Towards a longer assimilation window in 4D-Var - NASA · PDF fileTowards a longer assimilation window in 4D-Var ... Temperature zonal means, December 2010 ... 15 16 17. Y. Tr emolet

Towards a longer assimilation window in 4D-Var

Yannick Tremolet

Thanks to Paul Poli

ECMWF

October 2011

Y. Tremolet Long window 4D-Var October 2011

Outline

1 What we want to do

2 What we can do

3 ResultsModel Error Aspects24h Window: Operational System24h Window: Re-analysis System

4 Final Comments

Y. Tremolet Long window 4D-Var October 2011 1 / 21

Outline


2 What we can do


4 Final Comments


Weak Constraint 4D-Var

For Gaussian, temporally-uncorrelated model error, the weak constraint4D-Var cost function is:

J(x) =1

2(x0 − xb)TB−1(x0 − xb)

+1

2

n∑i=0

[Hi (xi )− yi ]TR−1

i [Hi (xi )− yi ]

+1

2

n∑i=1

[xi −Mi (xi−1)]TQ−1i [xi −Mi (xi−1)]

Do not reduce the control variable using the model and retain the 4D natureof the control variable.

Account for the fact that the model contains some information but is notexact by adding a model error term to the cost function.

This problem can be solved in parallel (saddle-point algorithm, no need forinverse of covariances, preconditioning is being investigated).


Longer is better

Theory says: long window weak constraint 4D-Var is equivalent to a full rankKalman smoother (Fisher et al., 2005, Menard and Daley, 1996).

Long window weak constraint 4D-Var works for simple systems (Lorenz 95,QG):

0 12 24 36 48 60 72 84 96 108 120 132 144 156 1680.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75Mean Analysis and First−Guess Error for Different Window Lengths

Length of the Assimilation Window (hours)

RM

S E

rror

for

Non

−di

men

sion

al S

trea

mfu

nctio

n

Initial quess using scaled qbarAnalysis using scaled qbar

H. Auvinen and M. Fisher


Long Window Weak Constraint 4D-Var

��

��

�

��

��

��

(1) Weak constraint 4D-Var (2) Extended window

��

��

� ��

��

��

� ��

(3) Initial term has converged (4) Assimilation window is moved forward

This implementation is an approximation of weak contraint 4D-Var with anassimilation window that extends indefinitely in the past...

...which is equivalent to a (full rank) Kalman smoother that has been runningindefinitely.

And B is a problem of the past! Only the error characteristics of thefundamental ingredients of the DA problem remain.


Outline


2 What we can do


4 Final Comments


4D-Var with Model Error Forcing

In practice, weak constraint 4D-Var is still difficult to implement (in the IFS).

Change of variable:

J(x0, η) =1

2

n∑i=0

[H(xi )− yi ]TR−1

i [H(xi )− yi ]

+1

2(x0 − xb)TB−1(x0 − xb) +

1

2

n∑i=1

ηTi Q−1

i ηi

with xi =Mi (xi−1) + ηi

ηi represents model error in a time step,

ηi has the same dimension as a 3D state.


4D-Var with Constant Model Error Forcing

Approximation: model error is constant.

J(x0, η) =1

2

n∑i=0


i [H(xi )− yi ]

+1

2(x0 − xb)TB−1(x0 − xb) +

1

2ηTQ−1η

with xi =Mi (xi−1) + η

η represents model error in a time step,

η has the same dimension as a 3D state.

The number of degrees of freedom doubles.


Weak Constraints 4D-Var for Systematic Model Error

For random model error, the 4D-Var cost function is:

J(x0, η) =1

2

n∑i=0


i [H(xi )− yi ]

+1

2(x0 − xb)TB−1(x0 − xb) +

1

2ηTQ−1η

For systematic model error:

J(x0, η) =1

2

n∑i=0


i [H(xi )− yi ]

+1

2(x0 − xb)TB−1(x0 − xb) +

1

2(η − ηb)TQ−1(η − ηb)

Test case: model bias in the stratosphere.


Model Error Covariance Matrix

Currently, tendency differences between integrations of the members of anensemble are used as a proxy for samples of model error.

Statistics of model drift (for systematic model error).

Use results from stochastic representation of uncertainties in EPS.

It is possible to derive an estimate of HQHT from cross-covariances betweenobservation departures produced from pairs of analyses with different lengthwindows (R. Todling).

Is it possible to extract model error information using the relationPf = MPaMT + Q?

