A study of the flow-dependence of background error covariances
based on the NMC method
Harald Anlauf, Werner Wergen, and Gerhard Paul
Deutscher Wetterdienst, Offenbach, Germany
E-Mail: harald.anlauf@dwd.de
The NMC method, i.e., using differences of short-range forecasts verifying at the same time but with
different starting dates as proxies for background error, was applied to the global model GME at DWD.
The aim was to explore the variability of the background error covariance matrix B with respect to region,
altitude, season, and flow pattern. The method was first applied univariately to the mass, humidity and
wind field and then extended to study multivariate aspects. We present results for the years 2003 and
2004 and discuss the implications our findings will have for the modeling of the B-matrix.
1. Motivation
Modern data assimilation systems such as the
4D-Var and the Ensemble Kalman Filter (EnKF)
use the dynamics of the forecast model to evolve
the covariance matrix of background error, B, in
time either implicitly (4D-Var) or explicitly (EnKF).
As a consequence, these schemes effectively use
flow-dependent structure functions, although at
correspondingly high computational costs.
Nevertheless, the initial B-matrix still has to be
specified, like for any simpler assimilation system
(e.g., OI, 3D-Var). The 4D-Var also does not
transfer the full dynamical background covariances
to subsequent assimilations.
2. Method
In order to explore the variability of the B-matrix
with respect to flow pattern explicitly, we applied
the NMC method at DWD to the global model GME
(Majewski et al. 2002). The NMC method (Parrish
and Derber 1992) uses differences of short-range
forecasts verifying at the same time but with
different starting dates as proxies for background
error. It assumes that the spatial structure of
background error does not strongly vary with
forecast time, so that correlations obtained from
forecast
differences
are
reasonable
approximations to the true background error
correlations, at least for the mid-latitudes.
covariances was performed with forecasts
interpolated to a regular grid and with zonal
averaging.
For an assessment of seasonal
variations we also used the forecasts for the
summer periods of 2003 and 2004.
Our investigation began with a study of the
isotropic part of the empirical horizontal correlation
of the 500 hPa geopotential height at 45°N using
the full sample of forecasts for winter 2003/2004,
shown as the blue curve in figure 1.
For the exploration of the flow-dependence, we
took the 24h forecast of the local 500 hPa height
as an indicator for the meteorological situation.
Based on the statistics of the 500 hPa geopotential
height at fixed latitude over the whole period, we
distributed the contributions into three classes of
approximately equal sample sizes.
For this
forecast sample and at 45°N, a data point was
associated with a region of “high” and “low”
pressure if z(500 hPa) > 5527 gpm and z(500 hPa)
< 5371 gpm, respectively, otherwise it was
assigned to the “neutral” class.
3. Results
The main results of the present study are based
on the differences of 48h and 24h forecasts
verifying at 00Z during the winter period from
Dec. 1, 2003, until Feb. 29, 2004. The forecast
fields were taken from the archived main runs of
the operational GME model. The evaluation of
Fig. 1: Flow-dependence of the 500 hPa horizontal
geopotential correlation at 45°N, winter 2003/2004.
The resulting horizontal correlation functions
obtained from contributions in the classes “high”
and “low” are shown in figure 1 as green and red
curves. Clearly the correlations for "highs" are
significantly broader than for "lows".
While the above result appears quite simple, its
implementation is actually non-trivial. Any attempt
to modify the B-matrix in order to take into account
flow-dependence has to respect symmetry and
positive-definiteness constraints (Riishøjgaard
1998). The dynamical B-matrix therefore cannot
be a function of the meteorological situation at only
one point, it must be a function of the flow at both
points. However, for the present study we ignored
the latter dependence.
A related quantity that depends on one point
only and still provides some insight is given by the
the standard deviation of the 48h-24h forecast
difference. Figures 2 and 3 show the meridional
distribution for the 500 hPa height, indicating that
the standard deviations at mid-latitudes on the
northern hemisphere are roughly 50% higher for
“lows” than for "highs" during the winter period
(fig. 2), and up to 60% higher during summer
(fig. 3). Analogous results are obtained on the
southern hemisphere for southern winter and
summer, resp. These findings also agree with the
higher variance of background error for lows than
for highs in the implicitly evolved covariances of
the ECMWF 4D-Var system (Thépaut et al. 1996).
Fig. 3: Same as fig.2, but for summer 2004.
Correlations obtained from forecast differences
of the horizontal wind at the same vertical level
show a similar flow-dependence.
Figure 4
compares the transverse wind correlations at 45°N
for contributions of "high” and "low” pressure, as
determined by the geopotential height of the 500
hPa surface. Again, the correlations are clearly
broader in the former case. The flow-dependence
of the correlations of longitudinal wind is
significantly smaller, see fig. 5.
Fig. 4: Flow-dependence of transverse wind correlations
at 500 hPa, 45°N, winter 2003/2004.
Fig. 2: Flow-dependence of the zonally averaged
standard deviation of the 48h-24h forecast differences of
the 500 hPa height for winter 2003/2004.
Fig. 5: Same as fig.4, but for longitudinal wind.
The same pattern is found in the crosscorrelation of geopotential height and transverse
wind (fig. 6). This result is expected because of
the geostrophic balance relationship that is well
satisfied at mid- and high latitudes. It also stresses
that a proper treatment of mass-wind balance is an
essential ingredient in a flow-dependent modeling
of background error covariance.
the local minimum of the correlation length scale is
shifted upwards because the tropopause level is
higher during these periods.
The flow-dependence of background error
covariances manifests itself also in the empirical
vertical correlations. Figure 8 shows the vertical
temperature correlation with the 700 hPa surface
for summer 2004. Again, the blue line represents
the result from taking all contributions into account,
while the green and red lines show the correlations
for "highs" and "lows", respectively.
Fig. 6: Flow-dependence of the cross-correlation of 500
hPa height and transverse wind.
We extended the above analyses of the
horizontal correlations also to other pressure
levels. In order to keep a uniform criterion for the
selection of meteorological situations, we always
determined the flow by the geopotential height of
the co-located 500 hPa surface.
Horizontal
correlation length scales were obtained from fits of
the empirical correlations to a second order
autoregressive function. The dependence of the
horizontal correlation scale on vertical level is
displayed in fig. 7. The data clearly demonstrate
that there is a considerable variation of correlation
scale with flow pattern also at other levels.
Fig. 8: Flow-dependence of the vertical temperature
correlation with 700 hPa, summer 2004.
4. Outlook
Investigations are underway how to perform a
consistent modeling of a symmetric and positive
definite B-matrix that incorporates the empirical
flow-dependence presented here. As an essential
ingredient, the simple local selection criterion for
the meteorological situation used here has to be
extended to a bi-local and level-dependent
formulation that automatically satisfies the
symmetry and positive-definiteness constraints.
References
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Fig. 7: Vertical profile of the length scale of the horizontal
geopotential correlation as a function of the co-located
500 hPa height, 45°N, winter 2003/2004.
Results for the summer seasons 2003 and
2004 were similar, although the vertical location of
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