Estimating cross-correlation parameters between forecast and observation errors within the ensemble transform Kalman filter with cross correlation (ETKFCC)

Abstract. The Kalman filter (KF) and the ensemble KF (EnKF) are formulated under the assumption of no cross-correlation between the forecast and observation errors. However, some data assimilation systems assimilate analysis products such as optimal interpolation analyses and satellite retrievals, which may contain errors correlated with the forecast errors. The authors’ previous study extended the KF and the ensemble transform KF (ETKF) to account for the cross-correlation (KFCC and ETKFCC, respectively) and demonstrated that the ETKFCC significantly outperforms the ETKF using the Lorenz-96 model. However, these experiments assumed that the cross-correlation parameters were perfectly known although they are unknown in practice. In this study, we extended the previous study by proposing a novel method to estimate the cross-correlation parameters from innovation statistics. We performed three experiments: (i) ETKFCC with estimated parameters, (ii) ETKFCC with prescribed true parameters, and (iii) ETKF. The results showed that the parameters were estimated well for positive cross-correlations, but not for unlikely cases of negative cross-correlations. For positive cross-correlations, the ETKFCC with the estimated parameters significantly outperforms the ETKF and is comparable to the ETKFCC with the prescribed true parameters when the cross-correlation is 0.2–0.7.