Model error is correlated in time: Q should account for time correlations.How?

How to account for flow dependence?


Outline


2 What we can do


4 Final Comments


Outline


2 What we can do


4 Final Comments


Weak Constraints 4D-Var with Cycling Term

01 15Jun

01 15Jul

01 15Aug

01 15Sep

01 15Oct

01 15Nov

01 15Dec

01 15Jun

01 15Jul

01 15Aug

01 15Sep

Weak constraints 4D-Var with cycling – MetOp-A AMSU-A Tb 13 N. Hemis – Model level 14

Weak constraints – MetOp-A AMSU-A Tb 13 N. Hemis – Model level 14

01 15Oct

01 15Nov

01 15Dec

0.6

0.4

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0.0

–0.2

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ΔT

(K

)

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–0.6

ΔT

(K

)

OBS-BG OBS-AN OBS Bias Increment Model Error

The short term forecast is improved with the model error cycling.Weak constraints 4D-Var can correct for seasonal bias (partially).


Observation Error or Model Error?

05 10 15 20 25 01 05 10 15 20 25 01 05 10 15 20 25 01 05 10 15 20 25 01 05 10 15 20

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�T (K)Weak constraints 4D-Var with cycling - Metop-A AMSU-A Tb 13 N.Hemis - Model level 14

OB-FGOB-ANObs BiasIncrementModel Error

05 10 15 20 25 01 05 10 15 20 25 01 05 10 15 20 25 01 05 10 15 20 25 01 05 10 15 20

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Dec

Jan

Feb

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�T (K)

CY35R2 - Metop-A AMSU-A Tb 13 N.Hemis - Model level 14

OB-FGOB-ANObs BiasIncrement

Observation error bias correction can compensate for model error.



Model Error (K/day) Model Drift (K/day)

90N/ 90S/ MetgraF-3.0-2.00-1.80-1.6-1.40-1.20-1.00-0.8-0.60-0.4-0.20-0.10.100.20.400.60.801.001.201.41.601.802.0

-1.8-1.4

-1.4

-1.0

-1.0

-0.6

-0.6

-0.6

-0.6

-0.2

-0.2

-0.2

-0.2

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-0.2

-0.2

-0.2

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0.1

0.1

0.1

0.1

0.1 0.1

0.1

0.1

0.1

0.4

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0.4

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0.4

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1.2

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90N/ 90S/ MetgraF-3.0-2.0-1.5-1.0-0.70-0.5-0.30-0.20-0.15-0.10-0.07-0.04-0.02-0.010.010.020.040.070.100.150.200.30.500.71.01.52.0

-0.3

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-0.1

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0.00.0 0.0 0.0

0.0

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141516171819202122

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2628303234363840424446485052545658606264687276808488

Temperature zonal means, December 2010



Mean (K/day) Standard Deviation (K/day)

90N/ 90S/ MetgraF-3.0-2.00-1.80-1.6-1.40-1.20-1.00-0.8-0.60-0.4-0.20-0.10.100.20.400.60.801.001.201.41.601.802.0

-1.8-1.4

-1.4

-1.0

-1.0

-0.6

-0.6

-0.6

-0.6

-0.2

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90N/ 90S/ MetgraF0.000.050.100.150.200.250.300.350.400.450.500.600.700.8

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Temperature zonal means, December 2010

Model error estimates vary rapidly in NH stratosphere.


Outline


2 What we can do


4 Final Comments


24h 4D-Var: Forecast Scores

-5

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0

1

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0 1 2 3 4 5 6 7 8 9 10 11

Forecast Day

00UTC | Confidence: 95.0 | Population: 42

Date: 20101120 00UTC to 20101231 00UTCN Hem Extratrop (lat 20.0 to 90.0, lon -180.0 to 180.0)

Correlation coefficent of forecast anomaly500hPa geopotential Overlaping 24h 4D-Var minus 12h 4D-Var

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Forecast Day

00UTC | Confidence: 95.0 | Population: 42

Date: 20101120 00UTC to 20101231 00UTCS Hem Extratrop (lat -90.0 to -20.0, lon -180.0 to 180.0)

Correlation coefficent of forecast anomaly500hPa geopotential Overlaping 24h 4D-Var minus 12h 4D-Var

Forecast scores for overlapping 24h 4D-Var with respect to 12h 4D-Var.


