Modelling of error covariances by four-dimensional variational data assimilation
LORENC, A.C.
The extended Kalman filter is presented as a good approximation to the optimal assimilation of observations into an NWP model, as long as the evolution as errors stays close to linear. The error probability distributions are approximated by Gaussians, characterised by their mean and covariance. The full nonlinear forecast model is used to propagate the mean, and a linear model (not necessarily tangent to the full model) the covariances. Since it is impossible to determine the covariances in detail, physically based assumptions about their behaviour must be made, for instance three-?dimensional balance relationships are used. The linear model can be thought of as extending the covariance relationships to the time dimension. Incremental 4D-Var is derived as a practical implementation of the extended Kalman filter, optimally using these modelled covariances for a finite time-window. It is easy to include a simplified model of forecast errors in the representation. This Kalman filter based paradigm differs from more traditional derivations of 4D-Var in attempting to estimate the mean, rather than the mode of the posterior pdf. The latter is difficult for an NWP system representing scales which exhibit chaotic behaviour over the period of interest. The...
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