ensemble strategies for state and parameters estimation in ... · problem statement da techniques...

Problem statement DA techniques Alternative algorithmic formulation Application: 1D ecosystem model Summary

Ensemble Strategies for State and ParametersEstimation in Ocean Ecosystem Models

– Joint, Dual, and OSA-based EnKF schemes –

Workshop on Meteorological Sensitivity Analysis and Data AssimilationRoanoke, West Virginia

M.E. Gharamti⋆†, L. Bertino⋆, B. Ait-El-Fquih‡, A. Samuelsen⋆, I. Hoteit‡

⋆Nansen Environmental and Remote Sensing Center, Bergen, NORWAY‡King Abdullah University of Science and Technology, Thuwal, KSA

†E-mail: [email protected]: http://www.nersc.no/group/ecosystem-modeling-group

June, 2015

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Context: Ocean Ecosystem ModelingCoupled models

• NorESM: Norwegian Earth System Model;coupled atm-land-ice-ocean(MICOM)-biogeochemistry(HAMOCC)

• TOPAZ-ECO: physics(HYCOM,GOTM)-biology(ECOSMO,NORWECOM)

Biological data

• Satellite: surface chlorophyll-a

• In-situ: Nutrients concentrations, pCO2, ..

DA framework and usage

• Combined state-parameters estimation (EnKF)

• Dimension, non-linearties (bloom), complexity

▶ Environmental monitoring – Fisheries

▶ Initialization for climate projections

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Surface silicate Anthropogenic Carbon

CHLa (mg/m3)

Biological data

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CHLa (mg/m3)

Biological data

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CHLa (mg/m3)

Outline of the Talk

Problem statement

Standard DA techniques

Alternative formulation of the state-parameters estimation problem

Application using a 1D ecosystem model

Conclusion

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Challenges and Motivation

Objective ⇒ Need to find p (xk , θk |y0:k) recursively in time using an EnKF

Issues of the joint-filtering problem:

▶ Positive variables (concentration of nutrients, ...)

▶ Poorly known parameters (e.g., grazing efficiency)

▶ Noisy, seasonal (sparse) data extracted from“sub-optimal” locations!

■ Gaussian anamorphosis:Transform both variablesand observations (Bertino et

al., 2003; Simon and Bertino 2009,

2012; Song et al., 2014)

A simple alternative: Truncate any negative variables after the analysis

▷ Possible depletion of the components of the ensemble “pdf deformation”▷ Parameters updated in wrong directions (spring bloom time)

Our Approach: Use simple truncation and propose a different and a more consistentformulation of the state-parameters estimation problem.

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2012; Song et al., 2014)

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2012; Song et al., 2014)

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2012; Song et al., 2014)

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2012; Song et al., 2014)

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State-Parameters Estimation (Standard Techniques)

Joint-EnKF: Classical state-space augmented form p (xk , θk |y0:k) → p (zk |y0:k).; Update both the state and parameters simultaneously:

p (xk−1, θk−1|y0:k−1)≡ p (zk−1|y0:k−1)

F p (xk , θk |y0:k−1)≡ p (zk |y0:k−1)

A p (xk , θk |y0:k)≡ p (zk |y0:k)

• Could yield to significant inconsistency (Wen and Chen, 2006)

• Might be subject to stability and tractability issues (Moradkhani et al., 2005; Wang et al., 2009)

Dual-EnKF: Separate the densities p (xk , θk |y0:k) → p (θk |y0:k) · p (xk |θk , y0:k).; Update the parameters before the state:

p (xk−1, θk−1|y0:k−1) Aθ p (θk |y0:k)

F p (xk |θk , y0:k−1)

p (xk , θk |y0:k)• More expensive: requires 2 forward model integrations!

