Accoun&ng for mul&-scale ver&cal error correla&on - - PowerPoint PPT Presentation

Accoun&ng for mul&-scale ver&cal error correla&on within ETKF through eigen-spectral covariance localiza&on

Daisuke Hotta1 and Craig H. Bishop2

1 MRI/JMA 2 Univ. Melbourne

7th International Symposium on Data Assimilation 2019/01/21 Kobe, Japan

1

Ba Backg kground: Variou

• us covariance loc
• caliza4on

4on schemes used in EnK EnKF (o (or EnV EnVar)

R-localiza)on:

• Inflate R for distant obs

in place of directly localizing B

• Pros: computationally

efficient à currently mainstream for LETKF Obs-space B-localization:

• Localize B but in obs-

space rather than in model-space:

K=(ρ1BHT){ρ2(H BHT) +R}-1

• Pros: easy to implement
• n serial DA

2

Model-space B-localization:

• Localize B directly in

model-space: Bloc= ρ B

• Pros: straighIorward

treatment of non-local

• bs
• Cons: can be expensive

à recently overcome by modulated ensemble approach (Bishop, Whitaker

and Lei, 2017, MWR)

• Cons: not evident how to treat non-local obs

(whose H depends on mulSple grid points)

Model-space B-localiza/on by modulated ensemble + Gain-form ETKF (Bishop, Whitaker and Lei 2017, MWR)

• Given noisy raw ensemble covariance,

we can construct a good filtered covariance w/o explicitly storing B

• provided that

1) true B has a localized structure

(in the sense that correlation between distant grid points are small), and

2) you know how to construct the appropriate localization matrix F (in practice, this is subject to manual tuning that depends the ensemble size K)

Questions: What if

• true B is not well-localized, or
• appropriate localization matrix

F is not evident?

3

From Bishop, Whitaker and Lei (2017)

• Goals of this study: To formulate a localization

scheme that is

• applicable even when the true B is not well-

localized

• less sensitive to tuning (or does not require it at all)

5

Wha What t is is lo localiz aliza,on? n? What do we exactly y do by y localiza,on?

• Common interpretation (as I understand…):
• 1. Error correlation between distant locations should be

weak (ß which is our a priori knowledge)

• 2. à if sampled correlation is large between distant

locations, that is a sign of sampling error, so that we should damp it. (ß which is just one way of imposing

• ur a priori knowledge)
• Extended interpretation:
• Impose our a priori knowledge about true B to constrain

the structure that a sampled B can take

• à Physical location (or distance) is not essential

6

Wha What t is is lo localiz aliza,on? n? What do we exactly y do by y localiza,on?

• Common interpreta,on (as I understand…):
• 1. Error correla,on between distant loca,ons should be

weak (ß which is our a priori knowledge)

• 2. à if sampled correla,on is large between distant

loca,ons, that is a sign of sampling error, so that we should damp it.(ß which is just one way of imposing

• ur a priori knowledge)
• Extended interpreta,on:
• Localiza,on imposes our a priori knowledge about true

B to constrain the structure that a sampled B can take

• à Physical loca,on (or distance) is not essen,al

7

Propo pose sed d metho hod: d: eigen-spe spectral localization

Assumed situa+on:

• Btrue stochas*cally changes day-to-day, Btrue is not well-localized
• We have an ensemble X of samples drawn from N(0,Btrue) and want to

es*mate Btrue

• Btrue is unknown, but we know its climatological mean Bclim.
• Each instance of Btrue may be dissimilar to Bclim, but we know that their

structures are similar. Key idea:

• Eigen-decomposi*on of Btrue : Btrue=VtrueΛtrueVtrueT
• allows us to represent B (or state vector x) in mode space:

B8true= VtrueTBtrueVtrue, x̂=VtrueTx

• In the mode space, true covariance is diagonal: B8true=Λtrue

àIgnoring the off-diagonal elements of B8ens = VtrueTBensVtrue should effec*vely reduce sampling noises

• But Btrue and hence Vtrue are unknown, so use Bclim=VΛclimVT in stead to

construct the mode space.

