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Ensemble Data Assimilation without Ensembles with Application to Ocean Data Assimilation or Flow-Adaptive Background-Error-Covariance Modeling for Data Assimilation using Information from a Single Model Trajectory or E unus pluribum

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Ensemble Data Assimilation without Ensembles with Application to Ocean Data Assimilation

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Flow-Adaptive Background-Error-Covariance Modeling for Data Assimilation using Information from a Single Model Trajectory

Christian Keppenne1, Michele Rienecker2, Robin Kovach1, Guillaume Vernieres1

1SSAI inc.

2NASA GSFC

6th WMO Symposium on Data Assimilation College Park, MD, October 7-11 2013

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E unus pluribum

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SLIDE 2
  • 2001-2008: Ocean EnKF in GMAO NSIPP-1 S/I Prediction system
  • Poseidon isopycnal OGCM
  • ARIES AGCM
  • ODAS EnKF run prior to AGCM coupling
  • 2009-present: GEOS iOdas used with GEOS-5 CGCM

iOdas focus

  • Seasonal-interannual-decadal forecast initialization
  • Retrospective ocean reanalysis (1950-present)
  • Coupled data assimilation
  • Global eddy resolving ocean data assimilation + coupled DA

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OGCM run CGCM forecast LSM-AGCM-OGCM coupling ODAS EnKF CGCM run CGCM forecast ADAS replay GEOS iOdas

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SLIDE 3

NASA GMAO Integrated Ocean Data Assimilation System

(GEOS iOdas)

GEOS-5 coupled modeling system

  • NASA GEOS-5 AGCM
  • NASA GOCART chemistry + aerosol + radiation model
  • GEOS-5 catchment LSM
  • GFDL MOM5 OGCM
  • LANL CICE
  • NASA NOBM ocean biology model

GEOS analysis system

  • GEOS DAS atmospheric analysis system
  • iOdas

Production ocean analysis with:

  • MOM ½° x ½°-⅙°x L40
  • CICE ½° x ½ x 5
  • AGCM + GOCART 1¼° x 1° x L72 (Ganymed 4)
  • 4 hours wallclock time/month to run CGCM on 360 cores
  • 7-8 hours wallclock time/month with ocean analysis

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SLIDE 4

iOdas background error-covariance modeling options

  • Multiple trajectory methods
  • hybrid particle filter EnKF (HPEnKF)
  • Steady state Ensemble (aka asymptotic EnKF, EnOI, SE*K)
  • Single trajectory methods
  • SAFE (Space Adaptive Forecast error Estimation)
  • FAST (Flow Adaptive error Statistics from a Time series
  • Ocean DA is about updating fields of unobserved variables
  • Need reliable cross-field covariances

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SLIDE 5

EnKF with particle pre-filter step (HPEnKF)

(Keppenne et al., 2013a: manuscript)

  • prefilter step
  • xp: ensemble member that minimizes RMS OMF
  • xp: ensemble mean
  • Dp = xp - xp
  • EnKF analysis step

xm Y Dp xp xm Y xp

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GEOS iOdas ocean EnKF

Forecast ensemble augmented with time lagged instances: (e.g., EnKF 16x11: current ensemble + 10 previous lags)

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SLIDE 6

control RMS OMF – EnKF-16x11 RMS OMF: z<200m control RMS OMF – HPEnKF-16x11 RMS OMF: z<200m control RMS OMF – EnKF-16x11 RMS OMF: z>200m control RMS OMF – HPEnKF-16x11 RMS OMF: z>200m

GEOS 5 CGCM

  • AGCM 288x181x72
  • OGCM 720x410x40
  • ARGO T assimilated (active)
  • ARGO S passive
  • Control: Univariate T OI
  • EnKF 16x11: T, S, u, v update
  • HPEnKF 16x11: T, S, u, v update

AMJ 2006 Vertically integated RMS (passive) ARGO S OMF in 3° x 3° bins:

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EnKF 16 better control better

