An OSSE-based evaluation of 4D-Ensemble-Var (and hybrid variants) for the NCEP GFS

Daryl Kleist1,2 and Kayo Ide2

1

CMOS 2012 Congress AMS 21st NWP and 25th WAF Conferences

Montréal, Québec, Canada -- 29 May to 01 June 2012

with acknowledgements to Dave Parrish, Jeff Whitaker, John Derber, Russ Treadon

1NOAA/NWS/NCEP/EMC2Univ. of Maryland-College Park, Dept. of Atmos. & Oceanic Science

Outline

• Introduction– Background on hybrid data assimilation

• Hybrid 3DVAR/EnKF experiments with GFS using an OSSE

• 4D-Ensemble-Var– Evaluation using OSSE framework– Hybridization with time invariant static B

2

3

IncrementalVariational Data Assimilation

J : Penalty (Fit to background + Fit to observations)

x’ : Analysis increment (xa – xb) ; where xb is a background

Bf : (Fixed) Background error covariance

H : Observations (forward) operator

R : Observation error covariance (Instrument + representativeness)

, where yo are the observations

Cost function (J) is minimized to find solution, x’ [xa=xb+x’]B is typically static and estimated a-priori/offline

yxHRyxHxBxx 1T1f

T3DVAR 2

1

2

1J

bo xyy H

4

Motivation from [Ensemble]Kalman Filter

TbbKF 1

1XXB

N

ab XX

• Problem: dimensions of AKF and BKF are huge, making this practically impossible for large systems (GFS for example)

• Solution: sample and update using an ensemble instead of evolving AKF and BKF explicitly

TaaKF 1

1XXA

N

Ensemble Perturbations

ba XX Forecast Step:

Analysis Step:

N is ensemble size

f & e: weighting coefficients for fixed and ensemble covariance respectively

xt’: (total increment) sum of increment from fixed/static B (xf’) and ensemble B

k: extended control variable; :ensemble perturbations

- analogous to the weights in the LETKF formulation

L: correlation matrix [effectively the localization of ensemble perturbations]5

Hybrid Variational-Ensemble

• Incorporate ensemble perturbations directly into variational cost function through extended control variable– Lorenc (2003), Buehner (2005), Wang et. al. (2007), etc.

ekx

yxHRyxH

LxBxx

t1T

t

1

1T

ef1

fT

fff

2

1

2

1

2

1 N

n

nnββ,J ααα

N

n

nn

1eft xxx α

Single Temperature Observation

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3DVAR

f-1=0.0 f

-1=0.5

Outline

• Introduction– Background on hybrid data assimilation

• Hybrid 3DVAR/EnKF Experiments with GFS using an OSSE

• 4D-Ensemble-Var– Evaluation using OSSE framework– Hybridization with time invariant static B

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Observing System SimulationExperiments (OSSE)

• Typically used to evaluate impact of future observing systems– Doppler-winds from spaced-based lidar, for example

• Useful for evaluating present/proposed data assimilation techniques since ‘truth’ is known– Series of experiments are carried out to test hybrid variants

• Joint OSSE– International, collaborative effort between ECMWF, NASA/GMAO,

NOAA (NCEP/EMC, NESDIS, JCSDA), NOAA/ESRL, others– ECMWF-generated nature run (c31r1)

• T511L91, 13 month free run, prescribed SST, snow, ice

Synthetic Observations

• Observations from (operational) 2005/2006 observing system developed– NCEP: ‘conventional’, sbuv ozone retrievals, GOES sounder radiances– NASA/GMAO: all other radiances (AMSUA/B, HIRS, AIRS, MSU)

• Older version of the CRTM

• Simulated observation errors developed by Ron Errico (GMAO/UMBC)– Horizontally correlated errors for radiances– Vertically correlated errors for conventional soundings

• Synthetic observations used in this study were calibrated by Nikki Prive (GMAO)– Attempt to match impact of various observation types with results from data

denial experiments

9*Thanks to Ron and Nikki for all of their help in getting/using the calibrated synthetic observations.

