## 5 Estimates of the Error Covariance Matrix for NWP Data Assimilation
A bending angle forward model developed by the STAR RO DSC (see [33]) was used to calculate a corresponding model background forecast bending angle for each RO observation by (i) interpolating the GFS 6 h forecast pressure, temperature, and water vapor fields to a grid column at the RO observation's latitude/longitude location and time; (ii) computing the forecast BA/refractivity from these fields; and then (iii) integrating the forecast BA/refractivity field's vertical gradient from the model top down to the observation impact height. The observation errors are estimated from the observation-minus-background sample variance (\(\sigma_{a}\)) and the model error variance estimate (\(\sigma_{b}\)) using Equation (1). Figure 19 shows the estimates of the four RO missions' bending angle observation errors (normalized by the mean BA profile, aka the relativeBA error (%)) based on one month of observations and the corresponding GFS forecasts from 45\({}^{\circ}\)N to 45\({}^{\circ}\)S over oceans (Figure 19a), land (Figure 19b), and both oceans and land (Figure 19c). The observation errors are less than 4% above 9 km and below 36 km. We also depict the relative BA errors over oceans for the mid-latitude southern hemisphere (from 20\({}^{\circ}\)S to 45\({}^{\circ}\)S, Figure 19d), the tropical region (from 20\({}^{\circ}\)N to 20\({}^{\circ}\)S, Figure 19e), and the mid-latitude northern hemisphere (from 45\({}^{\circ}\)N to 20\({}^{\circ}\)N, Figure 19f). Below 9 km, the observation errors increase with a decreasing height and peak at around 2 km. For all four missions, the lower troposphere peak value over oceans (~15%) is larger than that over land (~12%). These significant errors in the lower troposphere may result from a greater RO retrieval uncertainty relative to the middle and upper troposphere, due to multipath propagation and a smaller SNR over the mid-latitudes for COSMIC-2. Larger observation errors also exist in the upper stratosphere above 36 km. This likely results from larger GFS forecast field errors at these altitudes and RO retrieval uncertainty (i.e., residual ionospheric errors). The over-land observation errors (Figure 19b) are generally smaller than those over oceans. Owing to the significantly larger number of RO observations over oceans than over land, the full-sample observation errors (Figure 19c)  
Figure 19: Fractional BA observation errors (in %) estimated for the region within [45\({}^{\circ}\)N, 45\({}^{\circ}\)S] for (**a**) over oceans, (**b**) over land, and (**c**) over oceans and land, (**d**) over oceans [20\({}^{\circ}\)S, 45\({}^{\circ}\)S], (**e**) over oceans [20\({}^{\circ}\)N, 20\({}^{\circ}\)S], and (**f**) over oceans [45\({}^{\circ}\)N, 20\({}^{\circ}\)N]. One month (from 15 December 2020 to 15 January 2021) of COSMIC-2, KOMPSAT-5, GeoOptics, and Spire bending angle observations were used for generating these figures. are similar to the over-ocean observation errors shown in Figure 19a. We can implement the defined COSMIC-2, PAZ, Metop-C, KOMPSAT-5, and Sentinel-6 RO observation errors and updated forward operator (including QC settings) generated based on these results into STAR's offline RO data assessment system. The NCEP can also use these results for the global NWP through DA. The relative BA error over oceans is larger for the tropical region and mid-latitude summer (in the southern hemisphere) than that in the mid-latitude winter region (in the northern hemisphere) (see Figure 19d-f).