Sequential Data Assimilation for Nonlinear Dynamics: The Ensemble Kalman Filter
105
Clearly the observations d must be treated as random variables to get the measurement error covariance matrix into the expres sion.
6.3.4 Summary
We now have a complete system of equations which constitutes the ensemble
Kalman filter (EnKF), and the resemblance with the standard Kalman filter is
maintained. This is also true for the forecast step. Each ensemble member evolves
in time according to the model dynamics and the ensemble covariance matrix of
the errors in the model equations, Qe' converges to Q in the limit of an infinite
ensemble size. The ensemble mean then evolves according to an equation of the
form
'l'k+ 1 = j('I'k)
(26)
= j('I'k) + n.I.
where n.l. represents the terms which may arise ifjis non-linear. One ofthe advantages of the EnKF is that the effect of these terms is retained since each ensemble
member is integrated independently by the model.
The covariance of the ensemble evolves according to an equation
k+ 1
k T
Pe =FPeF+Qe+n.I.,
(27)
where F is the tangent linear model evaluated at the current time step. This is again
an equation ofthe same form as is used in the standard Kalman filter, except ofthe
extra terms n.l. that may appear ifjis non-linear. Implicitly, the EnKF retains these
terms also for the error covariance evolution. Thus, ifthe ensemble mean is used as
the best estimate, with the ensemble covariance P fa interpreted as the error covariance pf.a, and by defining the observation error covariance matrix Re = R and the
model error covariance Qe= Q , the EnKF and the standard Kalman filter become
identical. This discussion shows that there is a unique correspondence between the
EnKF and the standard Kalman filter (for linear dynamics) and that one can certainly interpret the ensemble covariances as error covariances while the ensemble
mean is used as the best guess trajectory.
The extended Kalman filter applies the evolution equations (26) and (27) with
the n.l. terms neglected. However, the ensemble Kalman filter includes the full
effect ofthese terms and there are no linearizations or closure assumptions applied.
In addition, there is no need for a tangent linear operator or its adjoint, and this
makes the EnKF very easy to implement for practical applications.
105
Clearly the observations d must be treated as random variables to get the measurement error covariance matrix into the expres sion.
6.3.4 Summary
We now have a complete system of equations which constitutes the ensemble
Kalman filter (EnKF), and the resemblance with the standard Kalman filter is
maintained. This is also true for the forecast step. Each ensemble member evolves
in time according to the model dynamics and the ensemble covariance matrix of
the errors in the model equations, Qe' converges to Q in the limit of an infinite
ensemble size. The ensemble mean then evolves according to an equation of the
form
'l'k+ 1 = j('I'k)
(26)
= j('I'k) + n.I.
where n.l. represents the terms which may arise ifjis non-linear. One ofthe advantages of the EnKF is that the effect of these terms is retained since each ensemble
member is integrated independently by the model.
The covariance of the ensemble evolves according to an equation
k+ 1
k T
Pe =FPeF+Qe+n.I.,
(27)
where F is the tangent linear model evaluated at the current time step. This is again
an equation ofthe same form as is used in the standard Kalman filter, except ofthe
extra terms n.l. that may appear ifjis non-linear. Implicitly, the EnKF retains these
terms also for the error covariance evolution. Thus, ifthe ensemble mean is used as
the best estimate, with the ensemble covariance P fa interpreted as the error covariance pf.a, and by defining the observation error covariance matrix Re = R and the
model error covariance Qe= Q , the EnKF and the standard Kalman filter become
identical. This discussion shows that there is a unique correspondence between the
EnKF and the standard Kalman filter (for linear dynamics) and that one can certainly interpret the ensemble covariances as error covariances while the ensemble
mean is used as the best guess trajectory.
The extended Kalman filter applies the evolution equations (26) and (27) with
the n.l. terms neglected. However, the ensemble Kalman filter includes the full
effect ofthese terms and there are no linearizations or closure assumptions applied.
In addition, there is no need for a tangent linear operator or its adjoint, and this
makes the EnKF very easy to implement for practical applications.