1. GSI-based Hybrid Data Assimilation:
Practical Implementation and impact on the NCEP GFS
Daryl Kleist
Environmental 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
Taiwan Central Weather Bureau

2. Variational Data Assimilation
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) operator
R : Observation error covariance (Instrument + representativeness)
yo’ : Observation innovations
Jc : Constraints (physical quantities, balance/noise, etc.)
B is typically static and estimated a-priori/offline 2

3. 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

4. Kalman Filter in Var Setting
Forecast Step
Extended Kalman Filter
Analysis
Analysis step in variational framework (cost function)
BKF: Time evolving background error covariance
AKF: Inverse [Hessian of JKF(x’)]

5. Ensemble Perturbations
Motivation from KF
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
Forecast Step:
Ensemble Perturbations
Analysis Step:

6. 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 6

7. Temperature observation near warm front
What does Be gain us?
Temperature observation near warm front Bf Be 7

8. Surface pressure observation near “atmospheric river”
What does Be gain us?
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. 8

9. What is “hybrid DA”?
Simply put, linear combination of fixed and ensemble based B:
Bf: Fixed background error covariance
Be: Ensemble estimated background error covariance
bf: Weighting factor for fixed contribution (0.25 means 25% fixed)
be: Weighting factor for ensemble contribution (typically 1- bf) 9

10. 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

11. Analysis Error (50% observation coverage)
3DVAR
bf = 0.7
bf = 0.3
ETKF
M (ensemble size) = 20, r (inflation factor) = 1.1
Hybrid (small alpha) as good as/better than ETKF (faster spinup)
Hybrid (larger alpha) in between 3DVAR and ETKF

12. Analysis RMSE (x10) over 1800 cases
Sensitivity to b
Analysis RMSE (x10) over 1800 cases bf
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
50% observation coverage (M = 20, r = 1.1)
Improvement a near linear function of weighting parameter
Small enough weighting (on static error estimate) improves upon ETKF

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.
bf & be: 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 formulation
L: correlation matrix [effectively the localization of ensemble perturbations]

14. Preconditioning Sidebar
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:

15. 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
*Acknowledgement: Dave Parrish for original implementation of extended control variable

16. Single Temperature Observation
3DVAR
bf-1=0.0
bf-1=0.5

17. Single Pressure Observation
3DVAR
bf-1=0.0
bf-1=0.5
Single ps observation (-2mb O-F, 1mb error) near center of Hurricane Ike

18. Why Hybrid? VAR (3D, 4D) EnKF Hybrid References x
Benefit from use of flow dependent ensemble covariance instead of static B x
Hamill and Snyder 2000; Wang et al. 2007b,2008ab, 2009b, Wang 2011; Buehner et al. 2010ab
Robust for small ensemble
Wang et al. 2007b, 2009b; Buehner et al. 2010b
Better localization (physical space) for integrated measure, e.g. satellite radiance
Campbell et al. 2009
Easy framework to add various constraints
Kleist 2012
Framework to treat non-Gaussianity
Use of various existing capabilities in VAR

19. 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 a
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.

20. Localization

21. So what’s the catch?
Need an ensemble that represents first guess uncertainty (background error)
This can mean O( ) 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.

22. Dual-Res Coupled Hybrid Var/EnKF Cycling
Generate new ensemble perturbations given the latest set of observations and first-guess ensemble
member 1
forecast
member 1
analysis
T254L64
EnKF
member update
member 2
forecast
member 2
analysis
recenter analysis ensemble
member 3
analysis
member 3
forecast
Ensemble contribution to background error covariance
Replace the EnKF ensemble mean analysis
GSI
Hybrid Ens/Var
high res
forecast
high res
analysis
T574L64
Previous Cycle
Current Update Cycle

23. TC Vitals from NHC/JTWC
GDAS Hybrid Cycle
Observations
Pre-processing
Analysis
Forecast
Post-Processing
QC Codes
“Prepbufr” Create
Guess TC Relocation
TC Vitals from NHC/JTWC
Surface Cycle
GFS HiRes
Ingest
Pressure Grib
Data Dump
Bufr Generation
GSI Hybrid 3DVAR
Graphics
Verification
GSI Pre-processor
Ensemble O-F
EnKF
Low Resolution
Ensemble
09hr forecasts
Interpolate Hybrid
Analysis
Recenter about
Hybrid Analysis

24. 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”)

25. 500 hPa Anom.Corr.
Northern Hemisphere
Southern Hemisphere

26. AC Frequency Distributions
Northern Hemisphere
Southern Hemisphere

27. Geopotential Height RMSE
Northern Hemisphere
Southern Hemisphere
Significant reduction in mean height errors

28. Improved fits to stratospheric observations
Stratospheric Fits
Improved fits to stratospheric observations

29. 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).

30. Forecast Fits to Obs (Tropical Winds)
Forecasts from hybrid analyses fit observation much better.

31. Global Data Assimilation System Upgrade
Implemented 22 May 2012
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

32. Operational Configuration
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* 32

33. Retrospective GFS Hurricane

34. 500mb Anomaly Correlation Real Time

35. Tropical Winds

40. 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

41. 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

42. Dual-Res Coupled Hybrid Var/EnKF Cycling
Generate new ensemble perturbations given the latest set of observations and first-guess ensemble
member 1
forecast
member 1
analysis
T254L64
EnKF
member update
member 2
forecast
member 2
analysis
recenter analysis ensemble
member 3
analysis
member 3
forecast
Ensemble contribution to background error covariance
Replace the EnKF ensemble mean analysis
and inflate
GSI
Hybrid Ens/Var
high res
forecast
high res
analysis
T574L64
Previous Cycle
Current Update Cycle

43. Post EnKF Inflation (Whitaker and Hamill 2012)
Multiplicative inflation factor, r, (function of reduction of spread by assimilation of observations):
Additive inflation extracts quasi-balanced pseudo-random perturbations from database of lagged forecast pairs
Inflation is an ad-hoc method for overcoming lack of consideration of model/system error within ensemble-filtering systems

44. Use of inflation and re-centering
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 ensemble
filter is dropped and an ensemble of analyses is created from simply additively inflating the central analysis?

45. Ensemble Member 01 increments

46. No Filter Experiment Settings
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)

47. Evolution of 6hr Spread
EnKF-based hybrid
Filter-Free hybrid

48. Spread within 9hr window
EnKF-based hybrid
Filter-Free hybrid

49. Cycled Results
3DVAR
EnKF-Hybrid
FilterFree-Hybrid

50. Cycled Results
3DVAR
EnKF-Hybrid
FilterFree-Hybrid

51. 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

52. Un(der)-represented error sources in an EnKF ensemble
Model error
Sampling error
Observation error
Boundary condition error
Forward operator error

53. 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 forecast differences (valid at same time), added to EnKF analysis ensemble.

54. 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: /2007JD009284)
All use stochastic random pattern generators to generate spatially and temporally correlated noise.

55. SPPT+SHUM+SKEB U
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.
SPREAD
SPRD – SPRD CTL
But is it “good” spread? Does it improve covariances in EnKF?
ERROR
ERR – ERR CTL

56. 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.

57. 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.

58. Single Observation (-3h) Example for 4D Variants
4DEnVar
H-4DVAR_AD
bf-1=0.25
H-4DEnVar
bf-1=0.25

59. 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.