Gradient of the 4D-Var cost function

  • Linearization of \({\cal M}_{0\to i}\) and \({\cal H}_i\) with respect to the initial conditions \(\underline{x}\):

    \({\cal H}_i {\cal M}_{0\to i}(\underline{x} + {\underline {\delta x}}) \; \approx\; {\cal H}_i{\cal M}_{0\to i} (\underline{x})+ \underline{\underline H}_i\; \underline{\underline M}_{0\to i} \; {\underline {\delta x}}\)

  • The linear tangent model is denoted by : \(\displaystyle \underline{\underline M}_{0\to i}\)

  • The linearized observation operators are: \(\displaystyle \underline{\underline H}_i\)

  • The gradient of the cost function is

    \(\underline {\rm grad\ } J = \underline{\underline B}^{-1} (\underline{x} - \underline{x}^b) + \sum_{i=0}^M\underline{\underline M}_{0\to i}^T \, \underline{\underline H}_i^T \, \underline{\underline R}_i^{-1}\left[ \underline{y}_i^o - {\cal H}_i \, {\cal M}_{0\to i}(\underline{x})\right]\)

  • It involves the adjoint of the model \(\underline{\underline M}_{0\to i}^T\) and of the observation operators \(\underline{\underline H}_i^T\)

  • NB: one obtains the 4D-FGAT model by replacing \(\underline{\underline M}_{0\to i}\) by  \(\underline{\underline I}\)