Ensemble Kalman Filtering with One-Step-Ahead (OSA) smoothing: - - PowerPoint PPT Presentation

Ensemble Kalman Filtering with One-Step-Ahead (OSA) smoothing: Application to State-Parameter Estimation and 1-Way Coupled Models

Naila Raboudi, Boujemaa Ait-El-Fquih, Ibrahim Hoteit King Abdullah University of Science and Technology Earth Science & Engineering Applied Mathematics and Computational Sciences

January, 2019

Context and Motivation

Consider a discrete-time dynamical system xn = Mn−1 (xn−1) + ηn−1; ηn−1 ∼ N(0, Qn−1) yn = Hnxn + εn; εn ∼ N(0, Rn) ,

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Context and Motivation

Consider a discrete-time dynamical system xn = Mn−1 (xn−1) + ηn−1; ηn−1 ∼ N(0, Qn−1) yn = Hnxn + εn; εn ∼ N(0, Rn) , Ensemble Kalman Filters (EnKFs)

Robust performance Reasonable computational cost Non-intrusive formulation Small ensembles in large scale applications Poorly known model error statistics Nonlinear dynamics ⇒ Limit the representativeness of EnKFs background covariances.

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Context and Motivation

Some auxiliary techniques

– Inflation (Anderson 2001) – Localization (Houtekamer and Mitchell 1998) – Hybrid formulation (Hamill and Snyder 2000) – Adaptive formulation (Song et al. 2010)

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Context and Motivation

Some auxiliary techniques

– Inflation (Anderson 2001) – Localization (Houtekamer and Mitchell 1998) – Hybrid formulation (Hamill and Snyder 2000) – Adaptive formulation (Song et al. 2010)

Our approach: Improve the background through a more efficient use of the data: Follow the One-Step-Ahead (OSA) smoothing formulation of the Bayesian filtering problem: → OSA adds a smoothing step with the future observation, within a Bayesian framework, to compute an ”improved” background

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Bayesian formulation of the OSA smoothing algorithm

The standard filtering path, which involves the forecast pdf when moving from the analysis pdf at n − 1 to the analysis pdf at the next time n, is not unique

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Bayesian formulation of the OSA smoothing algorithm

The standard filtering path, which involves the forecast pdf when moving from the analysis pdf at n − 1 to the analysis pdf at the next time n, is not unique Standard path p(xn−1|y0:n−1)

Forecast

− − − − → p(xn|y0:n−1)

Analysis

− − − − → p(xn|y0:n) OSA smoothing path p(xn−1|y0:n−1)

Smoothing

− − − − − − → p(xn−1|y0:n)

Analysis

− − − − → p(xn|y0:n)

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Bayesian formulation of the OSA smoothing algorithm

The standard filtering path, which involves the forecast pdf when moving from the analysis pdf at n − 1 to the analysis pdf at the next time n, is not unique Standard path p(xn−1|y0:n−1)

Forecast

− − − − → p(xn|y0:n−1)

Analysis

− − − − → p(xn|y0:n)

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Bayesian formulation of the OSA smoothing algorithm

The standard filtering path, which involves the forecast pdf when moving from the analysis pdf at n − 1 to the analysis pdf at the next time n, is not unique Standard path p(xn−1|y0:n−1)

Forecast

− − − − → p(xn|y0:n−1)

Analysis

− − − − → p(xn|y0:n) Corresponding KF algorithm xa

n−1

xf

n

xa

n

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Bayesian formulation of the OSA smoothing algorithm

The standard filtering path, which involves the forecast pdf when moving from the analysis pdf at n − 1 to the analysis pdf at the next time n, is not unique Standard path p(xn−1|y0:n−1)

Forecast

− − − − → p(xn|y0:n−1)

Analysis

− − − − → p(xn|y0:n) Corresponding KF algorithm xa

n−1

xf

n

xa

n

Forecastt(M

n − 1

)

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Bayesian formulation of the OSA smoothing algorithm

The standard filtering path, which involves the forecast pdf when moving from the analysis pdf at n − 1 to the analysis pdf at the next time n, is not unique Standard path p(xn−1|y0:n−1)

