| Home > Newsevents > Training > Rcourse_notes > DATA_ASSIMILATION > ASSIM_TECHNIQUES_4dVAR > |
Assimilation Techniques: 4dVar
April 2001
By Mike Fisher
European Centre for Medium-Range Weather
Forecasts.
Table of contents >>
Next Section >>
Previous Section >>
4 . Increments from a single observation
In my previous lecture, I demonstrated that for the simple case of a single observation of a model variable at a gridpoint, the analysis increment in 3dVar is proportional to a column of the background error covariance matrix. It is instructive to consider how 4dVar responds to the same observation.
As before, consider the non-incremental formulation, and suppose that the observation is at the gridpoint corresponding to the
element of the analysis vector. In 4dVar, we must also specify the
time of the observation. Denote this by 
. The 4dVar cost function is:
|
(4) |
, whereas the observation is compared to the
gridpoint value at time 
. We can eliminate 
by noting that 
is the result of a model integration with initial conditions 
. Let us write this as:
|
(5) |
|
(6) |
order and higher derivatives):
|
(7) |
represents the adjoint of the model
integration from time 
to 
. Multiplying through by
and rearranging gives, as for the 3dVar example, an expression for
the analysis increment (at the start of the 4dVar assimilation window, time
). Since we have just one observation, the expression 
is simply a scalar, and we find that, whereas in 3dVar,
the analysis increment was proportional to a column of 
, in 4dVar, it is proportional to a column
of 
:
|
(8) |
before rearranging:
|
(9) |
|
(10) |
So, the left hand side of equation 9 is the difference between the analysis and the forecast for time
with initial conditions given by the background at time
. In other words, the left hand side
of Eq. (9) is the difference between
the analysis trajectory and the background trajectory at the time of the
observations. This difference is proportional to a column of 
. Now, if
is the covariance matrix for errors in the background at time 
, then the matrix 
is the covariance matrix for errors in a forecast from time
to 
with initial conditions equal to the background. This is easy to
see: Under the tangent linear assumption, and for a perfect model, the background errors at time
and 
are related by:
|
(11) |
is:
|
(12) |
For an observation at the beginning of the 4dVar assimilation window (i.e. for
) the matrix 
is the identity matrix. (Integration of the tangent linear model
for no timesteps does nothing!) In this case, equation 9 is identical to
the 3dVar case. The analysis increments for an observation at the beginning
of the 4dVar assimilation window are the same as would be produced in 3dVar.
This illustrates the main shortcoming of 4dVar. Namely, that at each cycle
of assimilation the initial covariance matrix is the fixed, static and flow-independent
matrix 
. The flow-dependent covariances which are used implicitly during
the assimilation are not propagated to the next cycle. To propagate these
covariances, we must turn to the Kalman filter. This will be the subject
of my next lecture. 4.1 Examples
Fig. 1 shows analysis increments for 3 separate analyses for the same date. Each analysis had just a single observation of geopotential at 850hPa, 40N, 60W. In Fig. 1 (b), the observation was placed at the beginning of the 4dVar assimilation window, and a cross-section of the analysis increment at the beginning of the assimilation window is shown. The analysis increment in this case is determined entirely by the background error covariance matrix
, and is clearly almost symmetric and without any vertical tilt.
(The slight asymmetry is probably due to the effects of normal model initialization
of the increments.) The increment is the same as would be generated by 3dVar
for this observation. By contrast. if the same observation is placed in
the middle of the analysis window (Fig. 1 (c)) then the analysis
increment, also in the middle of the window, shows a marked vertical tilt
towards the jet. (Fig. 1 (a) shows a cross section
of the background zonal wind.) The tilt is more marked if the observation
is placed at the end of the assimilation window. Fig.
1 (d) shows the increment at the end of the analysis window in this
case. 
The increments in Fig. 1 are plotted for the same time as the observations. Fig. 2 shows increments at the middle of the assimilation window (i.e. at the nominal analysis time, 0z) for single observations at the beginning, middle and end of the window. (Again, three separate analyses are shown. Each analysis used a single observation.) This illustrates that 4dVar spreads the infomation provided by the observations in a dynamically consistent way throughout the analysis window.
Training Course Notes Front Page >>
Table of contents >>
Next Section >>
Previous Section >>
 |
06.06.2002 |
 |
© ECMWF |
