客观分析-四维同化与伴随方法

1

2000

2

Vol.1No.2

Apr.2000

4

JournalofPLAUniversityofScienceandTechnology

Ξ

(

,

211101)

,

;

;

P437

,

2

TowardtheObjectiveAnalysisFourDimensionalAssimilation

andAdjointMethod

WangXingbao

(DepartmentofAtmosphericScienceIM,PLAUST,Nanjing211101)

AbstractAtfirst,objectiveanalysisandfour2dimensionalassimilationarereviewedhistorically.Thenthe

applicationsofadjointmethodandvariationassimilationinmeteorologicalfieldsinrecentyearshavebeen

introduced.Theotherapplicationsofadjointmethodarealsodiscussed.

Keywordsobjectiveanalysis;four2dimensionalassimilation;adjointmethod

,

(Cressman1959),

(

),

,

,

,

,

,

,

,

,

,

,

,

,

;

,

,

,

,

,

,

,

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,

(

Assimilation)

(

Nudging),

:1999210214

,1962

Ξ

:

,

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,

.

68

1

,

,

(OI)

Gandin

1963

:

,

,

,

,

,

(

)

,

,

Cressman

,

,

OI

,

,

,

,

,

Morel

,

,

,

,

,

Rossby

,

,

,

,

,

,

,

,

,

,

,

,

,

,

1

1958

,

,

,

,

1969

Charney

,

,

(1970)

(

Smagorinsky

)

,

,

,

,

:

,

,

,

,

,

,

U(t,x)

,

,x

,t

Hilbert

,

2

8(

U(t,x)(t,x)

()

,

Utx

),

F88

:

1

69

F(U)=0

δ

dxdt

J(U)=

U-

U

2

δ

,U

8

,

F(U)=0

Lagrange

J(U)

,

F(U)=0

,

,

,

,

,

,

,

,

80

,

,

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,

,

,

(

)

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,

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J(

),

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,

J

,

(

)

J

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(

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,

)

,

,

Hoff2

man(1986)

,

,

,

,

,

,

(

)

2

(Adjoint)

50

,

70

,

Marchuk(1974)

,

SadokovShteinbok(1977InRussian)

80

80

,Cacuci

,

Hall

(1982)

80

V.V.PenenkoF.X.LeDimetOTalagrand

,

(

)

,

(

)

:

,

,

(

)

,

(

DescentStep)

,

,

70

1

2.1

,

Adjoint

Hibert

Cacuci(1981)

Hilbert

,

Adjoint

Hilbert

,

,

Hilbert

,

Hilbert

,

Euclidean

,

Hilbert

,

,

Hilbert

(1)

,g

Hilbert

,

(,),

vg(v)

,

F

F

F

∆

∆

:

g

v

v

∆

g=(

∆

,)

gv

(1)

v

,

g

g

v

F

v

g

v

i

v

55

(1)

,

gv

i

(2)

Hilbert

,

,,L

,

Hilbert

E

E

F

:

L3

F

E

(v,Lu)=

,

L3vu

(2)

u

E

,v

F

L3

L

E

F

,

,L

3

(

,

L3

)

L

L

L

(

)

uv=G(u),

g(v)=g[G(u)]

E

F

v

u

,g

(1)

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∆

∆

Gu

v=

(3)

(4)

(3)

(1)

G3,

:

G

G

E

F

G

,∆

gu

v

∆

∆

)=

gGu

g=(

,

G

3

v

,

g

u

g

u

g=

G

3

g

u

v

(4)

,

g

v

u

u

(

uv=G(u)

),

g

u

,

g

u

u

v=G(u)

,

g

u

,

(4)

,

,

w,

G3w

,

g

g

g(v)

g

v

u

v

,

wG3w

G(u)

u

g

G

u

(u)

,

,

u

2.2

,u

,

uv=G(u)

,v

g(v)

,

[,]

tt

01

v

:

dxdt=F(x)

(5)

(6)

Hilbert

E,

,

x

F

E

E

u,(5)

x(t)

x(t0)=u

:

g

t1Hxttt

((),)d

g=

t0

:

