Development of Kalman Filter Assimilation Package Based on QG 2-layer Model

Development of Kalman Filter Assimilation Package Development of Kalman Filter Assimilation Package Based on QG 2-layer Model Based on QG 2-layer Model

Collaborators:Collaborators: Hyo-Jong, SongHyo-Jong, Song Joo-Wan, KimJoo-Wan, Kim Nam-Gyu, NohNam-Gyu, Noh Gyu-Ho, LimGyu-Ho, Lim

School of Earth and Environmental Studies, SNUSchool of Earth and Environmental Studies, SNUKim Baek MinKim Baek Min

Today’s talk

1.1. Review the current status of EnKFReview the current status of EnKF

2.2. Exploration of world of EnKF with Lorenz modelExploration of world of EnKF with Lorenz model

3.3. Description of Lorenz QG 2-layer modelDescription of Lorenz QG 2-layer model

4.4. EnKF test with QG modelEnKF test with QG model

5.5. (If you are not still hungry or not still bored^^;) (If you are not still hungry or not still bored^^;) Introduction to New Efficient EKFIntroduction to New Efficient EKF

•Many centers still use 3D-Var though it can not represent time-varying error statistics.

•Several centers (ECMWF, UK, Canada) have switched to 4D-Var, better than 3D-Var.

•4D-Var was clearly better than 3D-Var, but EnKF was only comparable to 3D-Var (Mitchell and Houtekamer,2003).

•Whitaker and Hamill(2005) show EnKF better than NCEP’s 3D-Var for real data.

•Houtekamer and Mitchell(2005) show that EnKF is now as good as 4D-Var.

Current status of EnKF

EnKF Exploration with Lorenz ModelEnKF Exploration with Lorenz Model

1( ) ( )a f f T f T fobsP H HP H R d H

Analysis equationAnalysis equation

obs trued H

( )( )T Tf f true f true f fp

•Forecast covariance matrixForecast covariance matrix should be given apriori should be given apriori

H

( )( )T Tobs true obs true obs obsR d d

Traditional OITraditional OI

•There is no model for covarianceThere is no model for covariance

•Observation error is represented by:Observation error is represented by:

• transforms from model space to obs. spacetransforms from model space to obs. space


1f f Tt tP LP L

Analysis equationAnalysis equation(just same with OI)(just same with OI)

Traditional Kalman FilterTraditional Kalman Filter

•There is There is modelmodel for covariance for covariance

•Covariance is updated with the aid of Model LCovariance is updated with the aid of Model L

•Model L should be linear model in Kalman filterModel L should be linear model in Kalman filter

•Forecast is provided by integration of L with analysisForecast is provided by integration of L with analysis

( )f aL


1f f Tt tP MP M

Analysis equationAnalysis equation(just same with OI)(just same with OI)

Extended Kalman FilterExtended Kalman Filter

•There is There is modelmodel for covariance for covariance

•Covariance is updated with the aid of Covariance is updated with the aid of TLM TLM of NLof NL

•Model Model NN is nonlinear model in Kalman filter is nonlinear model in Kalman filter

•Forecast is provided by integration of Forecast is provided by integration of NN with analysis with analysis

( )f aN

RfP

1 2( , ) ( ) ( )a f a f aP d P P d

2 ( )aP d1( )a fP

a fd

( , )a fP d 1( )a fP 2 ( )aP d

Interpretation of OI analysisInterpretation of OI analysis

•Given observation( ) , forecast ( ),Given observation( ) , forecast ( ),

find the find the conditional probabilityconditional probability(a posteriori prob.) through Bayes theorem.(a posteriori prob.) through Bayes theorem.

•Then, we get final analysis( ) by taking point maximizing Then, we get final analysis( ) by taking point maximizing that pdfthat pdf..

d f

a•The conditional probability is given byThe conditional probability is given by

•The structure of is solely determined by The structure of is solely determined by

•The structure of is solely determined by The structure of is solely determined by •When both PDF of obs. And forecast are gaussian!

•When both obs. and forecast are unbiased!

1( ) ( )a f f T f T fj j j jP H HP H R d H

1 2( , ,...., )f f fN

•Consider ensemble of forecast vector and observation vector:Consider ensemble of forecast vector and observation vector:

1 2( , ,...., )Nd d d

•Apply Kalman filter eq. for each jth ensemble memberApply Kalman filter eq. for each jth ensemble member : :

Introducing Ensemble Introducing Ensemble

1

1( )( )

1

Nf f f f fe j j

j

PN

fP

•Flow dependent forecast covariance through the benefit from the wellFlow dependent forecast covariance through the benefit from the well distributed distributed forecast ensemble membersforecast ensemble members..

