Incremental cost function
When \(\cal G\) is nonlinear: \(\displaystyle J(\underline{x}) ={1\over 2} \left(\underline{x} - \underline{x}^b\right)^T \underline{\underline B}^{-1} \left(\underline{x} - \underline{x}^b\right)+ {1\over 2} \left[\underline{y}^o - {\cal G}(\underline{x})\right]^T \underline{\underline R}^{-1}\left[\underline{y}^o - {\cal G}(\underline{x})\right]\)
Linearization of \(\cal G\) around \(\underline x^b\): \(\displaystyle{\cal G}(\underline{x}^b + {\underline {\delta x}}) \; \approx\; {\cal G}(\underline{x}^b) + \underline{\underline G} \; {\underline {\delta x}}\)
Denoting \(\displaystyle\underline{d} = \underline{y}^o - {\cal G}(\underline{x}^b)\) the innovation, the incremental cost function reads:
\(J_{inc}(\underline{x}^b + {\underline {\delta x}}) ={1\over 2} \, {\underline {\delta x}}^T\; \underline{\underline B}^{-1} \; {\underline {\delta x}}+ {1\over 2}\left( \underline{d} - \underline{\underline G} \, {\underline {\delta x}} \right)^T\;\underline{\underline R}^{-1} \;\left( \underline{d} - \underline{\underline G} \, {\underline {\delta x}} \right)\)
Analysis: \(\displaystyle \underline{x}^a = \underline{x}^b + \underline{\underline K} \; \underline{d}\) where te gain matrix is \(\displaystyle\underline{\underline K} = \underline{\underline B} \, \underline{\underline G}^T \, \left( \underline{\underline G} \, \underline{\underline B}\, \underline{\underline G}^T +\underline{\underline R}\right)^{-1}\)
Analysis error covariance matrix: \(\displaystyle \underline{\underline A} = \left(\underline{\underline I} - \underline{\underline K} \; \underline{\underline G} \right) \, \underline{\underline B}\)