Advanced data assimilation methods-EKF and EnKF

Hong Li and Eugenia Kalnay

University of Maryland

17-22 July 2006

Recall the basic formulation in OI

• OI for a Scalar:

Optimal weight to minimize the analysis error is:

• OI for a Vector:

• B is statistically pre-estimated, and constant with time in its practicalimplementations. Is this a good approximation?

Ta= T

b+ w(T

0! T

b) 10 !! w

22

2

of

fw

!!

!

+=

B = !xb•!x

b

T !xb= x

b" x

t

True atmospheret=0 t=6ha

i

f

i M1!= xx

xi

a= x

i

b+W(y

i

o! Hx

i

b)

BWHIP ][ !=a

1][!

+= RHBHBHWTT

OI and Kalman Filter for a scalar

• OI for a scalar:

• Kalman Filter for a scalar:

• Now the background error variance is forecasted using the lineartangent model L and its adjoint LT

Ta= T

b+ w(T

0! T

b)

w =!b

2

!b

2+!

o

2; !

a

2= (1" w)!

b

2

Tb(ti+1) = M (Ta (ti ); !

b

2= (1+ a)!

a

2=

1

1" w!

a

2= const.

Tb(ti+1) = M (Ta (ti )); !

b

2(t) = (L!

a)(L!

a)T; L = dM / dT

Ta= T

b+ w(T

0! T

b)

w =!b

2

!b

2+!

o

2; !

a

2= (1" w)!

b

2

“Errors of the day” computed with the Lorenz 3-variablemodel: compare with rms (constant) error

Not only the amplitude, but also the structure of B is constant in 3D-Var/OI

This is important because analysis increments occur only within thesubspace spanned by B

!z

2= (z

b" z

t)2

=0.08

“Errors of the day” obtained in the reanalysis(figs 5.6.1 and 5.6.2 in the book)

Note that the mean error went down from 10m to 8m because ofimproved observing system, but the “errors of the day” (on a synoptic timescale) are still large.

In 3D-Var/OI not only the amplitude, but also the structure of B is fixedwith time

Flow independent error covariance

In OI(or 3D-Var), the scalar error correlation between two points in thesame horizontal surface is assumed homogeneous and isotropic. (p162 inthe book)

!!!!!

"

#

$$$$$

%

&

=

2

,2,1

,2

2

22,1

,12,1

2

1

...covcov

.........

covcov

cov...cov

nnn

n

n

B

'

'

'

If we observe only Washington, D.C,we can get estimate for Richmond andPhiladelphia corrections through the

error correlation (off-diagonal term in B).

Typical 3D-Var error covariance

In OI(or 3D-Var), the error correlation between two mass points in thesame horizontal surface is assumed homogeneous and isotropic.(p162in the book)

!!!!!

"

#

$$$$$

%

&

=

2

,2,1

,2

2

22,1

,12,1

2

1

...covcov

.........

covcov

cov...cov

nnn

n

n

B

'

'

'

Flow-dependence – a simple example (Miyoshi,2004)

This is not appropriateThis does reflects the flow-dependence.

There is a cold front inour area…

What happens in thiscase?

Extended Kalman Filter (EKF)

• Forecast step

• Analysis step

• Using the flow-dependent , analysis is expected to be improvedsignificantly

However, it is computational expensive. , n*n matrix, n~107

computing equation directly is impossible

Pi

b

xi

b= Mx

i!1

a

xia= xi

b+Ki (yi

o!Hxi

f)

Pi

a

n!n = [I "KiH]P

i

b= [(P

i

b)"1+H

TR

"1H]

"1

Ki= P

i

bH

T[HP

i

bH

T+ R]

!1

iL

* (derive it)

*Pi

b

Pi

b= L

i!1Pi!1aLi!1

T+Q

* *

Ensemble Kalman Filter (EnKF)

Although the dimension of is huge, therank ( ) << n (dominated by the errors ofthe day)

Using ensemble method to estimate

*

Pib !

1

m(xk

f

k=1

m

" # xt )(xkf # xt )T

!"mIdeally

K ensemble members, K<<n

f

iP

Pib !

1

K "1(xk

f

k=1

K

# " x f)(xk

f " x f)T

=1

K "1X

b•X

bT

Problem left: How to update ensemble ?i.e.: How to get for each ensemblemember? Using K times?

f

iPPi

b= L

i!1Pi!1aLi!1

T+Q

a

ix

*

**

“errors of day” are the instabilities ofthe background flow. Stronginstabilities have a few dominantshapes (perturbations lie in a low-dimensional subspace).

It makes sense to assume that largeerrors are in similarly low-dimensionalspaces that can be represented by alow order EnKF.

Physically,

Ensemble Update: two approaches1. Perturbed Observations method:An “ensemble of data assimilations”

It has been proven that an observationalensemble is required (e.g., Burgers et al.1998). Otherwise

is not satisfied.

