data assimilation schemes in numerical weather forecasting and their link with ensemble forecasting...
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![Page 1: Data assimilation schemes in numerical weather forecasting and their link with ensemble forecasting Gérald Desroziers Météo-France, Toulouse, France](https://reader035.vdocuments.mx/reader035/viewer/2022062308/56649d375503460f94a106d0/html5/thumbnails/1.jpg)
Data assimilation schemes in numerical weather forecasting
and their link with ensemble forecasting
Gérald Desroziers
Météo-France, Toulouse, France
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Outline
Numerical weather prediction
Data assimilation
A posteriori diagnostics: optimizing error statistics
Ensemble assimilation
Impact of observations on analyses and forecasts
Conclusion and perspectives
![Page 3: Data assimilation schemes in numerical weather forecasting and their link with ensemble forecasting Gérald Desroziers Météo-France, Toulouse, France](https://reader035.vdocuments.mx/reader035/viewer/2022062308/56649d375503460f94a106d0/html5/thumbnails/3.jpg)
Outline
Numerical weather prediction
Data assimilation
A posteriori diagnostics: optimizing error statistics
Ensemble assimilation
Impact of observations on analyses and forecasts
Conclusion and perspectives
![Page 4: Data assimilation schemes in numerical weather forecasting and their link with ensemble forecasting Gérald Desroziers Météo-France, Toulouse, France](https://reader035.vdocuments.mx/reader035/viewer/2022062308/56649d375503460f94a106d0/html5/thumbnails/4.jpg)
Global Arpège model : DX ~ 15 km
Numerical Weather Prediction at Météo-France
DX ~ 10 km
Arome : DX ~ 2,5 km
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Initial condition problem
Observations yo
État atmosphérique à t0 Prévision état à t0 + h
Ebauche xb = M (xa -)
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Outline
Numerical weather prediction
Data assimilation
A posteriori diagnostics: optimizing error statistics
Ensemble assimilation
Impact of observations on analyses and forecasts
Conclusion and perspectives
![Page 7: Data assimilation schemes in numerical weather forecasting and their link with ensemble forecasting Gérald Desroziers Météo-France, Toulouse, France](https://reader035.vdocuments.mx/reader035/viewer/2022062308/56649d375503460f94a106d0/html5/thumbnails/7.jpg)
Data coverage
05/09/03 09–15 UTC(courtesy J-.N. Thépaut)
Radiosondes Pilots and profilers Aircraft
Synops and ships Buoys
ATOVS Satobs Geo radiances
ScatterometerSSM/I Ozone
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Satellites
(EUMETSAT)
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No. of sources
1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009
Year
Number of satellite sources used at ECMWF
AEOLUSSMOSTRMMCHAMP/GRACECOSMICMETOPMTSAT radMTSAT windsJASONGOES radMETEOSAT radGMS windsGOES windsMETEOSAT windsAQUATERRAQSCATENVISATERSDMSPNOAA
Satellite data sources
(courtesy J-.N.Thépaut, ECMWF)
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General formalism Statistical linear estimation :
xa = xb + x =xb + K d = xb + BHT (HBHT+R)-1 d,
with d = yo – H (xb ), innovation, K, gain matrix, B et R, covariances of background and observation errors,
H is called « observation operator » (Lorenc, 1986),
It is most often explicit,
It can be non-linear (satellite observations)
It can include an error,
Variational schemes require linearized and adjoint observation operators,
4D-Var generalizes the notion of « observation operator » .
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Statistical hypotheses
Observations are supposed un-biased: E(o) = 0.
If not, they have to be preliminarly de-biased,
or de-biasing can be made along the minimization (Derber and Wu, 1998; Dee, 2005; Auligné, 2007).
Oservation error variances are supposed to be known ( diagonal elements of R = E(ooT) ).
Observation errors are supposed to be un-correlated : ( non-diagonal elements of E(ooT) = 0 ),
but, the representation of observation error correlations is also investigated (Fisher, 2006) .
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Implementation
Variational formulation: minimization of J(x) = xT B-1 x + (d-H x)T R-1 (d-H x)
Computation of J’: development and use of adjoint operators
4D-Var : generalized observation operator H : addition of forecast model
M.
Cost reduction : low resolution increment x (Courtier, Thépaut et Hollingsworth, 1994)
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9h 12h 15h
Assimilation window
JbJo
Jo
Jo
obs
obs
obs
analysis
xa
xb correctedforecast
« old »forecast
4D-Var : principle
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Outline
Numerical weather prediction
Data assimilation
A posteriori diagnostics: optimizing error statistics
Ensemble assimilation
Impact of observations on analyses and forecasts
Conclusion and perspectives
![Page 15: Data assimilation schemes in numerical weather forecasting and their link with ensemble forecasting Gérald Desroziers Météo-France, Toulouse, France](https://reader035.vdocuments.mx/reader035/viewer/2022062308/56649d375503460f94a106d0/html5/thumbnails/15.jpg)
A posteriori diagnostics
Is the system consistent?
