Case of correlated measurements
Correlation coefficient of two random variables: \(\rho= { \left< T_1' \, T_2'\right>/ ( \sigma_1 \, \sigma_2 )}\)
The analysis \(\color{red} T^a\) now minimizes
\(\displaystyle J(T) = {1\over 2} \;\begin{matrix}(T_1^o-T, & T_2^o-T)\end{matrix}\left(\begin{matrix}\sigma_1^2 & \rho\, \sigma_1 \, \sigma_2 \cr\rho\, \sigma_1 \, \sigma_2 & \sigma_2^2\end{matrix}\right)^{-1}\left(\begin{matrix}T_1^o-T \cr T_2^o-T\end{matrix}\right)\)
which involves the matrix \(\underline{\underline R}\) of the observation error covariance