Forecast

− − − − → p(xn|y0:n−1)

Analysis

− − − − → p(xn|y0:n) Corresponding KF algorithm xa

n−1

xf

n

xa

n

Forecastt(M

n − 1

) Analysist (yn)

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Bayesian formulation of the OSA smoothing algorithm

The standard filtering path, which involves the forecast pdf when moving from the analysis pdf at n − 1 to the analysis pdf at the next time n, is not unique OSA smoothing path p(xn−1|y0:n−1)

Smoothing

− − − − − − → p(xn−1|y0:n)

Analysis

− − − − → p(xn|y0:n)

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Bayesian formulation of the OSA smoothing algorithm

The standard filtering path, which involves the forecast pdf when moving from the analysis pdf at n − 1 to the analysis pdf at the next time n, is not unique OSA smoothing path p(xn−1|y0:n−1)

Smoothing

− − − − − − → p(xn−1|y0:n)

Analysis

− − − − → p(xn|y0:n) Corresponding KF-OSA algorithm xa

n−1

xf1

n

xs

n−1

xf2

n

xa

n

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Bayesian formulation of the OSA smoothing algorithm

The standard filtering path, which involves the forecast pdf when moving from the analysis pdf at n − 1 to the analysis pdf at the next time n, is not unique OSA smoothing path p(xn−1|y0:n−1)

Smoothing

− − − − − − → p(xn−1|y0:n)

Analysis

− − − − → p(xn|y0:n) Corresponding KF-OSA algorithm xa

n−1

xf1

n

xs

n−1

xf2

n

xa

n

Forecastt(M

n − 1

)

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Bayesian formulation of the OSA smoothing algorithm

The standard filtering path, which involves the forecast pdf when moving from the analysis pdf at n − 1 to the analysis pdf at the next time n, is not unique OSA smoothing path p(xn−1|y0:n−1)

Smoothing

− − − − − − → p(xn−1|y0:n)

Analysis

− − − − → p(xn|y0:n) Corresponding KF-OSA algorithm xa

n−1

xf1

n

xs

n−1

xf2

n

xa

n

Forecastt(M

n − 1

) Smoothing (y

n

)

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Bayesian formulation of the OSA smoothing algorithm

The standard filtering path, which involves the forecast pdf when moving from the analysis pdf at n − 1 to the analysis pdf at the next time n, is not unique OSA smoothing path p(xn−1|y0:n−1)

Smoothing

− − − − − − → p(xn−1|y0:n)

Analysis

− − − − → p(xn|y0:n) Corresponding KF-OSA algorithm xa

n−1

xf1

n

xs

n−1

xf2

n

xa

n

Forecastt(M

n − 1

) Reforecastt(M

n − 1

) Smoothing (y

n

)

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Bayesian formulation of the OSA smoothing algorithm

The standard filtering path, which involves the forecast pdf when moving from the analysis pdf at n − 1 to the analysis pdf at the next time n, is not unique OSA smoothing path p(xn−1|y0:n−1)

Smoothing

− − − − − − → p(xn−1|y0:n)

Analysis

− − − − → p(xn|y0:n) Corresponding KF-OSA algorithm xa

n−1

xf1

n

xs

n−1

xf2

n

xa

n

Forecastt(M

n − 1

) Reforecastt(M

n − 1

) Analysis (yn) Smoothing (y

n

)

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KF Vs KF-OSA algorithms

KF and KF-OSA use different paths to compute same analysis/forecast KF applies 1 update and 1 forecast step while KF-OSA applies 2 ”update” and 2 ”forecast” steps KF-OSA uses the observation twice within a consistent Bayesian framework (for the RIP of Kalnay and Yang (2010))

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Motivation behind EnKF-OSA

Why an EnKF-OSA would outperform a classical EnKF ?