•No more predefinition of forecast covariance except for the I.C.No more predefinition of forecast covariance except for the I.C.

•Now, forecast model has Now, forecast model has ability to produce its own error statisticsability to produce its own error statistics..

•Forecast PDF follows Focker–planck equation(FPE) theoretically.Forecast PDF follows Focker–planck equation(FPE) theoretically.

•Direct linear approximation to FPE = EKF (Need TLM)Direct linear approximation to FPE = EKF (Need TLM)

•Monte-carlo approximation to FPE = EnKF(No need for TLM)Monte-carlo approximation to FPE = EnKF(No need for TLM)

1

1 Nf f

jjN

Heart of EnKF(1)Heart of EnKF(1)

R1

1( )( )

1

N

e j jj

R d d d dN

•Not essential for obtaining best estimate(analysis).Not essential for obtaining best estimate(analysis).

•But, helps to improve the spreading of analysis variance of analysisBut, helps to improve the spreading of analysis variance of analysis(Burgers et al., 1998)(Burgers et al., 1998)

•If ensemble is small, however, this is source of errorneous analysisIf ensemble is small, however, this is source of errorneous analysis

1

1 N

jj

d dN

Heart of EnKF(2)Heart of EnKF(2)

1( ) ( )a f f T f T fe e eP H HP H R d H

•Finally, the ananlysis equation for EnKF is given by:Finally, the ananlysis equation for EnKF is given by:

a fd

eRfeP

( , )a fP d

1 2( , ) ( ) ( )a f fa aP d P P d

1( )a fP 2 ( )aP d

approximates second mom. ofapproximates second mom. of 2 ( )aP d

approximates second mom. ofapproximates second mom. of 1( )a fP

Interpretation of EnKF analysisInterpretation of EnKF analysis

Example of EnKF analysisExample of EnKF analysis

10( )

8

3

dxy x

dtdy

rx y xzdtdz

xy zdt

•Suppose we conduct EnKF assimilation applied to Lorenz modelSuppose we conduct EnKF assimilation applied to Lorenz model

•Lorenz model •Dimension of model space is three

•Dynamics of the model considerably differs depending on parameter r

•r=21:Stable point attractorr=21:Stable point attractor •r=28: Chaotic attractorr=28: Chaotic attractor

Application to Lorenz ModelApplication to Lorenz Model

1 2 100

1 2 100 1 2 100

1 2 100

...

( , ,...., ) ...

...

f f f

f f f f f f

f f f

x x x

A y y y

z z z

•Dimension of Model state is 3 and 100 ensemble members are used.Dimension of Model state is 3 and 100 ensemble members are used.

...

( , ,...., ) ...

...

f f f

f f f f f f

f f f

x x x

A y y y

z z z

'A A A

' '

1

Tfe

A AP

N

Matrix representation of EnKF(1) (Evensen, 2003)Matrix representation of EnKF(1) (Evensen, 2003)

1 2 100

1 2 100 1 2 100

1 2 100

...

( , ,...., ) ...

...

obs obs obs

obs obs obs

obs obs obs

x x x

D d d d y y y

z z z

...

( , ,...., ) ...

...

obs obs obs

obs obs obs

obs obs obs

x x x

D d d d y y y

z z z

'D D D

' '

1

T

e

D DR

N

Matrix representation of EnKF(2) (Evensen, 2003)Matrix representation of EnKF(2) (Evensen, 2003)

1' ' ( ' ' ' ' ) ( )a T T T T TA A A A H HA A H D D D HA

1( ) ( )a T T Te e eA A P H HPH R D A

1( ) ( )a f f T f T fj j e e e j jP H HP H R d H

1 0 0

0 1 0

0 0 1

H

for our experiment.for our experiment.

