Kalman Filtering for a Two-Layer, Two-Dimensional

Kalman Filtering for a Two-Layer, Two-Dimensional Shallow-Water Model N 3 a- 7 12 5 4

(YA'jA-C2-186741) KALMAN F I L T E K I h G FOR A T h3- 9 I MENS I OW AL SHALL OW- WATER T H9- L A Y F R MOCEL (California U n i v . ) 12 p

,

ilncl a s 00/47

a292230

Ricardo Todling and Michael Ghil

Department of Atmospheric Sciences and Institute of Geophysics and Planetary Physics, University of California, LO3

Angeles,

CA

90024-1565 (U.S.A.)

-7

m OJ

I

Preliminary D r a f t

September 1988

-

s--.

..

%-.

t

Introduction

1.

During the last decade interest in data assimilation for the atmosphere and oceans has increased. This interest can be attributed to the increase in the number and accurancy of data available. In meteorology many methods have been

used

for

data

(Bengtsson et al., used

in

assimilation;

statistical,

and

empirical

1981; Ghil, 1988). A particularly promising method, widely

engineering,

Ghil et al.

variational,

has been applied to numerical weather prediction by

(1981). This method, Kalman

(19601, provides the best procedure

for sequentially assimilating data into a dynamic model, under a set of reasonable assumptions (Jazwinski, 1970; Gelb 1974; Ghil, 1988).

In the study of Ghil et

al.

(19811, further refined by Cohn

(1982),

the model atmosphere was governed by a one-dimensional set of shallow-water equations. This model includes some

interesting features of the large-scale

atmospheric and oceanic system, in particular the two time-scale behavior of slow Rossby waves and fast Poincar; waves (Pedlosky, 1987; Ghil and Childress, 1987). In order for the sequential filter to provide an estimate of the slowlyevolving state of the system, the gain matrix was multiplied by a projection matrix onto the slow-wave subspace, giving rise to a modification of the standard Kalman filter. This modification is fairly general and does not depend on the model atmosphere being considered, but it might be too complicated to be applied to an operational numerical weather prediction model.

A

comparison

between the operational procedures of optimal interpolation (Of) and the modified Kalman filter was made. Cohn (1982) showed the advantages of the latter and suggested how to improve the 01 procedure so as to perform almost as well as the modified Kalman filter. A summary of this improved 01 is given in Cohn

*

8

er a l . (1981).

more

A

realistic

cczsidered a

model

two-dimensional

was

studied by

(2-D),

Parrish

single layer

and

Cohn

(1985).

shallow-water

model

They on an

f - p l a n e and showed t h a t t h e Kalman f i l t e r can be implemented i n two dimensions. Tke r e s u l t s f o r t h e f o r e c a s t e r r o r c o r r e l a t i o n s a r e markedly d i f f e r e n t from

t h o s e o b t a i n e d by 01, l e a d i n g t o a v e r y d i f f e r e n t weighting of o b s e r v a t i o n s .

P a r r i s h and Cohn (1985) u s e d an ingenuous idea t o make t h e implementation

of t h e f i l t e r f e a s i b l e f o r a system of flow e q u a t i o n s . I n what f o l l o w s , w e u s e t h e i r a l g o r i t h m f o r an even m o r e r e a l i s t i c model,

i. e.,

w e consider a two-

l a y e r , 2-D shallow-water model on a b e t a p l a n e .

Two-layer purposes.

shallow-water

Takacs

(19861,

models

have

been

widely

used

for

different

c o n s i d e r e d such a model on t h e s p h e r e i n order t o

compare i t s r e s u l t s w i t h t h o s e of t h e Goddard Laboratory f o r Atmospheres (GLA) fourth

order

General C i r c u l a t i o n Model,

S i n t o n and Mechoso (19841,

and better

understand

s t u d i e d t h e n o n l i n e a r e v o l u t i o n of

the

latter.

f r o n t a l waves

w i t h a s i m i l a r model.

The main f e a t u r e of t h i s model which i n t e r e s t s u s is its a b i l i t y t o exhibit

b a r o c l i n i c i n s t a b i l i t y . T h i s p r o p e r t y was s t u d i e d by Pedlosky (1963) 8 who gave i n s t a b i l i t y c o n d i t i o n s . The performance of t h e Kalman f i l t e r i n t h e p r e s e n c e

of numerical i n s t a b i l i t y i n t h e barotropic v o r t i c i t y e q u a t i o n w a s s t u d i e d by Miller (1986). T h e main question t o be answered h e r e i s weather t h e f i l t e r a l -

lcws

s u f f i c i e n t l y a c c u r a t e t r a c k i n g of t h e atmospheric s t a t e i n t h e p r e s e n c e

of rapid b a r o c l i n i c developments.

