Hybrid variational-ensemble data assimilation at NCEP - NOAA

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					   Hybrid variational-ensemble
    data assimilation at NCEP


           Daryl T. Kleist
       daryl.kleist@noaa.gov


National Monsoon Mission Scoping Workshop
     IITM, Pune, India 11-15 April 2011

                                            1
             Variational Data Assimilation

J Var      B   
           1 '
        x  x
         '

           2
                  T    1
                       Var
                                1 '
                             x  y o  Hx '
                               '

                                2
                                                          R y
                                                          T    1     '
                                                                      o         
                                                                           Hx '  J c

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.)

                                                                                    2
               B is typically static and estimated a-priori/offline
              Motivation from Var

• Current background error covariance (applied
  operationally) in VAR
  – Isotropic recursive filters
  – Poor handle on cross-variable covariance
  – Minimal flow-dependence added
     • Implicit flow-dependence through linearization in
       normal mode constraint (Kleist et al. 2009)
     • Flow-dependent variances (only for wind, temperature,
       and pressure) based on background tendencies
  – Tuned NMC-based estimate (lagged forecast pairs)
                                                               3
                    Current background error for GFS




•   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               xb  M xa     
                      B KF  M A KF M T  Q
                      x  x  K Hx  y
                       a     b
                                              b
                                                          Extended Kalman
                                                                Filter
 Analysis         K  BKFH RHB KFH
                                 T
                                                  T 1
                                                       
                       A KF  I  KH B KF

• Analysis step in variational framework (cost function)

 J KF      B   
           1 '
        x  x
         '

           2
                  T    1
                       KF
                               1 '
                            x  y o  Hx '
                             '

                               2
                                                        R y
                                                       T   1   '
                                                                o    Hx '   
• BKF: Time evolving background error covariance
• AKF: Inverse [Hessian of JKF(x’)]                                              5
                     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:   X X
                 a       b
                             B KF 
                                     1
                                    K 1
                                           
                                         Xb Xb
                                                 T

                                                      Ensemble
                                                     Perturbations
Analysis Step:   X X
                 b       a
                             A KF 
                                     1
                                    K 1
                                           
                                         Xa Xa
                                                 T



                                                               6
                                     Why Hybrid?

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

                                                                                   7
                 Hybrid Variational-Ensemble
• Incorporate ensemble perturbations directly into
  variational cost function through extended control
  variable
    – Lorenc (2003), Buehner (2005), Wang et. al. (2007), etc.

     
    J x ,   f x f B x f   e
         '
         f         
                1 ' T 1 '
                2
                               2
                                                                       
                                     L    1 y 'o  Hx 't T R 1 y 'o  Hx 't
                                 1 T 1
                                                 2
                                                                                      
                                                          
                                           K
                                x  x   k  xe
                                  '
                                  t
                                      '
                                      f         k
                                           k 1

                                      1        1
                                                     1
                                      f       e
f & e: weighting coefficients for fixed and ensemble covariance respectively
xt: (total increment) sum of increment from fixed/static B (xf) and ensemble B
                                  e
k: extended control variable; x k :ensemble perturbation
L: correlation matrix [localization on ensemble perturbations]                            8
            Experiments with toy model
                d xi
                       xi  2 xi 1  xi 1 xi 1  xi  F
                 dt
• 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                                     9
       Analysis Error (50% observation coverage)




 3DVAR
 f = 0.7
 f = 0.3
 ETKF



• M (ensemble size) = 20, r (inflation factor) = 1.1
   – Hybrid (small alpha) as good as/better than ETKF (faster spinup)
                                                                        10
   – Hybrid (larger alpha) in between 3DVAR and ETKF
                          Sensitivity to 


                   Analysis RMSE (x10) over 1800 cases

  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



• 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
                                                                         11
                  Importance of
           Ensemble Generation Method?

• GEFS (already operational)
   – 80 cycled members
   – ETR
       • Virtually no computational cost
       • Uses analysis error mask derived for 500 mb streamfunction
       • Tuned for medium range forecast spread and fast “error growth”
   – T190L28 version of the GFS model
   – Viable for hybrid paradigm?

