## 3 STAR Temperature and Water Vapor Profile Inversion Implementation  
### STAR 1D-Var Approaches  
Different from UCAR's 1D-Var algorithm [40], we developed an independent 1D-Var inversion algorithm in STAR to convert refractivity profiles into temperature and water vapor profiles. In a neutral atmosphere, the refractivity (\(N\)) profile is a function of pressure (\(P\)), temperature (\(T\)), and the partial pressure of water vapor (\(P_{W}\)) [2]:  
\[N=77.6\frac{P}{T}+3.73\times 10^{5}\frac{P_{W}}{T^{2}} \tag{1}\]  
The refractivity information is contributed by temperature and moisture in the troposphere, where moisture is the dominant contributor [1; 2; 3; 4]. In the stratosphere and upper troposphere, where the moisture is negligible, refractivity information mainly comes from the temperature. Here we use the dry hydrostatic equation and Equation (1) (with the second term removed) to solve the dry temperature and dry pressure (two equations to solve for two unknowns). The derived dry temperature shall be very close to the actual temperature.  
In the troposphere, where the water vapor amount is not negligible, the inversion of temperature and water vapor from one variable (N) is an ill-posed problem. Here we use the maximum likelihood method introduced by [44] to retrieve temperature and water vapor from refractivity at specific altitude levels. For a given RO refractivity value (the observation) at a particular height z for iteration \(i\) (i.e., \(Y_{OBS}=N(z)\)), the optimal estimation of retrieval vector \(X=(T(z_{i})\), \(P_{W}(z_{i}))\) is determined from the maximum likelihood inversion [44]:  
\[X_{i+1}=X_{0}+\left(K_{i}^{T}E^{-1}K_{i}+B^{-1}\right)^{-1}\times K_{i}^{T}E^{ -1}\{(Y_{OBS}-Y(X_{i}))+K_{i}(X_{i}-X_{0})\} \tag{2}\]  
where \(X_{0}=\left(T_{0},P_{W_{0}}\right)\) is the a priori profile for temperature (\(T_{0}\)) and water vapor pressure (\(P_{W_{0}}\)), \(K=\left(\frac{\partial N}{\partial T},\frac{\partial N}{\partial P_{W}}\right)\) is a Jacobian vector, \(B\) is the background state covariance matrix, and \(E\) represents the combined instrument noise and the forward model-error covariance matrix. The index \(i\) corresponds to the iteration number. The iteration procedures continue until the residual difference between the observed input refractivity and forward simulatedrefractivity (computed from Equation (1) using the retrieved temperature and water vapor profile as inputs) from the retrieved state satisfies the condition  
\[\left|N_{OBS}-N_{i}\right|/N_{OBS}\leq 0.001\ (i.e.,\ 0.1\ \%) \tag{3}\]  
We use the NOAA global forecast system data (GFS) 6 h forecast interpolated onto the RO locations and time as the a priori atmospheric state (vector \(X_{0}\) in Equation (2)) to initialize the optimal estimation retrievals. The four-dimensional (space and time) global fields of the atmospheric temperature and water vapor have been downloaded from the NOAA National Centers for Environment Information GFS website [[https://www.ncdc.noaa.gov/data-access/model-data/model-datasets/global-forcast-system-gfs](https://www.ncdc.noaa.gov/data-access/model-data/model-datasets/global-forcast-system-gfs)]([https://www.ncdc.noaa.gov/data-access/model-data/model-datasets/global-forcast-system-gfs](https://www.ncdc.noaa.gov/data-access/model-data/model-datasets/global-forcast-system-gfs)) (accessed on 26 September 2022).