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Assimilation techniques: 3dVar
April 2001
By Mike Fisher
European Centre for Medium-Range Weather
Forecasts.
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3 . The background cost function
The background error term of the cost function is crucial to the performance of the analysis system. A simple example suffices to show why.
3.1 Example
Suppose we have a single observation of the value of a model field (e.g. temperature) at one gridpoint, corresponding to the
element of the state vector. The observation operator is very simple
in this case, and is represented by the 
matrix whose 
element is equal to one, and whose other elements are all zero:
|
(4) |
|
(5) |
, and rearranging gives
|
(6) |
is simply equal to the 
column of 
. Also, since we have just a single observation, 
is simply the scalar value 
, where 
is the analysed gridpoint value corresponding to the observation,
and where 
is the variance of observation error. Thus:
|
(7) |
. In other words, the background covariance matrix controls how information
is spread out from the single observation, to provide statistically consistent
increments at the neighbouring gridpoints and levels of the model, and to
ensure that observations of one model variable (e.g. temperature) produce
dynamically consistent increments in the other model variables (e.g. vorticity
and divergence). 3.2 Formulation of the background cost function in the ECMWF 3dVar
The background error covariance matrix is enormous, typically
. This is much too large to fit into computer memory. Moreover, even
if this was possible, we don't have enough statistical information to determine
all its elements. We are forced to simplify things. One way to approach the construction of the background error covariance matrix is to build a matrix
for which the variable 
has covariance matrix equal to the identity matrix. The background
cost function may then be written as
|
(8) |
The background error covariance matrix is defined implicitly by
as 
. The effect of multplying background error by the matrix
is to remove the correlations between
its elements. (Remember that the correlation matrix for 
is the identity matrix. That is, the elements
of 
are uncorrelated.) A natural way to
build 
is as a sequence of steps, each of which
removes some correlation from the background error. The most obvious correlation in the background errors is the balance between mass errors and wind errors in the extra-tropics. We therefore define our
as 
, where the matrix 
takes the model variables (vorticity, divergence, temperature,
log(surface pressure), specific humidity, and ozone mixing ratio) and subtracts
from the temperature and log(surface pressure) components which are in balance
with the vorticity. A component is also subtracted from the divergence to
remove the correlation between divergence and vorticity due to Ekman pumping
near the surface, and compensating upper-tropospheric outflow. Similarly,
a balanced component is subtracted from the ozone to account for the correlation
between vorticity and ozone background errors. The balanced components of temperature, log(surface pressure), divergence and ozone are defined as:
|
(9) |
has the same form as the linear balance operator relating vorticity
to height, but has coefficients which are statistically determined. The
matrices 
, 
and 
account for the vertical correlation between "height" (as defined
by 
) and the balanced temperature, log(surface
pressure), divergence and ozone. These matrices are block diagonal, with
one full vertical matrix for each spectral component, whose coefficients
depend only on the total wavenumber 
. The balance operator,
, is intended to remove all correlations between different variables.
The remaining part of the change of variable, 
, is therefore block diagonal, with one block for each
of the variables (vorticity, temperature-and-log(surface pressure), divergence,
and ozone). Each block has the same form, for example the block corresponding
to vorticity is:
|
(10) |
divides the vorticity by its standard
deviation of background error; 
removes horizontal correlation; and 
removes vertical correlation. Division by background error standard deviation is performed in gridpoint space, to allow a spatial variation of background error. However, the horizontal correlations are assumed to be spatially homogeneous and isotropic. This makes
diagonal, with coefficients which depend only
on the total wavenumber 
. The matrix
is block diagonal, with one vertical matrix for each spectral component,
whose coefficients depend only on the total wavenumber. This structure allows
the vertical correlations to vary with horiziontal scale, so that large
horizontal scales have deeper vertical correlations than small horizontal
scales. It does not allow spatial variation of the vertical correlations.
Note, however, that the vertical correlations of temperature do vary
spatially. This is because in the tropics they are determined by the correlations
defined for the unbalanced temperature, whereas in middle latitudes they
are largely determined implicitly by the action of the balance operator
on the vorticity correlation matrix. Fig. 1 (a) shows the vertical correlation of temperature error with model level 18 of the ECMWF 31-level model (level 18 is at approximately 500hPa). The statistics were estimated using the NMC method (see below). Note that the vertical extent of the correlation is smaller in the tropics than in middle latitudes. Figure 1b shows the vertical correlations for temperature implied by the Jb formulation. Note that the main latitudinal variation in the vertical correlations is retained.
