The application of Kalman smoother theory to the estimation of 4DVAR error statistics

Modern atmospheric data assimilation theory is dominated by the four-dimensional variational (4DVAR) and Kalman filter/smoother approaches. Both generate analysis weights (explicitly or implicitly) which are dynamically determined by the assimilation model. A Kalman smoother is basically a generalization of the Kalman filter which can process future observations. In control theory, a generalization of 4DVAR called Pontryagin optimization can account for an imperfect assimilating model. Pontryagin optimization and the fu;ed-interval Kalman smoother are equivalent when both methods use the same statistical information. We use the equivalence between Pontryagin optimization and the Kalman smoother to examine the effect of the perfect model assumption on the error statistics and analysis weights of the 4DVAR algorithm. This is done by developing the Kalman smoother equations for a a very simple assimilating model. A procedure for diagnosing the effect of model error, based on the observational cost function, is also developed.