HPEnKF 16 better

control better EnKF 16 better control better

HPEnKF 16 better

control better

Difference with control T OI analysis

  • cean DA needs reliable cross-field covariances!
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SLIDE 7

ARGO T assimilation experiment AMJ 2006 Vertically integated RMS (passive) S OMF in 3°x3° boxes

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T OI control EnKF 16x11 HPEnKF 16x11 Production analysis with EnOI (50 forecast error anomaly EOFs)

[Vernieres et al. 2012: NASA tech. report 2012]

1993-2005 0-300m

1993-2005 300-1000m 2006-2012 0-300m 2006-2012 300-1000m

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SLIDE 8

What’s really nice about ensemble assimilation methods?

  • Adaptively estimates background errors amplitudes
  • Not when using covariance inflation
  • Adaptively estimates background error distribution for
  • Fields of observed quantities
  • Alternative methods can do that too
  • Fields of unobserved quantities
  • Yes, but large ensembles are needed for proper <t, s>

(or other ocean variable pairs) covariance estimates

Want something

  • faster than EnKF/HPEnKF
  • able to flow-adaptively update unobserved model fields
  • that works as well as EnOI

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SLIDE 9

Ensemble data assimilation without ensembles

Flow-adaptive error covariance estimation without ensemble integration

2 approaches built in GEOS iOdas

(Keppenne et al 2013b, Joatech, submitted)

  • Ensemble in time (FAST)
  • Ensemble in space (SAFE)
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SLIDE 10
  • Ensemble of lagged state instances sampled along same trajectory
  • High pass filtered ensemble
  • x: original trajectory
  • x0: low-pass filtered trajectory (SMA or IR filter)
  • Resampling to compensate window centering effect
  • bij: random weights

Flow Adaptive error Statistics from a Time series (FAST)

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 

) 11 ( , 1 ) 11 ( , 1 , , ,

1 ) ( ) (

b n a n j

n j j k k k j k k

   

     x x x x X 

 

) 12 ( , 1 , , ,     

 

n j

j k j k k

 x x X ) 13 ( , ) 1 (

1 

    

k k k

x x x

 

 

) 14 ( , 1 , , , ) 14 ( , 1 , , ,

1

b n j a n j

j k k n i i k ij j k k

              b      

    

  x x X x x X

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SLIDE 11

a) b) Active Argo T RMS OMF 2010-2011 (0-3000m) Passive Argo S (0-3000m)

EnOI 0 order 1st order 2nd order FAST EnOI 0 order 1st order 2nd order FAST

11 EnOI 20 leading EOFs last 20 state vectors last 20 1st order time differences last 20 2nd order time differences FAST 20 lags

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SLIDE 12
  • Treat state variables in nearby grid cells as if they were

from other ensemble members (spatial ensemble analogy)

  • Other computations essentially same as in EnKF except

distance-dependent weighted mean instead of arithmetic mean (done with Gaussian filter)

  • Coastlines introduce difficulties in the ocean

(handled with aforementioned Gaussian averaging)

Space Adaptive Forecast error Estimation (SAFE)

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SLIDE 13
  • 1: estimate background error variance of observed field v at each

gridpoint (Gaussian filter (Q) replaces arithmetic ensemble mean)

  • 2: assimilation increment Dv for observed field v
  • 3: estimate cross-field covariances <v, w> at every grid point
  • 4: project Dv onto fields of unobserved variables at every gridpoint

Space Adaptive Forecast error Estimation (SAFE)

, ], , [        

ww wv vw vv

P P P P P w v x

  ),

) ( ( ) (

2 2

v v P σ Q  Q  

vv vv

diag

  • bserved: v

unobserved: w

  ).

