Post on 06-Jan-2016
description
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
2
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)
5
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
6
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
10
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)
11
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)
12
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
13
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
14
Data and Model
year
longitude
The color shows SSH anomalies.
15
Filtered estimates with different θ
Large Qlargeh 大 )
Large R(large ) Which one should be
chosen?
16
System noise: magnitude
,,,
,max,
yL
xL
hl
yL
xLhp
l
17
System noise: zonal correlation length
,,,
,max,
yL
xL
hl
yL
hx
Lp
l
18
System noise: meridional correlation length
,,,
,max,
yL
xL
hl
xL
hy
Lp
l
19
Observation noise: magnitude
,,,
,,max
yL
xL
hl
yL
xL
hp
l
20
Estimates with MLE
2m, 20deg, 5deg, 20MLE
magnitude = (5.95cm)2, correlation lengths= (2.38, 2.52deg)
Filtered estimate Smoothed estimate
year
longitude
21
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)
22
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
23
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
24
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
25
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
26
Data and Model
year
longitude
The color shows SSH anomalies.
27
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
28
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
29
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
30
Iteration times
• Only 2-4 times• Small number of parameters requires large iteration numbers
Rt t
, ,1
diag rR rt m
R It t
31
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).