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Assimilation Techniques: 4dVar
April 2001
By Mike Fisher
European Centre for Medium-Range Weather
Forecasts.
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5 . A cautionary example
In view of the results presented above, it is tempting to ascribe the superior performance of 4dVar (relative to 3dVar) to the dynamical propagation of the covariance matrix. In this section, a simple example will be presented to show that there is more to 4dVar than covariance propagation.
Consider a 4dVar system with the following characteristics:
•  is an orthogonal matrix. (That is  .) |
•  |
•  |
• The entire state vector
is observed at some single time  , so that  . |
Now, the covariance matrix of analysis error at the beginning of the assimilation window ofa 4dVar system is the inverse of the Hessian matrix of the cost function (see, for example Rabier and Courtier, 1992).
For observations at a single time, a 4dVar analysis has:
|
(13) |
|
(14) |
|
(15) |
of analysis error implied by the corresponding 3dVar analysis.
At later times during the analysis window, the analysis error covariance
matrix is dynamically propagated:
|
(16) |
, and since 
we find that the covariance matrix is constant throughout the analysis
window, and equal to the corresponding 3dVar matrix. By the same argument,
the covariance matrix of background error is also constant throughout the
analysis window. For the particular system described above, there is no covariance propagation. The covariance matrices of both analysis and background error are identical to those of the corresponding 3dVar system. It might be imagined that the 3dVar and 4dVar analyses are identical. This is not the case. Moreover, 4dVar is superior to 3dVar. To see this, let us rewrite the 4dVar analysis equation (assuming a linear model and linear observation operators) in the form:
|
(17) |
by 
:
|
(18) |
, with the model state at time 
. Similarly, the function of the adjoint integration in 4dVar is
to propagate the information from the observation back in time to the beginning
of the analysis window. This is missing in 3dVar. Fig. 3 shows an example in which the state vector is a single wind vector
and the model dynamics corresponds to rotation of the vector through
an angle 
. The correct analysis at time 
lies between the observation and the background at time 
. I.e., the analysis should increase the wind speed,
but should not alter its direction. The 3dVar and 4dVar analyses at time
are also shown. 4dVar is optimal. Also shown is
a so-called 3dFGAT analysis, which compares the observation and background
at the correct time, but does not propagate the increment back in time to
. 3dFGAT is superior to 3dVar, but is not optimal. The ECMWF 40-year
reanalysis project uses a 3dFGAT system. It is worth reiterating that for this simple example, the differences between 4dVar, 3dFGAT and 3dVar are not due to covariance propagation, since all three systems have the same covarince matrices of background and analysis error. Rather, the differences are due to the different ways in which the model state is propagated to the time of the observation, and the increment is propagated to the time of the analysis. Of course, in a real 4dVar analysis it is likely that covariance propagation plays at least some part in explaining the differences between 3dVar and 4dVar.
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06.06.2002 |
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