GSI-based Hybrid Data Assimilation:Practical Implementation and impact on

the NCEP GFS

Daryl KleistEnvironmental Modeling Center

NOAA/NWS/NCEP

With acknowledgements to John Derber (EMC), Dave Parrish (EMC), Jeff Whitaker (NOAA/ESRL), Kayo Ide

(UMD), and Ricardo Todling (NASA/GMAO)13 November 2013

1Taiwan Central Weather Bureau

Variational Data Assimilation

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''1Var

T''Var 2

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2

1JJ yHxRyHxxBxx

J : Penalty (Fit to background + Fit to observations + Constraints)x’ : Analysis increment (xa – xb) ; where xb is a background

Bvar: Background error covariance

H : Observations (forward) operatorR : Observation error covariance (Instrument +

representativeness)yo’ : Observation innovations

Jc : Constraints (physical quantities, balance/noise, etc.)B is typically static and estimated a-priori/offline

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Current flow-dependence

• Although flow-dependent variances are used, confined to be a rescaling of fixed estimate based on time tendencies

– No multivariate or length scale information used

– Does not necessarily capture ‘errors of the day’

• Plots valid 00 UTC 12 September 2008

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Kalman Filter in Var Setting

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1TKF

TKF

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Forecast Step

Analysis

• Analysis step in variational framework (cost function)

Extended Kalman Filter

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• BKF: Time evolving background error covariance

• AKF: Inverse [Hessian of JKF(x’)]

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Motivation from KF

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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/BKF explicitly

TaaKF 1

1XXAA e

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Ensemble Perturbations

ba XX Forecast Step:

Analysis Step:

What does Be gain us?

• Allows for flow-dependence/errors of the day• Multivariate correlations from dynamic model

– Quite difficult to incorporate into fixed error covariance models

• Evolves with system, can capture changes in the observing network

• More information extracted from the observations => better analysis => better forecasts

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What does Be gain us?

Temperature observation near warm front

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Bf Be

What does Be gain us?

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Surface pressure observation near “atmospheric river”

First guess surface pressure (white contours) and precipitable water increment (A-G, red contours) after assimilating a single surface pressure observation (yellow dot)

using Be.

What is “hybrid DA”?

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yxHRyxH

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1

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Simply put, linear combination of fixed and ensemble based B:

Bf: Fixed background error covariance

Be: Ensemble estimated background error covariance

f: Weighting factor for fixed contribution (0.25 means 25% fixed)

e: Weighting factor for ensemble contribution (typically 1-f)

Experiments with toy model

• Lorenz ‘96– 40 variable model, F=8.0, dt=0.025 (“3 hours”)– 4th order Runge-Kutta

• OSSE: observations generated from truth run every 2*dt (“6 hours”)– [N(0,1)]

• Experimental design– Assimilate single time level observations every 6 hours, at appropriate

time, R=1.0– F=7.8 (imperfect model) for DA runs

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Analysis Error (50% observation coverage)

• M (ensemble size) = 20, (inflation factor) = 1.1– Hybrid (small alpha) as good as/better than ETKF (faster spinup)

– Hybrid (larger alpha) in between 3DVAR and ETKF

3DVAR

f = 0.7

f = 0.3

ETKF

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Sensitivity to

f 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

3DVAR 12.08

Hybrid 3.321 3.764 4.074 4.633 5.060 5.770 7.044 8.218 9.595

ETKF 3.871

Analysis RMSE (x10) over 1800 cases

• 50% observation coverage (M = 20, = 1.1)– Improvement a near linear function of weighting parameter

• Small enough weighting (on static error estimate) improves upon ETKF

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

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

ak: extended control variable; :ensemble perturbations

- analogous to the weights in the LETKF formulationL: correlation matrix [effectively the localization of ensemble perturbations] 13

GSI Hybrid [3D] EnVar(ignoring preconditioning for simplicity)

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

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Preconditioning Sidebar

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In default GSI minimization, double CG, inverses of B and L not needed and the solution is pre-conditioned by full B. Discussed later, this is why the GSI specifies the inverses of beta via the hybrid namelist.