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p (xk−1, θk−1|y0:k−1)≡ p (zk−1|y0:k−1)

F p (xk , θk |y0:k−1)≡ p (zk |y0:k−1)

A p (xk , θk |y0:k)≡ p (zk |y0:k)

p (xk−1, θk−1|y0:k−1) Aθ p (θk |y0:k)

F p (xk |θk , y0:k−1)

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p (xk−1, θk−1|y0:k−1)≡ p (zk−1|y0:k−1)

F p (xk , θk |y0:k−1)≡ p (zk |y0:k−1)

A p (xk , θk |y0:k)≡ p (zk |y0:k)

p (xk−1, θk−1|y0:k−1) Aθ p (θk |y0:k)

F p (xk |θk , y0:k−1)

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p (xk−1, θk−1|y0:k−1)≡ p (zk−1|y0:k−1)

F p (xk , θk |y0:k−1)≡ p (zk |y0:k−1)

A p (xk , θk |y0:k)≡ p (zk |y0:k)

p (xk−1, θk−1|y0:k−1) Aθ p (θk |y0:k)

F p (xk |θk , y0:k−1)

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p (xk−1, θk−1|y0:k−1)≡ p (zk−1|y0:k−1)

F p (xk , θk |y0:k−1)≡ p (zk |y0:k−1)

A p (xk , θk |y0:k)≡ p (zk |y0:k)

p (xk−1, θk−1|y0:k−1) Aθ p (θk |y0:k)

F p (xk |θk , y0:k−1)

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p (xk−1, θk−1|y0:k−1)≡ p (zk−1|y0:k−1)

F p (xk , θk |y0:k−1)≡ p (zk |y0:k−1)

A p (xk , θk |y0:k)≡ p (zk |y0:k)

p (xk−1, θk−1|y0:k−1) Aθ p (θk |y0:k)

F p (xk |θk , y0:k−1)

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p (xk−1, θk−1|y0:k−1)≡ p (zk−1|y0:k−1)

F p (xk , θk |y0:k−1)≡ p (zk |y0:k−1)

A p (xk , θk |y0:k)≡ p (zk |y0:k)

p (xk−1, θk−1|y0:k−1)

Aθ p (θk |y0:k)

F p (xk |θk , y0:k−1)

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p (xk−1, θk−1|y0:k−1)≡ p (zk−1|y0:k−1)

F p (xk , θk |y0:k−1)≡ p (zk |y0:k−1)

A p (xk , θk |y0:k)≡ p (zk |y0:k)

p (xk−1, θk−1|y0:k−1) Aθ p (θk |y0:k)

F p (xk |θk , y0:k−1)

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p (xk−1, θk−1|y0:k−1)≡ p (zk−1|y0:k−1)

F p (xk , θk |y0:k−1)≡ p (zk |y0:k−1)

A p (xk , θk |y0:k)≡ p (zk |y0:k)

p (xk−1, θk−1|y0:k−1) Aθ p (θk |y0:k)

F p (xk |θk , y0:k−1)

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p (xk−1, θk−1|y0:k−1)≡ p (zk−1|y0:k−1)

F p (xk , θk |y0:k−1)≡ p (zk |y0:k−1)

A p (xk , θk |y0:k)≡ p (zk |y0:k)

p (xk−1, θk−1|y0:k−1) Aθ p (θk |y0:k)

F p (xk |θk , y0:k−1)

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p (xk−1, θk−1|y0:k−1)≡ p (zk−1|y0:k−1)

F p (xk , θk |y0:k−1)≡ p (zk |y0:k−1)

A p (xk , θk |y0:k)≡ p (zk |y0:k)

p (xk−1, θk−1|y0:k−1) Aθ p (θk |y0:k)

F p (xk |θk , y0:k−1)

p (xk , θk |y0:k)

• More expensive: requires 2 forward model integrations!

