4D-Var method

  • The control vector \(\underline{x}\) is an initial condition

  • Evolution model from time 0 to time \(i\): \(\displaystyle {\cal M}_{0\to i}\)

  • Observation operator at time \(i\): \(\displaystyle {\cal H}_i\)

  • Observation error covariance matrix at time \(i\): \(\displaystyle {\underline{\underline R}}_i\)

  • Cost function:

    \(J(\underline{x}) = {1\over 2} \left(\underline{x} - \underline{x}^b\right)^T \underline{\underline B}^{-1} \left(\underline{x} - \underline{x}^b\right)\\+{1\over 2} \sum_{i=0}^M\left[ \underline{y}_i^o - {\cal H}_i \, {\cal M}_{0\to i} (\underline{x}) \right]^T\; \underline{\underline R}_i^{-1} \;\left[ \underline{y}_i^o - {\cal H}_i \, {\cal M}_{0\to i}\; (\underline{x}) \right]\)