Data assimilation schemes in numerical weather forecasting and their link with ensemble forecasting Gérald Desroziers Météo-France, Toulouse, France

Data assimilation schemes in numerical weather forecasting

and their link with ensemble forecasting

Gérald Desroziers

Météo-France, Toulouse, France

Outline

Numerical weather prediction

Data assimilation

A posteriori diagnostics: optimizing error statistics

Ensemble assimilation

Impact of observations on analyses and forecasts

Conclusion and perspectives

Outline


Data assimilation





Global Arpège model : DX ~ 15 km

Numerical Weather Prediction at Météo-France

DX ~ 10 km

Arome : DX ~ 2,5 km

Initial condition problem

Observations yo

État atmosphérique à t0 Prévision état à t0 + h

Ebauche xb = M (xa -)

Outline


Data assimilation





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

Satellites

(EUMETSAT)

0

5

10

15

20

25

30

35

40

45

50

55

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)

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 » .

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) .

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)

9h 12h 15h

Assimilation window

JbJo

Jo

Jo

obs

obs

obs

analysis

xa

xb correctedforecast

« old »forecast

4D-Var : principle

Outline


Data assimilation





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) .

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

∙∙

∙∙∙

∙∙∙

∙

∙

∙

∙∙∙∙∙∙∙

Outline


Data assimilation





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)

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

Background error standard-deviations

Over a month

Vorticity at 500 hPa

For a particular date08/12/2006 00H

Vorticity at 500 hPa

500 hPa vorticity error surface pressure

Ensemble assimilation:errors 08/12/2006 06UTC

850 hPa vorticity error (shaded)

sea surface level pressure (isoligns)

Ensemble assimilation:errors 15/02/2008 12UTC

(Montroty, 2008)

Outline


Data assimilation





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)

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)

Degree of Freedom for Signal (DFS)

01/06/2008 00H

Error variance reduction

% of error variance reduction for T 850 hPaby area and observation type

(Desroziers et al, 2005)

Outline


Data assimilation






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)

Documents

Data assimilation schemes in numerical weather forecasting and their link with ensemble forecasting Gérald Desroziers Météo-France, Toulouse, France