Finite-size ensemble Kalman filters (EnKF-N) - Iterative ensemble Kalman smoothers (IEnKS)

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Finite-size ensemble Kalman filters (EnKF-N) - Iterative ensemble Kalman smoothers (IEnKS)

Marc Bocquet

To cite this version:

Marc Bocquet. Finite-size ensemble Kalman filters (EnKF-N) - Iterative ensemble Kalman smoothers

(IEnKS). International Conference on Ensemble Methods in Geophysical Sciences, Nov 2012, Toulouse,

France. �hal-00947750�

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Finite-size ensemble Kalman filters (EnKF-N) Iterative ensemble Kalman smoothers (IEnKS)

Marc Bocquet

Universit´e Paris-Est, CEREA, joint lab ´Ecole des Ponts ParisTech and EdF R&D, France INRIA, Paris-Rocquencourt Research center, France

Collaborator: Pavel Sakov, BOM, Australia.

(bocquet@cerea.enpc.fr)

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Outline

1 The primal EnKF-N

2 The dual EnKF-N

3 The iterative ensemble Kalman filter & smoother

4 Conclusions

M. Bocquet Int. Conf. Ens. Meth. Geo. Sc., Toulouse, France, 12-16 November 2012 2 / 29

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Failure of the raw ensemble Kalman filter (EnKF)

◮With the exception of Gaussian and linear systems, EnKF fails to provide a proper estimation on most systems.

◮To properly work, it needs fixes: localisation andinflation.

0 1000 2000

Analysis cycle 0.1

1

RMSE analysis

EnKF without inflation EnKF with inflation λ=1.02 EnKF-N

Lorenz ’95 N=20 ∆t=0.05

◮EnKF relies for its analysis on the first and second-order empirical moments:

x= 1 N

∑

N n=1

x_k, P= 1 N−1

∑

N k=1

(xk−x) (xk−x)^T. (1) Yet,xandPmay not be the true moments of the true filtering distribution (assuming there is one!). Hidden true moments of the true filtering distribution: x_bandB.

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Getting more from the ensemble

◮Idea: even under Gaussian assumptions of the true distribution, the pdf p(x|x₁, . . . ,xN) extracts more information thanp(x|x,P).

◮Using Gaussian assumptions, and being only interested in the filtering problem, one can get (hierarchical reasoning):

p(x|x₁, . . . ,xN) = 1 p(x1, . . . ,x_N)

Z

dx_bdBp(x|x_b,B)p(x1, . . . ,xN|x_b,B)p(xb,B). (2)

◮p(x|x_b,B): the standard Gaussian prior but based on the true statistics.

◮p(x1, . . . ,xN|x_b,B) =∏^Nk=1p(xk|x_b,B)

◮p(xb,B): prior for the background statistics!

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Choosing priors for the background statistics

◮To progress, we need to make assumptions on the background statisticsp(x_b,B): the statistics of the error statisticsorhyperpriors.

A very simple choice is aweakly informative prior: the Jeffreys’ prior[Jeffreys 1961]with an additional assumption of independence forx_b andB:

p(xb,B)≡pJ(xb,B) =pJ(xb)pJ(B) and

p_J(x_b) = 1, p_J(B) =|B|⁻^M+1² .

◮With Jeffreys prior, it is possible to perform the integral (with additional complications due to rank-deficiency usually not dealt with by mathematicians).

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Principle of the EnKF-N

◮The prior of EnKF andthe prior of EnKF-N:

p(x|x,P)∝exp

−1

2(x−x)^TP⁻¹(x−x)

p(x|x₁,x₂, . . . ,x_N)∝(x−x) (x−x)^T+εN(N−1)P⁻

N

2 , (3)

withεN= 1(mean-trusting variant), orεN= 1 +_N¹ (original variant).

◮Ensemble space decomposition (ETKF version of the filters): x=x+Aw.