• Conditions the ensemble sampling with future information
• Provides an improved background which should help mitigating for the

sub-optimal character of EnKFs

• This should be particularly expected when the filter is not implemented

under ideal conditions Stochastic EnKF-OSA update equations Smoothing:

xs,i

n−1

=

xa,i

n−1+P xa n−1,yf1 n

P−1

yf1 n

• yi

n−Hnxf1,i n

• Analysis:

xa,i

n

=

xf2,i

n

+Ka

n(yi n−Hnxf2,i n

) Ka

n=Qn−1HT n (HnQn−1HT n +Rn)−1=P xf2 n ,yf2 n

P−1

yf2 n 6 ISDA 2019 6 / 21

Analysis step of EnKF-OSA

Analysis step of EnKF-OSA

– Should be related to the sampling step in the particle filter (PF) with

• ptimal proposal density (Doucet et al., 2001; Desbouvries et al., 2011)

Deriving a deterministic EnKF-OSA

– We derived SEIK-OSA by assuming uncorrelated pseudo-forecast and

• bservational errors

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Numerical experiments with Lorenz-96

Governing equations (L-96) dxi dt = (xi+1 − xi−2) xi−1 − xi + F Experimental setup – Twin experiments – 5-years simulation period Compare EnKF, EnKF-OSA, SEIK and SEIK-OSA 3 different observational scenarios: all (40), half (20), and quarter (10) of the variables Data are assimilated every 4 model steps (1 day) Inflation and local analysis are used

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Numerical experiments with Lorenz-96

1 1.1 1.2 1.3

EnKF, All Obs

Min = 0.44

Inflation

EnKF−OSA, All Obs

Min = 0.39

SEIK, All Obs

Min = 0.40

SEIK−OSA, All Obs

Min = 0.35

0.35 0.4 0.6 1

1 1.1 1.2 1.3

EnKF, Half Obs

Min = 0.79

Inflation

EnKF−OSA, Half Obs

Min = 0.69

SEIK, Half Obs

Min = 0.72

SEIK−OSA, Half Obs

Min = 0.62

0.6 1 2

10 20 30 40 1 1.1 1.2 1.3

EnKF, Quarter Obs

Min = 1.26

Inflation Localization

10 20 30 40

EnKF−OSA, Quarter Obs

Min = 1.07

Localization

10 20 30 40

SEIK, Quarter Obs

Min = 1.18

Localization

10 20 30 40

SEIK−OSA, Quarter Obs

Min = 0.99

Localization

1 1.5 2 3

Figure: Time-averaged RMSE as a function of the localization radius (x axis) and inflation factor (y axis) (20 members, DA every 4 model steps)

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Numerical experiments with Lorenz-96

10 20 40 80

Ensemble size

2 4 6 8 10 12 14 All Obs 10 20 40 80

Ensemble size

2 4 6 8 10 12 14 16 18 Half Obs

10 20 40 80 Ensemble size 5 10 15 20 25 Quarter Obs SEIK (Ne) /SEIK-OSA (Ne) SEIK (2Ne)/SEIK-OSA (Ne)

Figure: Percentages of relative improvement, in terms of RMSE, resulting from SEIK-OSA compared to SEIK using the same and half the ensemble size

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Numerical experiments with Lorenz-96

10 20 40 80

Ensemble size

2 4 6 8 10 12 14 All Obs 10 20 40 80

Ensemble size

2 4 6 8 10 12 14 16 18 Half Obs

10 20 40 80 Ensemble size 5 10 15 20 25 Quarter Obs SEIK (Ne) /SEIK-OSA (Ne) SEIK (2Ne)/SEIK-OSA (Ne)

Figure: Percentages of relative improvement, in terms of RMSE, resulting from SEIK-OSA compared to SEIK using the same and half the ensemble size Benefit of OSA is more pronounced when – less data are assimilated – filter implemented with small ensembles, neglected model error EnKF-OSA outperformed EnKF even with half the ensemble size (i.e., similar computational cost)

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Storm Surge Forecasting with ADCIRC and EnKF-OSA

