The Structure of Background-error Covariance in a Four-dimensional Variational Data Assimilation System： Single-point Experiment

A four dimensional variational data assimilation （4DVar） based on a dimension-reduced projection （DRP-4DVar） has been developed as a hybrid of the 4DVar and Ensemble Kalman filter （EnKF） concepts. Its good flow-dependent features are demonstrated in single-point experiments through comparisons with adjointbased 4DVar and three-dimensional variational data （3DVar） assimilations using the fifth-generation Pennsylvania State University-National Center for Atmospheric Research Mesoscale Model （MM5）. The results reveal that DRP-4DVar can reasonably generate a background error covariance matrix （simply B-matrix） during the assimilation window from an initial estimation using a number of initial condition dependent historical forecast samples. In contrast, flow-dependence in the B-matrix of MM5 4DVar is barely detectable. It is argued that use of diagonal estimation in the B-matrix of the MM5 4DVar method at the initial time leads to this failure. The experiments also show that the increments produced by DRP-4DVar are anisotropic and no longer symmetric with respect to observation location due to the effects of the weather trends captured in its B-matrix. This differs from the MM5 3DVar which does not consider the influence of heterogeneous forcing on the correlation structure of the B-matrix, a condition that is realistic for many situations. Thus, the MM5 3DVar assimilation could only present an isotropic and homogeneous structure in its increments.