**Whitaker J. S.**, and **T. M. Hamill**, 2002: Ensemble data
assimilation without perturbed observations. *Mon. Wea. Rev.*,
**130**, 1913-1924.

**ABSTRACT**

The ensemble Kalman filter (EnKF) is a data assimilation scheme based on the traditional Kalman-filter update equation. An ensemble of forecasts are to estimate the background-error covariances needed to compute the Kalman gain. It has been shown that if the same observations and the same gain are used to update each member of the ensemble, the ensemble will systematically underestimate analysis-error covariances. This will cause a degradation of subsequent analyses and may lead to filter divergence. For large ensembles, it has been shown that this problem can be alleviated by treating the observations as random variables, adding random perturbations to them with the correct statistics.

Two important consequences of sampling error in the estimate of analysis-error covariances in the EnKF are discussed here. The first results from the analysis-error covariance being a nonlinear function of the background-error covariance in the Kalman filter. Due to this nonlinearity, if the background-error covariance calculated from the ensemble is overestimated, there is a larger reduction in variance during the analysis than if it is underestimated. Hence, even if the ensemble background-error covariance estimates are unbiased, the analysis-error covariance estimates will negatively biased. This problem must be dealt with in any Kalman-filter based ensemble data assimilation scheme.

A second consequence of sampling error is particular to schemes like the EnKF that use perturbed observations. While this procedure gives asymptotically correct analysis-error covariance estimates for large ensembles, the addition of perturbed observations adds an additional source of sampling error related to the estimation of the observation-error covariances. In addition to reducing the accuracy of the analysis-error covariance estimate, this extra source of sampling error increases the probability that the analysis-error covariance will be underestimated. Because of this, ensemble data assimilation methods which use perturbed observations are expected to be less accurate than those which do not. Several ensemble filter formulations have recently been proposed which do not require perturbed observations.

Here we examine a particularly simple implementation which we call the "ensemble square-root filter", or EnSRF. The EnSRF uses the traditional Kalman gain for updating the ensemble mean but uses a "reduced" Kalman gain to update deviations from the ensemble mean. There is no additional computational cost incurred by the EnSRF relative to the EnKF when observations are processed serially. Using a hierarchy of perfect model assimilation experiments, we demonstrate that the elimination of the sampling error associated with the perturbed observations makes the EnSRF more accurate than the EnKF for the same ensemble size.