  In order to leverage the information embedded in the background state and
observations, covariance matrices modelling is a pivotal point in data
assimilation algorithms. These matrices are often estimated from an ensemble of
observations or forecast differences. Nevertheless, for many industrial
applications the modelling still remains empirical based on some form of
expertise and physical constraints enforcement in the absence of historical
observations or predictions. We have developed two novel robust adaptive
assimilation methods named CUTE (Covariance Updating iTerativE) and PUB
(Partially Updating BLUE). These two non-parametric methods are based on
different optimization objectives, both capable of sequentially adapting
background error covariance matrices in order to improve assimilation results
under the assumption of a good knowledge of the observation error covariances.
We have compared these two methods with the standard approach using a
misspecified background matrix in a shallow water twin experiments framework
with a linear observation operator. Numerical experiments have shown that the
proposed methods bear a real advantage both in terms of posterior error
correlation identification and assimilation accuracy.
