Advanced data assimilation methods-EKF and EnKF

Alghero, Lecture 4

Hong Li and Eugenia Kalnay

University of Maryland

Optimal Interpolation for a scalar

Ta= T

b+ w(T

0! T

b)

w =!b

2

!b

2+!

o

2

Tb= T

t+ !

b

Information available in OI

background or forecast !b= 0; !

b!b= "

b

2

To= T

t+ !

oNew observations !

o= 0; !

o!o= "

o

2

analysis=forecast+optimal weight * observational increment:

Optimal weight

!a

2= (!

b

"2+!

o

"2)"1= (1" w)!

b

2 Analysis error

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)

w =!b

2

!b

2+!

o

2

B = !xb•!x

b

T!x

b= 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 (T

a(ti); assume !

b

2= (1+ a)!

a

2=

1

1" w!

a

2= const.

Tb(ti+1) = M (T

a(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 the subspacespanned by B

!z

2= (z

b" z

t)2

=0.08

“Errors of the day” obtained in the NCEP 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

'

'

'

Background ~106-8 d.o.f.

Suppose we have a 6hr forecast (background) and new observationsSuppose we have a 6hr forecast (background) and new observations

The 3D-Var Analysis doesn’t know about the errors of the day

Observations ~105-7 d.o.f.

BR

Background ~106-8 d.o.f.

Errors of the day: they lieon a low-dim attractor

With Ensemble Kalman Filtering we get perturbations pointingWith Ensemble Kalman Filtering we get perturbations pointingto the directions of the to the directions of the ““errors of the dayerrors of the day””

3D-Var Analysis: doesn’t know about the errors of the day

Observations ~105-7 d.o.f.

Background ~106-8 d.o.f.

Errors of the day: they lieon a low-dim attractor

Ensemble Kalman Filter Analysis:correction computed in the low dimensemble space

Ensemble Kalman Filtering is efficient because Ensemble Kalman Filtering is efficient because matrix operations are performed in the low-dimensional matrix operations are performed in the low-dimensional

space of the ensemble perturbationsspace of the ensemble perturbations

3D-Var Analysis: doesn’t know about the errors of the day

Observations ~105-7 d.o.f.

Background ~106-8 d.o.f.

Errors of the day: they lieon the low-dim attractor

Observations ~105-7 d.o.f.

After the EnKF computes the analysis and the analysis error covarianceAfter the EnKF computes the analysis and the analysis error covarianceAA, the new ensemble initial perturbations are computed:, the new ensemble initial perturbations are computed:1

1

k

T

i i

i

! !+

=

=" a a A

i!a

These perturbations represent the analysis error covariance and areused as initial perturbations for thenext ensemble forecast

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 hugely 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 "K

iH]P

i

b= [(P

i

b)"1+H

TR

"1H]

"1

Ki= P

i

bH

T[HP

i

bH

T+ R]

!1

iL

*Pi

b

Pi

b= L

i!1Pi!1

aLi!1

T+Q

* *

!i

b= L

i"1!i"1

a+ !

i"1

m

*

Ensemble Kalman Filter (EnKF)

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

Using ensemble method to estimate

*

Pib!1

m(xk

f

k=1

m

" # xt)(xk

f# x

t)T

!"mIdeally

K ensemble members, K<<n

f

iP

Pib!

1

K "1(xk

f

k=1

K

# " xf)(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?

f

iPPi

b= L

i!1Pi!1

aLi!1

T+Q

a

ix

*

“errors of day” are the instabilities ofthe background flow. Strong instabilitieshave a few dominant shapes(perturbations lie in a low-dimensionalsubspace).

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, perturbing observationsintroduces a source of sampling errors(Whitaker and Hamill, 2002).

xi

a

(k )= x

i

b

(k )+K

i(y

i

o

(k )! Hx

i

b

(k ))

Pi

a

n!n= [I "K

iH]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)(x

k

b" x

b)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

a= x

i

b+K

i(y

i

o! Hx

i

b)

1

k !1X

aX

aT

= Pi

a

n"n= [I !K

iH]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)(x

k

b" x

b)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 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. Go to 1

LETKF algorithm summary, Hunt 2005Forecast step (done globally): advance the ensemble 6 hours, global model size n

xn

b(i )= M (x

n!1

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

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

Xb= X ! x

bY

b= H (X) ! H (x

b) " HX

b

in model space (mxm)

Ensemble analysis perturbations in model space (mxk)

in ensemble space (kxk)

Analysis increments in ensemble space (kx1)

xn

a= x

n

a+ X

bw

n

a Analysis (mean) in model space (mx1)

We finally gather all the analysis and analysis perturbations from each grid pointand construct the new global analysis ensemble (nxk) and go to next forecast step

xn

a= x

n

a+ X

a Analysis ensemble in model space (mxk)

(mxk) (sxk)

!Pa= (k !1)I + HX

b( )T

R!1HX

b"#

$%

!1

babTaaTaXPXXXP

~==

2/1)~( abaPXX =

)(~ 1 b

n

o

n

bTaa

nyyRYPw !=

!

References posted: www.atmos.umd.edu/~ekalnayGoogle: “chaos weather umd”, publications

Ott, Hunt, Szunyogh, Zimin, Kostelich, Corazza, Kalnay, Patil, Yorke, 2004: LocalEnsemble Kalman Filtering, Tellus, 56A,415–428.

Kalnay, 2003: Atmospheric modeling, data assimilation and predictability,Cambridge University Press, 341 pp. (3rd printing)

Hunt, Kalnay, Kostelich, Ott, Szunyogh, Patil, Yorke, Zimin, 2004: Four-dimensional ensemble Kalman filtering. Tellus 56A, 273–277.

Szunyogh, Kostelich, Gyarmati, Hunt, Ott, Zimin, Kalnay, Patil, Yorke, 2005:Assessing a local ensemble Kalman filter: Perfect model experiments with theNCEP global model. Tellus, 57A. 528-545.

Hunt, Kostelich and Szunyogh 2007: Efficient Data Assimilation forSpatiotemporal Chaos: a Local Ensemble Transform Kalman Filter.Physica D.

Whitaker and Hamill, 2002: Ensemble Data Assimilation Without PerturbedObservations. Monthly Weather Review, 130, 1913-1924.

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 forward orbackward to the analysis time

Disadvantage of EnKF:

Low dimensionality of the ensemble inEnKF introduces sampling errors in theestimation of . ‘Covariancelocalization’ can solve this problem.

Pb

Ensemble analysis

Observations

GCM

Obs.operator

Ensembleforecast

Ensemble forecastat obs. locations

LETKF