2/3/16

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AOSC615ProjectExercise&ProjectDescrip:on

§  ProjectI:FrameworkofDataAssimila:onSystem•  ObservingSystemSimula:onExperiments(OSSEs)

§  Exercise1.Implementa:onofOp:miza:onMethods§  Exercise2.Construc:onofCovarianceMatrix§  ProjectII:3DMethods•  Op:malInterpola:on•  3DVar•  BuildingofDiagnos:cModuleforValida:on

§  Exercise3.Precondi:oningforminimiza:on§  Exercise4.Valida:onofTangentLinearModel§  Exercise5.Diagnos:csinObserva:onSpace§  ProjectIII:ExtendedKalmanFilter§  Exercise6.4DFramework§  ProjectIV:4DVar§  ProjectV:EnsembleKalmanFilter§  ProjectVI:FinalProject 1

1.Introduc:onProjectI

2.BackgroundExercise1Exercise2

3.3DMethodsExercise3ProjectII

4.Uncertain:es5.4DMethods

Exercise4Exercise5ProjectIIIExercise6ProjectIVProjectV

6.AdvancedMethods7.SpecialTopics

FinalProjectVI

AOSC615 '16Spring

ClassOrganiza:onProjectSummary

AOSC615ProjectExercise&ProjectDescrip:on

ProjectEvalua;on§  Presenta:on(exceptProjectI)§  Report§  Codes(nodataunlessrequested)Exercise§  Tohelpimplementa:onandvalida:onoftheupcoming

projects§  Noneedforsubmission

2

1.Introduc:onProjectI

2.BackgroundExercise1Exercise2

3.3DMethodsExercise3ProjectII

4.Uncertain:es5.4DMethods

Exercise4Exercise5ProjectIIIExercise6ProjectIVProjectV

6.AdvancedMethods7.SpecialTopics

FinalProjectVI

AOSC615 '16Spring

ClassOrganiza:onProjectSummary

2/3/16

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AOSC615Project.ReportOutline

Typicalreportmayconsistof(butchangeasnecessaryfollow)•  Title:“AOSC615.ProjectNo.<…>”•  Name•  Abstract [foreachproject]•  Maintext:sample1. Introduc:on/Background2. Objec:ves [foreachproject]3. Dataassimila:onsystema. Assimila:onmethod(howforecastICwasgenerated,spin-up?,assimila:onwindowsize)

b. Valida:onapproach(howtoverifyyourcodesdowhattheyaresupposedtodo)

4. Experimentalset-up5. Results(includingvisualiza:on)a. Implementa:onValida:onb. Results,incomparisonwithpreviousresultsifavailable

6. Concludingdiscussion•  References

3AOSC615 '16Spring

AOSC615ProjectProject

ProjectSetup1. Background2. OSSEExperimentalframework3.  Projectevalua:on4.  Specificnotes

a) Modelb) Assimila:onwindow

4

1.Introduc:onProjectI

2.BackgroundExercise1Exercise2

3.3DMethodsExercise3ProjectII

4.Uncertain:es5.4DMethods

Exercise4Exercise5ProjectIIIExercise6ProjectIVProjectV

6.AdvancedMethods7.SpecialTopics

FinalProjectVI

AOSC615 '16Spring

ClassOrganiza:onProjectSummary

2/3/16

3

Schema:csofDataAssimila:onSystem

Assimila:onwindow

Step1.ModelForecast

Forecast(=background):xbk

Observa;on

Measurement:yok

yk=hk xk( ) : yk ∈ Lk

xk

b =mk ,k−1 xk−1a( ) : x ∈!N Step2.Analysis

(Integra:onofxbkandyok)

Analysis:xak=funcofxbkandyok

natureUnknowntruthxtk

Nota:on

Vectorsx:statey:observa:onFunc:onm:physicalmodelh:observa:on

Superscriptb:backgrounda:analysiso:observa:onst:truth(nature)Subscriptk::me

AOSC615 '16Spring 5

ChallengesinDataAssimila:on

ModelForecast:xb

Observa;onMeasurement:yo

yk ' =hk ' xk '( ) : yk ' ∈Lk '

xk =mk ,k−1 xk−1( ) : x ∈N

Assimila:oncycle

Assimila:onwindow:me

Analysis:xa

xk−1 xk

yk’is/maybe•  insufficienttodeterminexk•  notexactlyattk

yk '

hk’is/maybe•  nonlinear•  imperfect

ymaybelargeortoosmall

mk,k-1is•  nonlinear•  imperfect

xcanbelarge

AOSC615 '16Spring 6

Accuratep(xk)?

Accurate&efficientp(xk|yk)?

Accuratep(yk|xk)?

2/3/16

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AOSC615ProjectFramework

§  Setup:ObservingSystemSimula:onExperiments(OSSEs)•  Useofthesynthe:cNatureRun(NR)usingtheavailablefreemodelrun•  Evalua:onagainsttheknown,unknowntruth(=NR)for- DataAssimila:on(DA)method&implementa:on- Observingsystem

§  MainelementsA.“NatureRun”(NR=control/truth)»  Quiteolenundera“perfectmodelscenario”.

