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

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

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

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’)]

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:

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

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

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

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

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

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

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

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]

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:

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

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

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

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

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.

Localization

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.

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

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

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 Distributions Northern Hemisphere Southern Hemisphere

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

Improved fits to stratospheric observations 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.

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

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

Retrospective GFS Hurricane

500mb Anomaly Correlation Real Time

Tropical Winds

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

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

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

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?

Ensemble Member 01 increments

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)

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

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

Cycled Results 3DVAR EnKF-Hybrid FilterFree-Hybrid

Cycled Results 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

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

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.

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 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

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.

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.

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

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.