◮The variational principle of the analysis (in ensemble space):

J(w) =1

2(y−H(x+Aw))^TR⁻¹(y−H(x+Aw)) +N−1 2 w^Tw J(w) =1

2(y−H(x+Aw))^TR⁻¹(y−H(x+Aw)) +N 2ln

εN+w^Tw

. (4)

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EnKF-N: algorithm

1 Requires: The forecast ensemble{x_k}k=1,...,N, the observationsy, and error covariance matrixR

2 Compute the meanxand the anomaliesAfrom{x_k}k=1,...,N.

3 ComputeY=HA,δ=y−Hx

4 Find the minimum:

wa= min

w

n

(δ−Yw)^TR⁻¹(δ−Yw) +Nln

εN+w^Tw o

5 Computex^a=x+Awa.

6 ComputeΩa=

Y^TR⁻¹Y+N(εN+w_a^Twa)IN−2w_aw^T_a (εN+w^T_awa)²

₋1

7 ComputeW^a={(N−1)Ωa}^1/2U

8 Computexâ_k =xâ+A Wâ_k

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The Lorenz ’95 model

◮The toy-model[Lorenz and Emmanuel 1998]:

It represents a mid-latitude zonal circle of the global atmosphere.

M= 40 variables{xm}m=1,...,M. Form= 1, . . . ,M:

dxm

dt = (xm+1−xm−2)xm−1−xm+F, whereF= 8, and the boundary is cyclic.

Chaotic dynamics, topological dimension of 13, a doubling time of about 0.42 time units, and a Kaplan-Yorke dimension of about 27.1.

◮Setup of the experiment: Time-lag between update: ∆t= 0.05 (about 6 hours for a global model), fully observed,R=I.

0 100 200 300 400 500

0 5 10 15 20 25 30 35

-7.5 -5.0 -2.5 0.0 2.5 5.0 7.5 10.0 12.5

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Application to the Lorenz ’95 model

◮EnKF-N: analysis rmse versus ensemble size, for ∆t= 0.05.

5 10 15 20 25 30 35 40 45 50

Ensemble size

0.2 0.3 0.4 0.5 1 2 3 4 5

Average analysis rmse

ETKF (optimal inflation) ETKF-N ε_N=1+1/N ETKF-N ε_Ν=1

rank-deficient regime rank-sufficient regime

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Application to the Lorenz ’95 model

◮Local version: LETKF-N, withN= 10 (beware ∆t= 0.01 requires a correction).

0.05 0.10 0.15 0.20 0.30 0.40 0.60

0.01 0.02 0.03 0.80

Time interval between updates ∆t

0.20 0.30 0.40 0.50 0.60 0.70 0.80

Average analysis rmse

LETKF-N (optimal localisation) LETKF (optimal inflation and localization)

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Forced 2D turbulence model

◮Forced 2D turbulence model

∂q

∂t +J(q,ψ) =λq+ν∆²q+F, q= ∆ψ, (5) whereJ(q,ψ) =∂xq∂yψ−∂yq∂xψ,qis the vorticity 2D field,ψ is the current function 2D field,F is the forcing,λ amplitude of the friction,ν amplitude of the biharmonic diffusion, grid: 64×64 small enough to be in the sufficient-rank regime.

◮Setup of the experiment: Time-lag between update: ∆t= 2, fully observed,R= 0.1I.

0 10 20 30 40 50 60

0

10

20

30

40

50

60

Vorticity q, t =251

-3.2 -2.4 -1.6 -0.8 0.0 0.8 1.6 2.4 3.2

0 10 20 30 40 50 60

0

10

20

30

40

50

60

Vorticity q, t =252

-3.2 -2.4 -1.6 -0.8 0.0 0.8 1.6 2.4 3.2

0 10 20 30 40 50 60

0

10

20

30

40

50

60

Vorticity q, t =253

-3.2 -2.4 -1.6 -0.8 0.0 0.8 1.6 2.4 3.2

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Application to forced 2D turbulence

◮Comparison of: EnKF with uniform inflation, EnKF-N, adaptive inflation EnKF (EnKF-ML),N= 80 (rank-sufficient regime). Starting away or close from the truth.

