An iterative ensemble Kalman filter in presence of additive model - - PowerPoint PPT Presentation

▶
an iterative ensemble kalman filter in presence of
SMART_READER_LITE
LIVE PREVIEW

An iterative ensemble Kalman filter in presence of additive model - - PowerPoint PPT Presentation

Apr 30, 2023 394 likes •735 views

An iterative ensemble Kalman filter in presence of additive model error Marc Bocquet 1 , Pavel Sakov 2 , Jean-Matthieu Haussaire 1 (1) CEREA, joint lab Ecole des Ponts ParisTech and EdF R&D, Universit e Paris-Est, France Institut

slide-1
SLIDE 1

An iterative ensemble Kalman filter in presence of additive model error

Marc Bocquet1, Pavel Sakov2, Jean-Matthieu Haussaire1

(1) CEREA, joint lab ´ Ecole des Ponts ParisTech and EdF R&D, Universit´ e Paris-Est, France Institut Pierre-Simon Laplace (2) Environment and Research Division, Bureau of Meteorology, Melbourne, Australia (marc.bocquet@enpc.fr)

  • M. Bocquet

12th EnKF workshop, Os, Norway, 12-14 June 2017 1 / 32

slide-2
SLIDE 2

What this talk is about . . .

◮ Iterative ensemble Kalman smoother (IEnKS): exemplar of nonlinear four-dimensional EnVar methods. ◮ It propagates the error statistics from one cycle to the next with the ensemble (errors of the day). ◮ It performs a 4D-Var analysis at each cycle (within the ensemble subspace). ◮ Typical cycling (L = 6, S = 2):

tL−3 tL−2 yL−3 yL−2 tL−1 tL yL−1 yL tL+1 tL+2 yL+1 yL+2 tL−2 tL tL+2 S∆t S∆t L∆t

Variational analysis in ens. space → Posterior ens. generation → Ens. forecast

  • M. Bocquet

12th EnKF workshop, Os, Norway, 12-14 June 2017 2 / 32

slide-3
SLIDE 3

Cost functions

◮ General cost function over [t1,...,tL]; weak-constraint formalism: JL(x1,...,xL) = x1 −xf

12 (Pf

1)−1 +

L

∑

i=1

yi −Hi(xi)2

R−1

i

+

L

∑

i=2

xi −Mi(xi−1)2

Q−1

i .

◮ Configurations addressed in this talk: ◮ The case L = 1, S = 1, the so-called iterative ensemble Kalman filter → IEnKF: JL(x1,x2) = x1 −xf

12 (Pf

1)−1 +y2 −H2 ◦M2(x1)2

R−1

2 .

◮ The IEnKF but, now, with additive model error → IEnKF-Q : JL(x1,x2) = x1 −xf

12 (Pf

1)−1 +y2 −H2(x2)2

R−1

2 +x2 −M2(x1)2

Q−1

2 .

◮ The linearized case L+1 = S with additive model error, called the asynchronous ensemble Kalman filter → AEnKF.

  • P. Sakov, J.-M. Haussaire, and M. Bocquet, An iterative ensemble Kalman filter in presence of additive model error, Q.
  • J. R. Meteorol. Soc., 0 (2017), pp. 0–0. Submitted
  • M. Bocquet

12th EnKF workshop, Os, Norway, 12-14 June 2017 3 / 32

slide-4
SLIDE 4

The iterative ensemble Kalman filter (IEnKF)

Outline

1

The iterative ensemble Kalman filter (IEnKF)

2

Theory of the IEnKF-Q Formulation Decoupling Base algorithm

3

Numerics for the IEnKF-Q

4

Asynchronous EnKF with additive model error

5

Conclusions

6

References

  • M. Bocquet

12th EnKF workshop, Os, Norway, 12-14 June 2017 4 / 32

slide-5
SLIDE 5

The iterative ensemble Kalman filter (IEnKF)

Iterative ensemble Kalman filter: a Bayesian standpoint

◮ Gaussian assumption for the prior: p(x1) = n(x1|x1,P1). ◮ Forecast under perfect model assumption: p(x2|x1) ∝ δ {x2 −M2(x1)}. ◮ Likelihood used in the analysis: p(y2|x2) = n(y2 −H2(x2)|0,R2). ◮ (Full cycle) analysis of the initial condition x1: p(x1|y2) ∝p(y2|x1)p(x1) ∝p(y2|x2 = M2(x1))p(x1). ◮ Analysis (forecast!) of the filtering distribution: p(x2|y2) =

