Fast Implementation of

Localization for Ensemble DA

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

1LASG/IAP, Chinese Academy of Sciences

2CESS, Tsinghua University

DAOS-2012，Sep 20, 2012

EnKF The formula of EnKF without localization

obs

TTyrrOrbx 1)(

Localization:

obs

TTyrrρOrbρx 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 51, 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

TTyrrρOrbρx 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

TyrrOrbyROrbx 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

Filter m

atrices

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

mndxxexexw

L

mn if0

if1 )( )( )(

0

21210 0

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

L L

n

Expansion of filter function in localization

Filter function: Base function

Weighting function

Orth

ogo

nal

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

nn

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

nn

)~( 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)

FXXXXdt

dXjjjj

j 121 )(

jj XX 40

)40( , ,1 MMj Spatial coordinate:

Experiment with simple model

8F Forcing parameter:

Lorenz-96 model:

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

Step of assimilation cycle

RM

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

TEMP: Assimilating Radiosondes 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.

Comments, please.

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.