1

アンサンブルカルマンフィルターによる大気海洋結合モデルへのデータ同化On-line estimation of observation error covariance for ensemble-based filters

Genta UenoThe Institute of Statistical Mathematics

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Covariance matrix in DA

,1fx x G vt t t tt

y h x wt t tt

State space model

~ ,1

~ 0

~

,

, ,

0,

N xx

N Qvt t

Nwt

Bb

Rt

1 1 1, |1 2 : 1 11: 2 2 2

1 12 1

1

1

TJ x B xy Qx v x x v vT t tT t

tT

y yh x h xRt t t ttt tt

fx x Gt

b

t

b

vt t t

where

Cost function

3

Filtered estimates with different θ

Large Qlargeh 大 )

Large R(large ) Which one should be

chosen?

4

Ensemble approx. of distribution

Ensemble Kalman filter (EnKF),Particle filter(PF)

Non-Gaussian dist.

Ensemble approx./ Particle approx.

xt

xt V jtx jtN |,|

x jt |

V jt |xt

Gaussian dist.

Exactly represented

Kalman filter (KF)

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RtH tV ttH tH tV ttK t 1|

11|

GtQtGtF tV ttF tV tt

x ttF tx tt

1|11|

1|11|

Kalman filter (KF)

V ttH tK tIV tt

x ttH tytK tx ttx tt

1||

1|1||

V ttx ttN 1|,1| V ttx ttN |,| V ttx ttN 1|1,1|1

x tt 1|1 x tt 1| x tt |

V tt 1| V tt 1|1 V tt |

xt 1 xt xt

Kalm

an gain

y t

Simulation

Filtered dist. at t-1 Predicted dist. at t Filtered dist. at

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EnKF and PF

x tt)1(

1|1 x tt

)1(1|

x tt)1(|R

esampling

y t

x tt)1(

1|1 x tt

)1(1|

x tt)1(|

Approx. K

alman gain

y t

x nttytp )(

1||

EnKF PF

xt 1 xt xt

KF

| | 1 | 1n n n n

yx x w xHK tt ttt t t t t t

1

| 1 | 1 | 1V H H V H RK t t t t t t t t t t

Likelihood

Which is the most likely distribution that produces observation yobs ?

Likelihood L() = p(yobs|θ)

In this example, 3 is most likely.

|1

p y |2

p y |3

p y

yobs yobs yobs

, , , |1 2

, , ,| | , | , , | ,1 21 2 1 3 1 2 1

| ,1: 11

L p y y yT

yy yp p p py y y y y y y N TT

p y yt tt

Likelihood of time series

Find θ that maximizes L(θ).In practice, log-likelihood is easy to handle:

log1:

| ,1:

|

g1

1lo

Tp

p yT

y yt t

t

Likelihood of time series

log1

lo

| ,1: 1

| , | ,1

1: 1

g

Tp y y

t t

p y x x y dxt t t

tT

tp

tt

Observation model

Predicted dist.

,H xt t t

N R,f fx

tPt

N ~ 0,w

y H wt t t

N Rt t

Non-Gaussian dist.[due to nonlinear model]

If it were Gaussian,

likelihood

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Estimation of covariance matrix

Minimizing innovation [predicted error]

Bayes estimation

• Naive• Ensemble mean and covariance of state• Adjustment according to cost function• Matcing with innovation covariance

1. With assumption of Gaussian dist. of state

Maximum likelihood

• Ensemble mean of likelihood2. Without assumption of Gaussian dist. of state

This study

Covariance matching Ueno et al., Q. J. R. Met. Soc. (2010)

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Ensemble approx. of likelihood

( )

~ 0,

y h x wt t t

wt

t

N Rt

• Find θ that maximizes the ensemble approx. log-likelihood.

|

log1

log1

log1

log1

dim 1log 2 log

2 21

1log exp |

log1:

| ,1: 1

| , | ,1: 1

1| ,

| 11

1| ,

| 11

2

p yT

y yt t

p y x x y dxt t t t t

N np y x x x dx

t t t t t

Tp

tT

ptT

tT

tT yt Rt

t

y H tt

tN nN n

p y xt t tN n

t t

1 log1 | 11

N n nNyx xR H tt t t t

n

Observation model

Ensemble mean of likelihood of each member xt|t-1

(n)

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Regularization of Rt

NN

nx n

ttH tytRtx nttH tyt

T

tRt

ytl

log1

1|1

1|2

1explog

1log

2

12log

2

dim

12

Sample covariance(singular due to n<<p)

Regularization withGaussian graphical model

12 neighborhood

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Maximum likelihood

T

tN

N

nx n

ttH tytRtx nttH tytRt

ytl1

log1

1|1

1|2

1exploglog

2

12log

2

dim

,,,

yL

xL

h

,

22

2

22

2

exp2

L y

y jyi

Lx

x jxihqijQ

R

0.1, 0.2, 0.5,1, 2, 5,10

4, 8, 20, 40

1, 2, 5,10

1, 2, 5,10, 20, 50,100,200,500

hL

xL

y

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Data and Model

year

longitude

The color shows SSH anomalies.

