**Background error for NCEP\'s GSI analysis in regional mode**

*Wan-Shu **Wu***

National Centers for Environmental Prediction, Environmental Modeling
Center, Washington, D. C.

NOAA/NWS/NCEP, W/NP, Room 207, 5200 Auth Rd., Camp Springs, MD 20746,

wanshu.wu@noaa.gov

**Summary**

To apply NCEP's Gridpoint Statistical Interpolation (GSI) to the WRF-NMM
regional system, the regional background error was needed for the
unified 3D Var. The error statistics from a global system was setup as
the default values for applications to various location and resolutions.
The NMC method was also used to estimate the statistics from the WRF-NMM
forecasts of the central US domain. Spectral calculations were used to
find the stream function and velocity potential from the forecast
differences of the wind fields. The horizontal scales derived from the
derivatives of the non-hydrostatic model fields were very unstable and
change sharply from layer to layer. Auto correlation was found to be the
most robust and stable in estimating the horizontal structure. The
characteristics of the error statistics were different from those of the
default. It was shown that the multivariate projections from the two
systems produced similar results and the variances and the structure
functions from a different forecasts system could be tuned to produce
comparable results.

1.  **Introduction**

A 3D Var formulated in model grid space was proposed in Wu et.
al.(2002). The formulation allowed greater flexibility (inhomogeneity
and anisotropy) for background error statistics. The initial tests, a
simpler formulation: inhomogeneous only in latitude direction, in
NCEP\'s global data assimilation showed promising results. It was
straight forward to apply this physical-spaced 3D Var to a regional
domain. The unified grid-point statistical interpolation (GSI) analysis
in NCEP was constructed to include both global and regional
applications.

The objective of this work was to estimate the background error
statistics for the data assimilation system of NCEP\'s regional forecast
model NMM. Since the regional model could be applied to various domains
and resolutions, the error statistics of the global forecast system were
setup as default for the movable assimilation system. The statistics
were interpolated to the latitude and height of the regional domain in
GSI and the differences in the resolutions were taken into account. In
comparison the NMC method was also used to estimate the regional
background error statistics directly from the WRF-NMM forecasts.
Spectral calculations were used to find the stream function and velocity
potential from the wind fields. Derived with the method proposed in the
paper, the horizontal scales of the non-hydrostatic model fields were
unstable and change sharply from layer to layer. Auto correlation was
found to be the most robust and stable in estimating the horizontal
structure.

Since the global error statistics were interpolated to the regional
domain, the differences in the background errors from these two very
different models could be compared directly and the impact on the
forecasts from the initial conditions generated with the two sets of
error covariance was examined.

2.  **Estimating background error in regional domain**

The differences of the WRF-NMM forecasts on the central US domain with 8
km resolution from different lead time were used to estimate the
background error covariance. The error statistics included the
multivariate projection between the temperature, surface pressure,
velocity potential and the stream function, the variances and the scales
of the analysis variables. The analysis variables were stream function,
unbalanced velocity potential, unbalanced temperature, normalized
relative humidity (Holm *et al*. 2002) and unbalanced surface pressure.

The stream function and the velocity potential fields were calculated in
spectral space from the forecasts differences of the wind components.
The wind components were interpolated from E to A grid and filled to
Fast Fourier Transform acceptable grid numbers by tapering to zero near
the boundaries. The fields were transformed to the spectral space and
vorticity and divergence fields were calculated. Applied
$\nabla^{- 2}$to find stream function and velocity potential and
transformed the two fields back to physical space. The practice could be
verified by taking derivatives of the stream function and velocity
potential in physical space to find the wind components. The winds can
then be compared with the original wind fields.

The horizontal scales of the structure function derived from the
horizontal gradients of the forecast differences, as used in the
original paper, varied sharply from layer to layer. The problem of the
statistics might relate to the facts that the regional model was
non-hydrostatic and the resolution was high. Among the new methods
tested, auto correlation was found to be the most stable and robust in
estimating the horizontal structure. Applying auto correlation to the
entire horizontal domain, however, meant that the scale statistics were
only height dependent for the regional domain.

![](media/image1.jpeg){width="3.25625in" height="2.5708333333333333in"}

![](media/image2.jpeg){width="3.25625in" height="2.5708333333333333in"}

Fig 1. The error variances of the stream function derived from the
global system (above) and the regional NMM (below) as function of
latitude and vertical level.

The background error statistics of the global GFS were used as the
default for regional application of GSI. The variances and scales were
defined as function of latitude and height in physical units. The
statistics were interpolated to the user defined domain and resolution.
Separate tuning coefficients for variances and scales were defined for
each variable. The balance projection matrices were also interpolated.
Note that the projection matrix from stream function to temperature
required double interpolation in the vertical.

![](media/image3.jpeg){width="3.040277777777778in"
height="3.6569444444444446in"}

![](media/image4.jpeg){width="3.060416666666667in"
height="3.6708333333333334in"}

Fig 2. Horizontal scales of the structure function derived from the
global system (above) and from NMM forecasts (below) as function of
vertical level of the model. Black curves represent stream function,
green: velocity potential, yellow: temperature and red: normalized
relative humidity.

