## 3 STAR Temperature and Water Vapor Profile Inversion Implementation
### The Averaging Kernels for COSMIC-2 Temperature and Water Vapor  
To quantify how much information of the STAR 1D-Var retrievals come from the measurements rather than the a priori, we show the magnitude of the COSMIC-2 temperature and moisture averaging kernels in the troposphere in this Section. As described in [44], the retrieved profiles are contributed from the relative weight from a priori (I \(-\) A) and the averaging kernel (A), which is defined as  
\[A=\left(K^{T}E^{-1}K+B^{-1}\right)^{-1}K^{T}E^{-1}K \tag{5}\]  
With the predefined \(B\) and \(E\), and the Jacobian for temperature (\(\frac{\partial N}{\partial T}\)) and water vapor (\(\frac{\partial N}{\partial P_{W}}\)), we can generate the averaging kernels for temperature (Figure 6) and water vapor (Figure 7) at different latitudinal zones at each month. The magnitude of \(A\) at the vertical level close to one indicates the observation mainly contributes to the retrieval result. In contrast, the magnitude of \(A\) close to zero indicates that information on the atmospheric state is primarily from the a priori.  
Figure 4: The same as Figure 3 but for background temperature standard deviation.  
The magnitude of temperature \(A\) is significantly affected by the seasonable variation of water vapor amount at different heights. The magnitude of \(A\) for temperature is generally more significant when the water vapor amount is low. Figure 6 depicts that the magnitude of \(A\) for temperature is higher than 0.5 above 10.5 km for the tropics with low seasonal variability. The magnitude of \(A\) for the temperature above 7 km altitude for the northern and southern mid-latitudes is higher than 0.5. Mid-latitude regions show higher seasonal variability of the RO sensitivity to temperature than the tropical region. The magnitude of \(A\) for temperature is also more prominent in colder seasons. For example, the magnitude of \(A\) for temperature is higher during the northern hemisphere winter but lower during winter.  
Figure 5: The error covariance model (in %) used in the optimal estimation: (**a**) northern mid-latitudes 60\({}^{\circ}\)N to 20\({}^{\circ}\)N, (**b**) tropics 20\({}^{\circ}\)N to 20\({}^{\circ}\)S, (**c**) southern mid-latitudes 20\({}^{\circ}\)S to 60\({}^{\circ}\)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), and red lines for northern summer (June, July, and August), and yellow lines (September, October, and November) are for northern autumn.  
Figure 6: RO sensitivity to the atmospheric temperature: (**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 monthly profiles, seasonally colored.  
As expected, the RO sensitivity to atmospheric moisture is more significant in the lower troposphere in the summer than in winter during drier seasons (Figure 7). The magnitude of \(A\) for water vapor is high near the surface over the summer season, especially in tropical regions. The magnitude of \(A\) for water vapor is lower during winter when the atmosphere becomes dryer. Over the tropical area below 8 km altitude, the magnitude of \(A\) for water vapor is close to 100%, which indicates that the refractivity measurement is very sensitive to water-vapor variation.