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Data assimilation concepts
and methods
March 1999
By F. Bouttier and P.
Courtier
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10 . Four-dimensional variational assimilation (4D-Var)
4D-Var is a simple generalization of 3D-Var for observations that are distributed in time. The equations are the same, provided the observation operators are generalized to include a forecast model that will allow a comparison between the model state and the observations at the appropriate time.
Over a given time interval, the analysis being at the initial time, and the observations being distributed among n times in the interval, we denote by the subscript
the quantities at any given observation time 
. Hence, 
, 
and 
are the observations, the model and the true states
at time 
, and 
is the
error covariance matrix for the observation errors 
. The observation operator 
at time
is linearized as 
. The
background error covariance matrix 
is only defined
at initial time, the time of the background 
and of the analysis 
.10.1 The four-dimensional analysis problem
In its general form, it is defined as the minimization of the following cost function:
|
which can be proven, like in the three-dimensional case detailed previously, to be equivalent to finding the maximum likelihood estimate of the analysis subject to the hypothesis of Gaussian errors.
The 4D-Var analysis, or four-dimensional variational assimilation problem, is by convention defined as the above minimization problem subject to the strong constraint that the sequence of model states
must be
a solution of the model equations:
|
where
is a predefined model forecast operator from the initial
time to 
. 4D-Var is thus a nonlinear constrained
optimization problem which is very difficult to solve in the general case.
Fortunately it can be greatly simplified with two hypotheses: Causality. The
forecast model can be expressed as the product of intermediate forecast
steps, which reflects the causality of nature. Usually it is the integration
of a numerical prediction model starting with  as the initial condition. If the times i are sorted, with
 so that  is the identity, then by denoting  the forecast step from  to  we have  and by recurrence |
|
Tangent linear hypothesis.
The cost function can be made quadratic by assuming, on top of the
linearization of  , that
the  operator can be linearized, i.e. |
|
where  is the tangent
linear (TL) model, i.e. the differential of  . For a discussion of this hypothesis, refer to the section
on the tangent linear hypothesis, in which the remarks made on  apply similarly to  . It explains
that the realism of the TL hypothesis depends not only on the model,
but also on the general characteristics of the assimilation system,
including notably the length of the 4D-Var time interval. |
The two hypotheses above simplify the general minimization problem to an unconstrained quadratic one which is numerically much easier to solve. The first term
of the
cost function is no more complicated than in 3D-Var and it will be left
out of this discussion. The evaluation of the second term 
would seem to require 
integrations
of the forecast model from the analysis time to each of the observation
times 
, and even more for the computation
of the gradient 
. We are
going to show that the computations can in fact be arranged in a much more
efficient way.10.2 Theorem: minimization of the 4D-Var cost function
and 
, requires one
direct model integration from times 0 to 
and one
suitably modified adjoint integration made of transposes
of the tangent linear model time-stepping operators 
.Proof:
The first stage is the direct
integration of the model from  to  , computing successively at each observation time  : |
1) the forecast state  , |
2) the "normalized departures"
 which are stored, |
3) the contributions to the cost
function  |
4) And finally  . |
To compute  it is necessary
to perform a slightly complex factorization: |
|
| and the last expression is easily evaluated from right to left using the following algorithm: |
5) initialize the so-called adjoint
variable  to zero at final time:  |
6) for each time step  the variable  is obtained by adding the adjoint forcing
 to  and by
performing the adjoint integration by multiplying the result
by  , i.e.  |
7) at the end of the recurrence,
the value of the adjoint variable  gives the required
result. |
The terminology employed in the algorithm reflects the fact that the computations look like the integration of an adjoint model backward in time with a time-stepping defined by the transpose time-stepping operators
and an external
forcing 
, which depends on the distance
between the model trajectory and the observations. In this discrete presentation
it is just a convenient way of evaluating an algebraic expression1.10.3 Properties of 4D-Var
Figure 12 . Example of 4D-Var intermittent assimilation in a numerical forecasting system. Every 6 hours a 4D- Var is performed to assimilate the most recent observations, using a segment of the previous forecast as background. This updates the initial model trajectory for the subsequent forecast.
When compared to a 3-D analysis algorithm in a sequential assimilation system, 4D-Var has the following characteristics:
| • it works only under the assumption that the model is perfect. Problems can be expected if model error are large. |
• it requires the implementation
of the rather special  operators, the
so-called adjoint model. This can be a lot of work if the forecast
model is complex. |
• in a real-time system
it requires the assimilation to wait for the observations over the
whole 4D-Var time interval to be available before the analysis procedure
can begin, whereas sequential systems can process observations shortly
after they are available. This can delay2
the availability of  . |
•  is used
as the initial state for a forecast, then by construction of 4D-Var
one is sure that the forecast will be completely consistent with the
model equations and the four-dimensional distribution of observations
until the end of the 4D-Var time interval  (the cutoff time). This makes
intermittent 4D-Var a very suitable system for numerical forecasting
(Fig. 12 ). |
• 4D-Var is an optimal
assimilation algorithm over its time period thanks to the following
theorem. It means that it uses the observations as well as possible,
even if  is not perfect, to provide  in a much less expensive way than the equivalent Kalman Filter.
For instance, the coupling between advection and observed information
in illustrated in Fig. 13 . |
Figure 13 . Example of propagation of the information by 4D-Var (or, equivalently, a Kalman filter) in a 1-D model with advection (i.e. transport) of a scalar quantity. All features observed at any point within the 4D-Var time window
will be related to the correct upstream point of the control variable
by the tangent linear and adjoint model, along the characteristic lines
of the flow (dashed).10.4 Equivalence between 4D-Var and the Kalman Filter
Over a given time interval, under the assumption that the model is perfect, with the same input data (initial background and its covariances
, distribution of observations and their
covariances 
), the
4D-Var analysis at the end of the time interval is equal to the Kalman filter
analysis at the same time.This theorem is discussed in more details in the section about the Kalman filter algorithm, with a discussion of the pros and cons of using 4D-Var.
A special property of the 4D-Var analysis in the middle of the time interval is that it uses all the observations simultaneously, not just the ones before the analysis time. It is said that 4D-Var is a smoothing algorithm3.
Ref: Talagrand and Courtier 1987, Thépaut and Courtier 1991, Rabier and Courtier 1992, Lacarra and Talagrand 1988, Errico et al. 1993.
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1 In a continuous (in time) presentation, the concept of adjoint model could be carried much further into the area of differential equations. However, this is not relevant to real models where the adjoint of the discretized model must be used, instead of the discretization of a continuous adjoint model. The only relevant case is if some continuous operators have a simple adjoint: then, with a careful discretization that preserves this property, the implementation of the discrete transpose operators can be simplified.
2 Some special implementations of 4D-Var can partly solve this problem.
3 Equivalent to the Kalman smoother algorithm which is a generalization of the Kalman filter, but at a much smaller cost.
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03.12.2001 |
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