Inspired from precondi*oning in the ver*cal direc*on used in VAR schemes

8

• Eigenspectral localiza0on: procedure

1. xiN(0,Btrue), i=1,…,K are at our disposal. X=[x1,x2,…,xK] 2. Project X onto (truncated) mode space : X$=VTX=[VTx1, VTx2,…, VTxK] 3. Form the covariance in the mode space, but retain only the diagonal elements: B$ensdiag=diagm(B$ens) = IB$ens where B$ens = X$ X$T/(K-1) 4. Transform back to physical space by Bensloc=V B$ensdiag VT

• In implementa0on…
• Retaining all modes of Bclim=VΛclimVT is not feasible (nor necessary) so

truncate it by retaining only the leading O(10) modes (20 in the experiments shown later).

• Do not hold B$ensdiag explicitly; instead, represent it with ensemble

modula0on: B$ensdiag=diagm(B$ens) =I B$ens= (I IT)(X$ X$T) /(K-1)=(I▵X$)(I▵X$)T

9

Propo pose sed d metho hod: d: eigen-spe spectral localiza5on n (con5nued)

✱ No need to explicitly specify localization matrix à No tuning required for different #ens

Id Idea ealized ed experi erimen ent: Problem em setup

• Dimension size n=100.
• Shape of Btrue characterized by two scales L1

and L2.

• L1 and L2: fixed in each experiment.
• Each instance of Btrue is generated with

parameters b*L1 and b*L2

• where b is a stochasCc parameter uniformly

distributed over [0.2, 3.0].

• Model-space and eigen-spectral localizaCon

schemes both truncated with by retaining

• nly ntrunc=20 leading modes.

10

Defined by Eq.(3) of Bishop, Whitaker and Lei (2017, MWR; referred to as BWL17 hereafter) b*L1 b * L

2

Diagonal dominance of B/true =VclimTBtrueVclim

• Prerequisite for the success of eigen-spectral

localization is that the true covariance expressed in the climatological eigen-mode space VclimTBtrueVclim is close to diagonal.

• à Check if this prerequisite is met by plotting

VclimTBtrueVclim for different sets of (L1,L2) and for different b

11

Diagonal dominance of B/true = VTBtrueV: Well-localized cases

VTBtrueV is

• almost diagonal
• dominated by smaller

number of leading modes

12

20 40 60 80 100 20 40 60 80 100

Ptrue (L1,L2)=(1.0,=5.0) b=0.2

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 20 40 60 80 100 20 40 60 80 100

Pclim

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 20 40 60 80 100 20 40 60 80 100

Ptrue in eig- spc of Pclim

1 2 3 4 5 6 7 8 9 10 5 10 15 20 5 10 15 20

Ptrue in eig- spc of Pclim (leading modes enlarged)

1 2 3 4 5 6 7 8 9 10

VTBtrueV Btrue Bclim

model space mode space

Diagonal dominance of B/true = VTBtrueV: Well-localized cases

13

20 40 60 80 100 20 40 60 80 100

Ptrue (L1,L2)=(1.0,=5.0) b=0.8

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 20 40 60 80 100 20 40 60 80 100

Pclim

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 20 40 60 80 100 20 40 60 80 100

Ptrue in eig- spc of Pclim

1 2 3 4 5 6 7 8 9 10 5 10 15 20 5 10 15 20

Ptrue in eig- spc of Pclim (leading modes enlarged)

1 2 3 4 5 6 7 8 9 10

VTBtrueV Btrue Bclim

VTBtrueV is

• almost diagonal
• dominated by smaller

number of leading modes

model space mode space

Diagonal dominance of B/true = VTBtrueV: Well-localized cases

14

20 40 60 80 100 20 40 60 80 100

Ptrue (L1,L2)=(1.0,=5.0) b=1.3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 20 40 60 80 100 20 40 60 80 100

