Modelling of error covariances by 4D-Var data assimilation

The extended Kalman filter is presented as a good approximation to the optimal assimilation of observations into a numerical weather prediction (NWP) model, as long as the evolution of errors stays close to linear. The error probability distributions are approximated by Gaussians, characterized 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 four-dimensional variational (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 probability density function. The latter is difficult for a NWP system representing scales which exhibit chaotic behaviour over the period of interest. The covariance modelling assumptions often result in a null space of error modes with little variance. It is argued that this is as important as the variance and correlation structures usually examined, since the implied constraints allow optimal use of observations giving gradient and tendency information. Difficulties arise in the approach when the NWP system is capable of resolving significant structures (such as convective cells) not always determined by the observations.