) ( ) ( (

2

w w v v Q  Q  Q  vw approximation:

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SLIDE 14

SAFE

FAST EnOI T OI

a) b)

1600 1800 200 400 600 800 1000 1200 1400

  • 0.25 -0.20 -0.15 -0.10 -0.05 0.0

(ºC)

1600 1800 200 400 600 800 1000 1200 1400

  • 0.06 -0.04 -0.02 0.0 0.02 0.04 0.06

(PSU)

14 5-day lead passive Argo S RMS OMF 2010-2011 5-day lead active Argo T RMS OMF 2010-2011

SAFE

FAST EnOI T OI

RMS OMF reduction over no-assimilation control

  • observed T
  • unobserved S
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SLIDE 15

100 150 200 250 300 20S 10S EQ 10N 20N 20S 10S EQ 10N 20N 20S 10S EQ 10N 20N 100 150 200 250 300 170W 160W 150W 140W 130W 120W 110W 170W 160W 150W 140W 130W 120W 110W 170W 160W 150W 140W 130W 120W 110W 100 150 200 250 300 20S 10S EQ 10N 20N 20S 10S EQ 10N 20N 20S 10S EQ 10N 20N 170W 160W 150W 140W 130W 120W 110W 170W 160W 150W 140W 130W 120W 110W 170W 160W 150W 140W 130W 120W 110W

a) b) TS SAFE TS FAST TS EnOI c) d) e) f) g) h) i) j) k) l) TT SAFE TT FAST TT EnOI/T OI

100 150 200 250 300

0.0

  • 0.5

0.5 0.0

  • 0.05

0.05 W-E W-E W-E S-N S-N S-N W-E W-E W-E S-N S-N S-N

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SLIDE 16

0.0

  • 0.05

0.05 a) b) c) d) e) f) g) h) i) j) k) l) m) n)

  • )

TS SAFE TS FAST TS EnOI Jan 1 2010

100 150 200 250 300 100 150 200 250 300 100 150 200 250 300 100 150 200 250 300 100 150 200 250 300 170W 160W 150W 140W 130W 120W 110W 170W 160W 150W 140W 130W 120W 110W 170W 160W 150W 140W 130W 120W 110W

Apr 1 2010 Jul 1 2010 Oct 1 2010 Jan 1 2011 Jan 1 2010 Apr 1 2010 Jul 1 2010 Oct 1 2010 Jan 1 2011 Jan 1 2010 Apr 1 2010 Jul 1 2010 Oct 1 2010 Jan 1 2011

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SLIDE 17

a) b) c) d)

SAFE FAST EnOI T OI 17 2011 control run RMS S OMF - 5-day lead Argo S RMS OMF from runs with ARGO T assimilation (0-300m)

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SLIDE 18

a) b) c) d)

SAFE FAST EnOI T OI 18 2011 control run RMS S OMF - 5-day lead Argo S RMS OMF from runs with ARGO T assimilation (300-3000m)

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SLIDE 19
  • 1st generation system (used in GMAO ocean reanalysis)
  • Assimilation of NSIDC ice fraction (aice)
  • ice fraction innovation: NSIDC - CICE model state
  • SAFE to estimate <aice, T> and <aice, S> covariances

(can also use FAST, EnOI)

  • Incremental update of MOM T, S ocean fields
  • OGCM: MOM4.1 tripolar grid 720x410x40
  • CICE 4.1 720x410x5
  • AGCM: Fortuna 2.5 288x181x72
  • iOdas-503

Application to sea ice concentration assimilation

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SLIDE 20
  • 1st. Generation iODAS sea ice assimilation

01/01/2011 NSIDC control aice OI SAFE FAST EnOI

NH ice SH ice

CGCM MERRA replay SAFE FAST EnOI aice OI 20

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SLIDE 21

1st Gen. Sea Ice Assimilation

(used in GMAO production analysis) Global RMS OMF & OMA (5-day lead) NSIDC ice concentrations

  • Strong seasonality in misfit
  • Worst in Summer

SH NH

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SLIDE 22

Adaptive Sea Ice concentration Assimilation (ASIA)