Pclim

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 20 40 60 80 100 20 40 60 80 100

Ptrue in eig- spc of Pclim

1 2 3 4 5 6 7 8 9 10 5 10 15 20 5 10 15 20

Ptrue in eig- spc of Pclim (leading modes enlarged)

1 2 3 4 5 6 7 8 9 10

VTBtrueV Btrue Bclim

VTBtrueV is

• almost diagonal
• dominated by smaller

number of leading modes

model space mode space

Diagonal dominance of B/true = VTBtrueV: Well-localized cases

15

20 40 60 80 100 20 40 60 80 100

Ptrue (L1,L2)=(1.0,=5.0) b=1.9

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 20 40 60 80 100 20 40 60 80 100

Pclim

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 20 40 60 80 100 20 40 60 80 100

Ptrue in eig- spc of Pclim

1 2 3 4 5 6 7 8 9 10 5 10 15 20 5 10 15 20

Ptrue in eig- spc of Pclim (leading modes enlarged)

1 2 3 4 5 6 7 8 9 10

VTBtrueV Btrue Bclim

VTBtrueV is

• almost diagonal
• dominated by smaller

number of leading modes

model space mode space

Diagonal dominance of B/true = VTBtrueV: Well-localized cases

16

20 40 60 80 100 20 40 60 80 100

Ptrue (L1,L2)=(1.0,=5.0) b=2.4

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 20 40 60 80 100 20 40 60 80 100

Pclim

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 20 40 60 80 100 20 40 60 80 100

Ptrue in eig- spc of Pclim

1 2 3 4 5 6 7 8 9 10 5 10 15 20 5 10 15 20

Ptrue in eig- spc of Pclim (leading modes enlarged)

1 2 3 4 5 6 7 8 9 10

VTBtrueV Btrue Bclim

VTBtrueV is

• almost diagonal
• dominated by smaller

number of leading modes

model space mode space

Diagonal dominance of B/true = VTBtrueV: Well-localized cases

17

20 40 60 80 100 20 40 60 80 100

Ptrue (L1,L2)=(1.0,=5.0) b=3.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 20 40 60 80 100 20 40 60 80 100

Pclim

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 20 40 60 80 100 20 40 60 80 100

Ptrue in eig- spc of Pclim

1 2 3 4 5 6 7 8 9 10 5 10 15 20 5 10 15 20

Ptrue in eig- spc of Pclim (leading modes enlarged)

1 2 3 4 5 6 7 8 9 10

VTBtrueV Btrue Bclim

VTBtrueV is

• almost diagonal
• dominated by smaller

number of leading modes

model space mode space

Diagonal dominance of B/true = VTBtrueV: Less-localized cases

18

20 40 60 80 100 20 40 60 80 100

Ptrue (L1,L2)=(1.0,=40.0) b=0.8

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 20 40 60 80 100 20 40 60 80 100

Pclim

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 20 40 60 80 100 20 40 60 80 100

Ptrue in eig- spc of Pclim

1 2 3 4 5 6 7 8 9 10 5 10 15 20 5 10 15 20

Ptrue in eig- spc of Pclim (leading modes enlarged)

1 2 3 4 5 6 7 8 9 10

VTBtrueV Btrue Bclim

• Similarly VTBtrueV stays

close to diagonal.

• Variance concentrated

to very small number of leading modes

model space mode space

Diagonal dominance of B/true = VTBtrueV: Less-localized cases

19

20 40 60 80 100 20 40 60 80 100

Ptrue (L1,L2)=(1.0,=40.0) b=1.3

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 20 40 60 80 100 20 40 60 80 100

Pclim

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 20 40 60 80 100 20 40 60 80 100

Ptrue in eig- spc of Pclim

1 2 3 4 5 6 7 8 9 10 5 10 15 20 5 10 15 20

Ptrue in eig- spc of Pclim (leading modes enlarged)

1 2 3 4 5 6 7 8 9 10

VTBtrueV Btrue Bclim

• Similarly VTBtrueV stays

close to diagonal.

