Twin experiments for α and P
True state: \(({ \color{blue} \alpha^t, P^t })\)
Artificial measurements: \(\underline{y}^o={\color{blue}[H_1^o, ..., H_i^o, ..., H_M^o]}\) with a root mean square error \({\color{blue}\sigma_r }\)
Observation operator: \({\cal G}(\alpha, P) = [H_1, ...,H_i, ..., H_M]\)
Background \((\alpha_b, P_b)\) with a root mean square error matrix \(\underline{\underline B} =\left( \begin{matrix}\sigma^2_{\alpha^b} & \rho \, \sigma_{\alpha^b} \, \sigma_{P^b} \cr\rho \, \sigma_{\alpha^b} \, \sigma_{P^b} & \sigma^2_{P^b} &\end{matrix} \right)\)
The analysis minimizes the cost function:
\(\displaystyle J(\alpha,P) ={1\over 2} (\alpha-\alpha^b, P-P^b) \; \underline{\underline B}^{-1}\left(\begin{matrix}\alpha-\alpha^b\cr P-P^b\end{matrix}\right)+\sum_{i=1}^M{ \left[H_i^o- {\cal G}_i(\alpha,P)\right]^2 \over 2\, \sigma_r^2}\)
