Implementation and Testing of 3DEnVAR and 4DEnVAR Algorithms within the ARPS

Data Assimilation Framework

Chengsi Liu, Ming Xue, and Rong KongCenter for Analysis and Prediction of Storms

University of Oklahoma

Outline

• ARPS 4DEnVar framework design

• OSSE for a storm case – Only Vr– Vr + reflectivity

– Vr + reflectivity + mass continuity constraint

• Conclusions• On-going work

Hybrid En-4DVar (Lorenc 2003, Clayton et al 2012)

C C C

T 1t 0 t 0

1

1 1 1( , ) [ ] [ ]

2 2 2

IT T

t t t tt

J

v α v v α α H L x d R H L x d

s x Uv

TB UU

'0 1 2 1 2

1

( ' )N

s e bi ii

x x x Uv x C α

'

1

( ' )N

e bi ii

x x C α

Static B part Ensemble covariance part

Localization matrix

Analysis increment and cost function

Alpha control variable 0

0

C

α α

C

innovation

2 21 2 1

4DEnVar-NPC (Non-Propagation of alpha Control variable)

' ( ) ( )bit t bi t bM M L x x x

(1) Neglecting temporal propagation of alpha control variable by TLM

(2) Use nonlinear model ensemble forecasts to replace the temporal propagation of perturbations by the TLM

Two approximations

AJM avoided !!

TLM avoided !!

1 1 1

) ) )N N N

t e t bi t bii i it bii i i

L L ( L ( (LC α C α Cx = x x x α

T T 1t t

1 1

( ) ( ) R [ {( ) } ]i

m N

i i bi t t bi i tt i

J

α L Lα α C x H H x C α d

The hybrid 4DEnVAR DA system based on the ARPS variational DA framework

• 4DEnVar-NPC is adopted as ARPS hybrid variational algorithm because:

1. Adjoint and tangent linear model are not good approximations in convective scale DA.

2. High resolution observations, like Radar, are main observation source for convective DA.

3. The alternative observation-space-perturbation-based 4DEnVar algorithm(Liu et al 2008, 2009) is computationally more expensive.

The ARPS 4DEnVar characteristics

• 3D-recursive filter is used for ARPS-4DEnVar localization.

• The capabilities for convective-scale radar DA ( Vr and reflectivity)

• Physical constraint terms (e.g. mass continuity constraint) can be considered in the cost function.

Radar data operators

10

9 1.75

(Lin et al. 1983; Gilmore

sin cos cos cos sin

: elevation angle, : azimuth angle

:

10log

( ) 3.63

et al. 2004; DoRefle well ctivi et al. 201

1 ( )

1

0

t )y

r

e qr qs qh

dB e

r r

Radial Velocity

V u v w

Z Z Z Z

Z Z

Z q q

8 1.75

8 1.75

10 1.75

( ) 9.80 10 ( ) ( )

( ) 9.80 10 ( ) ( )

( ) 4.33 10 ( )

s s

s s

h h

Z q q drysnow

Z q q wetsnow

Z q q

EnKF-En4DVar Hybrid

Obs

t0

Obs

Obs

ObsObs

Obs

Obs

t1

EnKF

4D Assimilation Window – 1 cycle

En4DVar

ObsObs

OSSE for a storm case

• Tested with simulated data from a classic supercell storm of 20 May 1977 near Del City, Oklahoma

• Domain : 35 x 35 x 35 grids. 2km horizontal resolution

• 70-min length of simulation, 5-min cycle intervene

Smoothed Gaussian obs. error:

rdrstd_vr = 1m/s

rdrstd_zhh=3dBZ

Perturbations added to the sounding (added to whole

Error

field):

stdu = stdv = st

specificat

dw = 2.0 m/

ion:

s,

st

Assimilation related parameter

dptprt = 2.0K,

stdqv = stdqc = stdqr = stdqi = stdqs = stdqh = 0.6g/kg,

inflation factor 1.07 (i.e., 7%)

localization function (Gaspari and C

s

ohn

1999),

cut-off radius = 6km for hori/vert..

hradius = 1.643km for hori. /vert. influence radius for arps3dvar.