• Variance concentrated

to very small number of leading modes

model space mode space

Diagonal dominance of B/true = VTBtrueV: Less-localized cases

• Similarly VTBtrueV stays

close to diagonal.

• Variance concentrated

to very small number of leading modes

20

20 40 60 80 100 20 40 60 80 100

Ptrue (L1,L2)=(1.0,=40.0) b=1.9

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 20 40 60 80 100 20 40 60 80 100

Pclim

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 20 40 60 80 100 20 40 60 80 100

Ptrue in eig- spc of Pclim

1 2 3 4 5 6 7 8 9 10 5 10 15 20 5 10 15 20

Ptrue in eig- spc of Pclim (leading modes enlarged)

1 2 3 4 5 6 7 8 9 10

VTBtrueV Btrue Bclim

model space mode space

Diagonal dominance of B/true = VTBtrueV: Less-localized cases

21

20 40 60 80 100 20 40 60 80 100

Ptrue (L1,L2)=(1.0,=40.0) b=2.4

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 20 40 60 80 100 20 40 60 80 100

Pclim

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 20 40 60 80 100 20 40 60 80 100

Ptrue in eig- spc of Pclim

1 2 3 4 5 6 7 8 9 10 5 10 15 20 5 10 15 20

Ptrue in eig- spc of Pclim (leading modes enlarged)

1 2 3 4 5 6 7 8 9 10

VTBtrueV Btrue Bclim

• Similarly VTBtrueV stays

close to diagonal.

• Variance concentrated

to very small number of leading modes

model space mode space

Diagonal dominance of B/true = VTBtrueV: Less-localized cases

22

20 40 60 80 100 20 40 60 80 100

Ptrue (L1,L2)=(1.0,=40.0) b=3.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 20 40 60 80 100 20 40 60 80 100

Pclim

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 20 40 60 80 100 20 40 60 80 100

Ptrue in eig- spc of Pclim

1 2 3 4 5 6 7 8 9 10 5 10 15 20 5 10 15 20

Ptrue in eig- spc of Pclim (leading modes enlarged)

1 2 3 4 5 6 7 8 9 10

VTBtrueV Btrue Bclim

• Similarly VTBtrueV stays

close to diagonal.

• Variance concentrated

to very small number of leading modes

model space mode space

• The prerequisite (mode-space covarinace VTBtrueV is

close to diagonal) does hold for a wide range of parameters (L1,L2) and b.

• à Eigenspectral localiza@on likely to work well for a

wide range of parameters.

23

Instances of Bensloc (with comparison to FBens of BWL17) #ens=20, ntrunc=20, L1=1, L2=5 (wel ell-lo localiz alized cas ase) LocalizaDon matrix F constructed by eigen-truncaDng PBWL17(3*L1, 3*L2)

• When Btrue is

localized, distance- based model-space localiza6on appears to be7er control off- diagonal noises.

24

20 40 60 80 100 20 40 60 80 100

Ptrue

0.25 0.50 0.75 1.00 1.25 1.50 0.25 0.50 0.75 1.00 1.25 1.50 20 40 60 80 100 20 40 60 80 100

Pens rmse=14.154 corr= 0.664

0.25 0.50 0.75 1.00 1.25 1.50 0.25 0.50 0.75 1.00 1.25 1.50 20 40 60 80 100 20 40 60 80 100

Ploc_dist rmse= 4.817 corr= 0.903

0.25 0.50 0.75 1.00 1.25 1.50 0.25 0.50 0.75 1.00 1.25 1.50 20 40 60 80 100 20 40 60 80 100

Ploc_trunc rmse= 7.309 corr= 0.927

0.25 0.50 0.75 1.00 1.25 1.50 0.25 0.50 0.75 1.00 1.25 1.50

Btrue raw Bens FBens

V diagm(VT BensV)VT

model-space localization

eigenspectral localiza6on Draw 20 samples

Instances of Bensloc (with comparison to FBens of BWL17) #ens=20, ntrunc=20, L1=1, L2=40 (less ess-lo localiz alized cas ase) LocalizaDon matrix F constructed by eigen-truncaDng PBWL17(3*L1, 3*L2)

• In contrast, when

Btrue is not well- localized, eigenspectral localization appears to better control noises.