Joint project with Clint Dawson (UT-AUstin) ADCIRC: ADvanced CIRCulation model EnKF-OSA is tested with a realistic setting of ADCIRC configured for storm surge forecasting in the Gulf of Mexico during Hurricane Ike (2008) Pseudo-observations of sea surface levels from a network of buoys are assimilated Combine OSA with hybrid formulation for efficient implementation with small ensembles Compare 4 EnKFs: ETKF, ETKFHyb, ETKF-OSA and ETKFHyb-OSA Filters are tested under the same computational cost

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Storm surge forecasting using ETKF-OSA

25 50 100 200 500 1000 1 1.05 1.1 1.15 1.2 1.25

inflation factor

ETKF (Ne = 20) Min = 0.89

25 50 100 200 500 1000 1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45

ETKF-OSA (Ne = 10) Min = 0.68

0.5 1 1.5 25 50 100 200 500 1000

LA radius

1 1.05 1.1 1.15 1.2 1.25

inflation factor

ETKFHyb (Ne = 20) Min = 0.72

25 50 100 200 500 1000

LA radius

1 1.05 1.1 1.15 1.2 1.25 1.3 1.35

ETKFHyb-OSA (Ne = 10) Min = 0.65

0.5 1 1.5

Figure: Coastal-averaged RMSEs [m] of maximum water elevation forecast errors for different LA radii and inflation factors

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Main conclusions: OSA for ensemble state estimation

OSA exploits the future observation, which provides improved background EnKF-OSAs outperformed the standard EnKFs for comparable computational costs The improvements are particularly pronounced when the filter is implemented under challenging conditions The hybrid formulation enables a more efficient implementation of the OSA formulation The smoothing window should not be too large so that the linear (correlation-based) updates remain relevant, iterations may help

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State-parameter estimation with OSA

{xa,(i)

n−1(θ(i) |n−1)} Ne i=1

{xf1,(i)

n

}

Ne i=1

{xs,(i)

n−1(θ(i) |n )} Ne i=1

{xf2,(i)

n

}

Ne i=1

{xa,(i)

n

}

Ne i=1

EnKF-OSA scheme for state-parameter estimation

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State-parameter estimation with OSA

{xa,(i)

n−1(θ(i) |n−1)} Ne i=1

{xf1,(i)

n

}

Ne i=1

{xs,(i)

n−1(θ(i) |n )} Ne i=1

{xf2,(i)

n

}

Ne i=1

{xa,(i)

n

}

Ne i=1

EnKF-OSA scheme for state-parameter estimation

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State-parameter estimation with OSA

{xa,(i)

n−1(θ(i) |n−1)} Ne i=1

{xf1,(i)

n

}

Ne i=1

{xs,(i)

n−1(θ(i) |n )} Ne i=1

{xf2,(i)

n

}

Ne i=1

{xa,(i)

n

}

Ne i=1

M

n − 1

EnKF-OSA scheme for state-parameter estimation

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State-parameter estimation with OSA

{xa,(i)

n−1(θ(i) |n−1)} Ne i=1

{xf1,(i)

n

}

Ne i=1

{xs,(i)

n−1(θ(i) |n )} Ne i=1

{xf2,(i)

n

}

Ne i=1

{xa,(i)

n

}

Ne i=1

M

n − 1

Analysis θ Smoothing x (yn)

EnKF-OSA scheme for state-parameter estimation EnKF θ EnKF x

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State-parameter estimation with OSA

{xa,(i)

n−1(θ(i) |n−1)} Ne i=1

{xf1,(i)

n

}

Ne i=1

{xs,(i)

n−1(θ(i) |n )} Ne i=1

{xf2,(i)

n

}

Ne i=1

{xa,(i)

n

}

Ne i=1

M

n − 1

M

n − 1

Analysis θ Smoothing x (yn)

EnKF-OSA scheme for state-parameter estimation EnKF θ EnKF x

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State-parameter estimation with OSA

{xa,(i)

n−1(θ(i) |n−1)} Ne i=1

{xf1,(i)

n

}

Ne i=1

{xs,(i)

n−1(θ(i) |n )} Ne i=1

{xf2,(i)

n

}

Ne i=1

{xa,(i)

n

}

Ne i=1

M

n − 1

M

n − 1

Analysis x (yn)

