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### Transcript of Fast Implementation of Localization for Ensemble DA _Wang_6_3.pdf Fast Implementation of...

• Fast Implementation of

Localization for Ensemble DA

B. Wang1,2 and J.-J. Liu1

2CESS, Tsinghua University

DAOS-2012，Sep 20, 2012

• EnKF The formula of EnKF without localization

obs

TT yrrOrbx  1)(

Localization:

obs

TT yrrρOrbρx yyxy 

1))()( ( 

where ρxy, ρyy : Filtering matrices;

: Schür product operator 

  

 

 

 

),, , ,( 1

1

), , , ,( 1

1

21

21

yy yyyyr

xxxxxxb

n

n

n

n

1

1

n

1i

1

i

n

i

i

n

n

yy

xx

• , 0 , 0 0 , 0( / ) ( / ) ,

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

h h v v

i j i j i j

y x

C d d C d d

i m j m

  

 

5 4 3 2

5 4 3 2 1

0

1 1 5 5 1, 0 1

4 2 8 3

1 1 5 5 2 ( ) 5 4 , 1 2,

12 2 8 3 3

0, 2

r r r r r

C r r r r r r r r

r

         

          

 

The elements of filter matrix ρxy (similar for ρyy):

where the filtering function C0 is defined as (Gaspari

and Cohn, 1999)

and respectively represent the horizontal and

vertical distances between the i-th and j-th row vectors;

and are the horizontal and vertical Schür radius.

h

jid , v

jid ,

hd0 vd0

• Issues caused by Localization

After localization, huge computing time is required

for the solution:

obs

TT yrrρOrbρx yyxy 

1)) ( )( ( 

Before localization, it is easy to get the solution:

yx mm 

510~ym

710~xm

)10~( 2nnmx 

nmy  yy mm diagonal

obs

TT

obs

T yrrOrbyROrbx   11 ) ( ) (

• The extra multiplications

due to localization

1410~lnmmnmm yyyx 

Available solution to the issue

1) To split the observation vector into a combination

of low-dimension sub-vectors

2) To sequentially assimilate the low-dimension

observation sub-vectors

• Our new method is to split filter matrix in localization

  

 N

k

Tkk

1

)()(

yxxy ρρρ

  

 N

k

Tkk

1

)()(

yyyy ρρρ

F ilter m

a trices

my-dimension filter eigenvector

mx-dimension filter eigenvector

N is the number of the major modes of the filter matrix

•  

 1

2121 )()(),( n

nnn xexeaxxC

  

  mn

mn dxxexexw

L

mn if0

if1 )( )( )(

0

2121 0 0

2121 )( )( ),()()( dxdxxexexxCxwxwa nn L L

n  

Expansion of filter function in localization

Filter function: Base function

Weighting function

O rth

o go

n al

• N=10

N=20 N=15

)( )(),( 0 1

0 xexeaxxC n

N

n

nnN  

Green:

Black: )(0 rC

5 ,49 ,/ 0000  dxdxxrwhere

x

C

One-Dimension Filter

• x

y

N=10 N=15

N=20 True:

)( )( )( )(),;,( 0 1 1

0,00 yeyexexeayyxxC mm

N

n

N

m

nnmnN   

Two-Dimension Filter

)(0 rC

    5 ,49 ,49 ,/ 0000 2

0

2

0  dyxdyyxxr

• ) , , ,

, , , ,

, , , ,( 1~

1

21

2

2

2

1

2

1

2

1

1

1

xxρxxρxxρ

xxρxxρxxρ

xxρxxρxxρb

)(

x

)(

x

)(

x

)(

x

)(

x

)(

x

)(

x

)(

x

)(

x





 

n

NNN

n

n n







  

 N

k

n

i

i

k

n 1 1

)(

~ 1

xρx x  )~( Nnn 

obs

TT

nn yOrrOrIbx  

111 ~~

~)~~( ~

EnKF with extended samples

where

•   

 N

k

n

i

i

k

n 1 1

)(

~ 1

yρy y 

) , , ,

, , , ,

, , , ,( 1~

1

21

2

2

2

1

2

1

2

1

1

1

yyρyyρyyρ

yyρyyρyyρ

yyρyyρyyρr

)(

y

)(

y

)(

y

)(

y

)(

y

)(

y

)(

y

)(

y

)(

y





 

n

NNN

n

n n







)~( Nnn 

•  Solution scheme: 4th -order Runge-Kutta scheme

 Time stepsize: 0.05 time unit

 Periodic boundary conditions:

 Note: 0.05 time unit ~6h (Lorenz and Emanuel, 1998)

FXXXX dt

dX jjjj

j   121 )(

jj XX 40

)40( , ,1  MMj  Spatial coordinate:

Experiment with simple model

8F Forcing parameter:

Lorenz-96 model:

• Red： EnKF with 10 sample (with localization) Blue: EnKF with 500 sample (without localization)

Step of assimilation cycle

R M

SE

• Experiment based on an operational prediction model: AREM

H

CTRL: No assimilation with prediction as IC

ERA：Using ERA40 as IC, no assimilation

AIRS：Assimilating T & Q of AIRS only

AIRS+TEMP: With random perturbation

AIRS+TEMP—2: No random perturbation

RMSEs averaged from 850hPa to 100hPa

• T

V U

Q

• Summary

 Localization for EnKF can be economically implemented using an orthogonal expansion of filter function;

 The new implementation can be applied to other ensemble-based DA, e.g., En-3DVar and En-4DVar.

Thank you!

•  Observations of all model variables: “true” states

plus uncorrelated random noise with standard

Gaussian distribution (with zero mean and

variance of 0.16)

 “True” states: simulations during a period of time

unit after a long-term integration (e.g., 105 model

time steps) of the model from an arbitrary IC .

 Assimilation experiments: assimilation cycles

over a period of 200 days using the EnKF with

18-hr assimilation window and, respectively.

500 samples are used for the experiments so that

they have good representativeness.