Data assimilation schemes in numerical weather forecasting and their link with ensemble forecasting

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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

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Outline

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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 y^o

État atmosphérique à t₀ Prévision état à t₀ + h Ebauche x^b = M (x^a^-)

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Outline

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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

Scatterometer

SSM/I Ozone

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Satellites

(EUMETSAT)

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Satellite data sources

(courtesy J-.N.Thépaut, ECMWF)

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General formalism

 Statistical linear estimation :

x^a= x^b + x =x^b + K d = x^b + BH^T(HBH^T+R)^-1 d, with d = y^o – H (x^b), 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(^o^oT) ).

 Observation errors are supposed to be un-correlated : ( non-diagonal elements of E(^o^oT) = 0 ),

 but, the representation of observation error correlations is also investigated (Fisher, 2006) .

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Implementation

 Variational formulation:

minimization of J(x) = x^T 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

J

_b

J

_o

J

_o

J

_o

obs

obs analysis

x

_a

x

_b

corrected forecast

« old » forecast

4D-Var : principle

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Outline

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A posteriori diagnostics

 Is the system consistent?

 We should have

E[J(x^a) ] = p,

p = total number of observations,

 but also

E[Joi(x^a) ] = pi – Tr(Ri-1/2 H i A H_iTRi-1/2 ),

pi : number of observations associated with Joi

(Talagrand, 1999) .

 Computation of optimal E[Joi(x^a) ] 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[Jô_i (xâ)] = (E[Jô_i (xâ)])ôpt.

by adjusting the ^o_i

∙

∙ ∙

∙

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Outline

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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 x^a and ^a:

x^a= (I – KH) x^b + K x^o

^a= (I – KH) ^b + K ^o

The same equation also holds for the analysis perturbation:

p^a= (I – KH) p^b + K p^o

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Background error standard-deviations

Over a month Vorticity at 500 hPa

For a particular date 08/12/2006 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

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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 : riDFS = Tr(Ki Hi)

 Reduction of error projected onto a variable/area:

riP = Tr(P Ki Hi B P^T)

 Reduction of error evolved by a forecast model:

riPM = Tr(P M Ki Hi B MT P^T) = Tr(L Ki Hi B L^T) (Cardinali, 2003; Fisher, 2003; Chapnik et al, 2006)

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Randomized estimates of error reduction on analyses and forecasts

( L K

_i

H B L

^T

)

i

tr

r 

It can be shown that

( H

_i

B L

^T

L K ).

 tr

This can be estimated by a randomization procedure:

j o T

j i

T j o

i i

r

i

^  (  y ) R

^¹

H B L L K (  y )

where

(  y

^o

)

_j is a vector of observation perturbations and

j a

)

(  x

the corresponding perturbation on the analysis.

j a i i

j

o

) R H B B L L ( )

(  y

^¹ ¹^/² ¹^/² ^* ^'

 x





(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 hPa by area and observation type

(Desroziers et al, 2005)

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Outline

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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)