Ensemble Data Assimilation without Perturbed Observations

Abstract

The ensemble Kalman filter (EnKF) is a data assimilation scheme based on the traditional Kalman filter update equation. An ensemble of forecasts are used to estimate the background-error covariances needed to compute the Kalman gain. It is known 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 is known 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 backgrounderror covariance in the Kalman filter. Due to this nonlinearity, analysis-error covariance estimates may be negatively biased, even if the ensemble background-error covariance estimates are unbiased. 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 that use perturbed observations are expected to be less accurate than those which do not. Several ensemble filter formulations have recently been proposed that do not require perturbed observations. This study examines a particularly simple implementation called 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 the observations have independent errors and are processed one at a time. Using a hierarchy of perfect model assimilation experiments, it is demonstrated 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.

Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter

“…However, this results in an analysis ensemble whose sample covariance is smaller than the analysis covariance P a given by (11), unless the observations are artificially perturbed so that each ensemble member is updated using different random realization of the perturbed observations [5,20]. Ensemble square-root filters [1,45,4,43,36,37] instead use more involved but deterministic algorithms to generate an analysis ensemble with the desired sample mean and covariance. As such, their analyses coincide exactly with the Kalman filter equations in the linear scenario of the previous section.…”

“…Localization is generally done either explicitly, considering only the observations from a region surrounding the location of the analysis [29,20,30,1,36,37], or implicitly, by multiplying the entries in P b by a distance-dependent function that decays to zero beyond a certain distance, so that observations do not affect the model state beyond that distance [21,17,45]. We will follow the explicit approach here, doing a separate analysis for each spatial grid point of the model.…”

“…Many variations on the Ensemble Kalman Filter have been published in the geophysical literature, and this article draws ideas from a number of them [1,2,4,17,20,21,30,36,37,45,48]. These articles in turn draw ideas both from earlier work on geophysical data assimilation and from the engineering and mathematics literature on nonlinear filtering.…”

Assessment of a Nonlinear Ensemble Transform Filter for High-Dimensional Data Assimilation

“…This partly accounts for the nonlinear nature of the model (Anderson 2012). Stochastic EnKFs (Burgers et al 1998;Houtekamer and Mitchell 1998) introduce additional sampling errors (Whitaker and Hamill 2002). In contrast, deterministic EnKFs (Tippett et al 2003) transform the prior ensemble such that its first two moments exactly match the theoretical KF values.…”

“…The overlapping observation regions of different local domains ensure that the covariances in the ensemble covariance matrix are taken into account. Mathematically, the observation impact is reduced with distance by multiplying R 21 by an appropriate correlation function through the use of a Schur product (Whitaker and Hamill 2002;Kirchgessner et al 2014). This term appears in Eqs.…”