EnKF, 4D-Var and ECCO in a “toy” ocean-atmosphere model

EnKF, 4D-Var and ECCOEnKF, 4D-Var and ECCO in ain a““toytoy”” ocean-atmosphere modelocean-atmosphere model

Tamara Singleton1,2, Eugenia Kalnay1,Kayo Ide1 and Shu-Chih Yang3

1UMD, 2Johns Hopkins, 3Taiwan NCU

EnKF, 4D-Var and ECCOEnKF, 4D-Var and ECCO in ain a““toytoy”” ocean-atmosphere modelocean-atmosphere model

Tamara Singleton1,2, Eugenia Kalnay1,Kayo Ide1 and Shu-Chih Yang3

1UMD, 2Johns Hopkins, 3Taiwan NCU

Simple Coupled Ocean-Atmosphere Model (Peña and Kalnay, 2004)

Coupling strengthTropical Atmosphere

Tropical OceanExtra-tropical AtmosphereOcean is vacillatingbetween a “normal”

(lasts about 2-10 years)and “El Niño” state

(lasts about a 1 year)

We compare 4D-Var and EnKF with this simple coupled model

Questions explored:-- Which is more accurate: 4D-Var or EnKF?-- Should we use short or long windows?-- Is it better to do an ocean reanalysis separately, or as

a single coupled system?-- Should we use frequent atmospheric observations in a

coupled system?-- Would RIP/QOL be beneficial in a coupled system?

ECCO is a ocean version of 4D-Var where the initialstate and the surface fluxes are both control variables.

This allows ECCO to use very long windows (decades)and estimate the surface fluxes that give the bestanalysis.

ECCO provides a single, continuous reanalysis--Is ECCO the best approach?

Simple Coupled Ocean-Atmosphere System

Ocean

Tropical atmosphere

Extratropical atmosphere

Model Parameter Definitions

3 coupled Lorenz models: A slow “ocean”component strongly coupled with a fast“tropical atmosphere component”, in turnweakly coupled with a fast “extratropicalatmosphere” (Peña and Kalnay, 2004).

Model State:k1=10k2 = -11

Uncenteringparameters

k1,k2

σ=10,b=8/3, andr=28

Lorenzparameters

σ, b, and rτ = 0.1time scaleτ

c,cz = 1ce = 0.08

Couplingcoefficient

c,cz,ce

ValuesDescriptionVariables

!xe = ! (ye " xe ) " ce(xt + k1)!ye = rxe " ye " xeze " ce(yt + k1)!ze = xeye " bze

!xt = ! (yt " xt ) " c(X + k2 ) " ce(xe + k1)!yt = rxt " yt " xtzt + c(Y + k2 ) + ce(ye + k1)!zt = xt yt " bze + czZ

!X = !" (Y # X) # c(xt + k2 )!Y = !rX # !Y # !XZ + c(yt + k2 )!Z = !XY # !bZ + czzt

[xe, ye, ze, xt , yt , zt ,X,Y ,Z]T


Coupling strengthTropical Atmosphere

Tropical OceanExtra-tropical AtmosphereOcean is vacillatingbetween a “normal”

(lasts about 2-10 years)and “El Nino” state

(lasts about a 1 year)


Time series of the x-component


fast tropicalatmosphere

slow ocean“normalyears”

fastextratropicalatmosphere

Δt=0.01


slow ocean“El Niñoyears”

Data Assimilation Experiment Design

• Simple Coupled Ocean-Atmosphere Model (perfect model)– Used to create the “true” trajectory

• Observations– Generated from the nature run plus “random errors” with s.d.– Every 8 time steps of a simulation

• Perform coupled and uncoupled ocean data assimilationswith several EnKF, 4D-Var, and ECCO-4D-Var

• Compute RMS errors of the difference between the analysisand the true solution.

• Lengthen assimilation windows, from 8 to 320 steps

• Perform fully coupled data assimilation (ETKF, 4D-Var),and just ocean assimilation (LETKF, 4D-Var and ECCO)

2

EnKF-Based Methods

Atmos: Available atanalysis timeOcean: Availablethroughout anassimilation window

Available at analysistime

Available at analysistime

Available throughoutan assimilationwindow

Available at the endof a window (analysistime)

Observations

4-dimensional

Subsystemlocalization

Fast and slowvariables separately

4D-LETKF(Separate Ocean)

Subsystemlocalization

Fast and slowvariables separately

LETKF(Separate Ocean)

Uses quasi-outerloop to improve theinitial analysis mean

Fast and slowvariablessimultaneously

ETKF-QOL(Fully coupled)

4-dimensionalFast and slowvariablessimultaneously

4D-ETKF(Fully coupled)

Fast and slowvariablessimultaneously

ETKF(Fully coupled)

Special FeaturesAssimilatingMethod

Description of EnKF-based methods

Coupled ocean-atmosphere ensembles:ETKF, 4D-ETKF, ETKF-QOL

RMS error as a function of assimilation window length

The fully coupled ETKF data assimilations work well.The shortest assimilation window (8 steps) is the best.