1

71

[,],H[x,t]

tt

(

xE

t

t

x

t

0

1

),

(5)

x(t),

∆

u

u

u

∆

:

g

g

∆

g=

t1

t0

H(t),∆x(t)

x

(7)

dt

H(t)H[x,t]

(x(t),t)

,∆x(t)

∆

x(t)

∆

x

u

x

x

(t)

∆

x(t)

(

)

:

u

∆

()

∆

Ftx

d

xdt=

(8)

F(t)

x(t)

(8)

,

t,

F

x

,

t0

∆

x(t)=

(,∆

Rtt)

u

0

(,)

0

,

(8)

,

R(t,t),

Rtt

t0

t

,

tt

[,],

tt

0

1

R(t,t)=

(9a)

(9b)

I

t

,

I

E

5

5tR(t,t)=F(t)R(t,t)

,

tt

(7)

∆

t1

=

()

HtRtt

(

)∆u)

dt

g

,

,

x

0

t0

t1

∆

=

=

R

3

(t,t)

H(t),)dt

(10)

u

0

x

t0

t1R3(t,t)

∆

()d,)

Httu

x

0

t0

R(t,t0)

R3tt0

tt

(,),t[,].

0

1

(1),

(10)

g

u

g=

t1R3(t,t)

()

Htt

d

(11)

u

0

x

t0

(8)

-d∆

∆

x

xdt=F3(t)

(12)

(12)

∆x

E

,F

(t)

(

)

3

,

tt,

∆x∆x,

Ft

S

(t,t)

,

∆x(t)=S(t,t)∆x(t)

,

∆

d

x(t),∆x(t)

+

∆

d

∆x(t),dx(t)

∆

∆

x(t),x(t)=

dt

dt

()(),x(t)-

dt

∆

Ftxt

∆

∆

∆

(t)x(t)=0

=

(),

xtF

3

,t

y

(8)

t

(,),

Rtty

y

y

y

E

t

(12)

S(t,t)y,

(,),

Rttyy

:

t

=

,(,)

yStty

,

S(t,t)R(t,t)

(11)

,

y

y

t

t

t

t

g=

t1Stt

(,)

()

Htt

d

(13)

(14)

u

0

x

t0

-d∆

xdt=

∆

3

(t)x+

H(t)

x

F

72

1

∆

)∆x(t)+

t1

(

,

St

x(t)=

(,

Stt

1

1

t

xt)=0

Σ

)

ΣΣ

H()d,

x

∆

(

1

∆

t1St

x(t)=

(13)

Σ

(,)

ΣΣ

H()d(15)

x

t

g=∆()

xt0

u

u,

g

u

:

(1)

,

t0

u

(5)

t1,

x(t)

t0

tt1

(2)

∆x(t1)=0

,

t1

t0

(14),

(t)

H(t)

F

3

x

(5)

x(t)

∆x(t0)

g

u

(3)

g,

,

u

g

11984

4

26

00

(GMT)

500hPa

(

)

umin

Fig.1The500hPaheightfieldofNorthernHemisphereat00GMTApril

261984(withoutvariationalassimilation)

Θ

D

n

=

-

(16)

u

u

n+1

n

n

u

,

nD

(),

gu

n

g

n

u

(u),

,

,Θ

n

n-1

Θ,

n

u

umin

n

,

,

,

(

)

,

,

CPU

,

,

,

,

,

21984

4

26

00

(GMT)

500hPa

(

)

Fig.2The500hPaheightfieldofNorthernHemisphereat00GMTApril

261984(withvariationalassimilation)

,

,

Spinup

:

1

73

(

);

,

,

;

,

,

,

2.3

Courtier

1

Talagrand

,

1984

4

26

00

GMT

500hPa

,

2

24

500hPa

,

3,

3

500hPa

Fig.3Thedifferencebetween500hPaheightfieldswithandwithoutvari2

,

1

2

ationalassimilation.

,

24

,

2

,

3

,

,

,

,

,

,

(

)

,

,

,

10

,

,

,

Doppler

,

,

,

,

,

,

,

,

,

,

,

,

,

(Derber

1989)

,

,

74

1

,

(Optimalpertur2

bation),

Lorenz

,

Farrell,Lacarra

Talagrand

,

Optimalperturbation

MonteCarlo

,

,

(

Derber1989),

,

,

,

,

,

,

,

PC

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