A new 4-dimensional variational data assimilation system for WRF

A new 4-dimensional variational data assimilation system for WRF

Juan Zhao , Bin Wang

LASG, Institute of Atmospheric Physics, Chinese Academy of Sciences, Beijing

2008-07-01University Allied Workshop

Outline

Introduction to a new DA approach

(HSP-4DVAR)

Observing system simulation experiment (OSSE)

Summary

University Allied Workshop

Outline

Introduction to a new DA approach

(HSP-4DVAR)


Summary


4DVARCost function of 4DVAR (incremental form):

1 11 1( ) ( ) ( ( ) ) ( ( ) )

2 2T T

obs obsJ x x B x y x y O y x y

Introduction to HSP-4DVAR


a bx x x

0 0( ) ( ( , )) ( ( , ))

i it t b t t by x H M x x H M x

0( ( , ))

iobs obs t t by y H M x

1 T T 1( ) M H (HM )obsJ x B x O x y

Calculate by making the nonlinear optimal iteration:

adjoint

(ECMWF,2002)

1 2[ , , , ]TNx x x x N-dimensionalmodel space(N： 106~108)

N*N

An effective and efficient 4DVAR

The huge computing cost of the iterative procedure based on the adjoint technique greatly limits the wide applications of traditional 4DVAR !

1 2( , , , )y mP y y y

1 2( , , )x mP x x x

Historical Sample Projection (HSP)-4DVAR (Bin Wang et al, 2008)

HSP-4DVAR

11 1( ) ( ) ( )

2 2T T

y obs y obsJ B P y P y

1 1( )T Ta b x y y y obsx x P B P P P y

1obs obsy o y



new cost function:

calculate explicitly:

1 2[ , , , ]Tm

m~102

?

TO oom-dimensionalsample spaceAbandon adjoint model

Avoid making nonlinear optimal iteration

Estimation of B matrixIntroduction to HSP-4DVAR


1 2

1 2

1( , , , )

11

( )

T

m

m

B b b

bm

m

Utilize historical forecast samples to estimate B

1 2

1 2

1( , , , )

11

( )

T

m

m

B bb

b x x x x x xm

x x x xm

In model space

In sample space

Estimation of B matrixIntroduction to HSP-4DVAR


1 2

1 1 1

0 0 0

0 0 0, , ,

0 0 0

0 0 0

m

1 1 11

1 1 1 1 1

1

1 1 1 1

m m m

b m m mm

m m m

not full rank(rank = m - 1)

underestimation of B

Take Xb as one of the samples !



Estimation of B matrix

1 1 11

1 1 11 1 1

1 11 1 1

1 1 1 1

1 1 1

m m m

b m m mm

m m m

full rank(rank = m)

localizationIntroduction to HSP-4DVAR


Use the analysis as the only sample

Much more timesaving than EnKF localization

( : Schur filtering operator)

1 1( )T Ta b x y y y obsx x P B P P P y

( )Ta x y obsx P P y 1 1( )T T Ty y y yP B P P P

( )Ta x y obsx P P y

Purpose: to filter the false covariance between one point and another far point in B

Analysis——in the middle of window



Xa

00 0603

Xa

traditional 4DVARnew 4DVAR

3DVAR

00 0603

Xa

mean value theorem (Math)

Outline

Introduction to a new DA approach(HSP-4DVAR)


Summary


Experiment design


• Domain configuration: 189×89×29, 30km

• TRUE—— forecasts from ECMWF global analysis (2.50×2.50) in the beginning of the assimilation window

• CTL—— forecasts from background field; background field is produced from a 48h forecast with NCEP/NCAR reanalysis (10×10) at 48h prior to the beginning of the assimilation window

• ASS—— forecasts from analysis field

• Simulated obs: temperature (T) on model level 1, 10, 19, 28, interpolated from ‘TRUE’

OSSE—— experiment design


Experiment results

OSSE—— experiment results

00 03-03

assimilation window

06 12 2418

CTL

ASS_middle(ASS)

00mX

00bX

ASS_start 03sX

03bX

00sX

1

5

9

13

17

21

25

29

0.0 0.8 1.6 2.4 3.2 4.0 4.8

CTL ASS

perturbation potential temperature (K)

mod

el le

vel

1

5

9

13

17

21

25

29

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6

water vapor mixing ratio (k/kg)

mod

el le

vel

CTL ASS

1

5

9

13

17

21

25

29

1.6 2.4 3.2 4.0 4.8 5.6 6.4

zonal wind (m/s)

mod

el le

vel

CTL ASS

1

5

9

13

17

21

25

29

1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0

meridional wind (m/s)