This formulation differs from the UKMO and Canadians, who use a square root formulation and apply the weights directly to the increment:

N

n

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Hybrid with (global) GSI

• Control variable has been implemented into GSI 3DVAR*– Full B preconditioning

• Working on extensions to B1/2 preconditioned minimization options– Spectral filter for horizontal part of A

• Eventually replace with (anisotropic) recursive filters– Recursive filter used for vertical– Dual resolution capability

• Ensemble can be from different resolution than background/analysis (vertical levels are the exception)

– Various localization options for A• Grid units or scale height• Level dependent (plans to expand)

– Option to apply TLNMC (Kleist et al. 2009) to analysis increment

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*Acknowledgement: Dave Parrish for original implementation of extended control variable

Single Temperature Observation

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3DVAR

f-1=0.0 f

-1=0.5

Single Pressure Observation

Single ps observation (-2mb O-F, 1mb error) near center of Hurricane Ike

3DVAR

f-1=0.0 f

-1=0.5

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Why Hybrid?

VAR (3D, 4D)

EnKF Hybrid References

Benefit from use of flow dependent ensemble covariance instead of static B

x x Hamill and Snyder 2000; Wang et al. 2007b,2008ab, 2009b, Wang 2011; Buehner et al. 2010ab

Robust for small ensemble x Wang et al. 2007b, 2009b; Buehner et al. 2010b

Better localization (physical space) for integrated measure, e.g. satellite radiance

x Campbell et al. 2009

Easy framework to add various constraints

x x Kleist 2012

Framework to treat non-Gaussianity

x x

Use of various existing capabilities in VAR

x x Kleist 2012

GSI Hybrid Configuration/Options• Several hybrid related parameters controlled via GSI namelist

&hybrid_ensemble– Logical to turn on/off hybrid_ensemble option (l_hyb_ens)– Ensemble size (n_ens), resolution (jcap_ens, nlat_ens, nlon_ens)– Source of ensemble: GFS spectral, native model, etc.

(regional_ensemble_option)– Weighting factor for static contribution to increment (beta1_inv)– Horizontal and vertical distances for localization, via L on augmented

control variable (s_ens_h, s_ens_v)• Localization distances are the same for all variables since operating on

– Option to specify different localization distances as a function of vertical level (readin_localization)

• Instead of single parameters, read in ascii file containing a value for each layer• Example for global in fix directory (global_hybens_locinfo.l64.txt)

– Other regional options related to resolution, pseudo ensemble, etc.

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Localization

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So what’s the catch?

• Need an ensemble that represents first guess uncertainty (background error)

• This can mean O(50-100+) for NWP applications– Smaller ensembles have larger sampling error (rely more

heavily on Bf)– Larger ensembles have increased computational expense

• Updating the ensemble: In NCEP operations, we currently utilize an Ensemble Kalman Filter– EnKF is a standalone (i.e. separate) DA system that updates every

ensemble member with information from observations each analysis time using the prior/posterior ensemble to represent the error covariances. Google “ensemble based atmospheric assimilation” for a good review article by Tom Hamill.

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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

T254

L64

T574

L64

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

Previous Cycle Current Update Cycle

Observations Pre-processing

Ingest

Data Dump

Bufr Generation

QC Codes

“Prepbufr” Create

Guess TC Relocation

TC Vitals from NHC/JTWC

Analysis

Surface Cycle

GSI Hybrid 3DVAR

EnKF

GFS HiRes

Forecast Post-Processing

GSI Pre-processorEnsemble O-F

Recenter aboutHybrid Analysis

Low ResolutionEnsemble

09hr forecastsInterpolate Hybrid

Analysis

Pressure Grib

Graphics

Verification

GDAS Hybrid Cycle

Initial Hybrid Var-EnKF GFS experiment

• Model– GFS deterministic (T574L64; post July 2010 version – current operational version)– GFS ensemble (T254L64)

