Analysis Scheme in the Ensemble Kalman Filter

Gerrit Burgers Royal Netherlands Meteorological Institute, De Bilt, the Netherlands

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Peter Jan van Leeuwen Institute for Marine and Atmospheric Research Utrecht, Utrecht University, Utrecht, the Netherlands

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Geir Evensen Nansen Environmental and Remote Sensing Center, Bergen, Norway

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Abstract

This paper discusses an important issue related to the implementation and interpretation of the analysis scheme in the ensemble Kalman filter. It is shown that the observations must be treated as random variables at the analysis steps. That is, one should add random perturbations with the correct statistics to the observations and generate an ensemble of observations that then is used in updating the ensemble of model states. Traditionally, this has not been done in previous applications of the ensemble Kalman filter and, as will be shown, this has resulted in an updated ensemble with a variance that is too low.

This simple modification of the analysis scheme results in a completely consistent approach if the covariance of the ensemble of model states is interpreted as the prediction error covariance, and there are no further requirements on the ensemble Kalman filter method, except for the use of an ensemble of sufficient size. Thus, there is a unique correspondence between the error statistics from the ensemble Kalman filter and the standard Kalman filter approach.

Corresponding author address: Gerrit Burgers, Oceanographic Research Division, Royal Netherlands Meteorological Institute, P.O. Box 201, 3730 AE De Bilt, the Netherlands.

Email: burgers@knmi.nl

Abstract

This paper discusses an important issue related to the implementation and interpretation of the analysis scheme in the ensemble Kalman filter. It is shown that the observations must be treated as random variables at the analysis steps. That is, one should add random perturbations with the correct statistics to the observations and generate an ensemble of observations that then is used in updating the ensemble of model states. Traditionally, this has not been done in previous applications of the ensemble Kalman filter and, as will be shown, this has resulted in an updated ensemble with a variance that is too low.

This simple modification of the analysis scheme results in a completely consistent approach if the covariance of the ensemble of model states is interpreted as the prediction error covariance, and there are no further requirements on the ensemble Kalman filter method, except for the use of an ensemble of sufficient size. Thus, there is a unique correspondence between the error statistics from the ensemble Kalman filter and the standard Kalman filter approach.

Corresponding author address: Gerrit Burgers, Oceanographic Research Division, Royal Netherlands Meteorological Institute, P.O. Box 201, 3730 AE De Bilt, the Netherlands.

Email: burgers@knmi.nl

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