B.“Observa;ons”ofthe“naturerun”.»  QuiteolenaddingGaussiannoisetonaturerun

C.DataAssimila;onSystem»  Itera:veprocess»  Flexiblesothatonecanchangethemethods

D.Valida;onandDiagnos;cs»  Implementa:on»  Evalua:onoftheresults,includingvisualiza:on

7AOSC615 '16Spring

§  KeypointsforOSSEs•  Flexibleforimplemen:ngseveraldataassimila:onmethods•  Valida:on&verifica:onarecrucial.

AOSC615ProjectFramework

Step1.ModelForecast

Forecast(=background):xbk

Observa;on

Measurement:yok

yk=hk xk( ) : yk ∈ Lk

xkb=mk ,k−1 xk−1

a( ) : x ∈ N

Step2.Analysis(Integra:onofxbkandyok)

Analysis:xak=funcofxbkandyok

AOSC615 '16Spring 8

MainObjec:veofAOSC615

2/3/16

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§  Keypoints•  Flexibleforimplemen:ngseveraldataassimila:onmethods

AOSC615ProjectFramework

Step1.ModelForecast

Forecast(=background):xbk

Observa;on

Measurement:yok

yk=hk xk( ) : yk ∈ Lk

xkb=mk ,k−1 xk−1

a( ) : x ∈ N

Step2.Analysis(Integra:onofxbkandyok)

Analysis:xak=funcofxbkandyok

NatureTruth:xtk

xkt =mk ,k−1 xk−1

t( ) : x ∈ N

A

AOSC615 '16Spring 9

B

ForOSSEs,A.  NatureRunneedstobemadeB.  Observa:onneedstobesampledC.  Dataassimila:onsystemneedstobe

implementedD.  Valida:on&verifica:onarecrucial.

C

§  Keypoints•  Flexibleforimplemen:ngseveraldataassimila:onmethods

AOSC615ProjectFramework

Step1.ModelForecast

Forecast(=background):xbk

Observa;on

Measurement:yok

yk=hk xk( ) : yk ∈ Lk

xkb=mk ,k−1 xk−1

a( ) : x ∈ N

Step2.Analysis(Integra:onofxbkandyok)

Analysis:xak=funcofxbkandyok

NatureTruth:xtk

xkt =mk ,k−1 xk−1

t( ) : x ∈ N

A

B

C

Valida;on[Diagnos;cs]

D

AOSC615 '16Spring 10

2/3/16

6

AOSC615ProjectModels

§  Model:Usedforbothnaturerunandforecast•  Time-con:nuousform:Ordinary/Par:alDifferen:alEqua:ons(O/PDEs)

•  Time-discreteform::meintegra:onover[tk-1,tk]givenI.C.x(tk-1)

§  NatureRun(NR)•  Fordissipa:vesystems,1. Spin-uptoputthetrajectoryontothearractorfirst

1. SimulatetheNR

11

!!!dxdt

= f(x ,t)!!!∂x∂t

= f(x ,t)

!!!

x(tk )=x(tk−1)+ f(x ,τ )dτtk−1

tk∫ =mk ,k−1(x(tk−1))

xk =mk ,k−1(xk−1)

!!!x0t = xt (t0 )=x0

su+ f(x ,τ )dτtk−1

T su

∫AOSC615 '16Spring

AOSC615ProjectModels

§  Suggestedmodels•  Lorenz3-VariableModel-  Lorenz,E.N.,1963:Determinis:cnon-periodicflow.J.Atmos.Sci.,20,130-141.

-  Kalnay,K.andco-authors,2007:4-D-VarorEnsembleKalmanfilter?Tellus,59A,758-773.

•  Lorenz40-VariableModel-  Lorenz,E.N.andK.Emanuel,1998:Op:malSitesforSupplementaryWeatherObserva:ons:Simula:onwithaSmallModel,J.Atmos.Sci.45,399-414

-  Bocquet,M.andP.Sakov,2014:AnItera:veEnsembleKalmanSmoother,Q.J.RMS,140,1521-1535(Sec:on4.1forsetup)

•  Lorenz960-VariableModel-  Lorenz,E.N.,2005:DesigningChao:cModels,J.Atmos.Sci.62,1574-1587

•  PointVortexModelwithtracers-  Hassen,Aref,2007:Pointvortexdynamics-Aclassicalmathema:csplayground.J.Math.Phys.,48,065401.

