Control of the initial condition
\(\underline x\): the initial condition of a temporal evolution model
\(\underline x^b\): the last forecast of the model
\(\cal G\): 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
\(\displaystyle J(\underline{x}) ={1\over 2} \left(\underline{x} - \underline{x}^b\right)^T \underline{\underline B}^{-1} \left(\underline{x} - \underline{x}^b\right)+ {1\over 2} \left[\underline{y}^o - {\cal G}(\underline{x})\right]^T \underline{\underline R}^{-1}\left[\underline{y}^o - {\cal G}(\underline{x})\right]\)
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 initial state \(\underline x\) of some model leading to the artificial observation \(\underline y = {\cal G} (\underline x)\) to be compared with real observation \(\underline y^o\).