6 Sequential Data Assimilation for
Nonlinear Dynamics: The Ensemble
Kalman Filter
GEIR EVENSEN
Nansen Environmental and Remote Sensing Center, Eergen, Norway
6.1 .Introduction
Sequential data assimilation methods have proven useful for many applications
in meteorology and oceanography. For example are most operational weather prediction systems applying a sequential data assimilation technique where observations are "assimilated" into the model whenever they are available.
For linear dynamics the optimal sequential technique is the Kalman filter. In the
Kalman filter an additional equation for the second-order statistic al moment is integrated forward in time to predict error statistics for the model forecast. The error
statistics are then used to calculate a variance-minimizing estimate whenever measurements are available.
For nonlinear dynamics the extended Kalman filter may be applied, in which an
approximate linearized equation is used for the prediction of error statistics. The
implementation of the extended Kalman filter for data assimilation in a multilayer
quasi-geostrophic (QG) model has previously been discussed by Evensen (1992).
The main result from this work is the finding of an apparent closure problem in the
error covariance evolution equation. The extended Kalman filter applies a closure
scheme where third- and higher-order moments in the error covariance evolution
equation are discarded. This simple closure technique results in a unbounded error
variance growth caused by the linearization performed when higher-order moments
are neglected. Thus, it has been shown that the error covariance equation is based
on a too simplified closure approximation and this may lead to a nonphysical error
variance evolution (see e.g., Evensen (1992), Miller et al. (1994), Gauthier et al.
(1993), and Bouttier (1994».
The Ensemble Kalman Filter (EnKF) was introduced by Evensen (1994b) as an
alternative to the traditional Extended Kalman Filter (EKF). It was shown that if
the dynamical model is written as a stochastic differential equation, one can derive
the Fokker-Planck equation for the time evolution ofthe probability density function which contains all the information about the prediction error statistics. The
EnKF is a sequential data assimilation method where the error statistics are predicted by solving the Fokker-Planck equation using Monte-Carlo or ensemble
integrations. By integrating an ensemble of model states forward in time it is possible to calculate statistical moments like mean and error covariances whenever
such information is required. Thus, alI the statistical information about the pre-
Nonlinear Dynamics: The Ensemble
Kalman Filter
GEIR EVENSEN
Nansen Environmental and Remote Sensing Center, Eergen, Norway
6.1 .Introduction
Sequential data assimilation methods have proven useful for many applications
in meteorology and oceanography. For example are most operational weather prediction systems applying a sequential data assimilation technique where observations are "assimilated" into the model whenever they are available.
For linear dynamics the optimal sequential technique is the Kalman filter. In the
Kalman filter an additional equation for the second-order statistic al moment is integrated forward in time to predict error statistics for the model forecast. The error
statistics are then used to calculate a variance-minimizing estimate whenever measurements are available.
For nonlinear dynamics the extended Kalman filter may be applied, in which an
approximate linearized equation is used for the prediction of error statistics. The
implementation of the extended Kalman filter for data assimilation in a multilayer
quasi-geostrophic (QG) model has previously been discussed by Evensen (1992).
The main result from this work is the finding of an apparent closure problem in the
error covariance evolution equation. The extended Kalman filter applies a closure
scheme where third- and higher-order moments in the error covariance evolution
equation are discarded. This simple closure technique results in a unbounded error
variance growth caused by the linearization performed when higher-order moments
are neglected. Thus, it has been shown that the error covariance equation is based
on a too simplified closure approximation and this may lead to a nonphysical error
variance evolution (see e.g., Evensen (1992), Miller et al. (1994), Gauthier et al.
(1993), and Bouttier (1994».
The Ensemble Kalman Filter (EnKF) was introduced by Evensen (1994b) as an
alternative to the traditional Extended Kalman Filter (EKF). It was shown that if
the dynamical model is written as a stochastic differential equation, one can derive
the Fokker-Planck equation for the time evolution ofthe probability density function which contains all the information about the prediction error statistics. The
EnKF is a sequential data assimilation method where the error statistics are predicted by solving the Fokker-Planck equation using Monte-Carlo or ensemble
integrations. By integrating an ensemble of model states forward in time it is possible to calculate statistical moments like mean and error covariances whenever
such information is required. Thus, alI the statistical information about the pre-