# Forecast Error Covariance Matrix Inflation In Ensemble Kalman

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```					           Forecast Error Covariance Matrix Inflation
In Ensemble Kalman Filtering Assimilation

Xiaogu Zheng 郑小谷

College of Global Change and Earth System Study
Beijing Normal University, Beijing, China

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Acknowledgements
Kunio Tanabe （田边国仕）
Institute of Statistical Mathematics, Japan

Liang Xiao （梁晓 ）
College of Global Change and Earth System Study, Beijing
Normal University, China

Wu Guocan （吴国灿），Zheng Heng （郑珩）
School of Mathematics, Beijing Normal University, China

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Outlines
1. Ensemble Kalman Filter (EnKF Evensen 1994)
with linear observation
1.2. Inflation on forecast error covariance matrix
(our modification to EnKF)
1.3. Case study

2. EnKF with nonlinear observation
2.2. Inflation on forecast error covariance matrix
2.3. Case study

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Outlines
1. Ensemble Kalman Filter (EnKF Evensen 1994)
with linear observation
1.2. Inflation on forecast error covariance matrix
(our modification to EnKF)
1.3. Case study

2. EnKF with nonlinear observation
2.2. Inflation on forecast error covariance matrix
2.3. Case study

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Nonlinear dynamic and linear observation system

x it = M i − 1 ( x it − 1 ) + η i

y o = H i x it + ε i
i

t
x : true state vector
i
y o : observation vector
i

M i : forecast model                                     Hi : observational matrix

ηi : model error vector                                  ε i : observation error vector

Q i : model error                                        R i : observation error
covariance matrix                                        covariance matrix

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Ensemble Kalman Filter with linear observation
i = 1; x         a
0, j   , j = 1 ,..., m
Major purpose of
x    f
i, j   = M i −1 ( x       a
i −1, j   )
EnKF, See next slice                                                              ~ N (0, R i )
d i , j = y o + ε i , j − H i x if, j
i
T
1        f    m
1                      m
 f         1    m

Pi =      ∑1  x i , j − m
m − 1 j= 
∑x
k =1
f
i ,k    x i , j −
           m
∑x
k =1
f
i ,k   


+ Pi H (H i Pi H + R i ) d i , j
−1
x   a
i, j   =x    f
i, j
T
i
T
i

Yes                                              No
End                           Is i last obs time?                          i=i+1
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Knowing forecast error covariance matrix Pi
we can apply the following filtering assimilation

x     i
f
= M i −1 ( x   a
i −1     )    i=i+1

d i = y o − H i x if
i

x = x + Pi H (H i Pi H + R i ) d i
a        f              T              T       −1
i       i               i              i

Pi is assumed to be the covariance matrix of forecast error

η if ≡ x if − x it
m
1
Use x
a
i   or
m
∑ x ia, j as analysis state
j =1

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EnKF and 3DV (linear observation)
3DV: find state which minimize objective function

J ( x ti ) = ( x ti − x if ) Pi− 1 ( x ti − x if ) T + ( y io − H i x ti ) R i− 1 ( y io − H i x ti ) T

The state which minimize objective function is

x = x + Pi H (H i Pi H + R i ) d i
a        f          T            T           −1
i       i           i            i

The analysis state of EnKF

Success of DA is depend on the estimation of Pi R i

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Outlines
1. Ensemble Kalman Filter (EnKF Evensen 1994)
with linear observation
1.2. Inflation on forecast error covariance matrix
(our modification to EnKF)
1.3. Case study

2. EnKF with nonlinear observation
2.2. Inflation on forecast error covariance matrix
2.3. Case study

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Forecast error covariance inflation                                         Pi f = λ i Pi
d i = y o − H i x if = (y o − H i x it ) + H i (x it − x if
i                 i                                      )

~ N ( 0 , H i λ i Pi H T + R )
i
~ N (0, R i )          ~ N ( 0 , H i λ i Pi H T )
i

-2-log-likelihood of d i

[
L i ( λi ) = ln(det( H i λi Pi H T + R i ) + d T ( H i λi Pi H T + R i ) −1 d i
i             i               i                  ]
λ i is estimated by minimizing Li (λi )

