JUNE 1998 1719BURGERS ET AL.
q 1998 American Meteorological Society
Analysis Scheme in the Ensemble Kalman Filter
GERRIT BURGERS
Royal Netherlands Meteorological Institute, De Bilt, the Netherlands
PETER JAN VAN LEEUWEN
Institute for Marine and Atmospheric Research Utrecht, Utrecht University, Utrecht, the Netherlands
GEIR EVENSEN
Nansen Environmental and Remote Sensing Center, Bergen, Norway
(Manuscript received 17 January 1997, in final form 10 June 1997)
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.
1. Introduction
The ensemble Kalman filter (EnKF) was introduced
by Evensen (1994b) as an alternative to the traditional
extended Kalman filter (EKF), which has been shown
to be based on a statistical linearization or closure ap-
proximation that is too severe to be useful for some
cases with strongly nonlinear dynamics (see Evensen
1992; Miller et al. 1994; Gauthier et al. 1993; Bouttier
1994). If the dynamical model is written as a stochastic
differential equation, one can derive the Fokker–Planck
or Kolmogorov’s equation for the time evolution of the
probability density function, which contains all the in-
formation about the prediction error statistics. The
EnKF is a sequential data assimilation method, using
Monte Carlo or ensemble integrations. By integrating
an ensemble of model states forward in time, it is pos-
Corresponding author address: Gerrit Burgers, Oceanographic Re-
search Division, Royal Netherlands Meteorological Institute, P.O. Box
201, 3730 AE De Bilt, the Netherlands.
E-mail: burgers@knmi.nl
sible to calculate the mean and error covariances needed
at analysis times.
The analysis scheme that has been proposed in Ev-
ensen (1994b) uses the traditional update equation of
the Kalman filter (KF), except that the gain is calculated
from the error covariances provided by the ensemble of
model states. It was also illustrated that a new ensemble
representing the analyzed state could be generated by
updating each ensemble member individually using the
same analysis equation.
The EnKF is attractive since it avoids many of the
problems associated with the traditional extended Kal-
man filter; for example, there is no closure problem as
is introduced in the extended Kalman filter by neglecting
contributions from higher-order statistical moments in
the error covariance evolution equation. It can also be
computed at a much lower numerical cost, since usually
a rather limited number of model states is sufficient for
reasonable statistical convergence. For sufficient ensem-
ble sizes, the errors will be dominated by statistical
noise, not by closure problems or unbounded error vari-
ance growth.

1720 VOLUME 126MONTHLY WEATHER REVIEW
The EnKF has been further discussed and applied
with success in a twin experiment in Evensen (1994a)
and in a realistic application for the Agulhas Current
using Geosat altimeter data in Evensen and van Leeu-
wen (1996).
A serious point that will be discussed here and was
not known during the previous applications of the EnKF
is that for the analysis scheme to be consistent one must
treat the observations as random variables. This as-
sumption was applied implicitly in the derivation of the
analysis scheme in Evensen (1994b) but has not been
used in the following applications of the EnKF. It will
be shown that unless a new ensemble of observations
is generated at each analysis time, by adding pertur-
bations drawn from a distribution with zero mean and
covariance equal to the measurement error covariance
matrix, the updated ensemble will have a variance that
is too low, although the ensemble mean is not affected.
A similar problem is present in the ensemble smooth-
er proposed by van Leeuwen and Evensen (1996), al-
though there only the posterior error variance estimate
is influenced since the solution is calculated simulta-
neously in space and time.
There was also another issue pointed out in Evensen
(1994b): the error covariance matrices for the fore-
casted and the analyzed estimate, P
f
and P
a
, are in the
Kalman filter defined in terms of the true state as
fftftT
P 5 (c 2 c )(c 2 c ) , (1)
aatatT
P 5 (c 2 c )(c 2 c ) , (2)
where the overbar denotes an expectation value, c is
the model state vector at a particular time, and the su-
perscripts f, a, and t represent forecast, analyzed, and
true state, respectively. However, since the true state is
not known, it is more convenient to consider ensemble
covariance matrices around the ensemble mean c ,
ff f ff fT
P . P 5 (c 2 c )(c 2 c ) , (3)
e
aa aaaaT
P . P 5 (c 2 c )(c 2 c ) , (4)
e
where now the overbar denotes an average over the
ensemble. It will be shown that if the ensemble mean
is used as the best estimate, the ensemble covariance
can consistently be interpreted as the error covariance
of the best estimate.
This leads to an interpretation of the EnKF as a purely
statistical Monte Carlo method where the ensemble of
model states evolves in state space with the mean as
the best estimate and the spreading of the ensemble as
the error variance. At measurement times each obser-
vation is represented by another ensemble, where the
mean is the actual measurement and the variance of the
ensemble represents the measurement errors.
Ensembles of observations were used by Daley and
Mayer (1986) in an observations system simulation ex-
periment, and more recently by Houtekamer and De-
rome (1995) in an ensemble prediction system, and by
A. F. Bennett (1996) as well (personal communication)
as well to derive posterior covariances for the repre-
senter method. Recently, Houtekamer and Mitchell
(1998) have used ensembles of observations in the ap-
plication of an ensemble Kalman filter technique.
In the following sections we will present an analysis
of the consequences of using the ensemble covariance
instead of the error covariance, and then present a mod-
ification of the analysis scheme where the observations
are treated as random variables. Finally, the differences
between the analysis steps of the standard Kalman filter,
the original EnKF, and the improved scheme presented
here will be illustrated by a simple example. An appli-
cation of the improved scheme to a more complex ex-
ample, that of the strongly nonlinear Lorenz equations,
is treated in Evensen (1997).
2. The standard Kalman filter
It is instructive first to review the analysis step in the
standard Kalman filter where the analyzed estimate is
determined by a linear combination of the vector of
measurements d and the forecasted model state vector
c
f
. The linear combination is chosen to minimize the
variance in the analyzed estimate c
a
, which is then
given by the equation
c
a
5 c
f
1 K(d 2 Hc
f
). (5)
The Kalman gain matrix K is given by
K 5 P
f
H
T
(HP
f
H
T
1 W)
21
. (6)
It is a function of the model state error covariance matrix
P
f
, the data error covariance matrix W, and the mea-
surement matrix H that relates the model state to the
data. In particular, the true model state is related to the
true observations through
d
t
5 Hc
t
, (7)
assuming no representation errors in the measurement
operator H, while the actual measurements are defined
by the relation
d 5 Hc
t
1 e, (8)
with e the measurement errors. The measurement error
covariance matrix is defined as
T
W 5 ee
ttT
5 (d 2 Hc )(d 2 Hc )
ttT
5 (d 2 d )(d 2 d ) . (9)
As usual, we assume that (d 2 d
t
)(c 2 c
t
)
T
5 0.
The error covariance of the analyzed model state vec-
tor is reduced with respect to the error covariance of
the forecasted state as