• 80 ensemble members, EnKF update, GSI for observation operators

• Observations– All operationally available observations (including radiances)– Includes early (GFS) and late (GDAS/cycled) cycles as in production

• Dual-resolution/Coupled• High resolution control/deterministic component

– Includes TC Relocation on guess• Ensemble is recentered every cycle about hybrid analysis

– Discard ensemble mean analysis

• Satellite bias corrections – Coefficients come from GSI/VAR

• Parameter settings• 1/3 static B, 2/3 ensemble• Fixed localization: 800km & 1.5 scale heights

• Test Period– 15 July 2010 – 15 October 2010 (first two weeks ignored for “spin-up”)

500 hPa Anom.Corr.Northern Hemisphere Southern Hemisphere

AC Frequency DistributionsNorthern Hemisphere Southern Hemisphere

Geopotential Height RMSENorthern Hemisphere Southern Hemisphere

Significant reduction in mean height errors

Stratospheric Fits

Improved fits to stratospheric observations

Tropical Winds

Hybrid forecasts seem worse than control at certain levels, but ratio of forecast/analysis standard deviation reduced (analysis standard deviation increased in hybrid).

Forecast Fits to Obs (Tropical Winds)

Forecasts from hybrid analyses fit observation much better.

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Global Data AssimilationSystem Upgrade

• Hybrid system• Most of the impact comes from

this change• Uses ensemble forecasts to

help define background error

• NPP (ATMS) assimilated• Quick use of data after launch

• Use of GPSRO Bending Angle rather than refractivity• Allows use of more data

(especially higher in atmos.)• Small positive impacts

• Satellite radiance monitoring code• Allows quicker awareness of

problems (run every cycle)• Monitoring software can

automatically detect many problems

• Post changes• Additional fields requested

by forecasters (80m variables)

• Partnership between research and operations

Implemented 22 May 2012

Operational Configuration

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• 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*

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N

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Retrospective GFS Hurricane

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500mb Anomaly CorrelationReal Time

Tropical Winds

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What next?

• Prototype dual-resolution, two-way coupled hybrid Var/EnKF system outperforms standard 3DVAR

– Prior results with one-way hybrid not as good

• Localization (currently: global/fixed parameter)– Level dependent– Flow dependent, adaptive, anisotropic

• Weighting– What is optimal? Scale-dependent?– Adaptive

• How should EnKF members be used for ensemble forecasting?

• Comparisons with 4DVAR– Regular, hybrid, and ens4DVAR

What if you don’t have an EnKF?

• In principle, any ensemble can be used– However, ensemble should represent well the forecast

errors• GSI can ingest GFS (global spectral) ensemble to

update regional models (WRF ARF/NMM)– Has been highly successful in NAM, RR, HWRF applications

• 80 member GFS/EnKF 6h ensemble forecasts are archived at NCEP since May 2012– Real-time ensemble should also be publicly available

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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

T254

L64

T574

L64

Generate new ensemble perturbations given the

latest set of observations and first-guess ensemble

Ensemble contribution to background error

covarianceReplace the EnKF

ensemble mean analysisand inflate

Previous Cycle Current Update Cycle

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• Additive inflation extracts quasi-balanced pseudo-random perturbations from database of lagged forecast pairs

• Multiplicative inflation factor, (function of reduction of spread by assimilation of observations):

Post EnKF Inflation(Whitaker and Hamill 2012)

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Inflation is an ad-hoc method for overcoming lack of consideration of model/system error within ensemble-filtering systems

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Ensemble part of Hybrid DA includes: re-centering plus inflation

Evaluations in NASA/GEOS DAS suggest:•Hybrid approach provides noticeable improvements only when using additive inflation, i.e., EnKF alone doesn’t do it •Forecasts from EnKF analyses plus additive inflation result in mild spread within the background time window•It seems that much of the initial (analysis) spread can be simulated with additive inflation alone•Appreciable background spread is obtained in the latter case

Question: how does hybrid-DA perform when the ensemblefilter is dropped and an ensemble of analyses is created from simply additively inflating the central analysis?

