# Atmospheric GNSS RO 1D-Var in Use at UCAR: Description and Validation
## 3 Results  
### 1D-Var Response to Single-Level Perturbations of the Observation  
The analysis vector \(\textbf{x}_{a}\), which minimizes the cost function in the form of Equation (1), is given by the solution of \(\
abla_{\textbf{x}}\,J=0\)[57]:  
\[\textbf{x}_{a}=\textbf{x}_{b}+\textbf{B}\textbf{H}^{\text{T}}\Big{(}\textbf{ R}+\textbf{H}\textbf{B}\textbf{H}^{\text{T}}\Big{)}^{-1}\{\textbf{y}_{o}-H( \textbf{x}_{b})\}. \tag{27}\]  
Interpretation of Equation (27) is facilitated by considering a special case in which \(\textbf{x}_{b}\) and \(\textbf{y}_{o}\) each have only one element, \(x_{b}\) and \(y_{o}\), and \(y_{o}\) is a direct observation of \(x\) and collocated with \(x_{b}\). In that case, the solution \(x_{a}\) is:  
\[x_{a}=x_{b}+\frac{\sigma_{b}^{2}}{\sigma_{b}^{2}+\sigma_{o}^{2}}(y_{o}-x_{b})= \frac{\sigma_{b}^{2}y_{o}+\sigma_{o}^{2}x_{b}}{\sigma_{b}^{2}+\sigma_{o}^{2}}, \tag{28}\]  
where \(\sigma_{o}^{2}\) and \(\sigma_{b}^{2}\) are observation error variance and the background error variance, respectively. Equation (28 angle profile (ID: C004.2013.083.10.42.G12) observed at 58.3\({}^{\circ}\)N and 169.5\({}^{\circ}\)W at 10:40 UT on 24 March 2013. The red horizontal lines, marked b-d, indicate perturbations added to the observation, exclusive to each other. The size and location of these perturbations are 3, 4, and 6 milliradians at 8, 5, and 3 km of the impact height, respectively. Figure 4b-d show the response of 1D-Var (i.e., change in the solution) to these perturbations. The 1D-Var attempts to fit the perturbed observation by altering the vertical refractivity gradient at the immediate height of the perturbation. Because the perturbations are of positive sign, the refractivity must increase below the height of the perturbation, or vice versa. In this case, the 1D-Var tends to increase lower-level refractivity, compelled by the Abel transform that propagates the observation information downward. As can be inferred from Equation (2), an increase in refractivity accompanies a pressure rise, moistening, or cooling. In relative humidity, the 1D-Var analysis is wetter than the background by up to 16-24%, depending on the location of perturbed observation. The maximum cooling is 0.3-0.6 \({}^{\circ}\)C. It is noteworthy that a single-level observation perturbation influences broad areas through the background error correlation, e.g., the temperature and moisture responses appear in deep layers that are thicker than 1.5 km. In comparison, a diagonal **B** leads to pointwise responses, i.e., the temperature and moisture changes occur only at the immediate height that the observation is perturbated. The hydrostatic pressure response is far broader. Notably, observation perturbations induce sizable changes to lower-level pressures. For instance, the observation perturbed at 8 km causes 0.9 hPa pressure change at the lowest level. RO observations have numerous data points (samples) that affect the 1D-Var solution differently. In this circumstance, 1D-Var does not determine its solution at a given height based solely on the local observation, but through consensus among many data samples across a considerable height range. The across-sample consensus makes the 1D-Var robust against outlying data samples. Meanwhile, a low-level bending angle contains information about the upper atmosphere. Therefore, it is difficult for a 1D-Var to closely fit a low-level bending angle unless it can accurately recover the upper atmosphere. Hence, 1D-Var owes its effectiveness partly to the global observation fitting and the across-sample consensus. The broad and complicated response of the 1D-Var to a perturbed data sample underscores the importance of accurate error covariances. The 1D-Var likely facilitates an effective correction to the background using a complete observation profile through the consensus among overlapping responses to individual data samples. Figure 4: 1D-Var response to a bending angle profile perturbated at single levels.