Use ensemble error covariance information in GRAPES 3DVAR

Jiandong GONG, Ruichun WANG

NWP center of CMAOct 27, 2015

wx U

)UUHMU()UUHMU(2

1

2

1hvp

1hvp dwOdwwwJ TT

GRAPES 3/4DVar Cost function

)HM()HM(2

1

2

1 11 dxOdxxBxJ TT

Here

BT UU hvp UUUU

Incremental formulation (Courtier et al. 1994)

oyMxHd )

Add alpha control variable in GRAPES 3DVAR

Define “alpha” control variable with localization filter：

(3dimension) or (2D)

for horizontal localization, for vertical localization.

In detail, horizontal spectral filter and vertical EOF mode used:

(2D)

(3D)

use single to decrease size

Alpha control variable in GRAPES 3DVar

Cost-function with climate and alpha control variable

N for number of ensemble members

;

3D control variable

2D alpha variable

or 3D lower dimension alpha variable

Climate dimension vs Flow-dependent dimension

𝑑 𝑋 1=𝑈𝑃𝑈 𝐾 ( 1√𝑁−1

∑𝑖=1

𝑁

(𝑇 𝐾𝑇 𝑃 𝛿 𝑋 𝑖𝑓 )∘𝑈 h

𝛼𝑈𝑣𝛼𝑣 𝑖

𝛼)Psi unbalanced & Chi localization

Localization on horizontal (20member)No localizationLocal Scale=1500KM

Local Scale=150KM Local Scale=1000KM

GRAPES Hybrid 3DVar: 2D alpha-control variable

Clim 1.0Ens 0.0

Clim 0.9ENS 0.1

Clim 0.5Ens 0.5

Clim 0.1Ens 0.9

Localize in u&v space or psi&chi space

u&v space psi&chi space

Horizontal localization impact on balance

U & V localization Psi & Chi localization

Localization on vertical (First 8 EOF’s eigenvector, difference vertical correlation scale)

2D horizontal localization 3D localization, narrow local corr.

3D localization, middle local corr.

3D localization, broad local corr.

Localization on vertical

2D horizontal localization 3D localization, narrow local corr.

3D localization, middle local corr. 3D localization, broad local corr.

1 0.608p

g

C q z

Hydrostatic balance

Vertical localization impact on balance

ρ𝑙 ,𝑘=1

1.0+𝐾 𝑧 ( 𝑧𝑙−𝑧𝑘 )2

Background Error vertical correlation function:

20member 60member hybrid (All observation, 0.5/1.0L60 analysis + 1.0/1.0L60 ensemble)

Items 3DVARControl

Hybrid 3DVar (3D Loc, 20m)

Hybrid 3DVar (3D Loc, 40m)

Hybrid 3DVar (3D Loc, 60m)

Hybrid 3DVar (2D Loc, 60m)

Control variable

,,,q,,,q

+ 3D

,,,q

+ 3D

,,,q

+ 3D

,,,q

+ 2D

CV number4x58xgauss grid (ggrid)

(4x58+20x8)x ggrid

(4x58+40x8)x ggrid

(4x58+60x8)x ggrid

(4x58+60)x ggrid

Ratio to CV 1 1.68 2.38 3.07 1.26

Iterate step 65 71 69 66 66

CPU cores 8x32 8x32 8x32 8x32 8x32

Minim time 64s 131s 195s 250s 75s

CPU Time(inner loop) 152s 220s 279s 335s 180s

Cld wall Time(inner loop) 194s 272s 343s 412s 230s

CV Ratio to Cld wall time 1 1.4 1.76 2.12 1.18

Real observation data cycling run

Ensemble member generation (EDA) Perturb all observations in 3DVAR

• Perturb with Gaussian(0,1) PDF distribution• RH perturbation within [0%~100%] • No surface perturbation (SST,etc)• No Physics perturbation

Spin-up running for 4 days Increment Digital Filter Initialization (IDFI) for each member Spectral horizontal filter for sampling noise, wave cut at T106

(Massimo,2011) 5 grid 3rd order vertical smoother for noise Generate 20 to 60 ensemble members

Ensemble RMSE average and Climate RMSE

EnsembleRMSE average (12days),

inflate1.5 times

Climate background error

Ensemble RMSE for U & V wind

En3DVAR will have more impact on tropic region, and on upper troposphere

Real observation data cycling runControl run ( May 4 to 16, 2013)

All observation, climate Background error 0.5/1.0 L60 resolution

Hybrid experiment Extend alpha control variable, 3D localization (1700km,Lkz=0.5) First 6 vertical EOF’s eigenvector for vertical localization 20 ensemble members (computer resource) Ensemble error inflation 1.5, for small number of ensemble member Climate/Ensemble: 0.8/0.5, top to level 48 (tropopause), smooth damping

to zero, Moisture analysis use climate B Localization on Psi/chi variable for better mass-wind balance

Hybrid parameter

Vertical correlation matrix for EOFWeighting coefficient for climate and ensemble

Case study 1: (2013050812)

Contour: 3DVAR analysis 300hpa heightShared: Hybrid -3DVAR height difference

Case Study 2: (Tropical Storm Mahasen)

May 6, Tropical perturbation May 9, Tropical low pressure area May 10, Tropical cyclone May 11, Tropical strom May 16 Low pressure and low pressure area

Weaken Tropical storm

Larger ensemble divergence for Tropical Strom location and intensity

NH

TR EA

GRAPES Height analysis RMSE (Unit:m)

SH

NH

TR

GRAPES U-wind analysis RMSE (Unit:m/s)

SH

TR

Future plan

Hybrid GDAS System develop and tuning Horizontal de-correlation length increase with model level in

3DVAR, so for horizontal localization Direct estimate vertical error covariance, not use pre-defined

structure again, with short vertical correlation. Increase ensemble members (20m to 60m) Balance issue (eg. 4D-IAU) Combine with 4DVAR, to develop GRAPES EN4DVAR

New Global Ensemble member perturbation method (LETKF) Computer cost expensive for perturb obs in VAR system

Perturb land surface moisture and SST, to enlarge ensemble spread in low troposphere

Acknowledgement:

Yan LIU, Yongzhu LIU, Lin Zhang, Huijuan LU, Jincheng WANG

Fengfeng Chen, Jian Sun, Yong Su, …

Suggestion and comments?