25

20 40 60 80 100 20 40 60 80 100

Ptrue

0.25 0.50 0.75 1.00 1.25 1.50 0.25 0.50 0.75 1.00 1.25 1.50 20 40 60 80 100 20 40 60 80 100

Pens rmse=21.174 corr= 0.577

0.25 0.50 0.75 1.00 1.25 1.50 0.25 0.50 0.75 1.00 1.25 1.50 20 40 60 80 100 20 40 60 80 100

Ploc_dist rmse=25.745 corr= 0.793

0.25 0.50 0.75 1.00 1.25 1.50 0.25 0.50 0.75 1.00 1.25 1.50 20 40 60 80 100 20 40 60 80 100

Ploc_trunc rmse=18.562 corr= 0.912

0.25 0.50 0.75 1.00 1.25 1.50 0.25 0.50 0.75 1.00 1.25 1.50

Btrue raw Bens FBens

V diagm(VT BensV)VT

Draw 20 samples

model-space localiza@on

eigenspectral localization

Comparison to distance-based model-space localization (statistical scores, mean over 100 trials) (w (well-lo localiz alized cas ase)

• Correla'on score: cor(vec(B),vec(Btrue))
• RMSE score: Frobenius norm between

es'mated B and Btrue.

• BWL17’s model-space localiza'on with

localiza'on matrix F constructed using different length scale, truncated with 20 leading modes.

26

0.5 0.5 1.0 1.0 1.5 1.5 2.0 2.0 2.0 2.5 2.5 3.0 3.0 [spec.loc] [spec.loc] 0.5 0.6 0.7 0.8 0.9 1.0

Correlation score (L1,L2)=(1,5)

localization scale parameter

#ens 10 15 20 25 30 35 40 45 50 0.5 0.5 1.0 1.0 1.5 1.5 2.0 2.0 2.0 2.5 2.5 3.0 3.0 [spec.loc] [spec.loc] 5 10 15

RMSE score (L1,L2)=(1,5)

localization scale parameter

#ens 10 15 20 25 30 35 40 45 50

BWL17’s model-space localization with different localization scales

Proposed eigenspectral localiza=on

When Btrue is localized,

• Eigenspectral localiza'on tend

tp be beNer than best-tuned model-space localiza'on,

• especially when ensemble size

is small.

20 40 60 80 100 20 40 60 80 100

Ptrue

0.25 0.50 0.75 1.00 1.25 1.50 0.25 0.50 0.75 1.00 1.25 1.50

Typical Btrue for L1=1, L2=5

Comparison to distance-based model-space localization (statistical scores, mean over 100 trials) (l (less-lo localiz alized cas ase)

27

0.5 0.5 1.0 1.0 1.5 1.5 2.0 2.0 2.0 2.5 2.5 3.0 3.0 [spec.loc] [spec.loc] 0.5 0.6 0.7 0.8 0.9 1.0

Correlation score (L1,L2)=(1,40)

localization scale parameter

#ens 10 15 20 25 30 35 40 45 50 0.5 0.5 1.0 1.0 1.5 1.5 2.0 2.0 2.0 2.5 2.5 3.0 3.0 [spec.loc] [spec.loc] 10 20 30

RMSE score (L1,L2)=(1,40)

localization scale parameter

#ens 10 15 20 25 30 35 40 45 50

When Btrue is not well-localized,

• Eigenspectral localiza9on

seem to be even more advantageous

20 40 60 80 100 20 40 60 80 100

Ptrue

0.25 0.50 0.75 1.00 1.25 1.50 0.25 0.50 0.75 1.00 1.25 1.50

Typical Btrue for L1=1, L2=40

BWL17’s model-space localization with different localization scales

Proposed eigenspectral localiza<on

• Correla9on score: cor(vec(B),vec(Btrue))
• RMSE score: Frobenius norm between

es9mated B and Btrue.