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p (xk−1, θk−1|y0:k−1)≡ p (zk−1|y0:k−1)

F p (xk , θk |y0:k−1)≡ p (zk |y0:k−1)

A p (xk , θk |y0:k)≡ p (zk |y0:k)

p (xk−1, θk−1|y0:k−1) Aθ p (θk |y0:k)

F p (xk |θk , y0:k−1)

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One-Step-Ahead Smoothing-based Joint-EnKF

p (zk−1|y0:k−1) p (zk |y0:k)?

p(zk |y0:k−1)

Alternative formulation:

▶ The classical path that involves theforecast pdf p(zk |y0:k−1) when movingfrom the analysis pdf p(zk−1|y0:k−1) tothe analysis pdf at the next timep(zk |y0:k) is not unique!

▶ We resort to using the one-step-aheadsmoothing pdf p(zk−1|y0:k).

1. Smoothing Step: p(xk−1, θ|y0:k) is first computed using likelihood p(yk |xk−1, θ):

p(xk−1, θ|y0:k) ∝ p(yk |xk−1, θ)p(xk−1, θ|y0:k−1)

2. Analysis Step: p(xk |y0:k) is computed using posteriori transition p(xk |xk−1, θ, yk):

p(xn|y0:k) =∫

p(xk |xk−1, θ, yk)p(xk−1, θ|y0:k)dxk−1dθ

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p (zk−1|y0:k−1) p (zk |y0:k)?

p(zk |y0:k−1)

p(xn|y0:k) =∫

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p (zk−1|y0:k−1) p (zk |y0:k)?

p(zk |y0:k−1)

p(xn|y0:k) =∫

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p (zk−1|y0:k−1) p (zk |y0:k)?

p(zk |y0:k−1)

p(xn|y0:k) =∫

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Ensemble Implementation: Joint-EnKFOSA

Joint-EnKFOSA: Reversed order of the forecast-update steps. ; Smooth the state andupdate the parameters before performing the forecast:

p (xk−1, θk−1|y0:k−1) Aθ,Sx p (xk−1, θk |y0:k)

F p (xk , θk |y0:k)

1. Smoothing Step

yf ,(m)k = Hk

(Mk−1(x

a,(m)k−1 , θ

(m)|k−1) + u(m)

)+ v(m)

k ; v(m)k ∼ N (0,Rk)

xs,(m)k−1 = xa,(m)

k−1 + Pxak−1

,yfkP−1

(yk − yf ,(m)

)θ(m)|k = θ

(m)|k−1 + Pθ|k−1,y

fkP−1

(yk − yf ,(m)

2. Analysis Step

xa,(m)n = Mk−1

(xs,(m)k−1 , θ

)+ u(m)

k−1 ; u(m)k−1 ∼ N (0,Qk−1)

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p (xk−1, θk−1|y0:k−1)

Aθ,Sx p (xk−1, θk |y0:k)

1. Smoothing Step

yf ,(m)k = Hk

(Mk−1(x

a,(m)k−1 , θ

(m)|k−1) + u(m)

)+ v(m)

k ; v(m)k ∼ N (0,Rk)

xs,(m)k−1 = xa,(m)

k−1 + Pxak−1

,yfkP−1

(yk − yf ,(m)

)θ(m)|k = θ

(m)|k−1 + Pθ|k−1,y

fkP−1

(yk − yf ,(m)

2. Analysis Step

xa,(m)n = Mk−1

(xs,(m)k−1 , θ

)+ u(m)

k−1 ; u(m)k−1 ∼ N (0,Qk−1)

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1. Smoothing Step

yf ,(m)k = Hk

(Mk−1(x

a,(m)k−1 , θ

(m)|k−1) + u(m)

)+ v(m)

k ; v(m)k ∼ N (0,Rk)

xs,(m)k−1 = xa,(m)

k−1 + Pxak−1

,yfkP−1

(yk − yf ,(m)

)θ(m)|k = θ

(m)|k−1 + Pθ|k−1,y

fkP−1

(yk − yf ,(m)

2. Analysis Step

xa,(m)n = Mk−1

(xs,(m)k−1 , θ

)+ u(m)

k−1 ; u(m)k−1 ∼ N (0,Qk−1)

