1. INTRODUCTION
It is well
known that numerical weather predictions are sensitive to small changes in the
initial conditions, i.e., a rapid growth of the initial errors can lead in a
relatively short time to large forecast errors. During the last decades much
effort has been devoted to study and improve the methods used for the
preparation of the initial conditions for numerical atmospheric models (the
so-called analysis), as well as to understand the mechanisms involved in the
growth of the initial errors.
The
analysis is obtained as a statistical interpolation of short-range numerical
forecasts (known as background) with new observations. The weight given to each
of these contributions is essentially proportional to the inverse of their
error covariance. It follows that a good representation of the observation and
background error covariances is one of the major goals in the development of
data assimilation systems.
In
3D-Variational schemes the background error covariance matrix is statistically
derived from long-term statistical estimations and it is maintained constant
in time during the assimilation cycle. This implies that the large time
dependence of the errors (“errors of the day”) is neglected, despite its large
variability (Corazza et al, 2001).
There are several methods that try to account for the errors of the day in the forecast error covariance, including 4D-Var, Kalman Filtering, and the method of representers (e.g. Klinker et al., 2000; Bennet et al., 1996; Houtekamer and Mitchell, 1998; Hamill and Snyder, 2000). Unfortunately, these methods are computationally very expensive, and can only be implemented with substantial shortcuts, such as the use of a reduced rank background error covariance matrix in Kalman Filtering and lower model resolution in 4D-Var. It follows that it is important to develop new, low-cost methods aimed to improve the estimation of the background error covariance matrix.
Kalnay and Toth (1994) argued that the similarity
between breeding (Toth and Kalnay, 1993, 1997) and data assimilation suggests
that the background errors should have a structure similar to those of bred
vectors. They also proposed a simple method to include the information of the
bred vectors in the data assimilation cycle.
Here we
take advantage of the relationship between the bred vectors and the analysis
and background errors demonstrated by Corazza et al., 2001, for a simple
quasi-geostrophic model (Morss, 1999), and test whether it is possible to
augment the constant forecast error covariance used in 3D-Var with “errors of
the day” derived from the breeding method. We present results obtained with
different methods aimed to include in the data assimilation scheme (i.e., in
the representation of the background error covariance matrix) the information
given by the bred vectors.
The numerical model is a quasi-geostrophic (QG) mid-latitude flow in a channel discretized by finite differences both in horizontal and vertical directions. The simulated data assimilation is performed with an algorithm similar to the operational Spectral Statistical Interpolation (SSI) at NCEP (Parrish and Derber, 1992). “Rawinsonde observations” are generated every 12 hours by randomly perturbing the true state at fixed observation locations. Bred vectors are produced using a method similar to that adopted at NCEP (Toth and Kalnay 1993,1997), rescaling the difference between the perturbed runs and the control forecast every 12 hours.
Since this
is a simulation system, we can explicitly define the “true state of the
atmosphere” (by integrating the model from a given initial state) and therefore
study the analysis and forecast errors. A perfect model assumption is made so
that our conclusions are not necessarily valid for more complex models with
model errors, and similar tests have to be made with more general simulation
systems and with real forecast systems.
2. BRED VECTORS AND BACKGROUND ERROR
Corazza et
al. (2001) found that bred vectors in this QG simulation system are indeed
closely related to the background errors. Figure 1 shows a typical example of
the largest values of the midlevel background error in potential vorticity
(solid lines) against two arbitrarily chosen bred vectors (dotted and dashed
lines respectively). The patterns are in good agreement in some areas, and
agree less well in other areas. In particular, the agreement is higher in those
areas where the background error is larger. In general, these results are valid
also for the vertical structure of the fields and at different observational
densities [only 16 observations were used in Figure 1, with similar results
obtained with 32 and 64 “rawinsondes” observation locations].
We found
(Corazza et al., 2001) the following properties of the bred vectors:
· Convergence
to well organized structures in the bred vectors occurs within a few (3-5)
days. This indicates that it is possible to operationally use the information
given by the bred vectors without waiting an infinite time for asymptotic
convergence.
· Bred
vectors obtained using normalizations based on the potential vorticity and on
the stream function are virtually indistinguishable.
· Bred
vectors obtained using the “true” atmosphere are very similar to those obtained
using the “analysis” atmosphere. This is true even if we use a low density
observing network, suggesting that the bred vectors are not too sensitive to
the details of the flow and that the errors themselves are more likely
dependent on the large scale nature of the flow.
