Introduction to Data Assimilation for Scientists and Engineers

Here, the observation operateur \({\cal G}(\underline{x})=\underline{\underline G}\, \underline{x}\) is linear.

• The cost function reads

  \(J(\underline{x}) ={1\over 2} \left(\underline{x} - \underline{x}^b\right)^T \underline{\underline B}^{-1} \left(\underline{x} - \underline{x}^b\right)+ {1\over 2} \left(\underline{y}^o - \underline{\underline G} \, \underline{x} \right)^T\underline{\underline R}^{-1}\left(\underline{y}^o - \underline{\underline G} \, \underline{x}\right)\)

• Gain matrix: \(\displaystyle\underline{\underline K} = \underline{\underline B} \,\underline{\underline G}^T \, \left( \underline{\underline G} \, \underline{\underline B}\, \underline{\underline G}^T +\underline{\underline R}\right)^{-1}\)

• Innovation vector: \(\displaystyle \underline d= \underline{y}^o - \underline{\underline G} \; \underline{x}^b\)

• Analysis: \(\displaystyle \underline{x}^a = \underline{x}^b + \underline{\underline K}\; \underline d\)

• Analysis error covariance matrix: \(\displaystyle \underline{\underline A} = (\underline{\underline I} - \underline{\underline K} \; \underline{\underline G} ) \, \underline{\underline B}\)