The effect of the balance operator in accounting for correlation between geopotential height and wind is demonstrated by Fig. 2 , which shows the wind increments generated by a single geopotential height observation at 60N, 30W. In Fig. 2 (a), the observation is placed at 1000hPa, and wind increments are shown for the nearest model level to 1000hPa. Note that the wind increment includes a convergent component. This is generated by the inclusion in the balance operator of a correlation between divergence and vorticity. Fig. 2 (b) shows the wind increment near 300hPa for a height observation at 300hPa. In this case, the increment is slightly divergent.
Fig. 3 demonstrates the way in which information from an observation is spread in the vertical. It shows a cross section of the increment generated by a single temperature observation at 200hPa.
). The vertical axis for both plots
is model level for the 31-level model.
3.3 Calculation of the background-error correlations
The previous section showed that, by making some simplifying assumptions, the number of non-zero elements of the background error covariance matrix may be drastically reduced. The largest matrices are of of order the number of levels of the model, and may be estimated statistically from a sample of background error of size a few times the number of levels.
Unfortunately, we cannot calculate background error, since this would require knowledge of the true state of the atmosphere. There are three main approaches to get round the problem:
3.3 (a) The Hollingsworth and Lönnberg (1986) method
This method looks at the spatial covariance of differences between observations and the background. These differences are a combination of background and observation error. We can partition the error into background errors and observation errors by assuming that the observation errors are spatially uncorrelated. If we bin observation-minus-background as a function of distance from each observation, only the zero-distance bin will contain a contribution from the observation error.
The Hollingsworth and Lönnberg (1986) method has the advantage that it is a direct diagnosis of background error covariance. However, it requires a uniform set of unbiased observations with spatially uncorrelated error. This makes it unsuitable for calculating the global statistics required by 3dVar. In addition, the method produces statistics for observable quantities such as wind and temperature, whereas the background cost function requires statistics for vorticity and unbalanced components of temperature, etc.. Nevertheless, the method remains a valuable tool. In particular, it can be used to verify that the standard deviations of background and observation error are correctly specified.
3.3 (b) The NMC method (Parrish and Derber, 1992)
This method was used until recently in the ECMWF 3dVar and 4dVar systems. The method assumes that the statistical structure of forecast errors varies little over 48 hours. Under this assumption, the spatial correlations of backgound error should be similar to the correlations of differences between 48h and 24h forecasts verifying at the same time.
The advantage of the method is that it is straightforward to calculate the required global statistics. The disadvantage is that the underlying assumption that the statistical structure of 48h forecast error is similar to that of background error is difficult to justify.
3.3 (c) The analysis-ensemble method.
The method currently used at ECMWF to estimate background error statistics is to run an ensemble of independent analysis experiments. For each experiment, the observations are perturbed by adding random noise drawn from the assumed distribution of observation error. The effect of the perturbations is to generate differences in the analyses for each experiment. These are propagated to the next analysis cycle as differences in backgrounds. After a few days of assimilation, the statistics of differences between background fields for pairs of members of the ensemble equilibrate, and in principle become representative of the true statistics of background error. As a further refinement to the method, the effect of model error may be represented by introducing random perturbations to the physical parameterizations used in the assimilating model.
Figs. 4 and 5 show some statistics of background error calculated using the analysis-ensemble method (figure 4) and the NMC method (Fig. 5 ). The analysis-ensemble method gives background correlations which are smaller in horizontal and vertical scale than those produced by the NMC method. The background differences are also less balanced than the forecast differences used by the NMC method.
3.4 Calculation of background-error variances
The variance of background error is specified in gridpoint space. This allows the spatial variability of background error to be taken into account. (For example, background error variance is likely to be relatively smaller in areas of dense observational coverage, than in areas where there are few observations.) In practice, only the vorticity and specific humidity variances have a horizontal variation of background variance in the ECMWF 3dVar. (Note, however that the balance operator implies a horizontal variation of the balanced components of the other analysed variables.)
For specific humidity, background error variances are given by an empirical formula which expresses the relative humidity background error as a function of temperature and relative humidity. Additionally, background errors for humidity are reduced to very small values in the stratosphere, and are reduced at low levels over sea.
For vorticity, three-dimensional fields of background error standard deviation are calculated using a cycling algorithm (Fisher and Courtier, 1995) which applies an empirical error-growth model to a diagnostic estimation of the standard deviations of analysis error.
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