Following Patil et al. (2001) we
also computed a measure of the effective dimensionality of the space spanned by
the bred vectors called Bred Vector Dimension (BV-Dimension). For every point a
surrounding domain of 25 grid points (5´5) is selected. We consider the 25 dimensional vector
composed of the values of potential vorticity over the grid points of the
domain, which we refer to as a local vector. The BV-Dimension is a
measure of the degree of linear independence of the 
local vectors
(
being the number of bred vectors) and is defined as:
,
where 
, 
are the singular
values of the 
´
covariance matrix of the 25´
matrix formed by the local vectors. Note that 
.
We found
that for 
=10 the local dimension of the subspace is always smaller
than 6.5 and that in areas where the background errors are large it is usually
between 2 and 4. These results did not change when a larger number of bred
vectors was used, indicating that 10-15 vectors are enough (at least for a
quasi-geostrophic system) to give a complete representation of the space
spanned by the bred vectors.
We also
defined the local angle between the forecast error and the subspace of bred
vectors. We found that in most of the area the angle is confined to less than
10° (cosine larger than 0.985), suggesting that the background error is mostly
confined to the subspace of the bred vectors. As in the case of the BV-Dimension, the areas with an angle larger
than 10° generally had small background errors.
Similar
results were obtained computing the local pattern correlation over the same 25
points between each bred vector and the background error and then identifying
the maximum correlation at each point. We obtained high correlations in most of
the domain of the model, which indicate that at least one bred vector has a structure
similar to that of the error, in particular where the background error is
large.
These
results suggest that the bred vectors can indeed be useful in specifying the
part of the background error covariance matrix that corresponds to the “errors
of the day”. In the following section we discuss some computationally
inexpensive methods to take advantage of this information.
3. BRED
VECTORS IN DATA ASSIMILATION
The
original data assimilation cycle (referred to as the regular data
assimilation system) is based on the NCEP 3D-Var scheme, and is solved
iteratively for the analysis state 
(Morss, 1999). Given
the background state (or first guess - the 12 hour forecast from the previous
analysis) 
, and the set of observations 
, the equation can be written as follows:
where 
is the background
error covariance matrix, 
is the observation
error covariance matrix, 
is the observation
operator and 
,
are the matrices that represent the linearized
and its transpose respectively. The right part of the
equation is computed at the beginning of the process, then the equation is
iteratively solved for 
until the equation is
satisfied with an error smaller than a given threshold.
The easiest
way to introduce the bred vectors in this equation is to globally substitute 
with the ensemble average of the outer product of the bred
vectors. We can build a new background covariance matrix as 
where 
is the 
th bred vector defined over the entire domain. The
substitution of 
with the new matrix
can be done at a negligible computational cost; moreover, this implementation
allows to apply 
and 
simultaneously
(Hamill and Snyder, 2000) so that the data assimilation scheme can be
generalized to:
where 
is a number between 0
(for the regular system) and 1 (background error covariance matrix based fully
on bred vectors) and 
is a normalization
factor kept constant in the results presented here. It should be noted that the
covariances in 
were tuned to
optimize the regular 3D-Var.
In Figure 2
we show the squared analysis and forecast (up to 72 hours) errors in mid-level
potential vorticity for the optimized regular system (diamonds) and for the
simulation obtained using 10 bred vectors with 
=0.4 (squares). The results are obtained averaging the errors
over the horizontal domain every 12 hours for a one-year simulation. The use of
the bred vectors allows decreasing the squared error by a factor between 15 and
20% (around 8-10% in the error). Fig. 2 shows that the percentage improvement
continues throughout the 72 hour forecast, suggesting that the correction to
the analysis due to the bred vectors affects, at least in part, the growing
errors.
We want to
understand whether the bred vectors are able to represent the evolution of the
fast growing errors of the day associated with the short term forecast starting
from the previous analysis. The errors of the analysis are due to forecast
(background) as well as to observational errors, which constantly introduce
random errors. However, the “classical” formulation of the bred vectors takes
into account only the errors due to the forecast. For this reason, we modified
the method to generate the bred vectors including “reseeding” with random
“observational” errors. After every renormalization step (12 hours), we
generate random errors 
at the observation
locations simulating the impact of the
observational errors. Then we apply the transpose of the observation operator
to obtain 
, and add 
to the rescaled bred
vectors.