We should have E[J(xa) ] = p,
p = total number of observations, but also
E[Joi(xa) ] = pi – Tr(Ri-1/2 H
i A Hi
T Ri-1/2 ),
pi : number of observations associated with Joi
(Talagrand, 1999) .
Computation of optimal E[Joi(xa) ] by a Monte-Carlo procedure is possible. (Desroziers et Ivanov, 2001) .
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Application : optimisation of R
(Chapnik, et al, 2004; Buehner, 2005)
Optimisation of HIRS o
One tries to obtain
E[Joi (xa)] = (E[Joi (xa)])opt.
by adjusting the oi
∙∙
∙∙∙
∙∙∙
∙
∙
∙
∙∙∙∙∙∙∙
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Outline
Numerical weather prediction
Data assimilation
A posteriori diagnostics: optimizing error statistics
Ensemble assimilation
Impact of observations on analyses and forecasts
Conclusion and perspectives
![Page 18: Data assimilation schemes in numerical weather forecasting and their link with ensemble forecasting Gérald Desroziers Météo-France, Toulouse, France](https://reader035.vdocuments.mx/reader035/viewer/2022062308/56649d375503460f94a106d0/html5/thumbnails/18.jpg)
Ensemble of perturbed analyses
Simulation of the estimation errors
along analyses and forecasts.
Documentation of error covariances
– over a long period (a month/ a season),
– for a particular day.
(Evensen, 1997; Fisher, 2004; Berre et al, 2007)
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Ensembles Based on a perturbation of observations
The same analysis equation and (sub-optimal) operators K and H
are involved in the equations of xa and a:
xa = (I – KH) xb + K xo
a = (I – KH) b + K o
The same equation also holds for the analysis perturbation:
pa = (I – KH) pb + K po
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Background error standard-deviations
Over a month
Vorticity at 500 hPa
For a particular date08/12/2006 00H
Vorticity at 500 hPa
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500 hPa vorticity error surface pressure
Ensemble assimilation:errors 08/12/2006 06UTC
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850 hPa vorticity error (shaded)
sea surface level pressure (isoligns)
Ensemble assimilation:errors 15/02/2008 12UTC
(Montroty, 2008)
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Outline
Numerical weather prediction
Data assimilation
A posteriori diagnostics: optimizing error statistics
Ensemble assimilation
Impact of observations on analyses and forecasts
Conclusion and perspectives
![Page 24: Data assimilation schemes in numerical weather forecasting and their link with ensemble forecasting Gérald Desroziers Météo-France, Toulouse, France](https://reader035.vdocuments.mx/reader035/viewer/2022062308/56649d375503460f94a106d0/html5/thumbnails/24.jpg)
Measure of the impact of observations
Total reduction of estimation error variance:r = Tr(K H B)
Reduction due to observation set i :ri = Tr(Ki Hi B)
Variance reduction normalized by B :ri
DFS = Tr(Ki Hi)
Reduction of error projected onto a variable/area:ri
P = Tr(P Ki Hi B PT)
Reduction of error evolved by a forecast model:ri
PM = Tr(P M Ki Hi B MT PT) = Tr(L Ki Hi B LT)
(Cardinali, 2003; Fisher, 2003; Chapnik et al, 2006)
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Randomized estimates of error reduction on analyses and forecasts
)( LBHKL Tii trr
It can be shown that
).( KLLBH Titr
This can be estimated by a randomization procedure:
joT
ji
Tj
oi iir )()( 1 yKLLBHRy
where jo)( y is a vector of observation perturbations and
ja)( x the corresponding perturbation on the analysis.
ja
iij
jo LLBBHR )()( '*2/12/11 xy
(Fisher, 2003; Desroziers et al, 2005)
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Degree of Freedom for Signal (DFS)
01/06/2008 00H
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Error variance reduction
% of error variance reduction for T 850 hPaby area and observation type
(Desroziers et al, 2005)
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Outline
Numerical weather prediction
Data assimilation
A posteriori diagnostics: optimizing error statistics
Ensemble assimilation
Impact of observations on analyses and forecasts
Conclusion and perspectives
![Page 29: Data assimilation schemes in numerical weather forecasting and their link with ensemble forecasting Gérald Desroziers Météo-France, Toulouse, France](https://reader035.vdocuments.mx/reader035/viewer/2022062308/56649d375503460f94a106d0/html5/thumbnails/29.jpg)
Conclusion and perspectives
Importance of the notion of « observation operator » :- most often explicit,- rarely statistical
Large size problems :- state vector : ~ 10^7- observations : ~ 10^6
Ensemble assimilation:– estimation error covariances– measure of the impact of observations– link with Ensemble forecasting (~ 40 members of +96h forecasts)