10 20 40 80 160 320

Cycle 0.01

0.1 1

Vorticity rmse analysis

Reference EnKF λ=1.02 EnKF-ML EnKF-N EnKF λ=1 EnKF λ=1.02 EnKF-ML EnKF-N

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Also tested on . . .

◮EnKF-N also tested on:

Lorenz ’63 model,[Lorenz, 1963]

Kuramato-Sivashinski model,

[Kuramato, 1975; Sivashinski, 1977]

NEDyM economical model,

[Hallegate, Ghil and co-authors, 2008-2012] 0₀ ₅₀ ₁₀₀ ₁₅₀ ₂₀₀ ₂₅₀ 20

40 60 80 100

Credit:http://images.math.cnrs.fr

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Outline

1 The primal EnKF-N

2 The dual EnKF-N

4 Conclusions

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Lagrangian duality

◮TheprimalEnKF-N cost function:

J(w) =1

2(y−H(x+Xw))^TR⁻¹(y−H(x+Xw)) +N 2ln

εN+w^Tw

. (6)

◮Idea: Split the radial degree of freedom ofw, that is√

w^Tw, from its angular degrees of freedom, that isw/√

w^Tw.

◮Lagrangian:

L(w,ρ,ζ) =1

2(δ−Yw)^TR⁻¹(δ−Yw) +1 2ζ

w^Tw−ρ +N

2ln (εN+ρ), (7) whereδ=y−Hx.

◮Saddle point equations:

ζ^⋆=N/(εN+ρ^⋆)

ζ^⋆w^⋆=−Y^TR⁻¹(δ−Yw^⋆) ⇒

( ρ^⋆=N/ζ^⋆−εN

w^⋆= ζ^⋆+Y^TR⁻¹Y₋1

Y^TR⁻¹δ (8)

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Non-convex strong duality

◮Dualcost function defined forζ>0 by D(ζ) = inf

w sup

ρ≥0L(w,ρ,ζ)

= 1

2δ^T

R+Yζ⁻¹Y^T ₋1

δ+εNζ 2 +N

2lnN ζ −N

2. (9)

◮Dual and primal problems:

∆ = inf

ζ>0D(ζ) and Π = inf

wJ(w). (10)

◮Strong dualityresult (non quadratic, non-convex case!):

∆ = Π. (11)

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The dual EnKF-N scheme

1 Requires: The forecast ensemble{x_k}k=1,...,N, the observationsy, and error covariance matrixR

2 Compute the meanxand the anomaliesAfrom{x_k}k=1,...,N.

3 ComputeY=HA,δ=y−Hx

4 Find the minimum:

ζa= min

ζ∈]0,N/εN]

δ^T

R+Yζ⁻¹Y^T₋1

δ+εNζ+NlnN ζ −N

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5 Computewa= Y^TR⁻¹Y+ζa₋1

Y^TR⁻¹δ.

6 Computexa=x+Awa.

7 ComputeΩa=n

Y^TR⁻¹Y+ζa

2εN

N ζa−1o₋1

8 ComputeW^a={(N−1)Ωa}^1/2U

9 Computexâ_k =xâ+AWâ_k

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Outline

1 The primal EnKF-N

2 The dual EnKF-N

4 Conclusions

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Iterative ensemble Kalman filters

◮The iterative extended Kalman filter[Wishner et al., 1969; Jazwinski, 1970]IEKF

◮The iterative extended Kalman smoother[Bell, 1994]IEKS

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Iterative ensemble Kalman filters

✞

✝

☎ Much too costly + needs the TLM and the adjoint−→ensemble methods ✆

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Iterative ensemble Kalman filters

✞

✝

◮The iterative ensemble Kalman filter[Sakov et al., 2012; Bocquet and Sakov, 2012]IEnKF

◮The iterative ensemble Kalman smoother[This talk. . . ] IEnKS

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Iterative ensemble Kalman filters