(Dee and da Silva 1999 MWR; Zheng 2009 AAS)

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Calculating             ( H i [λ i ]Pi [λ i ]H T + R i ) −1
i

m
1.     H i Pi H iT =   ∑
k =1
zkzT
k      where

1    f        1        m

zk =           x i, j −
Hi                  ∑x       f
i, j


m             m        j =1          

j
2. Define A j = λi ∑ z k z k + R i It is easy to check
T

k =1

A1 1 = Ri−1 − λi (Ri−1z1z1 Ri−1 ) (1 + λi z1 Ri−1z1 )
−                       T                 T

A−+1 = A−1 − λi (A−1z j+1zT+1A−1 ) (1 + λi zT+1A−1z j )
j
1
j         j       j   j             j   j

−1   −1
therefore ( λi H i Pi H i + R i ) = A m
T

Generalized Sherman-Morrison-Woodbury formula (Sherman and Morrison 1950)

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Ensemble Kalman Filter with forecast error covariance inflation

i = 1; x       a
0, j   , j = 1 ,..., m

1)                                x    f
i, j   = M i −1 ( x      a
i −1, j     )
~ N (0, R i )
d i , j = y o + ε i , j − H i x if, j
i
T
2)           1        f        1m                       m
 f         1    m

Pi =      ∑1  x i , j − m
m − 1 j= 
∑x
k =1
f
i ,k    x i , j −
           m
∑x
k =1
f
i ,k   


3) Estimate inflation factor λ i by minimizing
[
L i ( λi ) = ln(det( H i λi Pi H T + R i ) + d T ( H i λi Pi H T + R i ) −1 d i
i             i               i                                                ]

+ λi Pi H i (H i λi Pi H + R i ) d i , j
−1
4)      x   a
i, j   =x    f
i, j
T
i                                  i=i+1
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Outlines
1. Ensemble Kalman Filter (EnKF Evensen 1994)
with linear observation
1.2. Inflation on forecast error covariance matrix
(our modification to EnKF)
1.3. Case study

2. EnKF with nonlinear observation
2.2. Inflation on forecast error covariance matrix
2.3. Case study

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Test using toy models
Simple, the true state x it can be calculated, so is

∑ (xi − xi )
1 n a      t 2
RMSE =
n i =1

The smaller RMSE, the better scheme

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Case study: Burgers’ equation model
∂u t ( x )           ∂u t ( x )    ∂ 2u t ( x )
+ ut ( x)            =v              ,
∂t                   ∂x             ∂x  2

over spatial domain (-∞,∞) (Zhu and Kamachi 2000)

Solution: substitute differentiation by difference

Time stepΔt = 15minits

Spatial resolution Δt = 5 km:

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True states
v = 1 .0 × 10 4 m 2 s −1 one forward step
u t+ ∆t ( x ) = u t ( x )
        ∂ut(x)     ∂ 2u t ( x ) 
− ∆ t ut( x )        + v              ,
          ∂x          ∂x   2


∂ut(x)           ut(x + ∆ x) − ut(x − ∆ x)
≈
∂x                      2∆x
∂ 2u t ( x )      ut(x + ∆ x) + ut(x − ∆ x)
≈
∂x   2
(∆ x )2

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Velocity profiles at hours 1, 4, 7, 10, 13, 16

-3
x 10
5

4.5

4

3.5

3

2.5

2

1.5

1

0.5

0
-100         -50      0   50   100   150

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Observations and model for assimilation

Model M i used for assimilation

v = 1.4 ×104 m2s−1

Model error SD 0.13m/s

Observations: true states + uncorrelated noise with SD 0.15m/s

at every 3 hours from hour 1and every 10 km

Observations are not available for some times and locations

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Results with the “true” observation error
variance (0.0225, i.e. SD 0.15)
Hour 1   Hour 4   Hour 7   Hour
10
EnKF                  RMSE m/s         0.044    0.108    0.105    0.098
-2-log-likelihood   －59.7    －61.0    －66.4 －58.9
EnKF with forecast            RMSE m/s         0.053    0.074    0.078    0.069
error covariance          -2-log-likelihood   －62.1    －64.2    －71.5 －62.7
inflation
Lamda         2.74     2.85     2.28     0.31

