# Atmospheric GNSS RO 1D-Var in Use at UCAR: Description and Validation
## 3 Results
### Tests Using Synthetic Data  
Verification of 1D-Var with real-world data is challenging. One reason is that the required input error covariances are never perfect. Another is that it is hard to acquire verification data that are accurate, of sufficient amount, and independent of and collocated with 1D-Var retrievals, and even if such data were available, they would still not represent \"truth.\" One way to overcome this difficulty is designing and using a controlled system in which the atmosphere, the observation, and a priori data conform perfectly to our current understanding. For this purpose, we generated synthetic observation and a priori profiles using a high vertical resolution radiosonde profile of the Global Climate Observing System (GCOS) Reference Upper-Air Network (GRUAN) [77]. The radiosonde observation took place at the Lamont site (36.61\({}^{\circ}\)S, 97.49\({}^{\circ}\)W) at 08:30 UT on 4 April 2012 (ID: SGP-RS-01_2_RS92-GDP_002_20120904T083000). We used ECMWF analysis and MSIS to specify the atmospheric state above the balloon-burst point. We applied a lowpass filtering to the radiosonde temperature, pressure, and moisture profiles to reduce the radiosonde instrument noise. These smoothed profiles represent the true atmosphere from which a perfect observation is derived. We simulated the perfect observation (red line in Figure 5a) by interpolating the true refractivity to the data levels of a nearby COSMIC profile (ID: C002.2012.248.07.24.G21). The interpolation furnishes a realistic observation density in the vertical direction. Next, we generated \"observed\" refractivity profiles by adding zero-mean, random Gaussian noise to the perfect observation at each data level. The noise equals the observation error in magnitude. By repeating this step with different profile sets of random noise, we obtained 100 realizations of the observation (represented by black dots in Figure 5a). Aside from the observation, we generated a priori profiles consistent with the diagnostic of the background error covariance described in Section 2.2.4. First, we interpolated the true temperature, pressure, and moisture profiles (red lines in Figure 5b) to ECMWF model levels. These model-level profiles are perfect, but only within the model's vertical resolution limit. In other words, the \"perfect\" a priori profiles cannot describe subgrid-scale  
Figure 5: Synthetic 1D-Var input data for a subtropical case generated based on a GRUAN radiosonde observation and a collocated COSMIC profile. (**a**) The true atmospheric refractivity (red line) and assumed realizations of the observation (black dots). (**b**) Perfect, error-free a priori temperature and specific humidity (inset) profiles (red lines) and corresponding realizations (black dots). For visualization purposes only 50 realizations are shown. See text for details. fluctuations in the radiosonde-based \"true\" atmosphere. We created a vector of a priori error in the control variables' space as follows:  
\[\mathbf{\epsilon}=\sum_{i=1}^{M}\varepsilon_{i}\,\Lambda_{i}^{1/2}\,\mathbf{\Sigma}_{i}. \tag{29}\]  
where \(\mathbf{\epsilon}\) is the error vector of size \(M\), \(\varepsilon_{i}\) is the random, Gaussian noise associated with \(i\)-th eigenmode of \(\mathbf{B_{v}}\) whose eigenvalue and eigenvector are \(\Lambda_{i}\) and \(\mathbf{\Sigma}_{i}\), respectively. What followed next was a scaling to satisfy \(\sum_{i=1}^{M}\mathbf{e}_{i}^{2}=\sum_{i=1}^{M}\Lambda_{i}\). We applied linearized physical transforms (hydrostatic integration preceded by conversion to specific humidity) to \(\mathbf{\epsilon}\), which yields a priori error profiles for temperature, pressure, and specific humidity. Finally, we attained model-level a priori profiles by combining these error profiles with the \"perfect\" a priori profiles. In doing so, we constrained the specific humidity to be positive and subsaturated. We repeated this procedure to produce 100 sets of a priori profiles (represented by black dots in Figure 5b).