Analysis θ Smoothing x (yn)

EnKF-OSA scheme for state-parameter estimation EnKF θ EnKF x

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State-parameter estimation with OSA

{xa,(i)

n−1(θ(i) |n−1)} Ne i=1

{xf1,(i)

n

}

Ne i=1

{xs,(i)

n−1(θ(i) |n )} Ne i=1

{xf2,(i)

n

}

Ne i=1

{xa,(i)

n

}

Ne i=1

M

n − 1

M

n − 1

Analysis x (yn)

Analysis θ Smoothing x (yn)

EnKF-OSA scheme for state-parameter estimation EnKF θ EnKF x Introduced by Gharamti et al. (2015) and Ait-El-Fquih et al. (2016) to derive a Bayesian framework for the ”dual-EnKF” (Moradkhani et al. 2005) ”Original” dual-EnKF missed the smoothing step of the state (x)

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State-parameter estimation with OSA

Subsurface hydrology state-parameter estimation We estimate the water head and the hydraulic conductivity

Observation Frequency (days)

1 3 5 10 15 30

AAE (log-m/s)

0.7 0.8 0.9 1 1.1 1.2

Conductivity Estimates

Joint-EnKF Dual-EnKF Dual-EnKF-OSA

Figure: Mean average absolute errors (AAE) of log-hydraulic conductivity, log(k), in terms of the observation frequency of hydraulic head data. Data are

• btained from 9 wells, every 1, 3, 5, 10, 15 and 30 days using Ne = 100

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State-parameter estimation with OSA

Subsurface hydrology state-parameter estimation We estimate the water head and the hydraulic conductivity

Months

6 12 18

AAE (m)

0.1 0.15 0.2

Hydraulic Head Estimates

Joint-EnKF, p = 15 Dual-EnKF, p = 15 Dual-EnKF-OSA, p = 15 Joint-EnKF, p = 25 Dual-EnKF, p = 25 Dual-EnKF-OSA, p = 25

Months

6 12 18

AAE (log-m/s)

0.8 1 1.2

Conductivity Estimates

Figure: Time series of AAE for hydraulic head (left) and conductivity (right). Data of hydraulic head are obtained from p = 15, 25 wells using Ne = 100

OSA improves the state and parameter estimation with EnKFs

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Filtering One-Way-Coupled (OWC) systems with OSA

Consider the discrete-time OWC dynamical system:        xn = Mx

n−1 (xn−1) + ηx n−1

zn = Mz

n−1 (zn−1, xn−1) + ηz n−1

yx

n

= Hx

nxn + εx n

yz

n

= Hz

nzn + εz n

,

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Filtering One-Way-Coupled (OWC) systems with OSA

Consider the discrete-time OWC dynamical system:        xn = Mx

n−1 (xn−1) + ηx n−1

zn = Mz

n−1 (zn−1, xn−1) + ηz n−1

yx

n

= Hx

nxn + εx n

yz

n

= Hz

nzn + εz n

, Two classical solutions

– Strong (Joint) formulation (EnKF-S): Applies the filtering scheme on the augmented state Xn =

• xT

n zT n

T Cross-correlations considered in the update Cross-correlations may not be well estimated with small ensembles and not very representative in the presence of strong nonlinearities – Weak formulation (EnKF-W) Applies the filtering scheme on each component separately Separate updates are more practical in real applications Loss of information from neglecting cross-correlations Lost of information from neglecting cross-correlations

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Filtering One-Way-Coupled (OWC) systems with OSA

Strong EnKF-OSA (EnKF-S-OSA) introduces:

→ an extra smoothing step for both state components using the future

• bservations of both variables (a joint smoothing update)

→ an analysis step of both states, each using its own observation The ”separate” analysis steps result from the uncorrelated model errors ηx

n

and ηz

n.

