On Variational Data Assimilation in Continuous Time

TL;DR

This paper revisits variational data assimilation in continuous time, connecting it with optimal control theory and extending weakly constrained 4DVAR to handle partial observations and model errors without assuming noise statistics.

Contribution

It introduces a dynamical formalism for continuous-time variational data assimilation that generalizes existing methods and studies the trade-offs between model and observational errors.

Findings

The approach can assimilate trajectories with partial data and model errors.

Solving the problem reduces to a two-point boundary value problem.

Allowing some dynamical error improves assimilation even with nearly perfect models.

Abstract

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.