Nonlinear conditional model bias estimation for data assimilation

In this study, we develop model bias estimators based on an asymptotic expansion of the model dynamics for small time scales and small perturbations in a model parameter, and then use the estimators to improve the performance of a data assimilation system. We employ the well-known Lorenz (1963) model so that we can study all aspects of the dynamical system and model bias estimators in a detailed way that would not be possible with a full physics numerical weather prediction model. In particular, we first work out the asymptotics of the Lorenz model for small changes in one of its parameters and then use statistics from cycled data assimilation experiments to demonstrate that the asymptotics accurately represent the behavior of the model and that the coefficients of the nonlinear asymptotical expansion can be reasonably estimated by solving a least squares minimization problem. In data assimilation, the background error covariance matrix usually estimates the uncertainty of the model background, which is then used along with the observation error covariance matrix to produce an updated analysis. If the uncertainty of the model background is strongly influenced by time-dependent model biases, then the development of nonlinear bias estimators that also vary with time could improve the performance of the assimilation system and the accuracy of the updated analysis. We demonstrate this improvement through the combination of a constant background error covariance matrix with a dynamically-varying matrix computed using the model bias estimators. Numerical tests using the Lorenz (1963) model illustrate the feasibility of the approach and show that it leads to clear improvements in the analysis and forecast accuracy