mod

el le

vel

CTL ASS

00h



RMSE

ASS : ASS_middle

1

5

9

13

17

21

25

29

0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4

CTL ASS


mod

el le

vel

1

5

9

13

17

21

25

29

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6

water vapor mixing ratio (g/kg)

mod

el le

vel

CTL ASS

1

5

9

13

17

21

25

29

2.4 3.2 4.0 4.8 5.6 6.4

zonal wind (m/s)

mod

el le

vel

CTL ASS

1

5

9

13

17

21

25

29

1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0


mod

el le

vel

CTL ASS

03h



RMSE

1

5

9

13

17

21

25

29

0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4

CTL ASS


mod

el le

vel

1

5

9

13

17

21

25

29

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6


mod

el le

vel

CTL ASS

1

5

9

13

17

21

25

29

1.6 2.4 3.2 4.0 4.8 5.6 6.4

Y Axis Title

mod

el le

vel

CTL ASS

1

5

9

13

17

21

25

29

1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2


mod

el le

vel

CTL ASS

06h



RMSE

1

5

9

13

17

21

25

29

0.0 0.8 1.6 2.4 3.2 4.0 4.8 5.6


mod

el le

vel

CTL ASS

1

5

9

13

17

21

25

29

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 4.0


mod

el le

vel

CTL ASS

1

5

9

13

17

21

25

29

1.6 2.4 3.2 4.0 4.8 5.6 6.4

zonal wind (m/s)

mod

el le

vel

CTL ASS

1

5

9

13

17

21

25

29

1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2


mod

el le

vel

CTL ASS

12h



RMSE

00h


RMSE

1

5

9

13

17

21

25

29

-1.6 -1.2 -0.8 -0.4 0.0 0.4

zonal wind (m/s)

mod

el le

vel

ASS_start ASS_middle

1

5

9

13

17

21

25

29

-2.0 -1.6 -1.2 -0.8 -0.4 0.0 0.4 0.8


mod

el le

vel


1

5

9

13

17

21

25

29

-1.6 -1.2 -0.8 -0.4 0.0 0.4 0.8


mod

el le

vel


1

5

9

13

17

21

25

29

-0.4 -0.3 -0.2 -0.1 0.0 0.1


mod

el le

vel


ASS_start = ASS_start — CTLASS_middle = ASS_middle — CTL

< 0 better > 0 worse

03h



RMSE

1

5

9

13

17

21

25

29

-1.2 -0.8 -0.4 0.0 0.4 0.8

zonal wind (m/s)

mod

el le

vel


1

5

9

13

17

21

25

29

-2.2 -1.8 -1.4 -1.0 -0.6 -0.2 0.2 0.6 1.0


mod

el le

vel


1

5

9

13

17

21

25

29

-1.2 -0.9 -0.6 -0.3 0.0 0.3 0.6


mod

el le

vel


1

5

9

13

17

21

25

29

-0.4 -0.3 -0.2 -0.1 0.0 0.1


mod

el le

vel


06h



RMSE

1

5

9

13

17

21

25

29

-1.0 -0.6 -0.2 0.2 0.6 1.0

zonal wind (m/s)

mod

el le

vel


1

5

9

13

17

21

25

29

-2.0 -1.6 -1.2 -0.8 -0.4 0.0 0.4 0.8 1.2


mod

el le

vel


1

5

9

13

17

21

25

29

-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4


mod

el le

vel


1

5

9

13

17

21

25

29

-0.3 -0.2 -0.1 0.0 0.1


mod

el le

vel


12h



RMSE

1

5

9

13

17

21

25

29

-0.6 -0.2 0.2 0.6 1.0

zonal wind (m/s)

mod

el le

vel


1

5

9

13

17

21

25

29

-1.6 -1.2 -0.8 -0.4 0.0 0.4 0.8 1.2


mod

el le

vel


1

5

9

13

17

21

25

29

-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4


mod

el le

vel


1

5

9

13

17

21

25

29

-0.3 -0.2 -0.1 0.0 0.1 0.2


mod

el le

vel


06h



precipitation

CTL

ASS

TRUE

ASS : ASS_middle

12h



CTL

ASS

TRUE

precipitation

18h



CTL

ASS

TRUE

precipitation

24h



CTL

ASS

TRUE

precipitation

Outline

Introduction to a new DA approach(HSP-4DVAR)


Summary


Summary (1)

The new WRF HSP-4DVAR system performs well

• abandon the adjoint technique• avoid making the nonlinear optimal iteration very time-saving

• B is flow-dependent implicitly in the assimilation window explicitly from window to window

A promising approach to be applied in operational NWPs


Summary (2)


Plans:

• More experiments to test the new DA system (conventional and unconventional obs data)

• Further improvement of B (analog prediction sample, EOF technique……)

Thank you!Comments and questions are

welcome!


Documents

A new 4-dimensional variational data assimilation system for WRF