# 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 produced 120 different MSIS realizations for a given geophysical location and month with varying combinations of the key control parameters of the empirical model: hour of the day, day of the month, 10.7 cm solar radio flux (F10.7), and the geomagnetic amplitude index (Ap). The last two parameters, F10.7 and Ap, relate to the strength of solar activity. The spread of these MSIS realizations is scaled to match the NWP-based \(\sigma_{b}\) in the uppermost 10 km of ECMWF forecasts. The size-adjusted spread is the background error in MSIS heights. The off-diagonal elements of **B** represent inter-element error correlations. The NMC method furnishes a complete **B** that includes cross (inter-variable) correlations and auto (self) correlations, which is a virtue of the NMC method considering that the off-diagonal elements (especially those of multivariate correlation) are otherwise difficult to estimate. We used ECMWF forecasts to build **B** because the model top and vertical resolution are higher than any other model forecasts available to us, permitting us to estimate **B** at high altitudes. In practical use, however, short-term ECMWF forecasts may not always be the source of a priori profile. In that case, it is desirable to allow the 1D-Var to have the flexibility to adjust \(\sigma_{b}\), which can be accomplished by constructing error correlation matrix **C**:  
\[c_{i,j}=\frac{b_{i,j}}{\sqrt{b_{i,i}}\sqrt{b_{j,j}}}=\frac{b_{i,j}}{\sigma_{i} \,\sigma_{j}}. \tag{15}\]  
where \(c_{i,j}\) (\(b_{i,j}\)) is the element of **C** (**B**) at \(i\)-th column and \(j\)-th row, and \(\sigma_{i}\) is the square root of \(b_{i,j}\). We assume that **C** is generic and not dependent on a priori data source. It is then possible to construct a new **B** by swapping \(\sigma_{i}\) values with known or desired values while retaining **C**. In fact, this is what allowed us earlier to adjust \(\sigma_{f}^{NMC}\) toward \(\sigma_{b}^{HL}\). Figure 2: Horizontal view of annual-mean background error estimates: (**a**) temperature (\({}^{\circ}\)C) at 5 km above the ground level, (**b**) surface pressure (hPa), (**c**) 2 km specific humidity (g kg\({}^{-1}\)), and (**d**) 2 km relative humidity (%). Figure 3 shows an example of **C** used in the 1D-Var, at 60\({}^{\circ}\)N for March. The error correlations widen with increasing height and differ significantly, e.g., the pressure error shows much broader autocorrelations than the temperature error (Figure 3b,c). In addition, they change signs with height and exhibit a fair degree of asymmetry with respect to height: e.g., the autocorrelation of temperature errors. These complex aspects make it challenging to model **C** accurately, especially the cross-correlations. As inadequate approximation or idealization of **C** results in loss of information, we decided to use the error correlation matrix in its complete form as provided by the NMC method. We believe that doing so allows the 1D-Var to benefit the most from the error correlation matrix to the extent that the NMC method remains realistic. To specify \(\textbf{B}^{1/2}\) which the 1D-Var needs, we first decompose **B** in the control variables' space, \(\textbf{B_{v}}\), as follows:  
\[\textbf{B_{v}}=\textbf{D}\;\textbf{C}\;\textbf{D} \tag{16}\]  
where **D** is the diagonal matrix whose diagonal elements correspond to the background error. What followed next is an eigen decomposition of **C** using the method proposed by [75]:  
\[\textbf{C}=\textbf{\Sigma}\;\boldsymbol{\Lambda}\;\boldsymbol{\Sigma}^{\text{ T}}=\textbf{\Sigma}\;\boldsymbol{\Lambda}^{1/2}\;\Big{(}\textbf{\Sigma}\; \boldsymbol{\Lambda}^{1/2}\Big{)}^{\text{T}} \tag{17}\]  
where columns of \(\boldsymbol{\Sigma}\) are eigenvectors of **C**, which are mutually orthogonal (\(\boldsymbol{\Sigma}^{\text{T}}\textbf{\Sigma}=\textbf{I}\), where **I** is the identity matrix), and \(\boldsymbol{\Lambda}\) is the diagonal matrix of eigenvalues.