Use of inflation and re-centering

nnner0a xxxx

Ensemble Member 01 increments

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n0x n

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No Filter Experiment Settings

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• 0.5o outer loops, 72 level, NASA GEOS• 32 member ensemble, 1.0o, 72 levels• GSI (3D) Hybrid, 50% ensemble/static• TLNMC applied to total increment• Standards vertical/horizontal localization for ensemble B• Additive perturbations from lagged forecast pairs (24-48h)• Experiment period: April 2012

• EnKF• 0.25 additive inflation, EnKF to update ensemble (cycled)

• Filter free• 0.6 additive inflation, cold started each cycle from hybrid analysis (no

EnKF, no cycling)

Evolution of 6hr Spread

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EnKF-based hybrid Filter-Free hybrid

Spread within 9hr window

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EnKF-based hybrid Filter-Free hybrid

Cycled Results

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3DVAR EnKF-Hybrid FilterFree-Hybrid

Cycled Results

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3DVAR EnKF-Hybrid FilterFree-Hybrid

Summary of Filter Free

• Relative to currently configured hybrid, filter free produces similar results

• Inexpensive way to generate/update ensemble

• Perhaps ideal (short-term) option for those wishing to pursue hybrid DA

• However, system error can be better approximated within the EnKF filtered system

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Model error

Sampling error

Observation error

Forward operator error

Boundary condition error

Un(der)-represented error sources in an EnKF ensemble

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NCEP operational 3D ensemble/Var system

• Cycling EnKF (T254) provides an ensemble-based estimate of Jb term in 3DVar.

• 3DVar with ensemble Jb updates a T574 control forecast. The EnKF analysis ensemble is recentered around the high-res analysis.

• A combination of multiplicative and additive inflation is used to represent missing sources of uncertainty in the EnKF ensemble.

• Additive inflation: random draws from a database of 48-24 forecast differences (valid at same time), added to EnKF analysis ensemble.

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Can we replace the additive inflation by adding stochastic

physics to the model?• Schemes tested:

– SPPT (stochastically perturbed physics tendencies)– SKEB (stochastic KE backscatter)– VC (vorticity confinement, deterministic and/or

stochastic) – SHUM (perturbed boundary layer humidity, based on

Tompkins and Berner 2008, DOI: 10.1029/2007JD009284)

• All use stochastic random pattern generators to generate spatially and temporally correlated noise.

SPPT+SHUM+SKEB U

SPREAD

ERROR

SPRD – SPRD CTL

ERR – ERR CTL

• Spread/error almost consistent. With a bit of tuning, should be able to get near perfect consistency.

• SKEB adds spread in mid-latitude jets.

• Ens mean error slightly reduced.

But is it “good” spread? Does it improve covariances in EnKF?

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Replacing additive inflation with stochastic physics (preliminary)

red: control (EnKF with additive inflation)blue: Turn off additive inflation, include stochastic physics

in model (only in ensemble forecast).

Should be able to replace additive inflation with stochastic physics in the EnKF. Further progress likely as stochastic physics schemes improve representation of model uncertainty in the ensemble. 56

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Hybrid 4D EnVar[No Adjoint]

The cost function can be easily expanded extend to 4D and 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.

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Single Observation (-3h) Examplefor 4D Variants

4DVAR

H-4DVAR_ADf

-1=0.25H-4DEnVarf

-1=0.25

4DEnVar

Summary

• The “hybrid” EnVar option in GSI uses perturbations from an ensemble of short term forecasts to better estimate the background error covariance term.– Added expense (mostly IO)– Added complexity

• Running/updating ensemble• Additional DA component (EnKF)?

• Although any ensemble can be used, should be representative of forecast/background error– Feel effects of observations

• More tuning parameters– Weights, localization, etc., which may be dependent upon

model, resolution, and observing system.59