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1. Smoothing Step

yf ,(m)k = Hk

(Mk−1(x

a,(m)k−1 , θ

(m)|k−1) + u(m)

)+ v(m)

k ; v(m)k ∼ N (0,Rk)

xs,(m)k−1 = xa,(m)

k−1 + Pxak−1

,yfkP−1

(yk − yf ,(m)

)θ(m)|k = θ

(m)|k−1 + Pθ|k−1,y

fkP−1

(yk − yf ,(m)

2. Analysis Step

xa,(m)n = Mk−1

(xs,(m)k−1 , θ

)+ u(m)

k−1 ; u(m)k−1 ∼ N (0,Qk−1)

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1. Smoothing Step

yf ,(m)k = Hk

(Mk−1(x

a,(m)k−1 , θ

(m)|k−1) + u(m)

)+ v(m)

k ; v(m)k ∼ N (0,Rk)

xs,(m)k−1 = xa,(m)

k−1 + Pxak−1

,yfkP−1

(yk − yf ,(m)

)θ(m)|k = θ

(m)|k−1 + Pθ|k−1,y

fkP−1

(yk − yf ,(m)

2. Analysis Step

xa,(m)n = Mk−1

(xs,(m)k−1 , θ

)+ u(m)

k−1 ; u(m)k−1 ∼ N (0,Qk−1)

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1. Smoothing Step

yf ,(m)k = Hk

(Mk−1(x

a,(m)k−1 , θ

(m)|k−1) + u(m)

)+ v(m)

k ; v(m)k ∼ N (0,Rk)

xs,(m)k−1 = xa,(m)

k−1 + Pxak−1

,yfkP−1

(yk − yf ,(m)

)θ(m)|k = θ

(m)|k−1 + Pθ|k−1,y

fkP−1

(yk − yf ,(m)

2. Analysis Step

xa,(m)n = Mk−1

(xs,(m)k−1 , θ

)+ u(m)

k−1 ; u(m)k−1 ∼ N (0,Qk−1)

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1. Smoothing Step

yf ,(m)k = Hk

(Mk−1(x

a,(m)k−1 , θ

(m)|k−1) + u(m)

)+ v(m)

k ; v(m)k ∼ N (0,Rk)

xs,(m)k−1 = xa,(m)

k−1 + Pxak−1

,yfkP−1

(yk − yf ,(m)

)θ(m)|k = θ

(m)|k−1 + Pθ|k−1,y

fkP−1

(yk − yf ,(m)

2. Analysis Step

xa,(m)n = Mk−1

(xs,(m)k−1 , θ

)+ u(m)

k−1 ; u(m)k−1 ∼ N (0,Qk−1)

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Computational Complexity

Table: Approximate cost assuming Ny << Nx

Algorithm Time-update Measurement-update Storage

Joint-EnKF NNe (Cx + Cθ) NNeCy + NN2e (Nx + Nθ) 2NNe (Sx + Sθ)

Dual-EnKF NNe (2Cx + Cθ) NNeCy + NN2e (Nx + Nθ) 2NNe (Sx + Sθ)

Joint-EnKFOSA NNe (2Cx + Cθ) NNeCy + NN2e (Nx + Nθ) 2NNe (Sx + Sθ)

xa,(m)k

Dual−EnKF= Mk−1

(xa,(m)k−1 , θ

correction term︷︸︸︷Pxf

k × µ(m)k

xa,(m)k

Joint−EnKFOSA= Mk−1

xa,(m)k−1 +

correction term︷︸︸︷Pxa

k−1,yf

k× ν

(m)k︸︷︷︸

xs,(m)k−1

, θ(m)|k

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Computational Complexity

Table: Approximate cost assuming Ny << Nx

Algorithm Time-update Measurement-update Storage

Joint-EnKF NNe (Cx + Cθ) NNeCy + NN2e (Nx + Nθ) 2NNe (Sx + Sθ)

