# 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
Using Equations (16) and (17), we can rewrite \(\textbf{B}_{v}^{1/2}\) as:  
\[\textbf{B}_{v}^{1/2}=\textbf{D}\;\textbf{C}^{1/2}=\textbf{D}\;\boldsymbol{ \Sigma}\;\boldsymbol{\Lambda}^{1/2} \tag{18}\]  
Finally, we can obtain \(\textbf{B}^{1/2}\) as follows:  
\[\textbf{B}^{1/2}=\textbf{P}\;\textbf{B}_{v}^{1/2} \tag{19}\]  
where **P** is algebraic operations that convert the control vector into the state vector. In the 1D-Var, **P** involves two physical transforms: conversion between moisture variables and hydrostatic reconstruction of the pressure profile. The 1D-Var uses the pseudo-relative humidity \(RH^{*}\) as the default moisture control variable. As a result, it receives a new estimate of the humidity increment \(\delta RH^{*}\) at each iteration of Equation (9). Thereupon, the 1D-Var converts the humidity increment into the specific humidity increment:  
\[\delta q=\delta RH^{*}\cdot q_{b}^{s} \tag{20}\]  
where \(q_{b}^{s}\) is the saturation specific humidity with respect to the background temperature \(T_{b}\) and pressure \(p_{b}\). The specific humidity relates to the water vapor pressure as follows:  
\[q=\frac{\epsilon e}{p-(1-\epsilon)\;e}. \tag{21}\]  
where \(\epsilon\) is a constant (=0.622). The saturation water vapor pressure \(e^{s}\) is determined in the following manner:  
\[e^{s}=\left\{ \begin{array}{ll}e_{i}^{s},&T\leq T_{i}\\ e_{i}^{s}+\big{(}e_{w}^{s}-e_{i}^{s}\big{)}\Big{(}\frac{T-T_{i}}{T_{0}-T_{i}} \big{)}^{2},&T_{i}<T<T_{0}\\ e_{w}^{s},&T\geq T_{0}\end{array} \right.. \tag{22}\]  
where \(e_{w}^{s}\) (\(e_{i}^{s}\)) is the saturation water vapor pressure over water (ice), \(T_{0}\) is 0 \({}^{\circ}\)C, and \(T_{i}\) is \(-23\)\({}^{\circ}\)C. For \(e_{w}^{s}\) and \(e_{i}^{s}\), the 1D-Var uses the formula proposed by [76] as the default option among other formulae available. The transition between \(e_{w}^{s}\) to \(e_{i}^{s}\) in the second line of Equation (22) represents the freezing inhibition effect of supercooled water. The lowest-level pressure is the default option for the pressure control variable, and the 1D-Var determines upper-level pressure increments through a hydrostatic balance operator. In a hydrostatically balanced atmosphere, the following relation holds:  
\[\frac{dp^{\prime}}{dz}=-\rho_{v}^{\prime}\,g. \tag{23}\]  
where \(\rho_{v}\) is the virtual air density and the prime denotes the perturbation from the base state. Linearization of the ideal gas law leads to:  
\[\frac{\rho_{v}^{\prime}}{\rho_{v}}=\frac{p^{\prime}}{p}-\frac{T_{v}^{\prime}}{ T_{v}}. \tag{24}\]  
where \(T_{v}=T\) (\(1+0.608\)\(q\)) is the virtual temperature. Accordingly, Equation (23) can be rewritten as:  
\[\frac{dp^{\prime}}{dz}=-\bigg{(}\frac{p^{\prime}}{p}-\frac{T_{v}^{\prime}}{T_{ v}}\bigg{)}\rho_{v}\,g=\bigg{(}\frac{p^{\prime}}{p}-\frac{T_{v}^{\prime}}{T_{v}} \bigg{)}\frac{dp}{dz}. \tag{25}\]  
The discrete form of Equation (25), the hydrostatic balance operator, used for the 1D-Var is:  
\[\delta p^{k+1}=\delta p^{k}\bigg{(}\frac{p^{k+1}}{p^{k}}\bigg{)}+\frac{\bigg{(} p^{k}-p^{k+1}\bigg{)}\,p^{k+\frac{1}{2}}}{p^{k}}.\frac{\delta T_{v}^{k+\frac{1}{2} }}{T_{v}^{k+\frac{1}{2}}}. \tag{26}\]  
where \(\delta\) indicates the increment, the variables without \(\delta\) correspond to the background state, the superscript stands for the vertical level (indexed from bottom to top), and the half-level index, \(k+\frac{1}{2}\), denotes the arithmetic mean of adjacent full levels, \(k\) and \(k+1\). Equation (26) looks a bit complicated because Equation (25) is implicit (i.e., \(p^{\prime}\) appearing on both sides), and that the 1D-Var uses a non-staggered grid on which all state variables (pressure, temperature, and moisture) are collocated. It is worthwhile to mention that \(g\) and \(dz\) are absent in Equation (26), which allows one to use Equation (26) in arbitrary vertical coordinates, notwithstanding the change of \(g\) with the geometric height. Figure 3: Background error correlation matrix at 60\({}^{\circ}\) N in March.