# Atmospheric GNSS RO 1D-Var in Use at UCAR: Description and Validation
## 2 Materials and Methods
### The UCAR 1D-Var
#### 2.2.3 A Priori Profiles  
At the outset, we provide the 1D-Var with a priori temperature, moisture, and pressure profiles. Typically, we extract these profiles from a trace of NWP model data at the observed time and along the trajectory of rays' tangent points. Because the tangent points drift horizontally with the height, this extraction requires an iterative procedure to accurately determine the intersections between the tangent points' trajectory and the horizontal surfaces of individual model levels. The profile extraction is carried out prior to the 1D-Var and is thus not a part of the observation operator. The 1D-Var prepares the background (profiles) by interpolating a priori profiles vertically to a computational grid on which the state vector's elements are located. The computational grid is defined in a generalized coordinate \(\eta\) as:  
\[\eta=\frac{z-z_{b}}{z_{t}-z_{b}}. \tag{12}\]  
where \(z_{b}\) and \(z_{t}\) indicate the grid's bottom and top, respectively. We set the top height to 150 km for the refractivity inversion, and the non-linear observation operator of this step uses MSIS internally to model the refractivity above the top height. The top height is set to 80 km for the retrieval step. The bottom of the computational grid varies from one RO event to another. It is set to the highest of the following: (1) the lowest tangent point of the observation, (2) the terrain height at the geophysical location, and (3) the top of the highest a priori super-refraction layer. In the rare case that \(z_{b}\) is lower than the bottom of a priori profiles, we use a simple extrapolation: the specific humidity is held constant down to \(z_{b}\) (assuming a well-mixed boundary layer),a standard lapse rate (6.5 \({}^{\circ}\)C km\({}^{-1}\)) is assumed for the temperature, and the pressure is obtained by integrating the hydrostatic equation downward. The computational grid is a stretched one, meaning that we configure \(\eta\) values in the way that the layer depth increases gradually with increasing height. The stretched grid is similar to the vertical grids of NWP models, which tend to widen at stratospheric heights. If a regular, nonstretched grid were used, the upper part of the computational grid would hold highly redundant information of the a priori profiles. In that case, the a priori profiles would overcontribute to the cost function, preventing the 1D-Var solution from being constrained adequately by the observation. We chose the number of \(\eta\) levels considering the vertical resolution of the observation. The diffraction-limited resolution, which applies to the bending angles obtained by the geometrical optics method, is known to be 100-1500 m [1,2]. However, when wave optics methods are employed, the resolution can be enhanced by a factor of 10 [70] or higher to a few tens of meters [71]. To this end, the 1D-Var uses 860 \(\eta\) levels on which the layer depth increases progressively from 30 m at the bottom (when \(z_{b}\) equals the mean sea level) to 3 km at the top for the refractivity inversion step. Another reason for using such a dense grid, roughly an order of magnitude denser than today's NWP model grids, is that it reduces the discretization error of the Abel transform. While interpolations to different grids result in different errors, the interpolation error is seldom sensitive to the output (computational) grid for smooth input data sets such as NWP profiles. Moreover, univariate data interpolation is a well-defined problem for which many well-tested methods exist. The point is that the interpolation error is generally much smaller than the discretization error of the Abel transform [38]. As a result, using a fine grid improves the accuracy of the discrete Abel transform. For the non-linear observation operator, the 1D-Var uses a monotone piecewise cubic interpolation, which prevents overshooting by preserving the monotonicity of the input data set. The interpolation assumes that the temperature varies linearly with height, while the pressure and the moisture vary log-linearly.