4D-ETKF (assimilating all the obs) is better than ETKF for longerwindows. ETKF-QOL has the best performance (short windows).

Variational Data AssimilationExperiments:

Fully Coupled 4D-VarOcean only 4D-Var

ECCO-like Ocean 4D-Var

Fully coupled 4D-Var : the Cost Function• In 4D-Var, a cost function is minimized to produce an optimal analysis.

– The cost function measures the distance between the model withrespect to the observations and with respect to the background state.

• The analysis is obtained by minimizing the cost function given by

J(x t0

) =12

x t0! x t0

b"# $%TB0

!1 x t0! x t0

b"# $% +12

H(x ti)! y ti

o"# $%TR ti

!1

i=1

N

& H(x ti)! y ti

o"# $%

Jo- “observation” cost functionJb - ”background” cost function

where the control variables are the initial 9 model variables:

x0 = xe0,ye

0,ze0,xt

0,yt0,zt

0,X0,Y0,Z0( )T

Initial model state for oceanInitial model state for tropical atmos.Initial model state for extratropical atmos

4D-Var: Quasi-static Variational DataAssimilation (QVA)

• For longer windows, multipleminima are a problem for 4D-Var minimization (Pires et al.,1996).

• Also for longer assimilationwindows, non-Gaussianperturbations of theobservation error andbackground error -> in non-quadratic cost functions

• Pires et al. (1996) proposedthe Quasi-static VariationalData Assimilation (QVA)approach.

x

t

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1 3 5 7 9

ETKF-QOL

LETKF

4D-LETKF

4D-Var

Obs. Error

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1 3 5 7 9

ETKF-QOL

LETKF

4D-LETKF

4D-Var

Obs. Error

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1 3 5 7 9

ETKF-QOL

LETKF

4D-LETKF

4D-Var

Obs. Error

Fully coupled 4D-Var (+QVA) and EnKF: shorter windows

8 16 24 32 40 48 56 64 72 80

assimilation window (time-steps)

8 16 24 32 40 48 56 64 72 80


8 16 24 32 40 48 56 64 72 80


Extratropics Tropics

OceanETKF-QOLprovides thebest analysisfor very short

windows

4D-Var competeswith EnKF-based

methods forlonger windows

Obs. Error Obs. Error

Obs. Error

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

1 2 3 4 5 6

ETKF-QOLLETKF4D-LETKF4D-VarObs. Error

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1 2 3 4 5 6


0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

1 2 3 4 5 6


Fully coupled 4D-Var (+QVA) vs. EnKF: longer windows

88 96 120 160 200 240

assimilation window (time-steps) 88 96 120 160 200 240


88 96 120 160 200 240


Extratropics

Tropics

OceanCoupled 4D-Var

and EnKFcompetitive forlonger windows

Fully coupled 4D-Var vs EnKF summary

• We developed fully coupled 4D-Var and EnKF systems for thesimple coupled ocean-atmosphere model

• Lengthening the assimilation windows and applying QVAimproves the 4D-Var analysis because 4D-Var “forgets” B.But longer windows are more expensive…

• Fully coupled EnKF are optimal for short windows. Shortwindows are less expensive…

• EnKF+QOL works best (short windows).

• The optimal configurations (short windows for EnKFand long windows for 4D-Var) have similar accuracy.

ECCO-like 4D-Var

• The Consortium for Estimating the Circulation and Climateof the Ocean (ECCO) is a collaboration of a group ofscientists from the MIT, JPL, and the Scripps Institute ofOceanography

• The main characteristic of ECCO is that they includesurface fluxes as control variables.– This allows them to have exceedingly long assimilation windows in

4D-Var (e.g. 10 years or even 50 years).– They used NCEP Reanalysis fluxes (Kalnay et al, 1996) as a first

guess.

• ECCO used 4D-Var to estimate the initial ocean state andsurface fluxes (Stammer et al., 2004; Kohl et al., 2007) ina 50-year reanalysis

Carton and Santorelli (2008) plot of the First Empirical Orthogonal Eigenfunction of monthly heat content anomalyin the latitude band 20N-60N Explained variance is shown on the title line. Lower panel shows the correspondingcomponent time series annually averaged along with the Pacific Decadal Oscillation Index of Mantua et al. (1997)in black.