The third
line (bottom - triangles) of Figure 2 shows the results obtained for this
system using the same data assimilation scheme as before. The squared error in
midlevel potential vorticity is now 40% smaller than that of the regular system
(22% in the error). The relative improvement decreases slightly within the
forecast, but it is still around 30% after 72 hours. More work has to be done
to better understand why adding random errors to the bred vectors leads to such
a large improvement. It may be due in part to the fact that adding random noise
does not allow bred vectors to collapse into too few directions, and therefore
to span a larger subspace that better represents the fast growing modes. The
fact that after 72 hours the improvement is still far better than in the
standard bred vector simulation indicates that the role of the random errors
evolved for 12 hours upon the bred vectors is not limited to a mere correction
of mostly decaying analysis errors uniformly distributed in phase space.
Figure 3
shows the relative improvement (with respect to the regular 3D-Var data
assimilation system) in the squared error in potential vorticity at midlevel,
for the simulation obtained using bred vectors reseeded with random errors, and
for different values of 
. There is a remarkable impact of the background error
covariance matrix derived from the bred vectors even for very small values of 
, indicating that the new matrix is able to focus the
corrections in the background state along the most unstable directions even
when it is not the dominant part of 
. It is interesting to note that for large values of 
, when the role of the statistically derived 
is small, the
augmented system is not able to maintain the error small. This indicates that
the space spanned by the bred vectors is not large enough to represent all the
error directions, and that the contribution of the regular part of the
assimilation scheme cannot to be neglected when using global methods to include
the bred vectors in the data assimilation cycle. This was also observed using
local methods (not shown).
4. SUMMARY AND DISCUSSION
We have
used a 3D-Variational data assimilation scheme for a quasi-geostrophic channel
model (Morss, 1999) to study the structure of the background error and its
relationship to the corresponding bred vectors. The results of Corazza et al.
(2001) show that the bred vectors have spatial and temporal characteristics
similar to those of the background “errors of the day”. They pointed out some
properties of the bred vectors, including fast convergence to well organized
structures, independence on the choice of the renormalization norm (potential
vorticity and stream function) and similarity between bred vectors generated
from the “truth” and from the analysis. We also showed that the subspace
spanned by the bred vectors is able to locally describe most of the structure
of the background error.
Starting
from these considerations we described a global approach to include the bred
vectors in the data assimilation scheme and we showed that this method is able
to reduce the squared error in the analysis and forecasts up to a factor of
40%. Moreover, we considered a modified version of the bred vectors aimed to
take into account the observational errors introduced in the analysis step.
Simulations using these vectors show a remarkable improvement of the
performance of the assimilation cycle with respect to the one based on the
standard bred vectors without random “reseeding”.
We are
presently testing new methods to locally use the bred vectors in the data
assimilation system. The use of local methods is desirable in order to optimize
the information given by two or more bred vectors, which may be, for example,
positively correlated in one area and negatively correlated in another area far
away. The background error correlations should vanish beyond a limited
horizontal extent, whereas our use of global bred vectors implies correlations
over the global domain. An example of local use of the bred vectors can be
derived as a generalization of the method proposed by Kalnay and Toth (1994).
If we assume for simplicity that the forecast error covariance is given locally
by a single bred vector, 
, that the observational error covariance is 
, and perform a preliminary analysis, then the Optimal
Interpolation solution
becomes simply
,
where 
is the projector
operator from the global domain to the local domain. This system has a very low
computational cost. A similar formula is valid for a set of locally
orthogonalized bred vectors. As long as the preliminary analysis remains
confined to the unstable subspace of the bred vectors, the computational cost
remains low.
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FIGURES
Fig. 1. Background
error (solid line) and two randomly chosen bred vectors (dotted and dashed
lines respectively) after 36 days from their initialization. The fields are
normalized by their root mean square and contour lines are plotted for the
values 2.3 and 3.
Fig. 2. Squared
errors in midlevel potential vorticity for the regular system, the standard
bred vector system (
=0.3) and the bred vectors + random noise system (
=0.4).
Fig. 3. Relative improvement with respect to the
regular data assimilation system in the analysis and forecast squared errors in
midlevel potential vorticity varying 
from 0 to 1, for the
simulation with bred vectors obtained adding random noise after every
normalization step.