Cost function
We present the generic formalism of data assimilation with a control space \(\underline x\in I\!R^n\) and an observation space \(\underline y\in I\!R^m\) with an observation operator \(\underline y = {\cal G } (\underline x)\). The background \(\underline x^b\) is a first guess for the analysis \(\underline x^a\) with minimizes the cost function \(J(\underline x)\).
The analysis \(\underline x^a\) minimizes the cost function
\(\displaystyle J(\underline{x}) ={1\over 2} \left(\underline{x} - \underline{x}^b\right)^T \underline{\underline B}^{-1} \left(\underline{x} - \underline{x}^b\right)+ {1\over 2} \left[\underline{y}^o - {\cal G}(\underline{x})\right]^T \underline{\underline R}^{-1}\left[\underline{y}^o - {\cal G}(\underline{x})\right]\)
\(\underline {\underline B}\) : background error covariance matrix
\(\underline {\underline R}\) : observation error covariance matrix