# Atmospheric GNSS RO 1D-Var in Use at UCAR: Description and Validation
## 2 Materials and Methods
### The UCAR 1D-Var  
#### 2.2.1 Theoretical Framework  
The COSMIC 1D-Var (hereafter, referred to as the 1D-Var) searches for the best likelihood thermodynamic states of the atmosphere for a given set of the observation, a priori profile, and their error covariances. Such a solution attains the minimum of a quadratic cost function when all the relevant errors are random with Gaussian probability density functions [57]:  
\[J(\mathbf{x})=\frac{1}{2}(\mathbf{x}-\mathbf{x}_{b})^{\mathrm{T}}\,\mathbf{B} ^{-1}\,\left(\mathbf{x}-\mathbf{x}_{b}\right)+\frac{1}{2}\{\mathbf{y}_{o}-H( \mathbf{x})\}^{\mathrm{T}}\,\mathbf{R}^{-1}\,\{\mathbf{y}_{o}-H(\mathbf{x})\}. \tag{6}\]  
where \(J\) is the cost function, \(\mathbf{x}\) is the state vector, \(\mathbf{x}_{b}\) is the background (a priori \(\mathbf{x}\)), \(\mathbf{y}_{o}\) is the observation vector, \(H\) is the observation operator (often called the forward model), \(\mathbf{B}\) is the error covariance matrix (ECM) of \(\mathbf{x}_{b}\), \(\mathbf{R}\) is the ECM of \(\mathbf{y}_{o}\), and superscripts \"T\" and \"\(-1\)\" denote the matrix transpose and inverse, respectively. The observation vector represents a RO profile which can either be of refractivity (as a function of geometric height) or of bending angle (as a function of impact parameter). The state vector comprises atmospheric profiles of temperature, pressure, and moisture and are required for the forward modeling. The observation operator converts the state variables into the RO observed physical quantity and interpolates it to the observed heights. The state and observation vectors and their ECMs can be large and the observation operator can be complex. Hence, it is generally impossible to acquire an analytic solution to the minimization problem. The 1D-Var searches for the minimum of \(J\) through a Newton-type iteration for which the gradient of \(J\) is calculated with the adjoint technique [20; 21], which provides an efficient means to evaluate the gradient. Practical applications do not often use Equation (6). Instead, mathematically equivalent yet computationally efficient forms are used. The 1D-Var makes use of the incremental formulation [58]:  
\[J(\delta\mathbf{x})=\frac{1}{2}\delta\mathbf{x}^{\mathrm{T}}\mathbf{B}^{-1} \delta\mathbf{x}+\frac{1}{2}(\mathbf{H}\delta\mathbf{x}-\mathbf{d})^{\mathrm{ T}}\mathbf{R}^{-1}(\mathbf{H}\delta\mathbf{x}-\mathbf{d}). \tag{7}\]  
where \(\delta\mathbf{x}=\mathbf{x}-\mathbf{x}_{b}\), \(\mathbf{d}=\mathbf{y}_{o}-H(\mathbf{x}_{b})\), and \(\mathbf{H}\) is the linearized version of the observation operator. The incremental formulation reduces the computational complexity of the minimization problem by circumventing the nonlinearity of \(H\). Taking one step further, the 1D-Var employs the control-variable transform [23; 59] which introduces a control vector \(\mathbf{v}\) to rewrite Equation (7) as follows:  
\[J(\mathbf{v})=\frac{1}{2}\mathbf{v}^{\mathrm{T}}\mathbf{v}+\frac{1}{2}( \mathbf{H}\delta\mathbf{x}-\mathbf{d})^{\mathrm{T}}\,\mathbf{R}^{-1}\,( \mathbf{H}\delta\mathbf{x}-\mathbf{d}). \tag{8}\]  
where \(\mathbf{v}=\mathbf{B}^{-1/2}\)\(\delta\mathbf{x}\) with \(\mathbf{B}^{1/2}\) being a square root of \(\mathbf{B}\). The control-variable transform turns \(\mathbf{B}\) into an identity matrix--which is trivial to deal with [60]--and lowers the condition number of the Hessian of \(J\), leading to faster convergence of the minimization [61]. To further accelerate the minimization, the 1D-Var reduces the elements of the control vector. Stratospheric humidity in the heights above 30 km is excluded. The pressure profile is reduced to a single-level (bottom) pressure. This is because the pressure profile contains redundant information to the extent that the hydrostatic balance is valid. In addition, pressure errors are highly correlated with themselves and with temperature and moisture errors, which impedes the search of the steepest descent of the cost function.