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RMSE of analysis states for EnKF (black) and EnKF
with forecast error covariance inflation (red)

∑ (xi − xi )
1 n a      t 2
RMSE =
n i =1

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Sensitivity to observation error

x = x + λi Pi H (Hi λi Pi H + R i ) d i
a     f           T     T       −1
i    i            i     i

[
Li ( λi ) = ln(det(Hi λi Pi HT + R i ) + dT (Hi λi Pi HT + R i ) −1 di
i            i            i                 ]

Question:
If observation error covariance matrix R
is not accurate, what is happened?

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Results with half of the “true” observation
error variance (0.0225/2)
Hour 1   Hour 4   Hour 7   Hour
10
EnKF                  RMSE m/s         0.041    0.102    0.100    0.094
-2-log-likelihood   －44.5    －44.4    －55.8 －38.1
EnKF with forecast            RMSE m/s         0.081    0.066    0.070    0.064
error covariance          -2-log-likelihood   －56.5    －58.8    －70.3 －53.4
inflation
Lamda         3.60     4.36     3.27     4.85

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Summary on EnKF with linear observation
• The major innovation of EnKF is to estimate a first
guess of forecast error covariance matrix

• The inflation on the first guess forecast error
covariance matrix could significantly improving the
assimilation

• Even observational error statistics R is not accurate,
the proposed scheme may still work.

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Outlines
1. Ensemble Kalman Filter (EnKF Evensen 1994)
with linear observation
1.2. Inflation on forecast error covariance matrix
(our modification to EnKF)
1.3. Case study

2. EnKF with nonlinear observation
2.2. Inflation on forecast error covariance matrix
2.3. Case study

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Nonlinear dynamic and nonlinear observation system

x it = M i − 1 ( x it − 1 ) + η i

y o = H i ( x it ) + ε i
i

t
x : true state vector
i                                              y o : observation vector
i

M i : forecast model                             Hi    : nonlinear observational operator

ηi : model error vector                          ε i : observation error vector

Q i : model error                                R i : observation error covariance
covariance matrix                                matrix

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(I) EnKF with nonlinear observation
i = 1; x       a
0, j   , j = 1 ,..., m

x    f
i, j   = M i −1 ( x      a
i −1, j   )
~ N (0, R i )
di , j = y o + εi , j − H i (xif, j )
i
T
1          f   1  m
f 
m
1         f 
m
Pi H ≡  T
i   ∑  xi, j − m ∑ xi,k  Hi (xi, j ) − m ∑ Hi (xi,k ) 
m − 1 j =1        k =1   
f

k =1         
T
1                m
1         f 
m
1         f 
m
Hi Pi H ≡T
i     ∑  Hi (xi, j ) − m ∑ Hi (xi,k )  Hi (xi, j ) − m ∑ Hi (xi,k ) 
m − 1 j =1 
f

k =1         
f

k =1         

+ Pi H (H i Pi H + R i ) d i , j
−1
x   a
i, j   =x    f
i, j
T
i
T
i                         i=i+1
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EnKF and 3DV (nonlinear observation)
3DV: find analysis state to minimize objective function
(                   )        (
J ( x ti ) = ( x ti − x if ) Pi−1 ( x ti − x if ) T + y io − H i ( x ti ) R i−1 y io − H i ( x ti )   )
T

by using Newton method
If H i ( x it ) ≈ H i ( x if ) + H i ( x if )( x ti − x if )

where H i ( x ) = ∇ x H i ( x )

then x = x + Pi H ( x ) (H i ( x ) Pi H ( x ) + R i ) d i
a          f            T       f              f         T         f           −1
i         i             i      i              i          i        i

minimizes            J ( x ti )

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(II) EnKF with tangent linear observation
x if, j = M i −1 ( x ia−1, j )

d i , j = y o + ε i , j − H i ( x if, j )
i
T
1        f        m
1                      m
 f         1    m

