A Singular Vector Perspective of 4d-var: Filtering and Interpolating

Four-dimensional variational data assimilation (4D-Var) combines the information from a time-sequence of observations with the model dynamics and a background state to produce an analysis. In this paper, a new mathematical insight into the behaviour of 4D-Var is gained from an extension of concepts that are used to assess the qualitative information content of observations in satellite retrievals. It is shown that the 4D-Var analysis increments can be written as a linear combination of the singular vectors of a matrix which is a function of both the observational and the forecast model systems. This formulation is used to consider the filtering and interpolating properties of 4D-Var using idealized case-studies with a simple model of baroclinic instability. The results of the 4D-Var case-studies exhibit the reconstruction of the state in unobserved regions, as a consequence of the interpolation of observations through time. The results also exhibit the filtering of components with small spatial scales that correspond to noise, and the filtering of structures in unobserved regions. The singular vector perspective gives a very clear view of this filtering and interpolating by the 4D-Var algorithm and shows that the appropriate specification of the a priori statistics is vital to extract the maximal amount of useful information from the observations.