Long Window 4D-Var Cycling


24h 4D-Var: Forecast Scores

0 1 2 3 4 5Forecast Range (days)

0

10

20

30

40

50

RM

SE

rror

(m)

12h 4D-Var 24h fc1 an1 24h fc1 an2 24h fc2 an2

With overlapping analysis windows, there are several analyses to start theforecast from and to verify against!

Warning: too few cases to draw conclusions from this figure.


24h 4D-Var: Observation Statisticsexp: fk5q / fk5q / DA (black) v. �ut / DA 2010120100 – 2010121512(12)

AIREP-T N. Hemisphere used T

nobsexp

132294

241448

257268

238009

135873

127386

395932

277642

234

0

+4374

+8280

+10298

+9255

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+14242

+10116

+7

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exp-refStandard deviation

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ssu

re (

hP

a)

Pre

ssu

re (

hP

a)

Bias

Background departure o-b (ref ) Background departure o-b

Analysis departure o-a (ref ) Analysis departure o-a

0 0.2–0.2–0.4–0.6–0.8–1 0.4 0.6 0.8 11.81.20.60 2.4 3

exp: fk5q / fk5q / DA (black) v. �ut / DA 2010120100 – 2010121512(12)

TEMP-T N. Hemisphere used T

nobsexp

325105503663501548164051937345328653813142106420623561437972353233129726934

2403

+82+929

+1813+1653+1196+1068

+937+1053+1240+1362+1197+1285+1122

+983+849

+71

1000850700500400300250200150100

7050302010

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ssu

re (

hP

a)

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ssu

re (

hP

a)

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exp-refStandard deviation Bias

Background departure o-b (ref ) Background departure o-b

Analysis departure o-a (ref ) Analysis departure o-a

0 0.2–0.2–0.4–0.6–0.8–1 0.4 0.6 0.8 12.41.60.80 3.2 4


Outline


2 What we can do


4 Final Comments


Ps-only Re-analysis

Background and Analysis fit to Observations2004-07-01 to 2005-04-09

0 3 6 9 12 15 18 21 24Time (h)

0.5

1.0

1.5

2.0

Ps

obse

rvat

ion

fit(h

Pa)

Overlapping 24h 4D-Var 24h 4D-Var 12h 4D-Var

0 3 6 9 12 15 18 21 24Time (h)

0.5

1.0

1.5

2.0

Ps

obse

rvat

ion

fit(h

Pa)



Ps-only Re-analysis

Forecast scores vs. operational analysisZ500, NH, 2004-07-01 to 2005-04-09

0 1 2 3 4 5 6 7 8 9 10Forecast Range (days)

20

30

40

50

60

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100

Ano

mal

yC

orre

latio

n(%

)


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orre

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n(%

)



Ps-only Re-analysis

Verification against independent (unused) observations:I confirms positive results with overlapping windows,I shows that 24h 4D-Var without overlap is slightly better than 12h 4D-Var.

24h 4D-Var system has not been tuned.I Results should improve.

Why is 24h 4D-Var better in Ps-only re-analysis context?I Model error is small relative to other errors,I Kalman smoother rather than Kalman filter (in part),I Not enough observations to fully constrain the analysis in 12h 4D-Var,I Full observing system constrains the analysis so tightly that the assimilation

algorithm is not as important.


Outline


2 What we can do


4 Final Comments


24h Weak Constraint 4D-Var

In the current formulation of weak constraints 4D-Var (model error forcing):I Background term to address systematic error,I 24h assimilation window.

Observation biases can be an issue.I Experiment with bias corrected aircraft observations is starting.

Investigate physical meaning of model error estimates.I For the first time, we might be looking at model error!

Weak Constraints 4D-Var requires better knowledge of the statisticalproperties of model error.

Very good results in Ps-only experiments (re-analysis).

Kalman smoother is better at least for re-analysis.


Long Window Weak Constraints 4D-Var

Weak constraint 4D-Var with a 4D state control variable:I Four dimensional problem with a coupling term between sub-windows is a

smoother over the whole assimilation period.

Practical implementation is very difficult in current ECMWF system (code,scripts, archiving...).

We are re-designing our data assimilation system to make it all possible:Object Oriented Prediction System (OOPS).

I High level algorithms in C++,I Improved scalability, reliability, flexibility,I New algorithms are implemented (saddle point).


Documents

Towards a longer assimilation window in 4D-Var - NASA · PDF fileTowards a longer assimilation window in 4D-Var ... Temperature zonal means, December 2010 ... 15 16 17. Y. Tr emolet