Use of the breeding technique to estimate the structure of the analysis "errors of the day"

A 3D-variational data assimilation scheme for a quasi-geostrophic channel model (Morss, 1998) is used to study the structure of the background error and its relationship to the corresponding bred vectors. The "true" evolution of the model atmosphere is defined by an integration of the model and "rawinsonde observations" are simulated by randomly perturbing the true state at fixed locations. Case studies using different observational densities are considered to compare the evolution of the Bred Vectors to the spatial structure of the background error. In addition, the bred vector dimension (BV-dimension), defined by Patil et al. (2001) is applied to the bred vectors. It is found that after 3-5 days the bred vectors develop well organized structures which are very similar for the two different norms (enstrophy and streamfunction) considered in this paper. When 10 surrogate bred vectors (corresponding to different days from that of the background error) are used to describe the local patterns of the background error, the explained variance is quite high, about 85-88%, indicating that the statistical average properties of the bred vectors represent well those of the background error. However, a subspace of 10 bred vectors corresponding to the time of the background error increased the percentage of explained variance to 96-98%, with the largest percentage when the background errors are large. These results suggest that a statistical basis of bred vectors collected over time can be used to create an effective constant background error covariance for data assimilation with 3D-Var. Including the "errors of the day" through the use of bred vectors corresponding to the background forecast time can bring an additional significant improvement.