# Atmospheric GNSS RO 1D-Var in Use at UCAR: Description and Validation
## 2 Materials and Methods
### The UCAR 1D-Var
#### 2.2.2 Observation Operators
The reason is that as the density becomes less and less with increasing height, the atmosphere responds more strongly to solar insolation. For this reason, we specify the temperature above the model top using the empirical model of the US Naval Research Laboratory (NRL), MSIS [68]. The rationale behind this approach is that the temperature variability in the mesosphere and thermosphere can be explained to a good degree by atmospheric tides, seasonal insolation change, and the solar cycle, for which MSIS can serve as an acceptable first-order model. The air pressure is hydrostatically reconstructed from the MSIS temperature in the bottom-to-top direction under the assumption of a constant air composition, as done by [69]. This applies only to non-linear forward modeling. The accuracy of the discrete Abel transform is dependent on the step size of the vertical integration. Because NWP data are available on finite model levels, Equation (11) can result in a considerable numerical error when evaluated only with the data points on those model levels. We reduce the discretization error by integrating the discrete Abel transform on a higher-resolution grid. Besides the difficulties discussed so far, there is another issue to consider. Typically, the bending angle observation operator includes the refractivity observation operator--using temperature, pressure, and humidity the observation operator calculates the refractivity from which the bending angle is evaluated. The observed bending angle is provided as a function of the impact parameter, while all other variables are functions of the height. The refractional radius (=nr) is the model equivalent of the impact parameter at the ray's tangent point. The problem is that the simulated refractional radius, let alone the modeled bending angle, contains a model error due to imperfect temperature, pressure, and humidity. In this so-called errors-in-variables problem, the error of the independent variable (i.e., refractional radius) emerges as the error of the dependent variable (bending angle). This extra forward modeling error leads to suboptimal use of bending angle observation [38]. To address this issue, we undertake two separate, sequential minimizations. The first is carried out in the observation space (impact parameter coordinate) and the other in the model space (height coordinate). The first step replaces the Abel inversion used in the standard RO data processing, Equation (1), with a variational inversion of Equation (10). The primary concern regarding Equation (1) is that the standard Abel inversion accumulates observation errors at different altitudes and propagates them downward. For instance, an erroneous spike in observed bending angle severely deteriorates the retrieved refractivity in lower altitudes, regardless of the local quality of the bending angle observation there. Furthermore, the standard Abel inversion builds up systematic observation error, e.g., negative observation bias in the lower troposphere. The variational inversion aims to overcome these difficulties by combining observation error mitigation and refractivity recovery into an estimation problem. It provides the best likelihood refractivity--in light of observed bending angle, a priori refractivity, and their error covariances--as a function of the refractional radius. In simplest terms, the best refractivity profile is the one that fits the observed bending angle profile the most closely. As can be inferred from Equation (10), no information propagates back to temperature, pressure, and water vapor pressure at this stage. This variational inversion offers many advantages over the Abel inversion [38]. The second step is a regular 1D-Var procedure which takes the optimal refractivity acquired from the previous step as observation to give temperature, moisture, and pressure profiles. Here, the prior step's analysis (retrieval) error serves as the observation error, and the observation's height is determined by dividing the refractional radius by the optimal refractive index.