SUBOPTIMAL SCHEMES FOR RETROSPECTIVE DATA ASSIMILATION BASED ON THE FIXED-LAG KALMAN SMOOTHER

The fixed-lag Kalman smoother was proposed recently by S. E. Cohn et a l. as a framework for providing retrospective data assimilation capabi lity in atmospheric reanalysis projects. Retrospective data assimilati on refers to the dynamically consistent incorporation of data observed well past each analysis time into each analysis. Like the Kalman filt er, the fixed-lag Kalman smoother requires statistical information tha t is not available in practice and involves an excessive amount of com putation if implemented by brute Force, and must therefore be approxim ated sensibly to become feasible for operational use. In this article the performance of suboptimal retrospective data assimilation systems (RDASs) based on a variety of approximations to the optimal fixed-lag Kalman smoother is evaluated. Since the fixed-lag Kalman smoother form ulation employed in this work separates naturally into a (Kalman) filt er portion and an optimal retrospective analysis portion, two suboptim al strategies are considered: (i) viable approximations to the Kalman filter portion coupled with the optimal retrospective analysis portion , and (ii) viable approximations to both portions. These two strategie s are studied in the context of a linear dynamical model and observing system, since it is only under these circumstances that performance c an be evaluated exactly. A shallow water model, linearized about an un stable basic flow, is used for this purpose. Results indicate that ret rospective data assimilation can be successful even when simple filter ing schemes are used, such as one resembling current operational stati stical analysis schemes. In this case, however, online adaptive tuning of the forecast error covariance matrix is necessary. The performance of this RDAS is similar to that of the Kalman filter itself. More sop histicated approximate filtering algorithms, such as ones employing si ngular values/vectors of the propagator or eigenvalues/vectors of the error covariances, as a way to account For error covariance propagatio n, lead to even better RDAS performance. Approximating both the filter and retrospective analysis portions of the RDAS is also shown to be a n acceptable approach in some cases.