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

Click here to load reader

  • date post

    25-Jul-2020
  • Category

    Documents

  • view

    4
  • download

    0

Embed Size (px)

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

    1LASG/IAP, Chinese Academy of Sciences

    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

    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.