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Background Error for NCEP's GSI Analysis in
regional mode
How different are the balance projections from a
global T 126 spectral model and from a regional
8km non-hydrostatic model?
Background
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, on
NCEPs global data assimilation showed promising
results. Its straightforward to apply this
physical-spaced 3D Var to a regional domain. The
unified grid-point statistical interpolation
(GSI) analysis in NCEP has an option to run in
regional mode.
Vertical cross sections of the U component
increments (color contours) and the corresponding
T increments (black contours) of a 1m/s westerly
wind observational residual are shown.
Objective
The objective is to estimate the background error
statistics for NCEPs regional forecasts model
WRF-NMM. Two sets of background statistics are
compared and tested in the assimilation system.
1 Global error statistics are setup as default
for applications to various location and
resolution. The statistics are interpolated to
the latitude and height of the defined regional
domain and the differences in the resolutions are
taken into account. 2 The NMC method is used to
estimate the balance projection, variances,
vertical and horizontal scales of stream
function, normalized relative humidity,
unbalanced part of velocity potential,
temperature, and surface pressure, from the
WRF-NMM forecasts on the central US domain.
z
z
y
y
Observation at 45 N and 250mb with global
background statistics interpolated to regional
domain.
Observation at 45 N and 250mb with regional
background statistics.
The balance projection in the upper layers are
similar with the projection matrices from two
very different systems.
Stream Function
Vertical cross sections of the T component
increments (color contours) and the corresponding
U increments (black contours) of a 1o temperature
observational residual are shown.
Stream function and velocity potential are
calculated in spectral space from the forecast
differences of the wind fields.

• E2a grid of U V
• Fill to FFT grid number taper zero
• FFT both X Y directions
• Find vorticity and divergence fields in spectral
  space
• Apply del-2 in spectral space to find Psi and Chi
• FFT Psi Chi back to physical space
• Derivatives of Psi and Chi to find U2 and V2,
  only U fields are shown.

y
z
z
x
y
y
UU2
Velocity Potential
Observation at 45 N and 1000mb with regional
background statistics.
Observation at 45 N and 1000mb with global
background statistics interpolated to regional
domain.
The balance projections in the near surface
layers are quite different which may again be
related to the differences in the vertical grid
structures of the two systems (see vertical
resolution plot in next panel).
Analysis results and forecast impact
y
y
North-South cross sections of the analysis
results at the center of the regional domain.
U increments with default stats
U increments with nmm stats
x
x

z
z
Background Error Statistics
Default background statistics The background
statistics of the global GFS are used as the
default for regional application of GSI. The
variances and scales are defined as function of
latitude and height in physical units. The
statistics are interpolated to the user defined
regional domain and resolution with global tuning
parameters for each variables. The balance
projection matrices are also interpolated. Note
that the projection matrix from stream function
to temperature requires double interpolation in
the vertical. Background error statistics of 8km
NMM The NMC method is used to estimate the
background error from forecasts of 8km WRF-NMM
over the central US domain. The horizontal
scales of the structure function derived from the
horizontal gradients of the forecast differences
are very unstable and change sharply from layer
to layer. The problem of the statistics may
relate to the high resolution and non-hydrostatic
model. Among the new methods tested, auto
correlation is found to be the most stable and
robust in estimating the horizontal structures.
Applying auto correlation to the entire
horizontal domain, however, means that the scale
statistics are now only height dependent
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y
T increments with global stats
T increments with nmm stats
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z
Horizontal scales of the structure function
nmm stats
default stats
Black stream function Green velocity
potential Yellow temperature Red Pseudo RH
Error variance of stream function
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y
The two sets of background error statistics
produce similar analysis results in the mid to
upper troposphere but in the near surface layers
the solutions are different from each other which
may be related to the differences in the vertical
grid structure of the 2 systems .
nmm stats
Vertical Resolution
Total penalty of 48-hr forecast
Black NMM Green global
z
meter
meter
The horizontal scales are very different between
the two system. Issues of method used to estimate
the scales and statistical robustness?
latitude
Vertical scales of the structure function
default stats
psfc
q
t
u
v
nmm stats
default stats
Forecast impact Data assimilation systems over
the central US domain using the default and the
NMM background error statistics are cycled for 12
hours (4 X 3hr) before the 48-hour free
forecasts. The total penalties of 48 hour
forecasts fit to conventional data are shown. The
experiment with the NMM statistics is superior
over all in this case study.
Black stream function Green velocity
potential Yellow temperature Red Pseudo RH
Sigma
There are 8 layers below sigma0.9 for global
system and 19 layers for WRF-NMM on central US
domain.
Summary
The problems and the solutions to find the
background error statistics of a regional model
are presented. The characters of the forecast
model showed in the horizontal, vertical scales
of the structure functions, the variances and the
balance projections are discussed It is shown
that the variances and the structure functions
from a different forecast system can be tuned to
produce comparable analysis results except where
the vertical grid resolutions are very different.
The differences in the two system tested create
some very different analysis increments near the
surface. The background error covariance derived
from the forecast system itself has a positive
impact on the forecasts.
latitude
Amplitudes are different but with similar
patterns.
grid unit
grid unit
The vertical scales are especially different near
the surface. May be related to the vertical
resolution in the forecast systems (see vertical
resolution plot to the right).
Fourth WMO International Symposium on
Assimilation of Observations in Meteorology and
Oceanography
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