# Comparison of the Sentinel-3A and B SLSTR Tandem Phase Data Using Metrological Principles
## 2 Materials and Methods
### Overview of Methodology
#### 2.2.2 Difference Uncertainty  
The comparison of SLSTR-A and B then entails an analysis of the differences between their gridded products, \(\Delta\langle L\rangle_{\text{G}}\). The uncertainty of a grid cell difference is given by,  
\[u^{2}(\Delta\langle L\rangle_{\text{G}})=u^{2}(\langle L_{A}\rangle_{\text{G }})+u^{2}(\langle L_{B}\rangle_{\text{G}})-2\ u(\langle L_{A}\rangle_{\text{G }},\langle L_{B}\rangle_{\text{G}})+u_{\text{comp.}}^{2}\, \tag{5}\]  
where \(u_{\text{comp. }}\) is the comparison uncertainty and \(u(\langle L_{A}\rangle_{\text{G}},\langle L_{B}\rangle_{\text{G}})\) is the covariance between the SLSTR-A and B gridded values. The comparison uncertainty, \(u_{\text{comp. }}\), is caused by match-up errors that occur on a pixel level. Since these are random from pixel to pixel, the gridding process has the effect of reducing this uncertainty as in Equation (3). Sensor-to-sensor covariance may be caused by, for example, common calibration errors due to both sensors being calibrated at the same laboratory. Since we are taking the difference of SLSTR-A and B values, a non-zero covariance has the effect of reducing the uncertainty associated with the difference, as errors common to them both should cancel out. No quantitative information, at the time of writing, is available on the covariance between SLSTR-A/B data at L1 or L2, so it is omitted from the uncertainty model, which is a limitation of the following results. In the case of the L1 analysis, this covariance is thought to be relatively small, but, for the SST analysis, the common components are expected to be significant, as the uncertainty is dominated by retrieval effects that are common to both sensors. Uncertainty Propagation for L1  
As previously stated, one of the reasons for choosing the L1 grid size was to minimise the impact of random noise and match-up errors. A filtering is then performed to select homogeneous grid cells to further reduce match-up errors. Figure 2 shows the absolute difference between SLSTR-A and B grid cells (S8, nadir view) as a function of the standard deviation of SLSTR-A grid cell pixel values, \(\text{std}(\{L_{i}\})\), for an example orbit of data. The filtering selects pixels where \(\text{std}(\{L_{i}\})<2\,\text{K}\), this is indicated by the black line on the plot. This filtered population of pixels has a standard deviation of SLSTR-A and B differences of \(\sim\)0.2 K. For SLSTR, typical instrument noise is of the order 30 mK, so random uncertainty in the grid cells differences are therefore driven by the match-up errors. Since a grid cell of 0.5\({}^{\circ}\) contains roughly 50 pixels \(\times\) 50 pixels, random errors are reduced by a factor of 50 compared to their per pixel values (Equation (3)), compared to the common component of uncertainty \(u_{\text{com. }}^{2}(\langle L\rangle_{\text{G}})\) which is of order of 0.05 K and does not reduce with averaging, this is small. Thus, we have,  
\[u^{2}(\Delta\langle L\rangle_{\text{G}})=\langle u_{\text{cal.}}^{2}(L_{\text {A},i})\rangle_{\text{G}}^{2}+\langle u_{\text{cal.}}^{2}(L_{\text{B},i}) \rangle_{\text{G}}^{2}\, \tag{6}\]  
again noting covariance is omitted. Therefore, a comparison of the gridded products should directly allow the validation of the SLSTR L1 calibration uncertainties. Not treated here are error effects that are locally correlated, which is not possible as these effects are not described in the available SLSTR uncertainty model [8]. This approximation is assumed reasonable since these locally correlated effects are small (\(\sim\)1 mK), so handling them separately would not change the evaluated grid cell uncertainty. Uncertainty Propagation for L2  
Since for the finer SST comparison grid resolution, we generally have \(N_{\text{G}}=1\), the overall grid cell uncertainty is simply the value of the total pixel-level uncertainty,  
\[u^{2}(\Delta\langle SST\rangle_{\text{G}})=u^{2}(SST_{\text{A},i})+u^{2}(SST _{\text{B},i})\, \tag{7}\]  
again noting covariance is omitted.