Chaos and quadri-dimensional data assimilation: a study based on the Lorenz model

Quadri-dimensional data assimilation aims at extracting all information from observations distributed over a finite time interval. In this paper, variational assimilation with the adjoint model technique is applied to the Lorenz model to illustrate how the performance of quadri-dimensional data assimilation can vary from one case to another. Observations are generated for two situations, one (the regular case) being more predictable than the other (the case with transition). An examination of the functional being minimized shows that although the regular case does not reveal any significant secondary minimum, there are in the case with transition for which the point of convergence was seen to be highly dependent on the first guess. It was also observed that to pick the first guess on the underlying attractor of this dynamical system does not insure convergence to the true minimum. In the adjoint model technique, the gradient of the functional is obtained through a time integration of the adjoint model using the difference between the solution of the direct model and the observations. It is shown how to relate the observational error covariance matrix to the gradient error covariance matrix. This method is applicable to any model once its adjoint is available and can be used to provide an estimate of the accuracy of the final analysis. Applying it to the Lorenz model, it is shown that due to the different local error growth rates, the same observational error can lead to very different accuracies for the gradient vector.