The horizontal scales were very different between the two systems.
Besides the characteristics of the forecast models, there were issues of
different methods used to derive the statistics and the statistical
robustness of the methods, i.e. sampling errors. The multi-Gaussian
setup in GSI made the background error spectrum wider than when a single
Gaussian was used so that the analysis response was less sensitive to
the tuning of horizontal scales. The variances of the stream function
derived from the two systems in Fig. 1 showed differences in amplitude
but with similar patterns. The variance from both set of statistics
increased with height and has maximums around tropopause. The amplitudes
also increased slightly with latitude. The vertical scales in Fig. 2
showed maximums between z=10 to 15 but the decrease in the lowest 5
levels were much sharper in the regional statistics than those in the
global. Interpolation from the lower resolution global system to the
higher resolution NMM introduced errors in the rapidly changing layers
near the surface. The vertical grid structures of the 42-layer global
was much coarser than that of the 60-layer regional model. There were 8
layers below sigma=0.9 in the global system and 19 layers in the
WRF-NMM.

To find out the differences of the balance projections from a global
spectral model and a regional 8km non-hydrostatic model, the analysis
results of one single *U* observation at 250 mb were examined. The
vertical cross sections of the *U* component increments and the
projected *T* increments from a 1m/s westerly wind innovation with
regional background statistics and global background error statistics
interpolated to the regional domain were shown in Fig. 3. The tuning of
the statistics resulted in similar analysis response in wind fields to
the innovation. The responses in temperature field, no separate tuning
involved, through balance projection in the upper layers were also
similar through the projection matrices from the two very different
systems. The vertical cross sections of the *T* increments and the
corresponding *U* increments (not shown) from a 1^o^ temperature
observational residual at 1000 mb showed that the temperature increments
extended much higher in the domain when the global statistics were used
which is consistent with the estimated vertical scales showed in Fig. 2.
The projected wind increments were opposite in sign between the analysis
results using the two background error covariance. However, the
projected wind increments near the surface were much weaker than those
in the higher levels. This difference was shown to have minor impact on
the forecasts.

**3. Analysis results and forecast impact**

North-south cross sections of the analysis increments of *U* at the
center of the regional domain were shown in Fig. 4. There are
differences between the analysis results from the two sets of background
error statistics. However the general patterns of the analysis
increments are similar. The differences are more significant near the
surface in temperature increments (not shown). The vertical scales in
the lower atmosphere are smaller with the NMM statistics.

![](media/image5.jpeg){width="3.25625in" height="2.5708333333333333in"}

![](media/image6.jpeg){width="3.25625in" height="2.5708333333333333in"}

Fig. 3. The vertical cross sections of the analysis increments of *U*
component wind (color contours) and the temperature (black contours)
with global (above) and with regional background statistics (below) from
a 1 m/s westerly wind innovation at 250mb.

Data assimilation over the central US domain using the global and the
NMM background error statistics were cycled for 12 hours (four analysis
cycles with 3-hour interval) before 48-hour free forecasts. The
normalized total penalties of the 2-day forecasts fit to conventional
data were shown in Fig.5. With all the differences in the background
errors, the experiments produced comparable forecasts and the one with
NMM statistics produced small positive impact on the forecasts.

![](media/image7.jpeg){width="3.25625in" height="2.5708333333333333in"}

![](media/image8.jpeg){width="3.25625in" height="2.5708333333333333in"}

Fig. 4. vertical cross section at the center of the domain of analysis
increments of the *U*-component wind using the default statistics
(above) and the NMM statistics( below).

**4. Summary**

The problems and the solutions to find the background error statistics
of a regional model were presented. It was found that the auto
correlation is most effective in estimating the horizontal scales for
the high resolution non-hydrostatic regional model. The characters of
the back ground error covariance from two very different forecast models
shown in the horizontal, vertical scales of the structure functions, the
variances and the multivariate projections were discussed. It was shown
that the multivariate projections in the upper troposphere were similar
with the projection matrices derived from a global T126 spectral model
and a regional 8km non-hydrostatic model. The variances and the
structure functions are different for the two systems but could be tuned
to produce comparable results in most area except where the vertical
resolutions were very different. The differences in the vertical grid
structures of the two systems created very different analysis increments
near the surface. However, using the background error covariance derived
from a very different system had only small negative impact on the 2-day
forecasts.

*Acknowledgments*

The author would like to thank David Parrish, Russ Treadon and Dayle
Kleist for many helpful discussions.

**References**

Wu, W.-S., R. J. Purser, and D. F. Parrish, 2002: Three dimensional
variational analysis with spatially inhomogeneous covariance. *Mon. Wea.
Rev*. **130**. 2905-2916.

Holm, E., E. Andersson, A. Beljaars, P. Lopez, J-F. Mahfouf, A.J.
Simmons and J-N. Thepaut, 2002: Assimilation and modeling of the
hydrological cycle: ECMWF's status and plans. *ECMWF Tech Memo 383*.

![](media/image9.jpeg){width="3.2631944444444443in" height="2.23125in"}

Fig. 5. Normalized total penalties of 48-hour forecasts fit to
conventional data initialized with 12-hour data assimilation with
default and NMM derived background error statistics.