Twin experiments for the control of α
True state: \(({ \color{blue} \alpha^t })\)
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: \({\cal G}(\alpha) = [H_1, ...,H_i, ..., H_M]\)
Background \(\alpha_b\) with a root mean square error \(\sigma_{\alpha^b}\)
The analysis minimizes the cost function: \(\displaystyle J(\alpha) ={1\over 2}{ (\alpha - \alpha^b )^2\over \sigma_{\alpha^b}^2 }+\sum_{i=1}^M{ \left[H_i^o- {\cal G}_i(\alpha)^2\right] \over 2\, \sigma_r^2}\)