• BWL17’s model-space localiza9on with

localiza9on matrix F constructed using different length scale, truncated with 20 leading modes.

Summary

• BWL17’s model-space B-localization is extended to eigen-spectral B-

localization, in hope of better accounting for not-well-localized covariance structure in the vertical direction.

• Inspired by preconditioning practiced in VAR systems, vertical

correlation is expressed in the eigen-mode space of climatological B.

• The resultant eigen-mode-space covariance is close to diagonal, so that

simply neglecting the off-diagonals serves as a good way to control sampling noises.

• Advantage: free from any tuning
• From idealized benchmark problems, the proposed scheme appears to

work as good as best-tuned model-space distance-based B-localization.

28

so far, so good !

Last but not least…

(perhaps a huge) limitation (which I did not realize until yesterday…):

• rank(V diagm(VT BensV) VT) = min{rank(V),rank(Bens)}

= #(retained modes)

• à The proposed scheme may not help to mitigate

the rank deficiency issue. Thank you!

29

Back up slides

30

Idealized experiment: precise problem setup

• Dimension of state vector is n=100.
• Two parameters L1 and L2 characterize the correlation length-scale of Btrue.
• In each experiment L1 and L2 are fixed to predetermined values.
• Btrue is a stochastic matrix parametrized by a single stochastic parameter b ~ µ(b)= U[0.2, 3.0]

(uniform over [0.2, 3.0]).

• Btrue = PBWL17(b*L1, b*L2) : Eq.(3) of Bishop, Whitaker and Lei (2017, MWR; referred to as BWL17

hereafter)

• Bclim = ∫ PBWL17(b*L1, b*L2) dµ(b).
• K-member ensemble X is generated as (Btrue)1/2 randn(n,K) where randn(n,K) is an n x K matrix

each element of which independently follows N(0,1).

• Btrue is estimated by the proposed eigen-spectral localization or by the distance-based

localization as in BWL17, both with mode truncation at ntrunc=20.

• For BWL17 localization, localization matrix is taken as PBWL17(a*L1, a*L2) where the parameter a

is tuned to give best results.

31

Comparison to distance-based model-space localiza3on (sta3s3cal scores, mean over 100 trials)

32

0.5 0.5 1.0 1.0 1.5 1.5 2.0 2.0 2.0 2.5 2.5 3.0 3.0 [spec.loc] [spec.loc] 0.5 0.6 0.7 0.8 0.9 1.0

Correlation score (L1,L2)=(10,40)

localization scale parameter

#ens 10 15 20 25 30 35 40 45 50 0.5 0.5 1.0 1.0 1.5 1.5 2.0 2.0 2.0 2.5 2.5 3.0 3.0 [spec.loc] [spec.loc] 10 20 30 40

RMSE score (L1,L2)=(10,40)

localization scale parameter

#ens 10 15 20 25 30 35 40 45 50

When Btrue is very broad, again

• For any ensemble size,

Eigenspectral localiza<on can be be=er than best-tuned model-space localiza<on.

20 40 60 80 100 20 40 60 80 100

Ptrue

0.25 0.50 0.75 1.00 1.25 1.50 0.25 0.50 0.75 1.00 1.25 1.50

Typical Btrue for L1=10, L2=40

BWL17’s model-space localiza4on with different localiza4on scales

Proposed eigenspectral localization

• Correlation score: cor(vec(B),vec(Btrue))
• RMSE score: Frobenius norm between

estimated B and Btrue.

• BWL17’s model-space localization with

localization matrix F constructed using different length scale, truncated with 20 leading modes.