Reduced rank static error covariance for high-dimensional applications
A method for creating static (e.g., stationary) error covariance of reduced rank for potential use in hybrid variational-ensemble data assimilation is presented. The choice of reduced rank versus full rank static error covariance is made in order to allow the use of an improved Hessian preconditioning in high-dimensional applications. In particular, this method relies on using block circulant matrices to create a high-dimensional global covariance matrix from a low-dimensional local sub-matrix. Although any covariance used in variational data assimilation would be an acceptable choice for the pre-defined full-rank static error covariance, for convenience and simplicity, we use a symmetric Topelitz matrix as a prototype of static error covariance. The methodology creates a square root covariance, which has a practical advantage for Hessian preconditioning in reduced rank, ensemble-based data assimilation. The experiments conducted examine multivariate covariance that includes the impact of cross-variable correlations, in order to have a more realistic assessment of the value of the constructed static error covariance approximation. The results show that it may be possible to reduce the rank of matrix to O(10) and still obtain an acceptable approximation of the full-rank static covariance matrix.