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Data assimilation concepts
and methods
March 1999
By F. Bouttier and P.
Courtier
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8 . Three-dimensional variational analysis (3D-Var)
The principle of 3D-Var is to avoid the computation (A2) of the gain
completely by looking for the analysis as an approximate
solution to the equivalent minimization problem defined by the cost function
in
(A5). The solution is sought iteratively by performing
several evaluations of the cost function
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and of its gradient
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in order to approach the minimum using a suitable descent algorithm. The approximation lies in the fact that only a small number of iterations are performed. The minimization can be stopped by limiting artificially the number of iterations, or by requiring that the norm of the gradient
decreases by a predefined amount during
the minimization, which is an intrinsic measure of how much the analysis
is closer to the optimum than the initial point of the minimization. The
geometry of the minimization is suggested in Fig.
11 .
Figure 11 . Schematic representation of the variational cost-function minimization (here in a two-variable model space): the quadratic cost-function has the shape of a paraboloid, or bowl, with the minimum at the optimal analysis
. The minimization works by performing several line-searches
to move the control variable 
to areas where
the cost-function is smaller, usually by looking at the local slope (the
gradient) of the cost-function.In practice, the initial point of the minimization, or first guess, is taken equal to the background
. This is not compulsory, however, so it is important
to distinguish clearly between the terms background (which is
used in the definition of the cost function) and first guess (which
is used to initiate the minimization procedure). If the minimization is
satisfactory, the analysis will not depend significantly on the choice of
first guess, but it will always be sensitive to the background.A significant difficulty with 3D-Var is the need to design a model for
that properly defines background error covariances
for all pairs of model variables. In particular, it has to be symmetric
positive definite, and the background error variances must be realistic
when expressed in terms of observation parameters, because this is what
will determine the weight of the observations in the analysis.The popularity of 3D-Var stems from its conceptual simplicity and from the ease with which complex observation operators can be used, since only the operators and the adjoints of their tangent linear need to be provided1. Weakly non-linear observation operators can be used, with a small loss in the optimality of the result. As long as
is strictly
convex, there is still one and only one analysis.In most cases the observation error covariance matrix
is block-diagonal, or even diagonal, because there is no reason
to assume observation error correlations between independent observing networks,
observing platforms or stations, and instruments, except in some special
cases. It is easy to see that a block-diagonal 
implies that 
is a sum
of 
scalar cost-functions 
, each one defined by a submatrix 
and the
corresponding subsets 
and 
of the observation operators and values:
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The gradient
can be similarly decomposed. The
breakdown of 
is a useful diagnostic tool of
the behaviour of 3D-Var in terms of each observation type: the magnitude
of each term measures the misfit between the state 
and the corresponding subset of observations. It
can also simplify the coding of the computations of 
and its gradient2.Another advantage is the ability to enforce external weak (or penalty) constraints, such as balance properties, by putting additional terms into the cost function (usually denoted
). However, this can make the preconditioning
of the minimization problem difficult.ref: Parrish and Derber 1992, Courtier et al. 1998.
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1 whereas OI requires a background error covariance model between each observed variable and each model variable.
2 Actually the whole
can be
decomposed into as many elementary cost functions as there are observed
parameters, by redefining the observation space to be the eigenvectors of
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03.12.2001 |
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