  • dx1 p(x2|x1,y2)p(x1|y2)

=

  • dx1 δ {x2 −M2(x1)}p(x1|y2).
  • M. Bocquet and P. Sakov, An iterative ensemble Kalman smoother, Q. J. R. Meteorol. Soc., 140 (2014), pp. 1521–1535
  • M. Bocquet

12th EnKF workshop, Os, Norway, 12-14 June 2017 5 / 32

slide-6
SLIDE 6

The iterative ensemble Kalman filter (IEnKF)

Iterative ensemble Kalman filter: a variational standpoint

◮ Analysis IEnKF cost function in state space p(x1|y2) ∝ exp(−J (x1)): J (x1) = 1 2y2 −H2 ◦M2(x1))2

R−1

2 + 1

2x1 −x12

P−1

1 .

◮ Reduced scheme in ensemble subspace, x1 = x1 +A1w1, where A1 is the normalized ensemble anomaly matrix:

  • J (w1) = J (x1 +A1w1).

◮ IEnKF cost function in ensemble space:

  • J (w1) = 1

2y2 −H2 ◦M2 (x1 +A1w1)2

R−1

2 + 1

2w12.

  • P. Sakov, D. S. Oliver, and L. Bertino, An iterative EnKF for strongly nonlinear systems, Mon. Wea. Rev., 140 (2012),
  • pp. 1988–2004
  • M. Bocquet and P. Sakov, Combining inflation-free and iterative ensemble Kalman filters for strongly nonlinear systems,
  • Nonlin. Processes Geophys., 19 (2012), pp. 383–399
  • M. Bocquet

12th EnKF workshop, Os, Norway, 12-14 June 2017 6 / 32

slide-7
SLIDE 7

The iterative ensemble Kalman filter (IEnKF)

Iterative ensemble Kalman filter: minimization scheme

◮ As a variational reduced method, one can use Gauss-Newton [Sakov et al., 2012], Levenberg-Marquardt [Bocquet and Sakov, 2012; Chen and Oliver, 2012], etc, minimization schemes (not limited to quasi-Newton). ◮ Gauss-Newton scheme: w(j+1)

1

= w(j)

1 −

H −1

(j) ∇

J(j)(w(j)

1 ),

x(j)

1 = x1 +A1w(j) 1 ,

∇ J(j) = w(j)

1 −YT (j)R−1 2

  • y2 −H2 ◦M2(x(j)

1 )

  • ,
  • H(j) = IN +YT

(j)R−1 2 Y(j),

Y(j) = [H2 ◦M2]′

|x(j)

1 A1.

  • M. Bocquet

12th EnKF workshop, Os, Norway, 12-14 June 2017 7 / 32

slide-8
SLIDE 8

The iterative ensemble Kalman filter (IEnKF)

Iterative ensemble Kalman filter: computing the sensitivities

◮ Sensitivities Y(p) computed by ensemble propagation without TLM and adjoint ([Gu

and Oliver, 2007; Liu et al., 2008; Buehner et al., 2010])

◮ First alternative [Sakov et al., 2012]: the transform scheme. The ensemble is preconditioned before its propagation using the ensemble transform T(j) =

  • IN +YT

(j)R−1Y(j)

−1/2 ,

  • btained at the previous iteration. The inverse transformation is applied after

propagation. ◮ Second alternative [Bocquet and Sakov, 2012]: the bundle scheme. It simply mimics the action of the tangent linear by finite difference: Y(j) ≈ 1 ε H2 ◦M2

  • x(j)1T +εA1
  • IN − 11T

N

  • .
  • M. Bocquet

12th EnKF workshop, Os, Norway, 12-14 June 2017 8 / 32

slide-9
SLIDE 9

Theory of the IEnKF-Q

Outline

1

The iterative ensemble Kalman filter (IEnKF)

2

Theory of the IEnKF-Q Formulation Decoupling Base algorithm

3

Numerics for the IEnKF-Q

4

Asynchronous EnKF with additive model error

5

Conclusions

6

References

  • M. Bocquet

12th EnKF workshop, Os, Norway, 12-14 June 2017 9 / 32

slide-10
SLIDE 10

Theory of the IEnKF-Q Formulation

IEnKF-Q: formulation

◮ Analysis cost function: J(x1,x2) = x1 −xa

12 (Pa

1)−1 +y2 −H (x2)2

R−1 +x2 −M (x1)2 Q−1.

