) with uncertainties for two locations of interest for several wavelengths. Figure 7: Remote sensing reflectance data from MODIS sensor (**Left**), corresponding standard uncertainties (**Center**), and relative uncertainty (**Right**) for several wavelengths over the gulf of Mexico on 4 March 2020. Elevated values near Cuba and the Bahamas are due to shallow waters where reflection from the ocean bottom should be included (ignored in the current version of OC-SMART). Figure 4 shows an RGB image of the eastern coast of the United States along with three locations of interest. ST1 and ST2 are located in optically complex waters off the  
Figure 8: **(Left panel)** Locations of interest off the east coast of China on 19 March 2016 (MODIS). ST6 (red) is located between Korea and western Japan in the Sea of Japan, ST7 (blue) is located in the East China Sea near the Korean peninsula. **(Right panel)**\(R_{\rm rs}\) with uncertainties for two locations of interest for several wavelengths. Figure 9: Remote sensing reflectance data from MODIS sensor **(Left)**, corresponding standard uncertainties **(Center)**, and relative uncertainty **(Right)** for several wavelengths near the Korean peninsula and sea of Japan on 19 March 2016.  
coast, with ST1, in particular, being located in the Chesapeake Bay region where waters are extremely turbid. This turbidity is reflected by the elevated \(R_{\text{rs}}\) in the green bands for ST1. We would expect to see larger uncertainties in bands with weaker signals because noise is more prevalent. As waters are primarily blue, these larger uncertainties due to noise should be more prevalent in the green and red bands. However, the right panel of Figure 4 tells a different story. Here, we see that the largest uncertainties are more closely associated with larger \(R_{\text{rs}}\) estimates. To explain this behavior, let us first examine the terms of the posterior error covariance matrix Equation (see Equation (7)).