Including Cross Correlation between Forecast and Observation Errors in an Ensemble Kalman Filter

Abstract The Kalman filter is an unbiased minimum variance estimator under the assumption of no cross correlation between the forecast and observation errors. Some data assimilation (DA) systems assimilate analyzed data as if they were independent observations, but they may contain errors correlated with the forecast errors. Examples include satellite retrievals used in atmospheric DA systems and optimally interpolated sea surface temperature (OISST) analysis data used in ocean DA systems. Even if the forecasts are not directly used in satellite retrievals or OISST, the model formulations and analysis methods are generally similar and could introduce correlated errors. This study brings to light the potential impacts of including the cross correlation in an ensemble Kalman filter using perfect-model twin experiments with the Lorenz-96 model. The observation errors are generated by mixing the forecast errors in the observation space with independent random noise. We formulate the ensemble transform Kalman filter (ETKF) with the cross correlation (ETKFCC) with only minor modifications to the ETKF and implement it with the Lorenz-96 model. We find nonnegligible impacts from the cross correlation. The results show that the ETKFCC is significantly more accurate than the standard ETKF at negligible additional computational cost.