• EnKF
   – 80 cycled members
   – Perturbations specifically designed to represent analysis and
     background errors
   – T254L64 version of the GFS
   – Extra computational costs worth it for hybrid?
                                                                          12
                     EnKF/ETR Comparison




   EnKF (green) versus ETR (red)         Surface Pressure spread normalized
spread/standard deviation for surface   difference (ETR has much less spread,
  pressure (mb) valid 2010101312               except poleward of 70N)
                                                                                13
                      EnKF/ETR Comparison




EnKF zonal wind (m/s) ensemble standard   ETR zonal wind (m/s) ensemble standard
      deviation valid 2010101312               deviation valid 2010101312

                                                                                   14
                   EnKF/NMC B Compare




EnKF zonal wind (m/s) ensemble standard   Zonal Wind standard deviation (m/s) from
      deviation valid 2010101312                      “NMC-method”

                                                                                     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

                                       ' K              
                                                        
                                x  C x f    k  x e 
                                  '
                                                       k
                                            k 1        
                                                                                                16
      *Acknowledgement: Dave Parrish for original implementation of extended control variable
  Single Observation




                                                 17
Single 850mb Tv observation (1K O-F, 1K error)
                   Single Observation




                                                                                           18
Single 850mb zonal wind observation (3 m/s O-F, 1m/s error) in Hurricane Ike circulation
                 Dual-Res Coupled Hybrid




                                                  recenter analysis ensemble
 member 1
                                                                               member 1
  forecast
                                                                                analysis

 member 2                EnKF                                                  member 2
  forecast            member update                                             analysis

                                                                               member 3
 member 3
                                                                                analysis
  forecast




   high res           GSI              high res
   forecast      Hybrid Ens/Var        analysis



Previous Cycle        Current Update Cycle
            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                                                                            20
     – 15 July 2010 – 15 October 2010 (first two weeks ignored for “spin-up”)
              500 hPa Anom.Corr.
Northern Hemisphere          Southern Hemisphere




                                                   21
       AC Frequency Distributions
Northern Hemisphere         Southern Hemisphere




                                                  22
         Geopotential Height RMSE
Northern Hemisphere                                   Southern Hemisphere




               Significant reduction in mean height errors                  23
   Stratospheric Fits




Improved fits to stratospheric observations   24
Forecast Fits to Obs (Tropical Winds)




    Forecasts from hybrid analyses fit observation much better.
                                                                  25
26
27
28
            HVEDAS (3D) for GDAS/GFS
• Prototype dual-resolution, two-way coupled hybrid Var/EnKF
  system outperforms standard 3DVAR in GFS experiments
   – 2010 Hurricane Season (August 15 through October 31 2010) run off-
     site
   – Emphasis on AC, RMSE, TC Tracks

• Plan underway to implement into GDAS/GFS operationally
   – Target: Spring 2012 (subject to many potential issues)
       • Porting of codes/scripts back to IBM P6
       • Cost analysis (will everything fit in production suite?)
       • More thorough (pre-implementation) testing and evaluation
           – More test periods (including NH winter)
           – Other/more verification metrics
       • Potential moratorium associated with move to new NCEP facility

• Issues
   – Weighting between ensemble and static B
   – Localization
                                                                          29
   – How should EnKF be used within ensemble forecasting paradigm?
                                Cost

• Analysis/GSI side
  – Minimal additional cost
     • Reading in ensemble (3-9 hour forecast from previous cycle)
         – Working on building an ensemble post-processor to prep files for GSI
     • Coding already complete/in place
         – Optimize localization (?)

• Additional “GDAS” Ensemble (T254L64 GDAS)
  – EnKF-based perturbation update
     • Cost comparable to current analysis [say 8-10 nodes, something
       <40 minutes]
         – Includes ensemble of GSI runs to get O-F and actual ensemble update
           step
     • Work ongoing to optimize coding and scripting
  – 9hr forecasts, needs to be done only in time for next (not
    current) cycle
                                                                             30
           HVEDAS Extensions and Improvements

• Expand hybrid to 4D
    – Hybrid within ‘traditional 4DVAR’ (with adjoint)
    – Pure ensemble 4DVAR (non-adjoint)
    – Ensemble 4DVAR with static B supplement (non-adjoint)*

• EnKF improvements
    – Explore alternatives such as LETKF
    – Adaptive localization and inflation

• Non-GFS applications in development
    –   Other global models (NASA GEOS-5, NOAA FIM)
    –   NAM /Mesoscale Modeling
    –   Hurricanes/HWRF
    –   Storm-scale initialization
    –   Rapid Refresh

• NCEP strives to have single DA system to develop, maintain, and run
  operationally (global, mesoscale, severe weather, hurricanes, etc.)
    – GSI (including hybrid development) is community code supported through DTC
    – EnKF used for GFS-based hybrid being expanded for use with other applications
                                                                                      31

				
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