◮ Ensemble subspace representation: x1 = xa

1 +Aa 1u,

Aa

1(Aa 1)T = Pa 1,

Aa

11 = 0,

x2 = M (x1)+Aq

2v,

Aq

2(Aq 2)T = Q,

Aq

21 = 0.

◮ Cost function in ensemble subspace: J(u,v) = uTu+vTv +y2 −H (x2)2

R−1 .

  • M. Bocquet

12th EnKF workshop, Os, Norway, 12-14 June 2017 10 / 32

slide-11
SLIDE 11

Theory of the IEnKF-Q Formulation

IEnKF-Q: formulation

◮ Compactification: w ≡ u v

  • =

⇒ J(w) = wTw +y2 −H (x2)2

R−1 .

◮ Condition of zero gradient: w −(HA)TR−1 [y2 −H (x2)] = 0, where A ≡ [MAa

1,Aq 2],

H ≡ ∇H (x2), M ≡ ∇M (x1). ◮ The cost function can be minimized using a Gauss-Newton method wi+1 = wi −Di∇J(wi), where the inverse Hessian is approximated as Di ≈

  • I+(HiAi)TR−1HiAi−1

.

  • M. Bocquet

12th EnKF workshop, Os, Norway, 12-14 June 2017 11 / 32

slide-12
SLIDE 12

Theory of the IEnKF-Q Formulation

IEnKF-Q: formulation

◮ Posterior anomalies: δx1 = Aa

1 δu,

δx2 = MAa

1 δu+Aq 2 δv,

◮ Updated perturbations over [t1,t2]: Aa

2(Aa 2)T = E[δx⋆ 2(δx⋆ 2)T] = A⋆E[w⋆(w⋆)T](A⋆)T = A⋆D⋆(A⋆)T,

which implies Aa

2 = A⋆(D⋆)1/2 = A⋆

I+(H⋆A⋆)T(R)−1H⋆A⋆−1/2 . ◮ Updated (smoothed) perturbations at t1: As

1(As 1)T = E[δx⋆ 1(δx⋆ 1)T] = Aa 1E[u⋆(u⋆)T](Aa 1)T,

which implies As

1 = Aa 1 (D⋆ 1:m,1:m)1/2.

  • M. Bocquet

12th EnKF workshop, Os, Norway, 12-14 June 2017 12 / 32

slide-13
SLIDE 13

Theory of the IEnKF-Q Decoupling

IEnKF-Q: Decoupling

◮ In all generality: J(x1,x2) = −2lnp(x1,x2|y2) = −2lnp(x2|x1,y2)p(x1|y2). ◮ If the observation operator H is linear: −2lnp(x1|y2) = x1 −xa

12 (Pa

1)−1 +y2 −H ◦M (x1)2

(R+HQHT)−1 +c1,

and −2lnp(x2|x1,y2) =x2 −M (x1)−QHT(R+HQHT)−1 [y2 −H ◦M (x1)]2

Q−1+HTR−1H

+c2, ◮ Then the MAP of J(x1,x2) can be computed in two steps: ◮ Minimize −2lnp(x1|y2) over x1 just like the IEnKF in the absence of model error but with R → R+HQHT. ◮ The MAP of −2lnp(x2|x⋆

1,y2) is then directly given by:

x⋆

2 = M (x⋆ 1)+QHT

R+HQHT−1 [y2 −H ◦M (x⋆

1)].

  • M. Bocquet

12th EnKF workshop, Os, Norway, 12-14 June 2017 13 / 32

slide-14
SLIDE 14

Theory of the IEnKF-Q Decoupling

IEnKF-Q: Decoupling

◮ This decoupling also implies the decoupling of (u,v): ui+1 −ui =Di

u

  • (HMiAa

1)T(Ri u)−1

×

  • y2 −H ◦M (xa

1 +Aa 1ui)

  • −ui

. v⋆ = D⋆

v(HAq 2)T(R⋆ v)−1 [y2 −H (x⋆ 2)+HM⋆Aa 1u⋆].

◮ However, this decoupling does not convey to the perturbations update! ◮ The same decoupling is used in particle filtering [Doucet et al., 2000] to build the optimal importance proposal particle filter.

