Cost function and analysis
\(\underline x\): state vector containing model parameters or initial condition
\(\underline x^b\): background state, a priori knowledge of \(\underline x\)
\(\cal G\): observation operateor, the link between the initial condition and the observations
\(\underline y\): the observations simulated by the model
\(\underline y^o\): the measurements performed on the real system
\(\underline x^a\): analysis, obtained as the minimum of the cost function
\(\displaystyle J(\underline{x}) ={1\over 2} \left(\underline{x} - \underline{x}^b\right)^T \underline{\underline B}^{-1} \left(\underline{x} - \underline{x}^b\right)+ {1\over 2} \left[\underline{y}^o - {\cal G}(\underline{x})\right]^T \underline{\underline R}^{-1}\left[\underline{y}^o - {\cal G}(\underline{x})\right]\)
\(\underline{\underline B}\): the background error covariance matrix
\(\underline{\underline R}\): is the observation error covariance matrix