Monday, 12 January 2004: 5:15 PM
Amelioration of bias in the Ensemble Transform Kalman Filter
The Ensemble Transform Kalman Filter (ETKF) has been extensively used in the field of targeted observations and, as an ensemble generation technique, has been shown to provide superior ensembles to the breeding technique at a similar computational expense. It is shown that mean Ensemble Transform Kalman Filter (ETKF) analysis increments and error covariance updates are far from optimal when the ensemble size K is significantly smaller than the rank r of the normalized observation space forecast error covariance matrix. These biases occur even when the ensemble represents an accurate sampling of the true forecast error covariance matrix. It is shown that these biases are significantly ameliorated by accounting for (a) the fact that the sample covariance of K forecast trials systematically overestimates the true error variance within the ensemble subspace whenever (K-1)<r, and (b) the frequency with which an ensemble based eigenvector lies parallel to a true eigenvector. The amelioration technique applies to both the error covariance update and the analysis increment, is eigenvector dependent and provides a theoretical basis for understanding how and why bias amelioration should change as the ensemble size is increased. The form of the ETKF considered here does not include error covariance localization. Although the bias amelioration technique is presented within the algebraic framework of the ETKF, it could also be applied to the wider class of ensemble filters to which the ETKF belongs. The amelioration technique suggests the existence of highly scaleable data assimilation algorithms suitable for use with huge ensembles (K>2,000).