-4-

4

2 . N o d e l Description

The atmosphere is t r e a t e d as a system w i t h two shallow l a y e r s of i n v i s c i d , constant-density

immiscible f l u i d s . I n t h e 2-D c a s e t h e e q u a t i o n s for such a

system are,

(2.l a )

k s l , 2.

for

vector,

the

upper

and

layers,

0 , is t h e g e o p o t e n t i a l , f i s

c o n s t a n t s , a l = l , a2= p,/p2,

These there

lower

equations

can

i s given by

the

velocity

t h e C o r i o l i s parameter and t h e

a's a r e

where p is d e n s i t y .

now

be

linearized

i s no wind i n t h e y - d i r e c t i o n ,

x-direction

is

respectively:

Uk=

4 (y),

about

i. e.,

for e a c h

Vkr

state

a basic 0,

layer.

in

and t h e wind

On

a

beta

4S0N, t h e 2-D e q u a t i o n s (2.1) a r e given e x p l i c i t l y by,

(2.2b)

which

i n the

plane

at

y p o i n t eastward and northward, r e s p e c t i v e l y , while U,,

where x and

v,

and

0,

are t h e perturbation f i e l d s .

In order t o w r i t e System ( 2 . 2 ) i n a vector-matrix the notation

+

W f

( U1,

V1,

81 , U t ,

V2,

)T

form one can introduce

and then,

( 2.3a)

where

4 8 and

c are

UIO

0

A

and

=

given by

a 1 0

0

1

0

0

0

UlO

0

0

UlO

$ , O

0

0

a 2 U , 0

1

0

0

0

0

y

0

0

0

0 , o

o y

I

(2.3b)

t

0

c c -

-(f-uly)

0

0

0

0

f

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

O

f

0

0

0

0

0

0

0

0

To s o l v e system (2.3)

(2.3d)

0

'(f'U2Y)

w e u s e t h e Richtmyer two-step v e r s i o n of t h e Lax-

k'er.2roff scheme (see Richtmyer and Morton, 1967; G h i l e t a l . , step

to

the

finite-difference

expression

for

arr, k

f

1981). The f i r s t

&[( i - 1) AX, ( j - 1) Ay, kAt )

(see Appendix C i n P a r r i s h and Cohn, 1985) a t an intermediate l o c a t i o n ,

for i=1,2* ...,I and j=1,2

with

X

and

,..., J-1,

Y being the

The averaging operators

where

east-west

px and

and

north-south

extents,

respectively.

cl, a r e g i v e n by

(2.6a)

and t h e o p e r a t o r Lj i s d e f i n e d a s

(2.81

w i t h Ax=

dl /AX and

a,=

AIIAy.

T h e second s t e p u s e s t h e

i n t e r m e d i a t e values t o compute t h e s t a t e v a r i a b l e s

a t t h e next t i m e l e v e l

for i = l , Z ,

... , I

and j = 2 , 3 , . . . ,J-1.

Based on (2.91 it is p o s s i b l e t o c o n s t r u c t

t h e dynamics m a t r i x Y, which is needed i n t h e Kalman f i l t e r scheme.

The bound-

a r y c o n d i t i o n s can be i n c l u d e d i n t h e formulation of Y. It i s i m p o t t a n t t o notice that,

i n o r d e r t o o b t a i n b a t o c l i n i c i n s t a b i l i t y t h e free upper s u r f a c e

and the i n t e r f a c e between t h e two l a y e r s of f l u i d must have nonzero mean slopes.

3.

The Kalman Filter

V e r y good and c l e a r p r e s e n t a t i o n s of t h e Kalman f i l t e r i n g p r o c e d u r e can

!

be

6

f o u n d i n many p l a c e s : Jazwinski ( 1 9 7 0 1 , g i v e s a good mathematical d e s c r i p t i o n w h i l e Gelb

(1974) or Brown

(19831, a r e more a p p l i c a t i o n - o r i e n t e d .

The Kalman

f i z t e r as a p p l i e d t o numerical weather p r e d i c t i o n i s presented by G h i l e t a l . (1981). I n t h i s s e c t i o n w e r e c a p i t u l a t e b r i e f l y t h e e s s e n t i a l s .

Let us consider a d i s c r e t e l i n e a r system, g i v e n i n v e c t o r n o t a t i o n by

for

the

discrete times

senting the

g(k)

is a

true

k

=

s t a t e of

random n-vector

1,2,

...,

where

t h e system, which

i#t(k)

Y is

i s white

the

in

i s t h e n-vector nxn

repre-

dynamics m a t r i x

and

t i m e and unbiased w i t h

co-

superscript

the

variance Q(k):

The

symbol

E

t r a n s p o s e and 6,,

denotes

the

expectation,

the

T

denotes

is t h e Kronecker delta.

D i s c r e t e l i n e a r o b s e r v a t i o n s a r e described by

w h e r e s ( k ) i s a p-vector

interpolations

ing

for

and

s t a t e variables;

of o b s e r v a t i o n s ,

and

go(k)

for

any

and H ( k )

conversion

is a random p-vector,

matrix R ( k ) , d e s c r i b i n g o b s e r v a t i o n e r r o r ,

is a pxn m a t r i x account-

between

observed

unbaised with

variables covariance

a n d assumed t o be u n c o r r e l a t e d with

-Y-

%

+

t h e model e r r o r bt ( k ) :

(3.4a,b)

The Kalman f i l t e r f o r t h e system described above i s t h e following set of

equations :

ef ( k )

=

Pf(k) =

K(k)

=

Y(k-l)da (k-1)

,

( 3 . Sa)

+

Y(k-l)Pa(k-l)q(k-l)

Q(k-1)

(3.5b)

I

,

P f ( k ) H T ( k ) [ H(k)Pf(k)HT(k) + R(k)]’l

(3.5c)

(3.5d)

da(k)

+

df(k)

... .

for

k

and

analyzed

quantities,

the

evolution

of

=

1,2#3#

observations,

the

do(k)

K(k)[

H(k)df(k)]

superscripts

The

that

physical

repectively.

