# Atmospheric GNSS RO 1D-Var in Use at UCAR: Description and Validation
## 2 Materials and Methods
### The UCAR 1D-Var
#### 2.2.4 Error Covariance Matrices
We assume that the scaling factor remains constant in the horizontal direction and does not depend on the state variables. Under these assumptions, we determine the scaling factor by comparing refractivity error estimates. We use the resulting scaling factor to adjust \(\sigma_{f}^{NMC}\) in temperature, humidity, and pressure. In the stratosphere, the scaling factor showed a gradual increase with height, which is consistent with the fact that the NMC method underestimates thebackground error in observation-sparse areas. We use a second-order polynomial fitting to approximate the global-mean height dependence. Figure 1 shows the annual- and zonal-mean \(\sigma_{b}\) in temperature and moisture. The temperature \(\sigma_{b}\) is prominent around the tropical tropopause (\(\sim\)17 km), shows a sudden break over the subtropics, and gradually descends poleward to become \(\sim\)10 km at the poles. The tropical stratosphere shows a significant temperature \(\sigma_{b}\), which is about 2 \({}^{\circ}\)C in the height range of 20-30 km and does not change much with latitude and height. The large stratospheric temperature \(\sigma_{b}\) in the tropics might be due to vertically propagating equatorial waves, which the forecast model cannot fully resolve. On the other hand, in the upper stratosphere and mesosphere, temperature \(\sigma_{b}\) is more prominent over polar regions, which might relate to the strong thermal gradients associated with polar vortices and sudden stratospheric warming events. Specific humidity \(\sigma_{b}\) (Figure 1b) peaks at about 2 km, corresponding to the average top height of cloud-topped marine boundary layers. The specific humidity \(\sigma_{b}\) decreases rapidly with increasing height and becomes insignificant above 10 km. Relative humidity \(\sigma_{b}\) (Figure 1c) shows an entirely different pattern. The moisture \(\sigma_{b}\) in relative humidity is no longer significant in the lower troposphere. Instead, the relative humidity \(\sigma_{b}\) is most prominent in the upper troposphere, a few kilometers below the tropopause. The difference between specific humidity and relative humidity in the background error presents the difficulty in prescribing the moisture \(\sigma_{b}\). In the troposphere, relative humidity \(\sigma_{b}\) is less variable than specific humidity \(\sigma_{b}\), which varies by a few orders of magnitude. Hence, the use of relative humidity \(\sigma_{b}\) eases the error specification, which is one of the reasons that the 1D-Var uses the (pseudo) relative humidity as the moisture control variable. Figure 2 shows the horizontal structure of the annual-mean background error. In the Northern Hemisphere, the temperature \(\sigma_{b}\) at the height of 5 km and the surface pressure \(\sigma_{b}\) are larger over the storm track regions in the Pacific and Atlantic Oceans. Over the Southern Ocean, they are larger along the circumpolar storm track. The specific humidity \(\sigma_{b}\) at 2 km is greater in warm latitudes and largest in the intertropical convergence zone (ITCZ). In contrast, relative humidity \(\sigma_{b}\) is more significant in higher latitudes. The background error exhibits strong geographical dependences related to spatially varying atmospheric  
Figure 1: Meridional cross-section of annual-mean background error estimates: (a) temperature (\({}^{\circ}\)C), (b) specific humidity (g kg\({}^{-1}\)), and (c) relative humidity (%). structures and processes, observation coverages, and model deficiencies. The 1D-Var uses background errors represented on a \(5^{\circ}\times 5^{\circ}\) (latitude--longitude) grid for each month of the year. The background error at heights above the model top relates to the uncertainty of MSIS. Following [69], the uncertainty of MSIS is parameterized by the spread of the MSIS states obtainable under diverse assumed conditions.