Experiment Design

• ARPS 3DVar and ARPS En3DVar with full ensemble covariance– Only Vr– Vr + Z

– Vr + Z + mass continuity constraint – Vr + Z + mass continuity constraint and model error

True reflectivity and wind at 850Hpa

Truth

3DVarVr

3DVarVr+Z

En3DVarVr

En3DVarVr+Z

Reflectivity

3DVarVr

3DVarVr+Z

En3DVarVr

En3DVarVr+Z

Truth

Reflectivity

Truth

3DVarVr

3DVarVr+Z

En3DVarVr

En3DVarVr+Z

Vertical Velocity

Truth

3DVarVr

3DVarVr+Z

En3DVarVr

En3DVarVr+Z

Pot. Temp. Pert

U V

WPT

RMSE for 3DVar-Vr, 3DVar-Vr+Z, En3dVar-Vr, En3DVar-Vr+Z

En3DVar-Vr+Z

En3dVar-Vr

3DVar-Vr+Z3DVar-Vr

Qv Qr Qh

Qs Qc Qi

RMSE for 3DVar-Vr, 3DVar-Vr+Z, En3dVar-Vr, En3DVar-Vr+Z

En3DVar-Vr+ZEn3dVar-Vr

3DVar-Vr+Z3DVar-Vr

Mass Continuity Constraint Test

Exp 1 : 3DVar Vr + Z Exp 2 : 3DVar Vr + Z + ConstraintExp 3 : En3DVar Vr + Z Exp 4 : En3DVar Vr + Z + Constraint

Truth

3DVar 3DVar-CS

En3DVar En3DVar-CS

w and T

U

PTW

V

RMSE for 3DVar-CS, 3DVar, En3DVar-CS, En3DVar

Qs

Qv Qr Qh

Qc Qi

RMSE for 3DVar-CS, 3DVar, En3DVar-CS, En3DVar

Divergence Constraint term testwith model error

• Model error is introducing by using a different microphysical scheme in DA– Truth simulation: Ice microphysics (LINICE)– DA: WRF WSM6 scheme (WSM6WR)

• Smaller and larger weights for constraint term are tested (CSSW, CSLW)

• Both Vr and Z data

Vertical Velocity and T

Truth

En3DVarCSSW

En3DVarCSLW

En3DVar

RMSE forEn3DVar, En3DVar-CSSW, En3DVar-CSLW

U WV

Summary• 3DEnVar/4DEnVar algorithms are being implemented within the ARPS

variational DA framework, coupled with the ARPS EnKF system;

• The systems can assimilate both radar Vr and Z data;

• Cycled DA OSSEs were performed to compare performances of 3DVar and En3DVar, assimilating Vr and both Vr and Z;

• Much better state analyses were obtained using En3DVar assimilating both Vr and Z data;

• Adding Z in 3DVAR improved very little (hydrometero classification of Gao and Stensrud 2012 should help – not shown);

• Mass continuity constraint hurts En3DVar analyses, and improves w analysis in 3DVar with a perfect model;

• Analysis errors are much larger in the presence of mirophysics-related model error – adding mass continuity improves results slightly.

• 3DEnVar and EnKF results similar for single time analysis; cycled results to be compared (using deterministic background forecasts in EnKF)

On-going Research• ARPS 4DEnVar is being tested with OSSEs;

• Time localization is being considered on 4DEnVar;

• The 4DEnVar results will be compared with 3DEnVar and 4DEnSRF;

• Potential benefits of EnVar for assimilating attenuated X-band reflectivity will be evaluated.

• Will include other data sources and test with real cases.

• The entire system will directly support WRF model also. The EnKF system already does.