Background and observation error covariance matrices information in the physical nudging equations.

In this contribution we show how to extend the deterministic physical nudging scheme in order to include two important ingredients, the background and observation error covariance matrices, that are common features of classical data assimilation schemes. The derivation exploits the relation between the Langevin and Fokker-Planck equations to formulate the data assimilation problem and retrieve the assimilation scheme in coordinate space. The background, $\mathbf{B}$, was naturally introduced noticing the role of the diffusion matrix in the Fokker-Planck equation related to the noise modulation in the correspondent Langevin equation. The observation error covariance matrix $\mathbf{R}$ was introduced solving a Kolmogorov backward problem related to our formulation noticing that the final condition, instead of being considered as a Dirac $\delta$ function, could be simply represented by a Gaussian function with a covariance matrix corresponding to $\mathbf{R}$. In this way the initial and final conditions of the DA problem are treated dynamically.