# 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  
Error Covariance Matrices (ECMs) play vital roles in modern data assimilation and retrieval by determining how the information is spread in space and among the state variables. Accurate specification of ECMs is of prime importance in 1D-Var because the limited (1D) spatial scope disallows 1D-Var the use of some useful resources available to multi-dimensional systems such as other horizontally nearby observations, and model dynamics and balance operators involving 3D atmospheric structures and flows. Accurate ECMs are also important for 1D-Var to tackle the underdetermined retrieval problem. The observation error used in the 1D-Var varies with the latitude, month of the year, and height. [PERSON] [38] describes the methodology and procedure used for the error estimation. Here, we give a brief summary. We estimated the observation error for bending angle and refractivity by applying the Hollingsworth-Lonnberg method [72] to \(\sim\)1.5 million nearby pairs of RO profiles available for seven years, April 2007-April 2014. This approach uses the innovation statistics and rests on the assumption that background errors are mutually correlated in space, whereas observation errors are uncorrelated with themselves and the background errors. If the assumption holds, the change of innovation covariance with the separation (distance between paired soundings) is exclusively attributable to the spatial correlation of the background error. The error estimation proceeds with a least-squares fitting of distance-binned covariance values to an idealized (e.g., Gaussian) covariance model. The best-fitting covariance model at zero separation gives the background error variance, the subtraction of which from the innovation variance provides the observation error variance. Henceforward the square root of the observation (background) error variance is referred to as the observation (background) error. The observation error has shown significant latitudinal, altitudinal, and seasonal variations. Consideration of these spatiotemporal variations is vital for 1D-Var to specify a statically precise (locally accurate) observation error at the location of individual RO profiles. The observation ECMs are assumed diagonal for the sake of computational simplicity. We estimated background ECMs using the NMC (National Meteorological Center) method [23], assisted by the Hollingsworth-Lonnberg method. The NMC method relates the background error to the difference between short- and long-term forecasts that are valid at the same time:  
\[\mathbf{B}_{NMC}=E\left(\delta\mathbf{f}\,\delta\mathbf{f}^{\mathrm{T}}\right), \tag{14}\]  
where \(\mathbf{B}_{NMC}\) is a provisional background ECM, \(\delta\mathbf{f}\) is the forecast difference, and \(E(\cdot)\) is the statistical expectation. The square roots of the diagonal elements of \(\mathbf{B}_{NMC}\) give a measure of the forecast error, \(\sigma_{f}^{NMC}\). The NMC method is opportune in that it offers the error estimate in all desired areas and times (e.g., on every model grid point) without being restricted by the coverage of observation networks. The resulting--otherwise challenging to obtain--error estimate is continuous in space and consistent among the model's state variables. The NMC method is mainly empirical and lacks a theoretical basis [73]. A major limitation is that the choice of forecast lead times, which determines the variance of forecast differences, is arbitrary. Consequently, the forecast differences must be adjusted in magnitude to provide a realistic estimate of the background error [59; 74]. We used 12 and 24 h operational forecasts from the European Center for Medium-Range Weather Forecasts (ECMWF). We adjust \(\sigma_{f}^{NMC}\) toward \(\sigma_{b}^{HL}\), the background error of proper magnitude estimated using the Hollingsworth-Lonnberg method. Specifically, \(\sigma_{f}^{NMC}\) is multiplied with a scaling factor, \(S=\sigma_{b}^{HL}\cdot(\sigma_{f}^{NMC})^{-1}\), to give the final background error \(\sigma_{b}\).