Integration Integration with modelwith model

10( )

8

3

dxy x

dtdy

rx y xzdtdz

xy zdt

Weighted Weighted MeanMean

DMeasurement of Obs.Measurement of Obs. Random generatorRandom generator

Gaussian, Normal Gaussian, Normal PDFPDF

aA

Ensemble Ensemble of 100 of 100

analysisanalysis

A

Ensemble Ensemble of 100 of 100

ForecastForecast

feP

Statistics Statistics of of

EnsembleEnsemble

D

Ensemble of 1Ensemble of 100 Obs.00 Obs.

eR

Statistics Statistics of of

Ensemble Ensemble

Summary of EnKF analysisSummary of EnKF analysis

FC = pgf90FC = pgf90FCFLAGS = -Mfree -O2 -r8FCFLAGS = -Mfree -O2 -r8

.SUFFIXES= .F .i .o .f.SUFFIXES= .F .i .o .f

.f.o:.f.o:$(FC) -c $(FCFLAGS) $*.f $(FC) -c $(FCFLAGS) $*.f

OBJS = m_multa.o random.o EnKF.o OBJS = m_multa.o random.o EnKF.o analysis.oanalysis.o lorzrk.o rk4.o lorzrk.o rk4.o

EnKF.exe: $(OBJS)EnKF.exe: $(OBJS)$(FC) -o $@ $(OBJS) $(FCFLAGS) ../../lib/$(FC) -o $@ $(OBJS) $(FCFLAGS) ../../lib/lapack_LINUX.alapack_LINUX.a ../.. ../..

/lib//lib/blas_LINUX.ablas_LINUX.a$(RM) $(OBJS)$(RM) $(OBJS)$(RM) *.mod$(RM) *.mod

clean:clean:$(RM) EnKF.exe $(RM) EnKF.exe

MakefileMakefile

•analysis.f is obtained from http://www.nrsc.no/Code/

•LAPACK, BLAS is obtained from http://netlib.org

http://www.nrsc.no/Code/

•Time step=0.01

•Analysis time=every 0.5(every fifty step)

•Chaotic model regime( r=28)

•Assume true trajectory (x=1.5,y=-1.5,z=25.5)

•Observations are simulated by adding std. dev. 1 perturbation (gauss pdf).

to true trajectory at every analysis time(OSSE)

Experiment 1(Chaotic regime)Experiment 1(Chaotic regime)

Result(Chaotic regime)Result(Chaotic regime)

How does error grow?How does error grow?

T=0.5sT=0.5s

T=1sT=1s

T=1.5sT=1.5s

T=2sT=2s

T=2.5sT=2.5s

T=3sT=3s

•Time step=0.01

•Analysis time=every 0.5(every fifty step)

•Stable model regime( r=21)

•Assume true trajectory (x=1.5,y=-1.5,z=25.5)

•Observations are simulated by adding std. dev. 5 perturbation (gauss pdf).

to true trajectory at every analysis time(OSSE)

•Model dynamics converges to stable equilibrium point. Henceforth, cloud of model ensemble should be shrink as analysis goes by…

•We still expect good result even though we provide quite bad obs(std. dev.=5).

Experiment 2(Stable regime)Experiment 2(Stable regime)

T=0.5sT=0.5s

T=1sT=1s

T=5sT=5s

T=10sT=10s

ResultResult

Lorenz QG 2-layer modelLorenz QG 2-layer model

•Beta plane, channel

•Fourier basis

•Exact calculation of nonlinear terms using interaction coefficient method(exact but slow compared to Transform method)

•Runge-Kutta 4th order time integration scheme

•Periodic boundary condition in west/east direction

•No mass flux across the lateral boundary

•Model can be as simple as possible to Lorenz 3variable model.

•Model can be run in a very high resolution mode.

CharacteristicsCharacteristics

Friction at the interface

Ekman friction

Radiative cooling

10000km

5000km

Radiative equilibrium mean potentialtemperature (θ*(x,y))

H

ρ1

ρ2

Dynamics Quasi-Geostrophic 2-layer

β-plane(mid-latitude)

Parameter-izations

Ekman damping ( k )

Friction at interface ( k´ )

Radiative cooling ( h´´ )

Latent heat release (α0 )

SchematicsSchematics

Variables and parametersVariables and parameters

2 2 2 20

2 2 2 ' 2 20 0

' *

1 1( , ) ( , ) , ( )

2 2

1 1( , ) ( , ) 2 , ( )

2 2

( , ) ( )

d

d d

d

f hJ J J k

t x H

f w f hJ J k J k

t x H H

wJ h

t H

A

Equation setEquation set

( )

( , )

( , )

2 cos( )

2cos( )sin( )

2sin( )sin( )

i

j j

j j

A P i

K M P j j

L H P j j

f P y

f M x P y

f H x P y

1,2, ,

1, 2, ,

1, 2, ,

i T

j T

j T

P Y

M X

H X

Basis functionBasis function

Baroclinic eddy simulationBaroclinic eddy simulation

From Master thesis of Joo-Wan, KimFrom Master thesis of Joo-Wan, Kim

Tangent Linear ModelTangent Linear Model

•For the development of EKF of QG model, TLM of QG model is needed.For the development of EKF of QG model, TLM of QG model is needed.

