Model and observation-error covariance matrix information in the physical nudging equations

In this work we show how to extend the deterministic physical nudging scheme in order to include two important ingredients, the model and observation-error covariance matrices, which are common features of classical data-assimilation schemes. The method exploits the relation between a stochastic differential equation and the evolution of its probability density via the Fokker–Planck equation. Observations are introduced by evolving the posterior probability density backward in time to obtain a so-called smoother. To obtain a computationally feasible scheme, we used the small-time approximation, resulting in an efficient nudging scheme built from first principles. We explored the capabilities of this new nudging method with the low-dimensional Lorenz 1963 model and a surface quasi-geostrophic turbulence model on a (Formula presented.) grid, with many degrees of freedom. We show that the new method is more accurate than a 3DVar at similar computational cost, and is accurate and easy to implement in the high-dimensional system. The new scheme has the potential to be used in extremely high-dimensional systems, because ensemble integrations and adjoint models are avoided.