✞

✝

✞

✝

☎ It’s TLM and adjoint free!✆

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Iterative ensemble Kalman filters

✞

✝

✞

✝

✞

✝

☎ Don’t want to be bothered by inflation tuning?✆

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Iterative ensemble Kalman filters

✞

✝

✞

✝

✞

✝

☎ Don’t want to be bothered by inflation tuning?✆

◮The finite-size iterative ensemble Kalman filter[Bocquet and Sakov, 2012]IEnKF-N

◮The finite-size iterative ensemble Kalman smoother[This talk. . . ] IEnKS-N

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Iterative ensemble Kalman filters

◮A fairly recent idea:

[Gu & Oliver, 2007]: The idea.

[Kalnay & Yang, 2010]: A step in the right direction.

[Sakov, Oliver & Bertino, 2011]: The “pi`ece de r´esistance”

[Bocquet & Sakov, 2012]: Correction of the bundle scheme + ensemble transform form.

◮IEnKF cost function in ensemble space:

Jf(w) = 1

2(y2−H2(M1→2(x1+A₁w)))^TR⁻₂¹(y2−H2(M1→2(x1+A1w))) +1

2(N−1)w^Tw. (13)

◮Gauss-Newton scheme:

w^(p+1)=w^(p)−Hf_(p)⁻¹∇Jf(w^(p))

∇Jf(p)=−Y^T_(p)R⁻₂¹

y₂−H₂M1→2(x+A₁w^(p))

+ (N−1)w^(p),

Hf(p)= (N−1)I_N+Y^T_(p)R⁻₂¹Y_(p), Y_(p)= [H2M2←1A1]^′_(p), (14)

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Iterative ensemble Kalman filters

◮SentivitiesY_(p)computed by ensemble propagation without TLM and adjoint.

◮Finite-size versions of the filter are just defined by substituting the prior:

N−1

2 w^Tw−→ N 2ln

εN+w^Tw

. (15)

◮As a variationalreducedmethod, one can use Gauss-Newton[Sakov et al., 2012], Levenberg-Marquardt[Bocquet and Sakov, 2012; Chen and Oliver, 2012], etc, minimisation schemes (not limited to quasi-Newton).

◮Essentially a lag-one smoother. Does the job of a lag-one 4D-Var, with dynamical error covariance matrix and withoutthe use of the TLM and adjoint! Very efficient in very nonlinear conditions if one can afford the multiple ensemble propagations.

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Finite-size iterative ensemble Kalman filters

◮Setup: Lorenz ’95,M= 40,N= 40, ∆t= 0.05−0.60,R=I.

◮Comparison of EnKF-N, EnKF (optimal inflation), IEnKF-N (bundle and transform), IEnKF (bundle and transform, optimal inflation)

0 0.1 0.2 0.3 0.4 0.5 0.6

Time interval between updates 0.15

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Analysis rmse _EnKF-N

EnKF (opt. infl.) IEnKF (bundle, opt. infl.) IEnKF-N (transform) IEnKF (transform, opt. infl.) IEnKF-N (bundle)

N=40

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Iterative ensemble Kalman smoothers

◮In a mildly nonlinear context (built on linear and Gaussian hypotheses)

Many earlier studies, see[Cosme et al., 2012]for a review, and[Cosme et al., 2010]for an application to oceanography.

◮In a non-sequential but very non-linear context

Many earlier studies, for instance[Evensen and van Leeuwen, 2000]

[Chen & Oliver, 2012]in the context of reservoir modelling

◮Sequential nonlinear context: [This talk]

The IEnKS cost function is just the extension of the IEnKF cost function for a temporal window ofLcycles.

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Finite-size iterative ensemble Kalman smoothers

◮Setup: Lorenz ’95,M= 40,N= 20, ∆t= 0.05,R=I.

◮Comparison of EnKF-N, IEnKS-N, Lin-IEnKS-N, EnKS-N, IEnKS-N with a large inflation (reduced-rank 4D-Var?), withL= 20.

