## 3 STAR Temperature and Water Vapor Profile Inversion Implementation
### Pre-Defining the Background Covariance Matrix and Error Covariance Matrix  
To solve Equation (2) iteratively, we must first define the background covariance matrix (\(B\)) and the error covariance matrix (\(E\)). We use NOAA's GFS 6 h forecast in 2018 to estimate the background covariance matrix. To take the seasonal and latitude variability of the Earth's atmosphere into account, we evaluated \(B\) and \(E\) for three latitude zones (North mid-latitudes: 45\({}^{\circ}\)N to 20\({}^{\circ}\)N, tropics: 20\({}^{\circ}\)N to 20\({}^{\circ}\)S, and South mid-latitudes: 20\({}^{\circ}\)S to 45\({}^{\circ}\)S) for each month of the year.  
We have 36 (3 \(\times\) 12) estimated background states using monthly mean profiles and standard deviation for atmospheric temperature and water vapor. The diagonal elements of the covariance matrix \(B\) are equal to the square of the corresponding standard deviation, while off-diagonal elements are set to zero. Averaging has been performed each month and in the three latitude zones separately. Figures 3 and 4 present monthly standard deviation profiles for specific humidity and temperature that have been used to define matrix \(B\), respectively. Figures 3 and 4 depict latitudinal and seasonal dependence of atmospheric state, with the highest seasonal variability in mid-latitudes and the lowest in the tropics.  
Figure 3: Background standard deviation of water vapor used in the optimal estimation: (**a**) northern mid-latitudes 45\({}^{\circ}\)N to 20\({}^{\circ}\)N, (**b**) tropics 20\({}^{\circ}\)N to 20\({}^{\circ}\)S, (**c**) southern mid-latitudes 20\({}^{\circ}\)S to 45\({}^{\circ}\)S. Each panel presents 12 months where blue lines (December, January, and February) are for northern winter, green lines are for northern spring (March, April, and May), red lines are for northern summer (June, July, and August), and yellow lines (September, October, and November) are for northern autumn.  
In this study, we define the error covariance matrix to stabilize the matrix inversion (see Equation (2)). The same year-long set of GFS background states was used to derive a corresponding set of simulated refractivity N, using RO forward model. The estimated error covariance was defined as a matrix with off-diagonal elements equal to zero and diagonal elements proportional to the variance of \(N\) obtained for each month and for each latitude zone where  
\[E_{ij}=\{\gamma_{Noise}\times\sigma(N_{i})\}^{2}\,i=j\]  
\[E_{ij}=0\,\ i\
eq j \tag{4}\]  
The \(\gamma_{Noise}\) is a scaling factor where small \(\gamma_{Noise}\) leads to the instability of inversion and high \(\gamma_{Noise}\) results in losing information obtained from measurements, and retrievals become close to the first guess. We defined \(\gamma_{Noise}\) as 0.1 to stabilize the matrix inversion and optimally use refractivity information in the retrieval. This ensures the retrieval results fit more to the observed refractivity than the a priori (see the averaging kernels in Section 3.3). Note that the error covariances described here do not reflect the combined forward model and receiver errors.  
Figure 5 presents the error covariance model we defined. The estimated monthly errors are less than 1% of the expected observed refractivity in all latitude zones.