Cost function
The analysis \(x_a\) minimizes the cost function:
\(\displaystyle J(\underline{x}) ={1\over 2} \left(\underline{x} - \underline{x}^b\right)^T \underline{\underline B}^{-1} \left(\underline{x} - \underline{x}^b\right)+ {1\over 2} \left[\underline{y}^o - {\cal G}(\underline{x})\right]^T \underline{\underline R}^{-1}\left[\underline{y}^o - {\cal G}(\underline{x})\right]\)
where
\(\underline{\underline B}\) is the background error covariance matrix
\(\underline{\underline R}\) is the observation error covariance matrix
\(\cal G\) is the observation operator
The cost function associates a real number to any vector \(\underline x\) of the control space, given a background state \(\underline x_b\) and a vector \(\underline y^o\) of the observation space. The artificial observation \(\underline y = {\cal G} (\underline x)\) are extracted from the vector \(\underline x\) through the observation operator \(\cal G\).