  • Forget about updating ocean T & S
  • Focus on ice volume vice = aice * hice

Ice concentration (aice) innovations

  • Daice & Dhice increments
  • Dvice = hice*Daice + aice*Dhice
  • target ice volume: Tvice =aice*hice + Dvice

<aice, aice> <aice, hice> SAFE At each time step in CICE: Adjust skin water (Tw) temperature to bring vice to target ice volume (Tvice) DTw = dTw/dvice*(Tvice – aice*hice)

  • dTw/dvice is iteratively adjusted to optimize convergence to Tvice

Application to sea ice concentration assimilation

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SLIDE 23

Adaptive Sea Ice Assimilation (ASIA) (using SAFE)

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NH RMS NSIDC aice OMFA

  • 1st. Gen. analysis

CGCM run with MERRA replay free CGCM run (no MERRA replay) ASIA analysis

SH RMS NSIDC aice OMFA ASIA initialized from free CGCM run

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SLIDE 24

NSIDC ice fraction during 2012 NH/SH Summer

  • ASIA initialized from free CGCM run on 01/01/2012

ASIA ASIA 1st GEN NSIDC 1st GEN NSIDC 02 2012

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08 2012

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SLIDE 25

ASIA dTw/dvice evolution

  • Max. in summer
  • Min in spring

01/2012 04/2012 07/2012 10/2012

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SLIDE 26
  • MOM5 0.1°x 0.1°x 40 levels (3840x1736x40)
  • GEOS-5 AGCM 0.25°x 0.25° x 72 levels (1152x721x72)
  • CICE 0.1°x 0.1°x 5 ice categories
  • SAFE (because most economical)
  • Multi-scale approach
  • 1: assimilate production (720x410x40) GMAO ocean analysis
  • 2: assimilate 0.25° Reynolds SSTs and NSIDC aice

(shortened localization scales)

Application to global high-resolution

  • cean data assimilation (initial testing)

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SLIDE 27

0.25ºReynolds SST 08/01/2007 Detail of 0.1º SST analysis for 08/01/2007 Corresponding Detail of 0.5º analysis

High Res. SAFE Assimilation Initial Testing

  • 0.1º OGCM (3840 1736 40)
  • 0.25º AGCM (1152 721 72)
  • Daily assimilation of SST and ice fraction for 6 months
  • Step 1: replay 0.5º ocean analysis into high res. CGCM
  • Step 2: small-scale 0.1ºassimilation of Reynolds SST + NSIDC aice

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SLIDE 28

0.25º Reynolds SST 08/01/2007 Detail of 0.1º SST analysis for 08/01/2007 Corresponding Detail of 0.5º analysis

High Res. SAFE Assimilation Initial Testing

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SLIDE 29

MODIS chlorophyll assimilation with GEOS iOdas SAFE

  • MOM resolution: 720x410x40
  • GEOS-5 MERRA forcing
  • 2008-2009 spin up
  • Assimilate daily MODIS chlorophyll
  • Observed variable:

log(chlorophyll) ≈ log(total phytoplankton)

  • Updated:
  • 5 phytoplankton species
  • N03 + NH4 + SiO2 + Fe
  • Error covariances estimated with SAFE

control update phytoplankton update phytoplankton + NO3 + NH4 + SiO2 + Fe

Update plankton Update plankton + chemistry control

MODIS chlorophyll RMS OMF

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SLIDE 30

Conclusions

E unus pluribum...

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SLIDE 31

31 Xue et al., 2012: J. Climate, 25, 6905–6929

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SLIDE 32

“Regularization” with Global SVD solver

SH NSIDC ice concentrations RMS OMFA

LU solver (filtering off) SVD solver (90% variance retention) GMAO ocean analysis (1st gen.) NH: Large differences between regularized and unregularized solutions 32

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SLIDE 33

“Regularization” with Global SVD solver

NH NSIDC ice concentrations RMS OMFA

LU solver (filtering off) SVD solver (90% variance retention) GMAO ocean analysis (1st gen.) NH: smaller differences between regularized and unregularized solutions 33