Control of parameters
Data assimilation can be used to control the initial condition of a model
The analysis \(\underline x^a\)minimizes the cost function:
\(\displaystyle J(\underline{x}) ={1\over 2} \left(\underline{x} - \underline{x}^b\right)^T \underline{\underline B}^{-1} \left(\underline{x} - \underline{x}^b\right)+ {1\over 2} \left[\underline{y}^o - {\cal G}(\underline{x})\right]^T \underline{\underline R}^{-1}\left[\underline{y}^o - {\cal G}(\underline{x})\right]\)
\(\underline x^b\) : a guess of the model parameters to control
\(\underline y^o\) : measurements on the real system
\(\cal G\) : the link between the model parameters the model observations
\(\underline {\underline B}\) : background error covariance matrix
\(\underline {\underline R}\) : observation error covariance matrix