ECCO is the only one of the analyses for whichneither the first nor second heating EOF resemblethe Pacific Decadal Oscillation Pattern

ECCO

ECCO

Motivation: Comparison of Ocean Analyses

ECCO-like 4D-Var: Cost Function includesall surface fluxes as control variables

J =

12

[x0,f ! xb,nfe ]T (B0,nfe )!1[x0,f ! xb,nfe ]+12

[Hx ti- y ti

o ]T(R ti

!1)[Hx ti- y ti

o ]i=1

N

"

x0,f = X0,Y0,Z0 f1

1, f21, f3

1 f12 , f2

2 , f32 ... f1

n, f1n, f1

n( )T

Initial model state Fluxes for first 8 time steps Fluxes for last 8 time steps

Background state for the ocean

NCEP-like flux estimates for first 8 time steps

NCEP-like flux estimates for last 8 time steps

xb,nfe = Xb,Yb ,Zb f1

nfe,1, f2nfe,1, f3

nfe,1 f1nfe,2 , f2

nfe,2 , f3nfe2 ... f1

nfe,n, f1nfe,n, f1

nfe,n( )T

where the controlvariables are:

B0,nfe =B 00 Q

!

"#

$

%&

Comparison of ECCO-like & Ocean 4D-VarObs. s.d. error = 1.41 for oceanQVA APPLIED

ECCO improves the 4D-analyses

0

1

2

3

4

5

6

7

8

9

10

1 6 11

4D-VarECCO 4D-VarObs. Error

8 16 24 32 40 48 56 64 72 80 120 160 200 240 320


RMSE : Ocean State

ECCO (ocean only) remains satisfactory

OCEAN ONLY

4D-Var (ocean only) fails

By using sfc fluxes as control variables, ECCO can use very long windows

Are the ECCO fluxes more accurate?

RMS Errors (Flux 3 Estimate)

0.51

1.52

2.53

3.54

4.55

5.56

1 6 11


NCEP-like FluxEstimatesECCO Flux Estimates

ECCO does not improve the flux estimates over the first guess

8 16 24 32 40 48 56 64 72 80 120 160 200 240 320


Answers to the Research Questions

Questions:-- Which is more accurate: 4D-Var or EnKF?Fully coupled EnKF (with short windows) and 4D-Var (with longerwindows) have about the same accuracy. Both can handlefrequent atmospheric observations.


Questions:-- Which is more accurate: 4D-Var or EnKF?Fully coupled EnKF (with short windows) and 4D-Var (with longerwindows) have about the same accuracy. Both can handlefrequent atmospheric observations.-- Is it better to do the ocean reanalysis separately, or as a singlecoupled system?Both EnKF and 4D-Var are similar and most accurate whencoupled, but uncoupled (ocean only) reanalyses are fairly good.


Questions:-- Which is more accurate: 4D-Var or EnKF?Fully coupled EnKF (with short windows) and 4D-Var (with longerwindows) have about the same accuracy. Both can handlefrequent atmospheric observations.-- Is it better to do the ocean reanalysis separately, or as a singlecoupled system?Both EnKF and 4D-Var are similar and most accurate whencoupled, but uncoupled (ocean only) reanalyses are fairly good.-- Is ECCO 4D-Var with both the initial state and the surfacefluxes as control variables the best approach?In our simple ocean model 4D-Var cannot remain accurate withvery long windows. Our “ECCO” reanalysis remained satisfactorywith very long windows but at the expense of less accuratefluxes.

Practical Conclusions for CoupledAssimilation

• Since EnKF is as accurate as 4D-Var, and is optimal for shortwindows, it is more efficient than 4D-Var, which requires verylong windows for maximum accuracy.

• Contrary to our expectations, the best results included frequentatmospheric observations.

• ECCO is 4D-Var including surface fluxes as control variables.This allows ECCO to have very long windows (decades).

• Since the estimated surface fluxes “adapt” in order to force theocean model to be close to the observations, they are notguaranteed to be more accurate than the background fluxes.

• In our toy coupled model, the estimated surface fluxes wereless accurate than the background fluxes.

Extra: no-cost LETKF smoother allows a comparison of EnKFinitial and final increments: the initial 4D-Var increments are

sensitive to the norm, the final increments are similar to EnKF

“Smoother” reanalysis

LETKF Analysisxna = xn

f + Xnfwn

aLETKF analysis

at time n

Smoother analysis at time n-1 !xn!1

a = xn!1f + Xn!1

f wna

This very simple smoother allows us to go backand forth in time within an assimilation window:it allows assimilation of future data in reanalysis

Initial and final analysis corrections(colors), with one Bred Vector (contours)

LETKF

4D-Var

Initial increments

Initial increments

Final increments

Final increments

LETKF

4D-Var

EnKF, 4D-Var and ECCO in a “toy” ocean-atmosphere model

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