Adapting reduced-size control subspace in hybrid data assimilation

International audienceObservations may be abundant in nowadays oceanography but they are restricted to some components of the state vector only. Consequently, a need for some prior information on the unobserved variables arises. In the standard 4D-Var it is accounted for via regularising properties of the background error covariance matrix. Secondly, feasibility requirements may result in reduction of the size of the control vector and confine data assimilation correction to a small-size subspace of the original state space. Hence the necessity of an appropriate definition of the control subspace which must be of a small dimension and, at the same time, preserve regularising properties of the background error covariance matrix. The question addressed in this study goes even further and tackles the problem of adapting the definition of the control subspace to account for additional information gained in the data assimilation procedure. The problem has been studied in the framework of a hybrid approach to data assimilation. Hybrid in this study refers to a method merging an incremental 4D-Var with an equivalent Kalman smoother, both in a reduced rank approximation. The skeleton of the hybrid is thus formed by the 4D-Var enriched with an admixture of the smoother delivering a recipe for the evolution of the error covariance matrix. Its update is made at each transition from one assimilation window to another, at both analysis and forecast step. The analysis update modifies the reduced size error covariance matrix accordingly to the quality of the measurements assimilated into the system. Following the system's trajectory evolution in the forecast step, the basis spanning the control subspace is also adjusted. A series of OSSEs implementing the hybrid method into a shallow water model in a wind driven double-gyre circulation has been performed. It has been opted for a definition of the reduced-size control subspace reflecting the directions of the largest variability of the system. These directions have been obtained via principal component analysis of several different samples of model trajectory. A number of tests have been performed in various configurations of twin experiments and the circumstances where the hybrid outperforms the standard 4D-Var have been identified. The general conclusion inferred from this study indicates that the propagation of the basis spanning the control subspace is capable of compensating for imperfect initialization of this subspace. Moreover, as the information contained in the observations is gradually being assimilated into the system's trajectory, the control subspace evolving accordingly