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Numerical experiments with OWC Lorenz-96

Governing equations (Coupled L-96) dxi dt = (xi+1 − xi−2) xi−1 − xi + F − hc b

K

• J=1

zj,i, i = 1, · · · , nx = 8 dzj,i dt = (zj−1,i − zj+2,i) cbzj+1,i − czj,i + hc b xi, j = 1, · · · , K = 16

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Numerical experiments with OWC Lorenz-96

Governing equations (OWC L-96) dxi dt = (xi+1 − xi−2) xi−1 − xi + F i = 1, · · · , nx = 8 dzj,i dt = (zj−1,i − zj+2,i) cbzj+1,i − czj,i + hc b xi, j = 1, · · · , K = 16

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Numerical experiments with OWC Lorenz-96

Governing equations (OWC L-96) dxi dt = (xi+1 − xi−2) xi−1 − xi + F i = 1, · · · , nx = 8 dzj,i dt = (zj−1,i − zj+2,i) cbzj+1,i − czj,i + hc b xi, j = 1, · · · , K = 16 Twin experiments (8 + 1) × 16 = 144 variables 3-years simulation period Time step: 0.005 Assimilation scenarios Every second state variable (from each component) is observed every 4 model steps (1 day) We use covariance inflation and correlation-based localization to address the different spatial scales between components (Luo et al., 2017)

18 ISDA 2019 18 / 21

Numerical experiments with OWC Lorenz-96

1 1.5 2 x-RMSE Inflation

EnKF-S

Min = 1.00

EnKF-S-OSA

Min = 0.69

EnKF-W

Min = 0.87

EnKF-W-OSA

Min = 0.74 0.65 0.8 1 1.2 1.5 1 1.5 2 z-RMSE Inflation

EnKF-S

Min = 0.207

EnKF-S-OSA

Min = 0.177

EnKF-W

Min = 0.195

EnKF-W-OSA

Min = 0.175 0.17 0.18 0.2 0.22 0.25

Figure: Time averaged RMSE as function of inflation factor, Ne = 40 OSA formulation is beneficial With OSA, EnKF-S outperforms EnKF-W EnKF-S requires large enough ensembles to outperform EnKF-W OSA more robust to inflation values

19 ISDA 2019 19 / 21

Main conclusions: OSA for OWC systems

OSA is beneficial for strong and weak OWC DA compared to the standard EnKF EnKF-W-OSA outperforms the other schemes with very small ensembles The benefit of EnKF-S-OSA is more pronounced with less data The smoothing window should not be too large so that the linear correlation-based updates remain relevant, iterations may help

20 ISDA 2019 20 / 21

Main conclusions: OSA for OWC systems

OSA is beneficial for strong and weak OWC DA compared to the standard EnKF EnKF-W-OSA outperforms the other schemes with very small ensembles The benefit of EnKF-S-OSA is more pronounced with less data The smoothing window should not be too large so that the linear correlation-based updates remain relevant, iterations may help

• We are working on implementing the EnKF-OSA within the Data

Research Testbed (DART) to test it with a high resolution MIT general circulation model (MITgcm) of the Red Sea

20 ISDA 2019 20 / 21

Thank you

References

• Gharamti, M., B. Ait-El-Fquih, and I. Hoteit, 2015: An iterative ensemble Kalman filter

with one-step-ahead smoothing for state-parameters estimation of contaminant transport

• models. J. Hydrol.
• Ait-El-Fquih, B., M. El Gharamti, and I. Hoteit, 2016: A Bayesian consistent dual

ensemble Kalman filter for state-parameter estimation in subsurface hydrology. Hydrology and Earth System Sciences.

• Raboudi, N. F., B. Ait-El-Fquih, and I. Hoteit, 2018: Ensemble kalman filtering with
• ne-step-ahead smoothing. Monthly Weather Review.
• Raboudi, N. F., B. Ait-El-Fquih, C.Dawson and I. Hoteit, 2018: Combining Hybrid and

One-Step-Ahead Smoothing for Efficient Short-Range Storm Surge Forecasting with an Ensemble Kalman Filter (Submitted).

• Raboudi, N. F., B. Ait-El-Fquih and I. Hoteit, 2018: An ensemble Kalman filter with
• ne-step-ahead smoothing for efficient data assimilation into one-way-coupled systems

(To be submitted).

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