General formalism
Data assimilation cost function in the general case:
\(\displaystyle J(\underline{x}) ={1\over 2} \left(\underline{x} - \underline{x}^b\right)^T \underline{\underline B}^{-1} \left(\underline{x} - \underline{x}^b\right)+ {1\over 2} \left[\underline{y}^o - {\cal G}(\underline{x})\right]^T \underline{\underline R}^{-1}\left[\underline{y}^o - {\cal G}(\underline{x})\right]\)
Opérateur observation
Optimal estimation of the clock time : no background
\(\displaystyle J(T)= {1\over 2} (T_1^o- T, T_2^o- T) \; \underline{\underline R}^{-1}\;\left(\begin{matrix}T_1^o- T \cr T_2^o- T\end{matrix}\right)\)
The observation operator \({\cal G}(T)=(T, T)\) is linear.
The "Best Linear Unbiased Estimator" (BLUE) \(\color{red} T^a\) minimizes this cost function