•Multi-variable taylor expansion is given by:

0 0 0

0 0 0

0 0 0

, , , ,

, , , , . .

, , , ,

F F FD

Dt F FF F FD

F F H OTDt

F FD F F FDt

F F FD

DtF F FD

DtD F F FDt

•TLM is defined as:TLM is defined as:

Basis expansion converts PDE ->ODE

2 2 2 201 1( , ) ( , ) , ( )

2 2 d

f hJ J J k

t x H

,

( ( ) ( ) ( ) ( )) ...iijk j k j k ij j

j k j

dA t t t t B

dt

,

( ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )) ...iijk j k j k j k j k ij j

j k j

dA t t t t t t t t B

dt

Example of Tangent LinearizationExample of Tangent Linearization

Preliminary EnKF Experiment(High resolution)Preliminary EnKF Experiment(High resolution)

Percentage of errors averaged in 192 grid points

0

20

40

60

80

100

120

140

160

0 1 2 3 4 5 6 7 8Analysis time (6hr interval)

Depart

ure

s o

f estim

ate

s o

r

analy

sis

fro

m a

tru

e (

%)

Perturbed observation

Unperturbed observation


0

20

40

60

80

100

120

140

160


Dep

art

ure

s o

f esti

ma

tes o

r a

na

lysis

fro

m a

tru

e (

%)

100 members

50 members

25 members

10 members

Preliminary EnKF Experiment(High resolution)Preliminary EnKF Experiment(High resolution)

Preliminary EnKF Experiment(Low resolution)Preliminary EnKF Experiment(Low resolution)


0

2

4

6

8

10

12

14

16

18

20


Depart

ure

s of

est

imate

s or

analy

sis

from

a t

rue (

%)

Perturbed observation

Unperturbed observation


02

46

810

1214

1618

20

0 1 2 3 4 5 6 7 8

Analysis time (6hr interval)

Dep

art

ures

of

estim

ate

s or

ana

lysi

s fr

om

a t

rue

(%)

100 members50 members25 members10 members

Preliminary EnKF Experiment(Low resolution)Preliminary EnKF Experiment(Low resolution)

Model HistoryModel History

•Kim(Baek Min) implemented QG-2layer model based on Cehelsky and Tung(1987).Kim(Baek Min) implemented QG-2layer model based on Cehelsky and Tung(1987). He used the model in his master thesis for the predictability study(2001).He used the model in his master thesis for the predictability study(2001).

•Kim(Joo Wan) made a TLM version and obtained a singular vector ofKim(Joo Wan) made a TLM version and obtained a singular vector of QG-2layer model in his master thesis(2003).QG-2layer model in his master thesis(2003).

•Noh(Nam Kyu) implemented EKF of QG-2layer model.Noh(Nam Kyu) implemented EKF of QG-2layer model. He compared 4Dvar and EKF in his master thesis using Lorenz model(2005).He compared 4Dvar and EKF in his master thesis using Lorenz model(2005).

•Song(Hyo Jong) implemented EnKF of QG-2layer model.Song(Hyo Jong) implemented EnKF of QG-2layer model.

Estimate of the Forecast Error Covariance with Governing Eigen-modes

• The forecast error covariance matrix consists of eigen-values and eigenvectors.

• To estimate a variance of each eigen-mode, we need statistically 100 ensemble members.

• The number of eigen-modes, which can be detected by forecast ensemble, is equal to that of ensemble members.


Rank of the analysis error covariance matrix

02468

1012141618

0 1 2 3 4 5 6 7 8

Analysis time (6hr interval)

Num

ber

of

go

vern

ing

eig

enm

od

es

• If the number of governing eigen-modes is smaller than 100, the forecast error covariance may be estimated by another method to reduce computational cost.


100.an smaller th is modes-eigen governing ofnumber

theifcost low with covarianceerror forecast theestimatecan We).( of form

expansion Talyor in first than sorder termhigher discards ly,Particular linear.

tangent and model a does and also state, model a and covarianceerror an

denotes and true,and analysis forecast,for stand and , tssuperscrip where

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)( }))({(

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interval, analysis

anfor validis covarianceerror forecast theion toapproximatlinear tangent If

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T2

111

T

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TT111

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Documents

Development of Kalman Filter Assimilation Package Based on QG 2-layer Model