Unbiased ensemble square root filters

There is an established framework that describes a class of ensemble Kalman filter algorithms as square root filters (SRFs). These algorithms produce analyses by updating a state estimate and a square root of the ensemble covariance matrix. The matrix square root of the forecast covariance is post-multiplied by another matrix to give a matrix square root of the analysis covariance. The choice of post-multiplier is not unique and can be multiplied by any orthogonal matrix to give another scheme that is also a SRF. Not all filters of this type bear the desired relationship to the forecast ensemble: the analysis ensemble mean may not be equal to the analysis state estimate and consequently there may be an accompanying shortfall in the spread of the analysis ensemble as expressed by the ensemble covariance matrix. This points to the need for a restricted version of the notion of an ensemble SRF, which we call an unbiased ensemble SRF. This paper provides a generic set of necessary and sufficient conditions for the scheme to be centred on the analysis state estimate (unbiased). A few of these results have already been published elsewhere in the literature; this paper brings these together with new results and provides simple proofs and examples, as well as a mathematical description of the set of unbiased ensemble SRFs. Many (but not all) published ensemble SRF algorithms satisfy the criteria that we establish. While these conditions are not a cure-all and cannot deal with independent sources of bias such as model and observation errors, they should be useful to designers of ensemble SRFs in the future.