On Variational Data Assimilation in Continuous Time

Variational data assimilation in continuous time is revisited. The central techniques applied in this paper are in part adopted from the theory of optimal nonlinear control. Alternatively, the investigated approach can be considered as a continuous time generalisation of what is known as weakly constrained four dimensional variational assimilation (WC–4DVAR) in the geosciences. The technique allows to assimilate trajectories in the case of partial observations and in the presence of model error. Several mathematical aspects of the approach are studied. Computationally, it amounts to solving a two point boundary value problem. For imperfect models, the trade off between small dynamical error (i.e. the trajectory obeys the model dynamics) and small observational error (i.e. the trajectory closely follows the observations) is investigated. For (nearly) perfect models, this trade off turns out to be (nearly) trivial in some sense, yet allowing for some dynamical error is shown to have positive effects even in this situation. The presented formalism is dynamical in character; no assumptions need to be made about the presence (or absence) of dynamical or observational noise, let alone about their statistics.

💡 Research Summary

The paper revisits variational data assimilation (VDA) in a continuous‑time setting and shows how techniques from optimal nonlinear control can be employed to formulate a continuous‑time analogue of weakly constrained four‑dimensional variational assimilation (WC‑4DVAR). The authors start by defining a cost functional that penalises both the misfit between observations and the model‑predicted state and the deviation of the trajectory from the governing dynamics. Formally, for a state vector (x(t)), observation operator (H), model dynamics (f), observation error covariance (R) and model‑error covariance (Q), the functional reads

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