Random perturbations are added to theobservations to obtain observations foreach independent cycle

However, it introduces a source ofsampling errors when perturbingobservations.

xi

a

(k )= x

i

b

(k )+K

i(y

i

o

(k )! Hx

i

b

(k ))

Pi

a

n!n = [I "KiH]P

i

b

xi

b

(k )= Mx

i!1

a

(k )

Ki= P

i

bH

T[HP

i

bH

T+ R]

!1

Pi

b !1

K "1(x

k

b

k=1

K

# " xb )(xk

b " xb )T

noiseo

ik

o

i+= yy

)(

Ensemble Update: two approaches

2. Ensemble square root filter(EnSRF) Observations are assimilated to

update only the ensemble mean.

Assume analysis ensembleperturbations can be formed bytransforming the forecast ensembleperturbations through a transformmatrix

xi

b= x

i

b+K

i(y

i

o! Hx

i

b)

1

k !1X

aX

aT

= Pi

a

n"n = [I !KiH]P

i

b= [I !K

iH]

1

k !1X

bX

bT

xi

b= Mx

i!1

a

Ki= P

i

bH

T[HP

i

bH

T+ R]

!1

Pi

b !1

K "1(x

k

b

k=1

K

# " xb )(xk

b " xb )T

Xi

a= T

iX

i

b

xi

a= x

i

a+ X

a

xi

a= x

i

b+K

i(y

i

o! Hx

i

b)

Xi

a= T

iX

i

b

Several choices of the transform matrix

EnSRF, Andrews 1968, Whitaker and Hamill, 2002)

EAKF (Anderson 2001)

ETKF (Bishop et al. 2001)

LETKF (Hunt, 2005)Based on ETKF but perform analysissimultaneously in a local volumesurrounding each grid point

X

a= (I ! %KH)X

b, %K = "K

Xa= X

bT

Xa= AX

b

one observation at a time

An Example of the analysis corrections from3D-Var (Kalnay, 2004)

An Example of the analysis corrections fromEnKF (Kalnay, 2004)

Summary of LETKF (Hunt, 2005)Forecast step (done globally): advance the ensemble 6 hours

xn

b(i )= M (x

n!1

a(i )) i = 1,...,k

Analysis step (done locally). Local model dimension m, locally used s obs

X = xb(1)| ... | x

b(k )!" #$ Matrix of ensembles (mxk)

Xb= X ! x

b Matrix of ensemble perturbations (mxk)

Yb= H (X) ! H (x

b) " HX

b Matrix of ens. obs. perturbations (sxk)

Pb= (k !1)

!1X

bTX

b in model space, so %Pb= (k !1)

!1I in ensemble space (kxk)

(Pa)!1= (P

b)!1+H

TR

!1H in model space, so that

( %P

a)!1= (k !1)I + (HX

b)TR

!1(HX

b) in ensemble space (kxk)

Summary of LETKF (cont.)Forecast step (done globally): advance the ensemble 6 hours

xn

b(i )= M (x

n!1

a(i )) i = 1,...,k

Analysis step (done locally). Local model dimension m, locally used obs sX

b= X ! x

bY

b= H (X) ! H (x

b) " HX

b

X

a= X

b( %P

a)1/2

in model space (mxm) Pa= X

aTX

a= X

bT %PaX

b

Ensemble analysis perturbations in model space (mxk)

( %P

a) = (k !1)I + (HXb

)TR

!1(HX

b)"# $%

!1in ensemble space (kxk)

w

n

a= %PaYbTR!1

(yn

o! y

n

b) Analysis increments in ensemble space (kx1)

xn

a= x

n

a+ X

bw

n

a Analysis in model space (mx1)

We finally gather all the analysis and analysis perturbations from each grid pointand construct the new global analysis ensemble (nxk)

Summary steps of LETKF

Ensemble analysis

Observations

GCM

Obs.operator

Ensembleforecast

Ensemble forecastat obs. locations

LETKF

1) Global 6 hr ensemble forecaststarting from the analysis ensemble

2) Choose the observations used foreach grid point3) Compute the matrices of forecastperturbations in ensemble space Xb

4) Compute the matrices of forecastperturbations in observation space Yb

5) Compute Pb in ensemble spacespace and its symmetric square root

6) Compute wa, the k vector ofperturbation weights

7) Compute the local grid pointanalysis and analysis perturbations.

8) Gather the new global analysisensemble.

EnKF vs 4D-Var

EnKF is simple and modelindependent, while 4D-Var requires thedevelopment and maintenance of theadjoint model (model dependent)

4D-Var can assimilate asynchronousobservations, while EnKF assimilateobservations at the synoptic time.

Using the weights wa at any time 4D-LETKF can assimilate asynchronousobservations and move them forwardor backward to the analysis time

Disadvantage of EnKF:

Low dimensionality of the ensemblein EnKF introduces sampling errors inthe estimation of . ‘Covariancelocalization’ can solve this problem.

Pb

Ensemble analysis

Observations

GCM

Obs.operator

Ensembleforecast

Ensemble forecastat obs. locations

LETKF