## Abstract

The fixed-lag Kalman smoother was proposed recently by S. E. Cohn et al. as a framework for providing retrospective data assimilation capability in atmospheric reanalysis projects. Retrospective data assimilation refers to the dynamically consistent incorporation of data observed well past each analysis time into each analysis. Like the Kalman filter, the fixed-lag Kalman smoother requires statistical information that is not available in practice and involves an excessive amount of computation if implemented by brute force, and must therefore be approximated 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 formulation employed in this work separates naturally into a (Kalman) filter portion and an optimal retrospective analysis portion, two suboptimal 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 strategies are studied in the context of a linear dynamical model and observing system, since it is only under these circumstances that performance can be evaluated exactly. A shallow water model, linearized about an unstable basic flow, is used for this purpose.

Results indicate that retrospective data assimilation can be successful even when simple filtering schemes are used, such as one resembling current operational statistical 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 sophisticated approximate filtering algorithms, such as ones employing singular values/vectors of the propagator or eigenvalues/vectors of the error covariances, as a way to account for error covariance propagation, lead to even better RDAS performance. Approximating both the filter and retrospective analysis portions of the RDAS is also shown to be an acceptable approach in some cases.

* Current affiliation: General Sciences Corporation (a subsidiary of Science Applications International Corporation), Laurel, Maryland.

*Corresponding author address:* Dr. Ricardo Todling, Data Assimilation Office, NASA/GSFC, Code 910.3, Greenbelt, MD 20771.

Email: todling@dao.gsfc.nasa.gov