  • M. Bocquet

12th EnKF workshop, Os, Norway, 12-14 June 2017 14 / 32

slide-15
SLIDE 15

Theory of the IEnKF-Q Base algorithm

IEnKF-Q: algorithm

1: function [E2] = ienkf cycle(Ea

1, Aq 2, y2, R, M ,H )

2: xa

1 = Ea 1 1/m

3: Aa

1 = (Ea 1 − xa 11T)/√m −1

4: D = I, w = 0 5: repeat 6: x1 = xa

1 +Aa 1w1:m

7: T = (D1:m,1:m)1/2 8: E1 = x11T +Aa

1T√m −1

9: E2 = M (E1) 10: HA2 = H (E2)(I−11T/m)T−1/√m −1 11: HAq

2 = H (E211T/m +Aq 2

mq −1)(I−11T/mq)/mq −1 12: HA = [HA2,HAq

2]

13: x2 = E21/m+Aq

2wm+1:m+mq

14: ∇J = w −(HA)TR−1[y2 −H (x2)] 15: D = [I+(HA)TR−1HA]−1 16: ∆w = −D∇J 17: w = w +∆w 18: until |∆w| < ε 19: A2 = E2 (I−11T/m)T−1 20: A = [A2/√m −1,Aq

2]D1/2

21: A2 = SR(A,m)√m −1 22: E2 = x21T +(1+δ)A2 23: end function

  • M. Bocquet

12th EnKF workshop, Os, Norway, 12-14 June 2017 15 / 32

slide-16
SLIDE 16

Numerics for the IEnKF-Q

Outline

1

The iterative ensemble Kalman filter (IEnKF)

2

Theory of the IEnKF-Q Formulation Decoupling Base algorithm

3

Numerics for the IEnKF-Q

4

Asynchronous EnKF with additive model error

5

Conclusions

6

References

  • M. Bocquet

12th EnKF workshop, Os, Norway, 12-14 June 2017 16 / 32

slide-17
SLIDE 17

Numerics for the IEnKF-Q

IEnKF-Q: numerical experiments

◮ Experiments performed on the Lorenz-95 model. Fully observed: H = I, R = I. We choose mq = 41, so that Q is full rank. ◮ Random mean-preserving rotations of the ensemble anomalies are sometimes applied to the IEnKF-Q, typically in the very weak model error regime. ◮ Comparisons with EnKF + accounting for Q and IEnKF + accounting for Q: ◮ [Rand] Stochastic approach: Af

2 = MAa 1 +Q1/2Ξ,

◮ [Det] Deterministic approach: Af

2 = A

  • I+A†Q(A†)T1/2, with A = MAa

1.

  • E. N. Lorenz and K. A. Emanuel, Optimal sites for supplementary weather observations: simulation with a small model, J.
  • Atmos. Sci., 55 (1998), pp. 399–414
  • P. N. Raanes, A. Carrassi, and L. Bertino, Extending the square root method to account for additive forecast noise in

ensemble methods, Mon. Wea. Rev., 143 (2015), pp. 3857–38730

  • M. Bocquet

12th EnKF workshop, Os, Norway, 12-14 June 2017 17 / 32

slide-18
SLIDE 18

Numerics for the IEnKF-Q

Test 1: nonlinearity

1 2 3 4 5 6 7 8 9 10 11 12

T

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.5 2.0

RMSE EnKF-Det EnKF-Rand IEnKF-Det IEnKF-Rand IEnKF-Q

◮ Q = 0.01TI, m = 20.

  • M. Bocquet

12th EnKF workshop, Os, Norway, 12-14 June 2017 18 / 32

slide-19
SLIDE 19

Numerics for the IEnKF-Q

Test 1: nonlinearity (non-diagonal Q)

1 2 3 4 5 6 7 8 9 10 11 12

T

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.5 2.0

RMSE EnKF-Det EnKF-Rand IEnKF-Det IEnKF-Rand IEnKF-Q

◮ [Q]ij = 0.05T(exp[−d2(i,j)/30])+0.1δij, m = 30

  • M. Bocquet

12th EnKF workshop, Os, Norway, 12-14 June 2017 19 / 32

slide-20
SLIDE 20

Numerics for the IEnKF-Q

Test 2: model noise magnitude

10−4 10−3 10−2 10−1

q

0.17 0.2 0.25 0.3 0.4 0.5 0.6 0.7 0.8 1.0 1.2 1.5

RMSE EnKF-Det EnKF-Rand IEnKF-Det IEnKF-Rand IEnKF-Q IEnKF-Det m = 41 IEnKF-Q m = 41

◮ Q = qTI, T = 1, m = 20.

