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Estimation of background error statistics in ARPEGE 4D-var. Margarida Belo Pereira (Instituto de Meteorologia, Lisboa). Loïk Berre (Météo-France, Toulouse). Importance of background error estimative. - The analysis field results from a combination of - PowerPoint PPT Presentation
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6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings
Estimation of background error statistics Estimation of background error statistics in ARPEGE 4D-varin ARPEGE 4D-var
Margarida Belo Pereira(Instituto de Meteorologia, Lisboa)
Loïk Berre(Météo-France, Toulouse)
6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings
Importance of background error Importance of background error estimativeestimative
- The analysis field results from a combination of observations and background (short range forecast)- The weights given to the observations and to the background depend on error statistics- The background errors statistics determines the way as the information from observations is spread spatially- How to estimate the background error statistics?
6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings
NMC method NMC method (operational in ARPEGE 4D-VAR)(operational in ARPEGE 4D-VAR)
1236tX
024tX
3636tX24
24tX0tX
ObsObsObsObs
024
2424 tta XXx 12
3636
36 ttf XXx
Analysis error
Forecast error
6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings
Alternative to NMC method?Alternative to NMC method?
Ensemble Analysis MethodEnsemble Analysis Method
Perturbed observations (5)
Perturbed analysis (5)6h forecast
Data assimilation
Random numbers (5)+
observations
Perturbed Perturbed background (5)background (5)
Experiments Ensemble with five 4D-VAREnsemble with five 4D-VAR
cycles of the non-stretchedcycles of the non-stretched
version of ARPEGE modelversion of ARPEGE model
with T299 and 41 levelswith T299 and 41 levels
PeriodPeriod
1 of February to 24 of March1 of February to 24 of March
of 2002of 2002
Background Background differences->Bdifferences->B
6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings
Level 21 (500hPa)
Standard deviation (normalized) of vorticity background error
Ensemble Method
Truncation T42
Level 32 (850hPa)
6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings
Impact of geographical variation of standard deviation of background errors
Forecast against ECMWF analysis Forecast against observations
Geopotential (anomaly correlation)
6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings
Impact of geographical variation of standard deviation of background errors
Forecast against ECMWF analysis Forecast against observations
Wind speed (anomaly correlation)
6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings
Ensemble Method versus NMC MethodEnsemble Method versus NMC Method
Spectral Space
6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings
Spectra of vorticity background errorSpectra of vorticity background error
Ensemble Method: the errors for the wind field have a bigger contribution from the mesoscale and subsynoptic scales than with the NMC method
Ensemble method
This result is valid also for the other variables, except for divergence
6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings
How to estimate the length scale of autocorrelation How to estimate the length scale of autocorrelation function?function?
Which assumptions are made?
Background error covariances are assumed to be
0~
2
22
~
x
x xd
dL
- stationary
- isotropic (horizontal length scale doesn’t depend on direction)
- spatially homogeneous
Definition of length scale (L) of the autocorrelation function
(Daley, 1991) for the one-dimensional case
L is a measure of the inverse curvature of the autocorrelation
function at the origin
For a sharp autocorrelation function, the curvature is large, so L is small. So, L gives an idea about
how the autocorrelation function decays with distance, from its initial value. In pratice, L is a measure
of the influence radius of one observation.
autocorrelation function
6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings
Surface pressure
Autocorrelation of background errors
Length scale of autocorrelation
The autocorrelation tends to zero faster in
Ensemble method than in NMC method
The length scale of vorticity is smaller than
the one of temperature, this difference is smaller
in Ensemble method than in NMC method
Length scale of background errors
are smaller in Ensemble method
than in NMC method
6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings
Ensemble Method NMC Method
North-South variation of vertical correlation of background error
NMC Method
Vorticity
Ensemble Method Temperature
6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings
Vertical profile of standard deviation of background error
On operational ARPEGE 4D-VAR, the vertical profiles of total standard deviation of the background errors are rescaled by a factor of 0.9
To use the statistics from Ensemble Method it is need to rescale the vertical profile?
to account for mismatch between the magnitudes of the 12/36-hours forecast differences and the 6-hour forecast errors
what is the best factor?
1.5 (green curve)
Standard deviation
Mo
del
leve
ls
NMC x 0.9
Vorticity
6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings
Impact of background covariances estimated by Ensemble method (against NMC method)
Geopotential Wind
Forecast against ECMWF analysis
6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings
Autocorrelation function in gridpoint spaceAutocorrelation function in gridpoint space
E
N
Isotropic: Lx= Ly Lx < Ly Lx > Ly
oLx
Ly
OR
6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings
2
2
1
121 xxvv
)()( 2121
2
2
22
)(
)(
xx
Lx
xv
Covariance of v between 2 points =>
Length scale of autocorrelationLength scale of autocorrelation
Helmholtz’s theorem => Rotational component of meridional wind
Covariance of stream function =>
standard deviation of background error
autocorrelation
Zonal length scale of autocorrelation =>
Meridional length scale of autocorrelation => 2
2
22
)(
)(
yy
Ly
6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings
Ensemble Method versus NMC MethodEnsemble Method versus NMC Method
Gridpoint Space
Length scale of autocorrelation
6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings
Horizontal length scale of vorticity
Ensemble LAND
Ensemble SEA
NMC SEA
Lx
Lx
Ly
Ly
Ensemble EURATL
Ensemble GLOBAL
NMC EURATL
NMC SEA
Ly is larger than Lx,
this difference is larger in EURATL region
6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings
Horizontal length scale of geopotential
Horizontal length scale of temperature
Lx
LxLy
Ly
Both Lx and Ly in Ensemble method
are smaller than in NMC method
Lx is smaller over
land than over sea,
mainly in Ensemble method
Ly is larger than Lx,
also for temperature
and geopotential
6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings
Ensemble Method
Horizontal length scale of wind
(Zonal and meridional length scale of zonal and meridional wind)
LAND SEA
EURATL
Lx (u) > Ly(u), except in PBL
Ly (v) > Lx(v), mainly in EURATL region
=>
v is more anisotropic in this region
u is more isotropic
in EURATL region
u is more anisotropic over
sea than over land, except near surface
Lx (v)
Ly (v)
Lx (u)
6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings
North-South variation of horizontal length scale of wind background error
Lx(u) and Lx (v) in Ensemble method
are smaller than in NMC method and both
are larger in the tropics
Lx(u) is larger than Lx (v) in both method
6-9 October 2003, Lisbon 25th EWGLAM and 10th SRNWP Meetings
ConclusionsConclusions
• In Ensemble method the length scale of autocorrelation is shorter than in NMC method
• This difference has a positive impact on forecasts• The meridional length scale is larger than the
zonal length scale for all variables, except zonal wind
• The meridional length scale is more homogeneous than the zonal length scale
• The zonal length scale is larger over the tropics