Assimilation of Standard and Targeted Observations within the Unstable Subspace of the Observation–Analysis–Forecast Cycle System

Abstract

In this paper it is shown that the flow-dependent instabilities that develop within an observation?analysis?forecast (OAF) cycle and that are responsible for the background error can be exploited in a very simple way to assimilate observations. The basic idea is that, in order to minimize the analysis and forecast errors, the analysis increment must be confined to the unstable subspace of the OAF cycle solution. The analysis solution here formally coincides with that of the classical three-dimensional variational solution with the background error covariance matrix estimated in the unstable subspace. The unstable directions of the OAF system solution are obtained by breeding initially random perturbations of the analysis but letting the perturbed trajectories undergo the same process as the control solution, including assimilation of all the available observations. The unstable vectors are then used both to target observations and for the assimilation design. The approach is demonstrated in an idealized environment using a simple model, simulated standard observations over land with a single adaptive observation over the ocean. In the application a simplified form is adopted of the analysis solution and a single unstable vector at each analysis time whose amplitude is determined by means of the adaptive observation. The remarkable reduction of the analysis and forecast error obtained by this simple method suggests that only a few accurately placed observations are sufficient to control the local instabilities that take place along the cycle. The stability of the system, with or without forcing by observations, is studied and the growth rate of the leading instability of the different control solutions is estimated. Whereas the model has more than one positive Lyapunov exponent, the solution of the OAF scheme that includes the adaptive observation is stable. It is suggested that a negative exponent can be considered a necessary condition for the convergence of a particular OAF solution to the truth, and that the estimate of the degree of stability of the control trajectory can be used as a simple criterion to evaluate the efficiency of data assimilation and observation strategies. The present findings are in line with previous quantative observability results with more realistic models and with recent studies that indicate a local low dimensionality of the unstable subspace.