  • M. Bocquet

12th EnKF workshop, Os, Norway, 12-14 June 2017 20 / 32

slide-21
SLIDE 21

Numerics for the IEnKF-Q

Test 2: model noise magnitude

10−4 10−3 10−2 10−1

q

0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.2 1.4 1.6 1.8 2.0 2.5 3.0

RMSE EnKF-Det EnKF-Rand IEnKF-Det IEnKF-Rand IEnKF-Q IEnKF-Det m = 41 IEnKF-Q m = 41

◮ Q = qTI, T = 10, m = 20.

  • M. Bocquet

12th EnKF workshop, Os, Norway, 12-14 June 2017 21 / 32

slide-22
SLIDE 22

Numerics for the IEnKF-Q

Test 3: ensemble size

15 18 20 25 30 35 40

m

0.3 0.35 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2

RMSE EnKF-Det EnKF-Rand IEnKF-Det IEnKF-Rand IEnKF-Q

◮ Q = 0.01TI, T = 1.

  • M. Bocquet

12th EnKF workshop, Os, Norway, 12-14 June 2017 22 / 32

slide-23
SLIDE 23

Numerics for the IEnKF-Q

Test 3: ensemble size

15 18 20 25 30 35 40

m

0.6 0.7 0.8 0.9 1.0 1.5 2.0 3.0

RMSE EnKF-Det EnKF-Rand IEnKF-Det IEnKF-Rand IEnKF-Q

◮ Q = 0.01TI, T = 10.

  • M. Bocquet

12th EnKF workshop, Os, Norway, 12-14 June 2017 23 / 32

slide-24
SLIDE 24

Asynchronous EnKF with additive model error

Outline

1

The iterative ensemble Kalman filter (IEnKF)

2

Theory of the IEnKF-Q Formulation Decoupling Base algorithm

3

Numerics for the IEnKF-Q

4

Asynchronous EnKF with additive model error

5

Conclusions

6

References

  • M. Bocquet

12th EnKF workshop, Os, Norway, 12-14 June 2017 24 / 32

slide-25
SLIDE 25

Asynchronous EnKF with additive model error

Asynchronous data assimilation for the EnKF

◮ How to simply and efficiently assimilate observations in between two update steps of the EnKF (linear order)?

  • B. R. Hunt, E. J. Kostelich, and I. Szunyogh, Efficient data assimilation for spatiotemporal chaos: A local ensemble

transform Kalman filter, Physica D, 230 (2007), pp. 112–126

  • P. Sakov, G. Evensen, and L. Bertino, Asynchronous data assimilation with the EnKF, Tellus A, 62 (2010), pp. 24–29

◮ How to do so in presence of additive model error (linear order, L → k)? J(x0,...,xk) =x0 −xa

02 (Pa

0)−1 +

k

∑

i=1

yi −Hi(xi)2

R−1

i

+

k

∑

i=1

xi −Mi−1→i(xi−1)2

Q−1

i

.

  • P. Sakov and M. Bocquet, Asynchronous data assimilation with the EnKF in presence of additive model error, Tellus A, 0

(2017), pp. 0–0. in preparation

  • M. Bocquet

12th EnKF workshop, Os, Norway, 12-14 June 2017 25 / 32

slide-26
SLIDE 26

Asynchronous EnKF with additive model error

Asynchronous data assimilation for the EnKF

◮ Ensemble subspace representation, for i = 1,...,k: x0 = xa

0 +Aa 0w0,

Aa

0(Aa 0)T = Pa 0,

Aa

01 = 0

xi = Mi−1→i(xi−1)+Aq

i wi,

Aq

i (Aq i )T = Qi,

Aq

i 1 = 0

◮ Compactification: w ≡ vec(w0,...,wk). ◮ Cost function in ensemble subspace: ˜ J(w) = wTw +y −H (x)2

R−1.

  • M. Bocquet

12th EnKF workshop, Os, Norway, 12-14 June 2017 26 / 32

slide-27
SLIDE 27

Asynchronous EnKF with additive model error

Asynchronous data assimilation for the EnKF

◮ Linearization (Gauss-Newton implied): x = xf +Aw +O(w2), with xf ≡ vec

  • {M0→i(xa

0)}i=0,...,k

  • ,

and A ≡ vec(A0,...,Ak), Ai ≡      [Aa

0,0],

i = 0 [Mi−1→iAi−1,Aq

i ,0],

i = 1,...,k −1, [Mk−1→kAk−1,Aq

k],

i = k.

