Introduction to Data Assimilation for Scientists and Engineers

One notices that the analysis \(\color{red} T^a = {C_1 \, T_1^o + C_2 \, T_2^o \over C_1 + C_2}\) minimizes the "cost function":

\(\displaystyle J(T) ={(T_1^o-T)^2\over 2\, \sigma_1^2} +{(T_2^o-T)^2\over 2\, \sigma_2^2} ={C_1 \over 2} \; (T_1^o-T)^2 + {C_2 \over 2} \; (T_2^o-T)^2\)