0 2 4 6 8 10 12 14 16 18 20

Lag (number of cycles) 0.10

0.18 0.20

0.16 0.14 0.12 0.22

0.08

Re-analysis rmse

EnKF-N Lin-IEnKS-N EnKSIEnKS-N λ=10 IEnKS-N

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Finite-size iterative ensemble Kalman smoothers

◮Setup: Lorenz ’95,M= 40,N= 20, ∆t= 0.30,R=I.

◮Comparison of EnKF-N, IEnKF-N, IEnKS-N, ETKS-N, withL= 10.

◮Lin-IEnKS-N has (understandably) diverged.

0 1 2 3 4 5 6 7 8 9 10

0.15 0.20 0.25 0.30 0.40 0.50 0.60 0.70 0.90 0.80

Re-analysis rmse

EnKF-N EnKS-N IEnKS-N IEnKF-N

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Finite-size iterative ensemble Kalman smoothers

◮Setup: 2D turbulence, 64×64,N= 40, ∆t= 2,R= 0.1I.

◮Comparison of EnKF-N, IEnKF-N, IEnKS-N, ETKS-N, withL= 20.

0 2 4 6 8 10 12 14 16 18 20

0.004 0.005 0.006 0.007 0.008 0.009 0.010 0.011 0.012 0.013 0.014 0.015 0.016

Vorticity re-analysis rmse

EnKF-N Lin-IEnKS-N EnKS-N IEnKS-N

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Outline

1 The primal EnKF-N

2 The dual EnKF-N

4 Conclusions

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Conclusions

A new prior for the ensemble forecast meant to be used in an EnKF analysis has been built. It takes into accountsampling errors.

It yields a new class of filtersEnKF-N, thatdoes not seem to require inflation supposed to account for sampling errors.

Local variants (both LA and CL) available.

Dual variant EnKF-N is an EnKF with built-inoptimalinflation (accounting for sampling errors).

Almost linear regime more problematic because of Jeffreys’ prior. Another hyperprior is needed.

The iterative ensemble Kalman filter has been generalised to an iterative ensemble Kalman smoother (IEnKF). It is anEn-Varmethod.

It istangent linear and adjoint free. It is, by construction,flow-dependent.

Though based on Gaussian assumptions, it can offer better retrospective analysis than standard Kalman smoothers in midly nonlinear conditions.

When affordable, it beats other Kalman filter/smoothers in strongly non-linear conditions.

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Main references I

Bocquet, M., 2011. Ensemble Kalman filtering without the intrinsic need for inflation. Nonlin.

Processes Geophys. 18, 735–750.

Bocquet, M., Sakov, P., 2012. Combining inflation-free and iterative ensemble kalman filters for strongly nonlinear systems. Nonlin. Processes Geophys. 19, 383–399.

Chen, Y., Oliver, D. S., 2012. Ensemble randomized maximum likelihood method as an iterative ensemble smoother. Math. Geosci. 44, 1–26.

Cosme, E., Brankart, J.-M., Verron, J., Brasseur, P., Krysta, M., 2010. Implementation of a reduced-rank, square-root smoother for ocean data assimilation. Mon. Wea. Rev. 33, 87–100.

Cosme, E., Verron, J., Brasseur, P., Blum, J., Auroux, D., 2012. Smoothing problems in a bayesian framework and their linear gaussian solutions. Mon. Wea. Rev. 140, 683–695.

Evensen, G., van Leeuwen, P. J., 2000. An ensemble Kalman smoother for nonlinear dynamics. Mon.

Wea. Rev. 128, 1852–1867.

Gu, Y., Oliver, D. S., 2007. An iterative ensemble Kalman filter for multiphase fluid flow data assimilation. SPE Journal 12, 438–446.

Sakov, P., Oliver, D., Bertino, L., 2012. An iterative EnKF for strongly nonlinear systems. Mon. Wea.

Rev. 140, 1988–2004.