-

is,

system

itself above

(3.5e)

and

quantities

introduced

K

”f”

,

“a” that

and

is

stand

are

those

for

due

simply

dependent

the Kahn

forecast

gain

on

to the

matrix,

which r e f e r s t o t h e weight w i t h which o b s e r v a t i o n s and f o r e c a s t are combined

to g i v e t h e f i n a l r e s u l t , and P i s t h e e r r o r c o v a r i a n c e matrix,

Pfta(k)

=

E

[

[ df a ( k )-Qt ( k ) ] [ df a ( k ) -dt e

f o r t h e f o r e c a s t and analyzed f i e l d s , t e p e c t i v e l y .

( k )]

] .

(3.6)

AcknowledgeItEtnt8. S p e c i a l thanks go t o D r . Dale Boggs who wrote a streaml i n e d code f o r t h e s i n g l e - l a y e r v e r s i o n .

Dick Dee

i n v a l u a b l e h e l p with t h e t e c h n i c a l d e t a i l s .

and Stephen Cohn p r o v i d e d

T h i s work was s u p p o r t e d by NASA

g r a c t s NAG 5-947 and NAG 5-713.

References M. G h i l and E. Kallen (eds.) , 1981: Dynamic Wteorology:

Bengtsson, L.,

Data Assimilation Methods. Springer-Verlag,

Brown, R. G.,

N e w York, 330 pp.

1983: Introduction t o Random Signal Analysis and lCalmap

Filtering, John Wiley, New York, 347 pp.

Cohn, S. E.,

1982: Methods of Sequential Estimation for Determining

I n i t i a l Data in Numericallteather Prediction. Ph.D. t h e s i s , Courant I n s t i t u t e of Mathematical Sciences, NYU, New York, NY 10012, 183 pp.

Cohn, S. E., M. Ghil and E. Isaacson, 1981: Optha1 Interpolation and

the It.-

Filter. F i f t h Conference on Numerical Weather Pre-

d i c t i o n , p r e p r i n t volume, h e r . Meteor. SOC., Boston, pp. 3642.

G e l b , A.,

1974: Applied Optimal Estimation.

374 pp.

Cambridge, MA, MIT P r e s s ,

I

Ghil, M., S. Cohn, J. Tavantzis, K. Buke and E. Isaacson, 1981: Applications of estimation theory to nmaerical weather prediction.

Dynamic Meteorology: Data Assimilation Methods, L. Bengtsson, M. Ghil and E. Kallen (eds.), Springer-Verlag, New York, PP.

139-224.

Ghil, M., and S. Childress, 1987: Topics in GeOphy8iCal Fluid

m d C 8 :

Atmospheric Dynamics, Dynamo Theory and C l i m a t e Dynamics. Springer

Verlag, New York, 485 pp.

Ghil, M., 1988: Xeteorological Data Assimilation for Oceanographers. Part I: Description and Theoretical Framework. Dyn. Atmos.

Oceans, submitted.

Jazwinski, A. H., 1970: Stochastic Processes and Filtering Theory. Academic Press, New York, 376 pp.

K a h n , R. E., 1960: A New Approach to Linear Filtering and Prediction Problems. Trans. ASME, Ser. D, J. Basic Eng., 82, 35-45.

Parrish, D. F. and S. E. Cohn, 1985: AKalman Filter for

8

--Dimen-

aional Shallow-Water Model: Formulation and Preliarinrry E q e rimenta. Office Note 304, National Meteorological Center,

Washington DC 20233, 64 pp.

Uiller, R. M., 1986: Toward the Application of the Xalman Filter t o Regional Open Ocean Modelling.

J. Phys. Oceanogr., 16, 72-86.

..

.e

Pedlosky, J., 1963: B a r o c l i n i c Instability in Two-Layer Systems. Tellus, 15, 20-25.

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Richtmayer, R. D. and K. W. Morton, 1967: D i f f e r e n c e Methods for InitialValue P r o b l m , (2nd ed.). Wiley-Interscience, New York, 405pp.

Sinton, D. M. and C. R. Mechoso, 1984: Nonlinear Evolution of Frontal Waves. J. Atmos. Sci., 14, No. 24, pp. 3501-3517.

Takacs, L. L., 1986: Documentation of the Goddard Laboratory for Atmospheres F o u r t h - O r d e r --Layer

S h a l l o w Water Modal.

Goddard Modelling and Simulation Branch, Goddard Space Flight Center, Greenbelt, Maryland 20771, 79 pp.