# 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 observation types admissible to the 1D-Var are neutral atmospheric bending angle and refractivity. The 1D-Var obtains these observations from a lower-level data product of CDAAC (atmPrf). [PERSON] et al. [66] and [PERSON] et al. [67] describe the data processing leading to the atmPrf. The observation operator involves the forward observation modeling, which comprises the conversion of the state variables into the observed physical quantity and the vertical interpolation to the data points of the observation. Considering that the interpolation is trivial, we focus on the physical conversion. In addition, the refractivity represents a local state of the atmosphere which can be computed with Equation (2). The bending angle can be modelled using the forward Abel transform, the inverse of Equation (1):  
\[\alpha(a)=-2a\int_{a}^{\infty}\frac{d\ln(n)}{dx}\frac{dx}{\sqrt{x^{2}-a^{2}}}. \tag{10}\]  
The forward Abel transform provides the bending angle profile corresponding to a refractivity profile given as a function of the height. Unfortunately, there are a few practical difficulties to overcome: a singularity at the lower limit, an infinite upper bound, and discretization error. One can avoid the singularity at \(x=a\) by using quadrature rules that do not require the integrand evaluation at the endpoint. The default discrete Abel transform used in the 1D-Var, amongst many options available, is a form based on the relation \(\int\frac{dx}{\left(x^{2}-a^{2}\right)^{1/2}}=ln\left\{x+\left(x^{2}-a^{2} \right)^{1/2}\right\}\)[16]:  
\[\alpha(a=a_{j})=-2a_{j}\sum_{i=1}^{j-1}\frac{ln\left(\frac{n_{i+1}}{n_{i}} \right)}{x_{i+1}-x_{i}}\ ln\left\{\frac{x_{i+1}+\left  
occaional overshooting when adjacent levels come too close in the refractional radius, i.e., \(x_{i}-x_{i+1}\to 0\). The overshooting occurs when the refractivity gradient approaches the critical refraction threshold, \(-157\) km\({}^{-1}\), regardless of the layer's physical (altitudinal) depth, \(z_{i}-z_{i+1}\). In addition, the refractivity gradient becomes more difficult to evaluate accurately when \(x_{i}-x_{i+1}\to 0\). Moreover, the computed refractivity gradient (and the physical interpretation) becomes dubious when the refractional radius is inverted, i.e., \(x_{i+1}>x_{i}\). In the case of a super-refraction, the modeled bending angle is not a monotonic function of the height. To circumvent these difficulties, we check the monotonicity of the refractional radius before evaluating the forward Abel transform. To do so, we examine the refractivity gradient downwards, starting from 5 km. When the refractivity gradient negatively exceeds \(-150\) km\({}^{-1}\) for the first time, super-refraction is assumed to occur within the layer. In this circumstance, we truncate a priori profiles at the layer's top and exclude the lower part from being used for the discrete forward Abel transform. Using a finite upper bound for the infinite integral can cause significant modeling error at high altitudes. For instance, when Equation (10) is evaluated with NWP data and the integrand above the model top is disregarded, the computed bending angle is to be smaller than the observation. The modeling error can be reduced by raising the upper bound. To do so, however, the atmospheric structure above the model top must be specified by some means. A simple approach that can serve the purpose is employing the assumption that the refractivity above the model top continues to decrease exponentially, which is widely exercised in data assimilation where no information is available about the atmosphere outside of the model domain. However, the assumption of exponentially varying refractivity is equivalent to assuming that the temperature above the model top remains constant, which hardly ever occurs in the real atmosphere. For instance, the temperature in the thermosphere often exceeds 1000 \({}^{\circ}\)C, far greater than the typical stratospheric temperature.