Dual-EnKF NNe (2Cx + Cθ) NNeCy + NN2e (Nx + Nθ) 2NNe (Sx + Sθ)

Joint-EnKFOSA NNe (2Cx + Cθ) NNeCy + NN2e (Nx + Nθ) 2NNe (Sx + Sθ)

xa,(m)k

Dual−EnKF= Mk−1

(xa,(m)k−1 , θ

correction term︷︸︸︷Pxf

k × µ(m)k

xa,(m)k

Joint−EnKFOSA= Mk−1

xa,(m)k−1 +

correction term︷︸︸︷Pxa

k−1,yf

k× ν

(m)k︸︷︷︸

xs,(m)k−1

, θ(m)|k

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Testing with 1D Ecosystem Model (NPZ)

Experimental setup

• Cycles of phytoplankton blooms in awater column (Eknes and Evensen 2002)

• 4-Years simulation period, 20 layers

• Layer depth: 10m, Time step: 1day

DA framework

• (Stochastic) EnKF, 80 members

• Twin experiments

• State variables: Nutrients (N),Phytoplankton (P), Zooplankton (H)

• Parameters: Metabolic Loss Rate (r),Grazing Efficiency (f ), Loss toCarnivores (g)

Nutrient N

0 1 2 3 4200

Phytoplankton P

0 1 2 3 4200

Zooplankton H

0 1 2 3 4200

Fig: Reference run solution

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System Configuration and ScenariosInitialization

• Reference run is initialized from the output of a spin-up solution (5 years)

• The parameters are log-normally distributed in space around specified originalvalues with 50% error

• The state members are assumed to follow a Gaussian distribution

Observations

• Observe the concentration of N, P, and H every 5 days

• 3 different observation networks: from all layers (20), half (10), and quarter (5)

• Observational error: ϵk ∼ N (0, σ = 0.3× yk)

Assimilation scenarios

• 4-Years assimilation period

• Experiments repeated 20 times for robustness

• Diagnostics (RMS, ...) averaged over the experiments

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State Estimates: Time-evolution RMS

0 1 2 3 40

0.3Nutrients

Joint-EnKF Dual-EnKF Joint-EnKF-OSA

0 1 2 3 4

1.5Phytoplanktons

0 1 2 3 40

1Zooplanktons

Figure: Time-evolution of RMS; observing all

layers.

• RMS errors for the nutrients arecomparable

• Most improvements of theproposed Joint-EnKFOSA aregiven by the estimates ofPhytoplanktons and Zooplanktons

• The standard joint and dualschemes behave poorly during thespring bloom

• Similar behavior is observed whenassimilating half and quarter ofthe observations

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State Estimates: Average RMS

Scenario Joint-EnKF Dual-EnKF EnKF-OSA Imp. JE Imp. DE

All 0.0753 0.0722 0.0614 18% 15%Half 0.0889 0.1011 0.0765 14% 24%Quarter 0.1184 0.1207 0.0893 25% 26%

All 0.0498 0.0517 0.0282 43% 45%Half 0.0578 0.0604 0.0332 43% 45%Quarter 0.0658 0.0643 0.0381 42% 41%

All 0.0299 0.0307 0.0135 55% 56%Half 0.0347 0.0363 0.0162 53% 55%Quarter 0.0378 0.0375 0.0180 52% 52%

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All 0.0753 0.0722 0.0614 18% 15%Half 0.0889 0.1011 0.0765 14% 24%Quarter 0.1184 0.1207 0.0893 25% 26%

All 0.0498 0.0517 0.0282 43% 45%Half 0.0578 0.0604 0.0332 43% 45%Quarter 0.0658 0.0643 0.0381 42% 41%

All 0.0299 0.0307 0.0135 55% 56%Half 0.0347 0.0363 0.0162 53% 55%Quarter 0.0378 0.0375 0.0180 52% 52%