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Filtered estimates with different θ

Large Qlargeh 大 )

Large R(large ) Which one should be

chosen?

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System noise: magnitude

,,,

,max,

yL

xL

hl

yL

xLhp

l

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System noise: zonal correlation length

,,,

,max,

yL

xL

hl

yL

hx

Lp

l

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System noise: meridional correlation length

,,,

,max,

yL

xL

hl

xL

hy

Lp

l

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Observation noise: magnitude

,,,

,,max

yL

xL

hl

yL

xL

hp

l

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Estimates with MLE

2m, 20deg, 5deg, 20MLE

magnitude = (5.95cm)2, correlation lengths= (2.38, 2.52deg)

Filtered estimate Smoothed estimate

year

longitude

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Summary for the first half

2m, 20 , 5 , 20MLE

• Maximum likelihood estimation can be carried out even for non-Gaussian state distribution with ensemble approximation

• Applicable for ensemble-based filters such as EnKF and PF

• Estimated parameters:

,,,

yL

xL

h

• … Tractable for just four parameters?

Ueno et al., Q. J. R. Met. Soc. (2010)

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Motivation for the second half

• The output of DA (i.e. “analysis”) varies with prescribed parameter θ, where θ = (B, Q1:T, R1:T)

B: covariance matrix of the initial state (i.e. V0|0)Qt: covariance matrix of system noiseRt: covariance matrix of observation noise

• My interest is how to construct optimal θ for a fixed dynamic model• Only four parameters so far …

• We allow more degree of freedom on R1:T

• (dim yt)2/2 elements at maximum

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Likelihood of Rt

Current assumption , , ,1: 1 2

R R R RT T

, ,1 :

12

,1

R RT t

tR

t

TR

Log-likelihood

Rt

1: 1R

t

,1: 1t t

R R

dim 1log 2 log

2 2

1 1lo

whe

g ex

re1:

p | 1 | 121

log

yt Rt

N n ny yh x h xRt ttt tt t t t

n

Rt t

N

　

1:R

T

and are fixed1:

B QT

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Estimation design

• Use ℓt(R1:t) for estimating Rt only• It is of course that R1:t-1 are parameters of ℓt(R1:t)• But they are assumed to have been estimated with former log-likelihood,

ℓ1(R1), …, ℓt-1(R1:t-1) , and to be fixed at current time step t.

• Rt is estimated at each time step t.

Bad news:• The estimated Rt may vary significantly between different time steps.• A time-constant R cannot be estimated within the present framework.

Rt

1: 1R

t

,1: 1t t

R R

dim 1log 2 log

2 2

1 1log exp | 1 | 12

1:

1

log

yt Rt

N n ny yh x h xRt ttt tt t t t

n

R

N

t t

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Experiment

case 1: 20 (control)

case 3 : , ,1

case 4

case

:

2 :t

t t

R rt m

R

R

R

diag r

It t

•　 Assumed structure of Rt

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Data and Model

year

longitude

The color shows SSH anomalies.

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Estimate of Rt (Temporal mean)

20Rt R

t t , ,

1diag rR r

t m R I

t t

varcov

•Case t similar output for •Case diagonal: large variance near equator, small variance for off-equator•Case tuniform variance with intermediate value

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Estimate of Rt (Spatial mean)

20Rt R

t t , ,

1diag rR r

t m R I

t t

var

• Case t: small variance for first half, large for second half• Case diagonal: large variance around 1998• Case t: similar for the diagonal case

1992- year -2002

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Filtered estimates20R

t R

t t , ,

1diag rR r

t m R I

t t

•Case t: false positive anomalies in the east

•Case t: negative anomalies in the east, but the equatorial Kelvin waves unclear •Case diagonal: negative anomalies and equatorial Kelvin reproduced

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Iteration times

• Only 2-4 times• Small number of parameters requires large iteration numbers

Rt t

, ,1

diag rR rt m

R It t

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Summary of the second half

• An on-line and iterative algorithm for estimating observation error covariance matrix Rt.• The optimality condition of Rt leads a condition of Rt in a closed form.•Application to a coupled atmosphere-ocean model•Only 4-5 iterations are necessary•A diagonal matrix with independent elements produces more likely estimatesthan those of scalar multiplication of fixed matrices ( or I).