Control of the model parameters
\(\underline x\): the model parameters to be fitted
\(\underline x^b\): an a priori knowledge of the parameters
\(\cal G\): the link between the model parameters and the observations
\(\underline y\): the observations simulated by the model
\(\underline y^o\): the measurements performed on the real system
\(\displaystyle J(\underline{x}) ={1\over 2} \left(\underline{x} - \underline{x}^b\right)^T \underline{\underline B}^{-1} \left(\underline{x} - \underline{x}^b\right)+ {1\over 2} \left[\underline{y}^o - {\cal G}(\underline{x})\right]^T \underline{\underline R}^{-1}\left[\underline{y}^o - {\cal G}(\underline{x})\right]\)
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
One can wish to control the vector of parameters \(\underline x\) of some model leading to the artificial observation \(\underline y = {\cal G} (\underline x)\) to be compared with real observation \(\underline y^o\).