-  Kuznetsov,L.,K.Ide,CKRTJones,2003:AMethodforAssimila:ngLagrangianData.MWR,131,2247-2260 12AOSC615 '16Spring

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§  Model(N=3) • Observa:onEx)observa:onofthe3rdvariable

AOSC615ProjectModel.LorenzModel(1963)

dx1dt

= σ (x2 − x1)

dx2dt

= x1(ρ − x3) − x2

dx3dt

= x1x2 − βx3

xk = mk ,k−1 xk−1( ) yk =hk xk( )

yk =x3 tk( ) = 0 0 1⎡⎣ ⎤⎦

x1 tk( )x2 tk( )x3 tk( )

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

= H3x tk( )

H3 x tk( )

xk = mk ,k−1 xk−1( ) & yk =hk xk( )

AOSC615 '16Spring 13

AOSC615ProjectModel:Lorenz40

§  Lorenz40-VariableModel(1995)•  Model •  Observa:on

-  Ex)2observa:onsta:ons

AOSC615 '16Spring 14

yk =hk x( )

land1

land2

ocean1

ocean2

land1 land2 ocean2ocean1

x11:overocean1

x1:overland1

h(x) = 1 ... 0 ... ...

0 ... 1 0 ...

⎡

⎣⎢

⎤

⎦⎥ =

x1

x11

⎛

⎝⎜

⎞

⎠⎟

Inland1 Inocean1

xk = mk ,k−1 xk−1( ) & yk =hk xk( )

!!

dxndt

= −xn−2xn−1 + xn−1xn+1 − xn +F

xk = mk ,k−1 xk−1( )

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n n+1n-2n-1xn

AOSC615ProjectModel:Lorenz960

§  Model:3:erextensionofLorenz’96

•  Advec:on:[x,x]Kwithlength-scaleK&:me-scaleb

•  Damping:-x•  Forcing:f

!!!

x ≈ [xL ,xL ]K − xL + f !!!!!!!!!!!:!large+scale!&!low+frequency

+b b[xS ,xS ]1 − xS !{ } !!!!!!:!small+scale!&!high+frequency

+c[xS ,xL ]1 !!!!!!!!!!!!!!!!!!!!!!!:!scale!interaction

xL = gL(x)=GxxS = gS (x)= x−gL(x)= (I−G)x

!!

[x,x]K ,n = −wn−2Kwn−K +1K

wn−K+ j xn+K+ jj=− J'!J∑

wm = 1K

xm−ii=− J'!J∑ = 1

Kxm−J + xm+J

2+ xm+ii=− J+1

J−1∑⎧⎨⎪

⎩⎪

⎫⎬⎪

⎭⎪

n n+K n+2Kn-2K n-K

Xn+K+jWn-K+j

Wn-KWn-2K

AOSC615 '16Spring 15

AOSC615ProjectModel:PointVortex

§  Pointvortex&tracermodel•  Model •  Observa:on

-  Ex)Observa:onoftracers xk = mk ,k−1 xk−1( ) yk =hk xk( )

xk = mk ,k−1 xk−1( ) = xk−1 +

ddt

xdttk −1

tk

∫ddtxv,n

ddtyv,n

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=Γv,k

2πk=1k≠n

Nv

∑ 1xv,n − xv,k

2

− yv,n − yv,k( )xv,n − xv,k( )

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

ddtxt ,n

ddtyvt ,n

⎛

⎝

⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟

=Γv,k

2πk=1

Nv

∑ 1xt ,n − xv,k

2

− yt ,n − yv,k( ) xt ,n − xv,k( )

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟

Vortex1Vortex2

Tracer1

(Nv,NT)=(2,1)

yk =ytracer ,k = Htracer xk( )

Htracer = 0 I⎡⎣ ⎤⎦

vortex

tracer

tracer

x =xV

xT

⎛

⎝⎜⎜

⎞

⎠⎟⎟

AOSC615 '16Spring 16

2/3/16

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§  Implementa:onframework•  Flexibleforimplemen:ngseveraldataassimila:onmethods•  Star;ngfrom3D,butkeep3DFGATand4Dinmind

AOSC615Project:Assimila:onWindow

Step1.ModelForecast

Forecast(=background):xbk

Observa;on

Measurement:yok

yk=hk xk( ) : yk ∈ Lk

xkb=mk ,k−1 xk−1

a( ) : x ∈ N

Step2.Analysis(Integra:onofxbkandyok)

Analysis:xak=funcofxbkandyok

NatureTruth:xtk

xkt =mk ,k−1 xk−1

t( ) : x ∈ N

A

B

C

Valida;on[Diagnos;cs]

D

AOSC615 '16Spring 17

AOSC615Project:Assimila:onWindow

§  3D

AOSC615 '16Spring 18

PreviouswindowCurrentwindow

ICxb

dobRo

yo

xa

2/3/16

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AOSC615Project:Assimila:onWindow

§  3DwithFirstGuessatAppropriateTime(FGAT)

AOSC615 '16Spring 19

PreviouswindowCurrentwindow

ICxb

dkRo

kyok

xbk

xa

AOSC615Project:Assimila:onWindow

§  4D

AOSC615 '16Spring 20

PreviouswindowCurrentwindow

ICxb1

dkRo

k

yok

xbk

xa