Availability of SimulatedObservations [00z 24 July]

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AMSUA/MSU AIRS/HIRS

AMSUB GOES SOUNDER

SURFACE/SHIP/BUOY

AIRCRAFT

SONDES

AMVS

SSMI SFC WIND SPDPIBAL/VADWND/PROFLR

3D Experimental Design

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• Model• NCEP Global Forecast System (GFS) model (T382L64; post May 2011 version – v9.0.1)

• Test Period• 01 July 2005-31 August 2005 (3 weeks ignored for spin-up)

• Observations• Calibrated synthetic observations from 2005 observing system (courtesy Ron Errico/Nikki

Privi)

• 3DVAR• Control experiment with standard 3DVAR configuration (

increment comparison to real system)

• 3DHYB• Ensemble (T190L64)

• 80 ensemble members, EnSRF update, GSI for observation operators• Additive and multiplicative inflation

• Dual-resolution, 2-way coupled• High resolution control/deterministic component• Ensemble is recentered every cycle about hybrid analysis

– Discard ensemble mean analysis

• Parameter settings• f

-1=0.25, e-1=0.75 [25% static B, 75% ensemble]

• Level-dependent localization

3DVAR Time MeanIncrement Comparison

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U

T

Ps

REAL OSSE(BACK)

• Full B preconditioned double conjugate gradient minimization

• Spectral filter for horizontal part of L• Eventually replace with

(anisotropic) recursive filters

• Recursive filter used for vertical• 0.5 scale heights

• Same localization used in Hybrid (L) and EnSRF

• TLNMC (Kleist et al. 2009) applied to total analysis increment*

3DHYB Details

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N

n

nn

1eft xxCx α

*TLNMC is also applied to increment in 3DVAR experiment.

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EnKFmember update

member 2 analysis

high resforecast

GSIHybrid Ens/Var

high resanalysis

member 1 analysis

member 2 forecast

member 1 forecast

recenter analysis ensemble

Dual-Res Coupled HybridVar/EnKF Cycling

member 3 forecast

member 3 analysis

Previous Cycle Current Update Cycle

T19

0L64

T38

2L64

Generate new ensemble perturbations given the latest set of observations and first-guess ensemble

Ensemble contribution to background error

covarianceReplace the EnKF

ensemble mean analysis

Time Series of Analysis andBackground Errors

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500 hPaU

850 hPaT

Solid (dashed) show background (analysis) errors

3DHYB background errors generally smaller than 3DVAR analysis errors (significantly so for zonal wind)

Strong diurnal signal for temperature errors due to availability of rawinsondes

3DVAR and 3DHYBAnalysis Errors

163DVAR 3DHYB 3DHYB-3DVAR

U

T

Q

Zonal Wind BackgroundErrors

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3DVAR 3DHYB

Bf BEN

3D Follow-on Experiments

• To further understand some of the degradation, two-additional 3D experiments are designed

– 3DENSV• 3DHYB re-run, but with no static B contribution (f

-1=0.0)

• Analogous to a dual-resolution 3D-EnKF (but solved variationally)

– 3DHYB_RS (Reduced Spread)• Inflation parameters reduced to produce a better match between

ensemble spread and actual F06 error

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3DENSV Analysis Error

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U

T

Q

3DENSV-3DVAR

Same problem areas from 3DHYB (error increase) reappear

Ensemble covariances improve tropics

New problem areas arise (static B really helps, especially with imperfect ensemble)

3DHYB_RS

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U

T

Q

New (RS) hybrid experiment almost uniformly better than 3DVAR

Inflation still a necessary evil (move to adaptive?)

3DHYB_RS-3DVAR

Summary of 3D Experiments

• 3DHYB generally better than 3DVAR– Hybrid background errors smaller than 3DVAR analysis errors

– Significant improvement in tropics (smaller improvements in extratropics)

• Quality of ensemble can have impact on quality of analysis– Need for adaptive inflation

– Stochastic physics, localization, ensemble size

– Hybrid helps mitigate (some) problems with imperfect ensemble

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Outline

• Introduction– Background on hybrid data assimilation

• Hybrid 3DVAR/EnKF Experiments with GFS using an OSSE

• 4D-Ensemble-Var– Evaluation using OSSE framework– Hybridization with time invariant static B

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Hybrid ensemble-4DVAR[H-4DVAR_AD]

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K

kkkkkkkkJ

10

1T00

1f

T00 2

1

2

1yxMHRyxMHxBxx

Incremental 4DVAR: bin observations throughout window and solve for increment at beginning of window (x0’). Requires linear (M) and adjoint (MT) models

K

kkkkkkkk

N

n

nn,J

10

1T0

1

1T

ef1

fT

fff

2

1

2

1

2

1

yxMHRyxMH

LxBxx ααα

Can be expanded to include hybrid just as in the 3DHYB case

N

n

nn

1ef0 xxx α

With a static and ensemble contribution to the increment at the beginning of the window

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4D-Ensemble-Var[4DENSV]

As in Buehner et al. (2010), the H-4DVAR_AD cost function can be modified to solve for the ensemble control variable (without static contribution)

Where the 4D increment is prescribed exclusively through linear combinations of the 4D ensemble perturbations

Here, the control variables (ensemble weights) are assumed to be valid throughout the assimilation window (analogous to the 4D-LETKF without temporal localization). Note that the need for the computationally expensive linear and adjoint models in the minimization is conveniently avoided.

