Singular-vector-based covariance propagation in a quasigeostrophic assimilation system

Variational data assimilation systems require the specification of the covariances of background and observation errors. Although the specification of the background-error covariances has been the subject of intense research, current operational data assimilation systems still rely on essentially static and thus flow-independent background-error covariances. At least theoretically, it is possible to use flow-dependent background-error covariances in four-dimensional variational data assimilation (4DVAR) through exploiting the connection between variational data assimilation and estimation theory. This paper reports on investigations concerning the impact of flow-dependent background-error covariances in an idealized 4DVAR system that, based on quasigeostrophic dynamics, assimilates artificial observations. The main emphasis is placed on quantifying the improvement in analysis quality that is achievable in 4DVAR through the use of flow-dependent background-error covariances. Flow dependence is achieved through dynamical error-covariance evolution based on singular vectors in a reduced-rank approach, referred to as reduced-rank Kalman filter (RRKF). The RRKF yields partly dynamic background-error covariances through blending static and dynamic information, where the dynamic information is obtained from error evolution in a subspace of dimension k (defined here through the singular vectors) that may be small compared to the dimension of the model's phase space n, which is equal to 1449 in the system investigated here. The results show that the use of flow-dependent background-error covariances based on the RRKF leads to improved analyses compared to a system using static background-error statistics. That latter system uses static background-error covariances that are carefully tuned given the model dynamics and the observational information available. It is also shown that the performance of the RRKF approaches the performance of the extended Kalman filter, as k approaches n. Results therefore support the hypothesis that significant analysis improvement is possible through the use of flow-dependent background-error covariances given that a sufficiently large number (here on the order of n/10) of singular vectors is used.