Pi =      ∑1  x i , j − m
m − 1 j= 
∑x         f
i ,k    x i , j −     ∑x         f
i ,k   
k =1                        m   k =1             

+ Pi H ( x ) (H i ( x ) Pi H ( x ) + R i ) d i , j
−1
x   a
i, j   =x    f
i, j
T
i    i
f
i
f            T
i    i
f

Tangent operator of H
i=i+1

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Outlines
1. Ensemble Kalman Filter (EnKF Evensen 1994)
with linear observation
1.2. Inflation on forecast error covariance matrix
(our modification to EnKF)
1.3. Case study

2. EnKF with nonlinear observation
2.2. Inflation on forecast error covariance matrix
2.3. Case study

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Forecast error covariance inflation                                                 Pi f = λ i Pi

i                                   [
d i = y o − H i ( x if ) = (y o − H i ( x it ) ) + H i ( x it ) − H i ( x if )
i                                                                                      ]
i                          [
≈ (y o − H i ( x it ) ) + H i ( x if ) (x it − x if          )]
Tangent operator
~ N (0, R i )          ~ N ( 0 , H i ( x if ) λ i Pi H T ( x if ))
i

-2-log-likelihood of d i                   conditioned on x if

[
L i ( λi ) = ln(det( H i ( x if ) λi Pi H T ( x if ) + R i ) + d T ( H i ( x if ) λi Pi H T ( x if ) + R i ) −1 d i
i                      i                        i                           ]
λ i is estimated by minimizing Li ( λi )

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(III) EnKF with tangent linear observation
and forecast error covariance inflation
1)                                x if, j = M     i −1 ( x ia− 1 , j )

2)                                   d i , j = y o + ε i , j − H i ( x if, j )
i
T
1        f       m
1                         m
 f         1     m

Pi =      ∑1  x i , j − m
m − 1 j= 
∑x      f
i ,k    x i , j −      ∑x      f
i ,k   
k =1                     m    k =1          

3) Estimate inflation factor λ i by minimizing
[
L i ( λi ) = ln(det( H i ( x if ) λi Pi H T ( x if ) + R i ) + d T ( H i ( x if ) λi Pi H T ( x if ) + R i ) −1 d i
i                      i                        i                           ]
+ λi Pi H ( x ) (H i ( x )λi Pi H ( x ) + R i ) d i , j
−1
4) x
a
i, j   =x    f
i, j
T
i    i
f
i
f                 T
i   i
f
i=i+1

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Outlines
1. Ensemble Kalman Filter (EnKF Evensen 1994)
with linear observation
1.2. Inflation on forecast error covariance matrix
(our modification to EnKF)
1.3. Case study

2. EnKF with nonlinear observation
2.2. Inflation on forecast error covariance matrix
2.3. Case study

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Burgers’ equation with nonlinear Observations

Model M i used for assimilation

v = 1.4 ×104 m2s−1

Nonlinear observations:
true states+ true states^3 + uncorrelated noise with SD 0.15m/s
at every 3 hours from hour 1 and every 10 km, i.e.

y = x + (x ) + εi
o
i
t
i
t 3
i

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RMSE of analysis states for Scheme I (black),
scheme II (red) and scheme III (blue)

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Hour 1   Hour 4   Hour 7   Hour 10

Assimilation           RMSE m/s         0.053    0.197    0.226    0.206
scheme (I)
-2-log-likelihood   832.1    40694    4675      2063

RMSE m/s         0.053    0.134    0.233    0.202
Assimilation
scheme (II)         -2-log-likelihood   1176.4   9135     4560      1963

RMSE m/s         0.053    0.108    0.151    0.116
Assimilation
-2-log-likelihood   342.3    4937     2279      1048
scheme (II)
Lamda          9.78    9.81     9.74      8.65

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Summary on EnKF with nonlinear observation
• For nonlinear observation, EnKF may be poorly
performed.

• Using tangent linear observational operator and
inflation on forecast error covariance matrix would
improve the assimilation with nonlinear observation.

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