K

kkkkkkkk

N

n

nnJ1

1T

1

1T

2

1

2

1yxHRyxHL ααα

N

n

n

kn

k1

exx α

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Hybrid 4D-Ensemble-Var[H-4DENSV]

The 4DENSV cost function can be easily expanded to include a static contribution

Where the 4D increment is prescribed exclusively through linear combinations of the 4D ensemble perturbations plus static contribution

Here, the static contribution is considered time-invariant (i.e. from 3DVAR-FGAT). Weighting parameters exist just as in the other hybrid variants.

K

kkkkkkkk

N

n

nn,J

1

1T

1

1T

ef1

fT

fff

2

1

2

1

2

1

yxHRyxH

LxBxx ααα

N

n

n

kn

k1

ef xxx α

Single Observation (-3h) Examplefor 4D Variants

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4DVAR

H-4DVAR_ADf

-1=0.25H-4DENSVf

-1=0.25

4DENSV

Time Evolution of Increment

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t=-3h

t=0h

t=+3h

H-4DVAR_AD H-4DENSV

Solution at beginning of window same to within round-off (because observation is taken at that time, and same weighting parameters used)

Evolution of increment qualitatively similar between dynamic and ensemble specification

** Current linear and adjoint models in GSI are computationally unfeasible for use in 4DVAR other than simple single observation testing at low resolution

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4D OSSE Experiments

• To investigate the use of 4D ensemble perturbations, two new OSSE based experiments are carried out.

• The original (not reduced) set of inflation parameters are used.

• Exact configuration as was used in the 3D OSSE experiments, but with 4D features

– 4DENSV• No static B contribution (f

-1=0.0)

• Analogous to a dual-resolution 4D-EnKF (but solved variationally)

• To be compared with 3DENSV

– H-4DENSV • 4DENSV + addition of time invariant static contribution (f

-1=0.25)

• This is the non-adjoint formulation

• To be compared with 4DENSV

Impact on Analysis Errors

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U

T

Q

4DENSV-3DENSV H-4DENSV-4DENSV

Dynamic Constraints

• One advantage of the variational framework is ease in which one can apply dynamic constraints on the solution

• Several constraint options have been developed and explored for use with the 4DENSV algorithm– Tangent linear normal mode constraint

–Weak constraint digital filter

– Combined normal mode/digital filter30

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Comparison of 3DHYB andH-4DENSV (with dynamic constraints)

U

T

Q

4DHYB-3DHYB

Something in the 4D experiments is resulting in more moisture in the analysis, triggering more convective precipitation

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Summary of 4D Experiments

• 4DENSV seems to be a cost effective alternative to 4DVAR

• Inclusion of time-invariant static B to 4DENSV solution is beneficial for dual-resolution paradigm

• Extension to 4D seems to have larger impact in extratropics (whereas the original introduction of the ensemble covariances had largest impact in the tropics)

• Moisture constraint and/or observations contributing to increased convection in 4D extensions

• Original tuning parameters for inflation were utilized. Follow-on experiments with tuned parameters (reduced inflation) and/or adapative inflation should yield even more impressive results.

Summary and Conclusions

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• Analysis errors are reduced when using a hybrid algorithm relative to 3DVAR

• Inclusion of a static B contribution proved beneficial to 3D and 4D variants when using a dual-resolution configuration

• Extension to 4D ensemble has larger impact in extratropics whereas the introduction of the ensemble covariance estimate had largest impact in the tropics

• Quality of analyses sensitive to quality of ensemble. Care needs to be taken in specifying certain aspects such as inflation parameters

Future Work

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• Comparison of 4DENSV variants with 4DVAR in OSSE context

• Follow-on OSSE experiments to better understand what is leading to increase in moisture (therefore exciting more convection) in the 4D experiments relative to 3D

• Testing of H-4DENSV (with constraint) using real observations. This configuration is what I envision being a prototype for future operational implementation

• Temporal (and variable?) localization in hybrid

• Hybrid Weighting Variations()– Piecewise scale-dependent (already coded, preliminary testing compled)

– Fully adaptive (weights as a control variable?)

• Hybridization of EnKF update

• LETKF instead of EnSRF (?)

• Adaptive Inflation

• Observation impact estimation within hybrid DA

• QOL/RIP within 4DENSV variants

Operational (3D) Hybrid

• Implemented into GDAS/GFS at 12z 22 May 2012

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3DHYB-3DVAR

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