## 5 Estimates of the Error Covariance Matrix for NWP Data Assimilation  
The observation error matrix is an important component of any NWP DA system because it modulates the relative weighting assigned to individual observations and the background state when generating the analysis solution, regardless of whether the algorithm is variational, an ensemble Kalman filter (EnKF), or hybrid-flavored. RO profiles from different receiver satellite platforms may have systematic differences in their observation uncertainty, which must be specified in the RO DA system. Based on the calibration and validation results, we can provide RO data quality information to NWP centers and collaborate to optimize RO data assimilation. We also use NWP diagnostic tools to analyze the RO data usage from our forecast experiment datasets and provide feedback to NCEP for assessing the RO DA impacts on the current NCEP data assimilation system. Accurate estimates of RO observation errors and their dependence on the platform, observation location, and meteorological fields are needed for the optimal usage of the RO data assimilated into a NWP model. In the operational NCEP Global Forecast System (GFS) model's Gridpoint Statistical Interpolation (GSI) DA system, BA, instead of refractivity data, are assimilated [52]. The RO observation error sources associated with signal measurement and processing into bending angle profiles can be grouped into measurement and calibration errors [53]. These measurement errors include the SNR performance, open-loop tracking error, clock instability, and local multipath propagation; calibration errors include the residual ionospheric effects, orbit determination accuracy, and clock error removal. In an NWP framework, \"representativeness errors\" arising from uncertainties in simulating observations from the background forecast field (i.e., forward operator errors) provide an additional observation error source. The perturbation method can theoretically predict the observation error for individual error sources and then assemble it to establish an overall accuracy estimate. The differences (aka the apparent or perceived error) between the observations and their corresponding values estimated from the background forecast contain contributions from observation and model forecast errors. One method for estimating observation error variance uses observation innovation (i.e., observation-minus-background) and short-term forecast error statistics, as [53] described. Equation (1) shows that, under the assumption that observation errors are uncorrelated with forecast errors, the apparent error variance (\(\sigma_{a}^{2}\)) can be divided into model forecast error variance (\(\sigma_{b}^{2}\)) and observation error variance (\(\sigma_{o}^{2}\)) components:  
\[\sigma_{a}^{2}=\sigma_{b}^{2}+\sigma_{o}^{2} \tag{1}\]  
Therefore, if the model forecast error can be reasonably well estimated, one can estimate the observation error by subtracting the model forecast error variance from the apparent error variance. Our work calculates the model forecast error using the National Meteorological Center (NMC) method [53; 54], which approximates \(\sigma_{b}^{2}\) using the differences between sets of 12 h and 24 h GFS forecasts, verifying simultaneously. The apparent error is calculated using the difference between the RO observations and 6 h GFS forecasts. A recent STAR RO DSC study estimated the bending angle observation error for four RO missions, COSMIC-2, KOMPSAT-5, GeoOptics, and Spire, from 15 December 2022 to 15 January 2023. Note that, because the Spire quality does not change at different Delivery Orders, the estimated errors from DO3 are very similar to those from other Delivery Orders, and are not repeated here.