A better model initial condition combining the information from observations can improve the skill of weather or ocean forecasts dramatically. To this end, data assimilation schemes are proposed to transport information from sparse observations to the gridded model fields. To connect observations with state variables, variational schemes, and Kalman filter schemes have been widely adopted. Under similar statistical interpretations, most data assimilation schemes rely on Gaussian processes and optimal linear estimation theory due to their simplicity in reality. Background error covariances not only indicate the relationship between state variables, but also control the propagations of information through each grid points, thus playing a significant role in the data assimilation schemes (C. Liu et al., 2008). Despite its importance, and due to its huge dimensions and complex dynamics of state variables, covariances cannot be always easily estimated. In variational data assimilation schemes, the background error covariance is static. Traditionally, the error covariances are assumed to be a function of vertical and horizontal distance. The parameters of these functions can be inferred by using an innovation method (Xu et al. 2001, 2002), NMC method or ensemble methods (Buehner, 2005; Wang et al., 2014). Bannister (2008a, 2008b) discussed these numerical methods in details. As an efficient approximation of a Gaussian filter, recursive filters, and diffusion filters can be employed to set up inhomogeneous and anisotropic background error covariances (Derber et al., 2003; Zhang et al.2015). A more effective construction has been done through

 A data-driven prediction model based on empirical orthogonal function, complete ensemble empirical mode decomposition and artificial neural networks is proposed. Effectively considers the correlations not only of different spatial points but also of different ocean variables. Spatial domain prediction of sea surface multivariate for 30 days.

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The analytical four-dimensional ensemble variational (A-4DEnVar) data assimilation scheme inherits the advantages of the conventional four-dimensional variational (4D-Var) data assimilation scheme and removes the adjoint model. However, compatible operational improvements such as the reduction of the computational costs and the localization method should be considered when it is used in realistic systems. In this paper, the computational complexity of calculating the inverse of background error covariance (the B matrix) is decreased by a precondition transform method, i.e., introducing a new state variable whose product with the B matrix is the original state variable to be optimized in the cost function. Furthermore, an independent point (IP) scheme is combined to construct an implicit localization method and further decreases the computational cost. Based on the Princeton Ocean Model with the generalized coordinate system (POMgcs), the operational improved A-4DEnVar is applied to optimize the spatially varying bottom friction coefficients (BFCs) of the M2 constituent in the Bohai and Yellow seas. A twin experiment with idealized observations is designed to validate the effectiveness of the proposed method. In practical experiments, with no more than 10 IPs, the algorithm can assimilate observations from the National Astronomical Observatory (NAO) dataset and obtain a good simulation. The experimental performances increase with the increase of either the IPs or observations, which indicates the efficacy of the proposed algorithm.

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