  • M. Bocquet

12th EnKF workshop, Os, Norway, 12-14 June 2017 27 / 32

slide-28
SLIDE 28

Asynchronous EnKF with additive model error

Asynchronous data assimilation for the EnKF

◮ Cost function expansion: ˜ J(w) = wTw +

  • y −H (xf )−Yw +O(w2)
  • 2

R−1 ,

where Y ≡ vec

  • {HiAi}i=1,...,k
  • .

◮ Linear order analysis (AEnKF): x⋆ = xf +Aw⋆, A⋆ = AT, T = D−1/2U, w⋆ = D−1YTR−1 y −H (xf )

  • ,

D ≡ I+YTR−1Y. ◮ The computation of Ai and Y can also be extrapolated to mild nonlinearity.

  • P. Sakov and M. Bocquet, Asynchronous data assimilation with the EnKF in presence of additive model error, Tellus A, 0

(2017), pp. 0–0. in preparation

  • M. Bocquet

12th EnKF workshop, Os, Norway, 12-14 June 2017 28 / 32

slide-29
SLIDE 29

Conclusions

Outline

1

The iterative ensemble Kalman filter (IEnKF)

2

Theory of the IEnKF-Q Formulation Decoupling Base algorithm

3

Numerics for the IEnKF-Q

4

Asynchronous EnKF with additive model error

5

Conclusions

6

References

  • M. Bocquet

12th EnKF workshop, Os, Norway, 12-14 June 2017 29 / 32

slide-30
SLIDE 30

Conclusions

Conclusions

We have extended the iterative ensemble Kalman filter (IEnKF) to iterative ensemble Kalman filter in presence of model error (IEnKF-Q). It consistently outperforms ad hoc schemes that incorporate model error into the IEnKF with the L95 model, and any other EnKF-based scheme. We have extended the asynchronous ensemble Kalman filter (AEnKF) to the asynchronous ensemble Kalman filter in presence of model error (AEnKF-Q). In practice, one would have to estimate Q on top of these developments. A currently flourishing topic!

  • M. Bocquet

12th EnKF workshop, Os, Norway, 12-14 June 2017 30 / 32

slide-31
SLIDE 31

Conclusions

Final word

Thank you for your attention!

  • M. Bocquet

12th EnKF workshop, Os, Norway, 12-14 June 2017 31 / 32

slide-32
SLIDE 32

References

References

[1]

  • M. Bocquet and P. Sakov, Combining inflation-free and iterative ensemble Kalman filters for strongly nonlinear systems, Nonlin. Processes

Geophys., 19 (2012), pp. 383–399. [2]

  • M. Bocquet and P. Sakov, An iterative ensemble Kalman smoother, Q. J. R. Meteorol. Soc., 140 (2014), pp. 1521–1535.

[3]

  • B. R. Hunt, E. J. Kostelich, and I. Szunyogh, Efficient data assimilation for spatiotemporal chaos: A local ensemble transform Kalman filter,

Physica D, 230 (2007), pp. 112–126. [4]

  • E. N. Lorenz and K. A. Emanuel, Optimal sites for supplementary weather observations: simulation with a small model, J. Atmos. Sci., 55

(1998), pp. 399–414. [5]

  • P. N. Raanes, A. Carrassi, and L. Bertino, Extending the square root method to account for additive forecast noise in ensemble methods,
  • Mon. Wea. Rev., 143 (2015), pp. 3857–38730.

[6]

  • P. Sakov and M. Bocquet, Asynchronous data assimilation with the EnKF in presence of additive model error, Tellus A, 0 (2017), pp. 0–0.

in preparation. [7]

  • P. Sakov, G. Evensen, and L. Bertino, Asynchronous data assimilation with the EnKF, Tellus A, 62 (2010), pp. 24–29.

[8]

  • P. Sakov, J.-M. Haussaire, and M. Bocquet, An iterative ensemble Kalman filter in presence of additive model error, Q. J. R. Meteorol. Soc.,

0 (2017), pp. 0–0. Submitted. [9]

  • P. Sakov, D. S. Oliver, and L. Bertino, An iterative EnKF for strongly nonlinear systems, Mon. Wea. Rev., 140 (2012), pp. 1988–2004.
  • M. Bocquet

12th EnKF workshop, Os, Norway, 12-14 June 2017 32 / 32