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Data assimilation concepts
and methods
March 1999
By F. Bouttier and P.
Courtier
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14 . The extended Kalman filter (EKF)
The Kalman Filter and its extended version (EKF) are developments of the least-squares analysis method in the framework of a sequential data assimilation, in which each background is provided by a forecast that starts from the previous analysis. It is adapted to the real-time assimilation1 of observations distributed in time into a forecast model
.The analysis equations of the linear Kalman Filter are exactly the ones already described in the least-squares analysis theorem. The notation is the same, except that the background (i.e. forecast) and analysis error covariance matrices are now respectively denoted
and 
. The
background state 
is a forecast denoted 
.14.1 Notation and hypotheses
They are the same as in the least-squares analysis theorem, except that:
• the background and analysis
error covariance matrices  and  are respectively
replaced by  and  to denote
the fact that the background is now a forecast. |
• The time index  of each quantity is denoted by the suffix  . The model forecast operator from dates  to  is denoted by  |
• forecast errors:
the deviation of the forecast prediction from the true evolution,
 , is called the model error2 and we assume that it is not biased3
and that the model error covariance matrix  is known. |
• uncorrelated
analysis and model errors: the analysis errors  and model errors
of the subsequent forecast  are assumed
to be mutually uncorrelated. |
• linearized forecast
operator: the variations of the model prediction in the vicinity
of the forecast state are assumed to be a linear function of the initial
state: for any  close enough
to  ,  ,where  is a linear
operator. |
14.2 Theorem: the KF algorithm
Under the specified hypotheses the optimal way (in the least squares sense) to assimilate sequentially the observations is given by the Kalman filter algorithm defined below by recurrence over the observation times  :
and the analyses are the sequences of  . |
Proof:
The forecast equation (KF1) just translates the fact that we
use the model  to evolve the model state, starting
from the previous optimal analysis  . The
equation (KF2) is obtained
by first subtracting  from (KF1)
and using the linearity of the forecast operator: |
|
Multiplying it on the right by
its transpose and taking the expectation of the result yields, by
definition,  on the left-hand side, and on the right-hand side four
terms. Two of these are  and  by definition. The remaining two terms are cross-correlations between
the analysis error  and the model
error for  , which are assumed to be zero. This means
that  provided by (KF2) is the background error covariance matrix
for the analysis at time  . |
The equations (KF3), (KF4) and (KF5) are simply the least-square analysis equations (A2), (A1) and (A4) that were proven above, using
as background
errors, and assuming that 
is computed
optimally.14.3 Theorem: KF/4D-Var equivalence
Over the same time interval  assuming that
 (i.e. the model is perfect), and that both algorithms
use the same data (notably,  is the initial
background error covariance matrix), then there is equality between
This theorem means that the KF verifies the four-dimensional least-squares optimality theory expressed by the 4D-Var cost function, although it is defined by a sequence of 3-D analyses, whereas 4D-Var solves the 4-D problem globally. |
produced by the above Kalman filter algorithm, and
.14.4 The Extended Kalman Filter (EKF)
The Kalman filter algorithm can be generalized to non-linear
and 
operators,
although it means that neither the optimality of the analysis nor the equivalence
with 4D-Var hold in that case. If 
is non-linear, 
can be defined as its tangent linear
in the vicinity of 
, as discussed
in a previous section. Similarly, if 
is non-linear, which is the case of most meteorological and
oceanographical models, 
can be defined
as the tangent linear forecast model in the vicinity of 
, i.e. we assume that for any likely initial state
(notably 
),
|
and the realism of this hypothesis must be appreciated using physical arguments, as already discussed about the observation operator and 4D-Var. If
and/or 
are non-linear,
the algorithm written above is called the Extended Kalman Filter.
Note that the linearization of 
interacts with the model errors in a possibly complicated way,
as can be seen from the proof of Eq. (KF2) above. If non-linearities are important,
it may be necessary to include empirical correction terms in the equation,
or to use a more general stochastic prediction method such as an ensemble
prediction (or Monte Carlo) method, which yields an algorithm
known as the Ensemble Kalman Filter.14.5 Comments on the KF algorithm
The input to the algorithms is: the definition of the model and the observation operator, the initial condition for
when the recurrence of the filter is started4, the sequence of observations 
, and the sequence of model and observation error covariance
matrices 
. The output is the sequence of estimates 
of the model state and its error covariance matrix. The organization
of the KF assimilation looks like a coupled stream of estimations of model
states and error covariances (Fig.
16 ).
Figure 16 . The organization of computations in a KF or EKF assimilation.
The variational form of the least-squares analysis can be used in the analysis step of the Kalman filter, instead of the explicit equations written above.
The numerical cost of the KF or EKF is that of the analysis itself, plus the estimation of the analysis error covariances, discussed in a specific section, plus the (KF2) covariance forecast equation which requires n forecasts of the tangent linear model (
being the dimension of the model state) to build the operator
. The storage cost itself is significant, since each 
matrix is 
(only a half can be stored since
they are symmetric) and in (KF5) the 
matrix must be evaluated and stored too (unless the variational
form is used, in which case 
evaluations
of the gradient of the cost function must be performed to build the Hessian
which must then be inverted). It means that the cost of the KF is much larger
than 4D-Var, even with small models. The algorithm should rather be regarded
as a reference in the design of more approximate assimilation algorithms
which are being developed nowadays. It is still not clear what is the best
way to approximate the KF, and the answer will probably be application-dependent.There are many similarities between 4D-VAR and the EKF and it is important to understand the fundamental differences between them:
| • 4D-VAR can be run for assimilation in a realistic NWP framework because it is computationally much cheaper than the KF or EKF. |
| • 4D-VAR is more optimal than the (linear or extended) KF inside the time interval for optimization because it uses all the observations at once, i.e. it is not sequential, it is a smoother. |
• unlike the EKF, 4D-VAR
relies on the hypothesis that the model is perfect (i.e.  ). |
| • 4D-VAR can only be run for a finite time interval, especially if the dynamical model is non-linear, whereas the EKF can in principle be run forever. |
• 4D-VAR itself does not
provide an estimate of  , a specific procedure to estimate
the quality of the analysis must be applied, which costs as much as
running the equivalent EKF. |
Ref: Ghil 1989, Lacarra and Talagrand 1988, Errico et al. 1993.
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1 The word filter characterizes an assimilation techniques that uses only observation from the past to perform each analysis. An algorithm that uses observations from both past and future is called a smoother. 4D-Var can be regarded as a smoother. Observation smoothing can be useful for non-real time data assimilation, e.g. reanalysis, although the idea has not been used much yet. The Kalman filter has a smoother version called Kalman smoother.
2 Or modelling error.
3 This is equivalent to assuming that the background errors are unbiased, so it is not really a new hypothesis.
4 Note that it is not well known whether, after a long time, the analysis ceases or not to depend significantly on the way the KF is initialized.
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03.12.2001 |
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© ECMWF |