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All 0.0753 0.0722 0.0614 18% 15%Half 0.0889 0.1011 0.0765 14% 24%Quarter 0.1184 0.1207 0.0893 25% 26%

All 0.0498 0.0517 0.0282 43% 45%Half 0.0578 0.0604 0.0332 43% 45%Quarter 0.0658 0.0643 0.0381 42% 41%

All 0.0299 0.0307 0.0135 55% 56%Half 0.0347 0.0363 0.0162 53% 55%Quarter 0.0378 0.0375 0.0180 52% 52%

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State Estimates: Spread

0 1 2 3 40

0.4Nutrients

0 1 2 3 4Avera

0.5Phytoplanktons

Years0 1 2 3 4

0.2Zooplanktons

• The proposed scheme suggestssmaller ensemble spreads; largerconfidence in the resulting estimates

• Unlike the standard schemes, lessover-shooting is observed at thebloom time

• Better maintaining of the ensemblespread over time:

0 1 2 3 4

5Zooplanktons

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State Estimates: Spread

0 1 2 3 40

0.4Nutrients

0 1 2 3 4Avera

0.5Phytoplanktons

Years0 1 2 3 4

0.2Zooplanktons

• The proposed scheme suggestssmaller ensemble spreads; largerconfidence in the resulting estimates

• Unlike the standard schemes, lessover-shooting is observed at thebloom time

• Better maintaining of the ensemblespread over time:

0 1 2 3 4

5Zooplanktons

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Parameter Estimates: Plant metabolic loss (r)

• All layers are observed

• Quick convergence towards the target value

• No significant difference between the schemes

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Years0 1 2 3 4

0.11Metabolic Loss Rate (r)

Joint-EnKF

Dual-EnKF

Joint-EnKF-OSA

Target-Value

Parameter Estimates: Grazing efficiency (f)

• 10 layers are observed

• First bloom: Joint and Dual-EnKFs impose large corrections in opposite direction

• Significant improvement is obtained using the proposed Joint-EnKFOSA scheme

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Years0 1 2 3 4

0.75Grazing Efficiency (f)

Joint-EnKF

Dual-EnKF

Joint-EnKF-OSA

Target-Value

Parameter Estimates: Loss to carnivores (g)

• The Dual-EnKF performs better than the Joint-EnKF

• Bloom times: Joint and Dual-EnKFs impose corrections in different directions

• The proposed Joint-EnKFOSA scheme is the most accurate with quick convergence

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Years0 1 2 3 4

0.12Loss to Carnivors (g)

Joint-EnKF Dual-EnKF Joint-EnKF-OSA Target-Value

Parameter Estimates: All assimilation runs

• 20 runs: The proposed Joint-EnKFOSA scheme is robust and much more accuratethan the other schemes

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Joint-EnKF

0.060.4

Dual-EnKF

0.060.4

Joint-EnKF-OSA

0.060.4

Target Values Initial Values Assimilation

Impact of truncation on the estimation

0 1 2 3 4

uncate

Truncation (Phytoplanktons)

• 5 layers are observed, one assimilation run

• High truncation observed using the Joint and the Dual-EnKFs: Depletion of theherbivores ensemble; experience large correction on parameters in wrong directions

• The proposed scheme shows less truncation thanks to its dynamically moreconsistent updating algorithm

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Concluding Remarks

• Data assimilation in ocean ecosystem models is challenging given its highlynonlinear character and the poorly known parameters

• Standard assimilation techniques might become inconsistent under complexscenarios

• We propose a smoothing-based joint ensemble Kalman filter in which the

measurement and the time update steps are reversed

▷ More accurate state and parameter estimates▷ More robust to assimilation scenarios: less truncation of “unphysical”

ensemble variables

• Currently being employed in the atlantic system assimilating real physical andbiological data

• Future research: work with different ensemble sizes for the state and parameters!

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