Reduced-cost ensemble Kalman filter for front-tracking problems
2 likes1,218 views
The document presents a reduced-cost ensemble Kalman filter (pc-EnKF) designed for parameter estimation in front-tracking problems, particularly applied to wildfire spread forecasting. It introduces an algorithm that optimally combines observations and model forecasts with a focus on uncertainty quantification to enhance performance in estimating dynamic systems. The methodology demonstrates its potential through controlled fire experiments, emphasizing the need for further development to address spatially-varying parameters and enhance estimation accuracy.
![INTRODUCTION ●●●●
Data assimilation: why? how?!
➙ Key idea: “optimal combination of observations and forward model”!
Ensemble Kalman filter (EnKF)
• Forecast step ➙ uncertainty propagation
- Explicit propagation of the error statistics
- Nonlinear extension of the Kalman filter
• Analysis step ➙ Kalman filter update equation
!
reality
model forecast
Diagnostic!
measurements
analysis
Time!
Sequential approach
=
+
K
[
-‐
]
Distance to observations!
G( )
Kalman gain matrix!
Stochastic characterization
Estimation of error
covariance matrices
Control variables!
3 !Rochoux et al. – UNCECOMP 2015 – MS-10!](https://image.slidesharecdn.com/rochouxuncecomp15slides-150902075419-lva1-app6891/85/Reduced-cost-ensemble-Kalman-filter-for-front-tracking-problems-3-320.jpg){kind=tracking}
![INTRODUCTION ●●●●
Uncertainty quantification!
➙ Challenging idea: Use uncertainty quantification to overcome the slow
convergence rate and sampling errors of the Monte Carlo-based EnKF!
!
reality
model forecast
Diagnostic!
measurements
analysis
Time!
Sequential approach
Npc
X
k=1
ˆck k( )
●
Basis functions
4 !Rochoux et al. – UNCECOMP 2015 – MS-10!
=
+
K
[
-‐
]
G( )
Control variables!
Hybrid Ensemble Kalman filter (PC-EnKF)
• Forecast step ➙ uncertainty propagation
- Use of surrogate model to compute model trajectories
- Polynomial Chaos (PC) expansion
• Analysis step ➙ Kalman filter update equation
!](https://image.slidesharecdn.com/rochouxuncecomp15slides-150902075419-lva1-app6891/85/Reduced-cost-ensemble-Kalman-filter-for-front-tracking-problems-4-320.jpg){kind=tracking}
![ALGORITHM ●●●
Standard EnKF!
Cxy = Pt
f
Gt
T
xt
a,(k)
= xt
f,(k)
+Cxy (Cyy + R)−1
(yt
o
+ξo,(k)
− yt
f,(k)
)
Prior
parameters
Prior
fire fronts
Posterior
parameters
xt
f,(1)
xt
f,(2)
xt
f,(Ne )
yt
f,(1)
yt
f,(2)
yt
f,(Ne )
Covariance matrices
EnKF
update
Cyy = GtPt
f
Gt
T
xt
a,(1)
xt
a,(2)
xt
a,(Ne )
yt
o
+ξo,(1)
Ke
t
Posterior
fire fronts
yt
a,(1)
yt
a,(2)
yt
a,(Ne )
EnKF
prediction
FORECAST ANALYSIS
EnKF
prediction
yt
o
+ξo,(Ne )
yt
o
+ξo,(2)
Gt Gt
➙ Key idea: 3D-Var approach with stochastically-based estimation of the error
covariance matrices over the assimilation cycle [t-1, t]
Specificities!!
• Random walk model for
parameter evolution
• Data randomization
➙ Burgers et al. 1998
• Limitations
➙ Slow convergence
rate (large number of
members)
➙ Sampling errors (Li
2008) – local & global
7 !Rochoux et al. – UNCECOMP 2015 – MS-10!](https://image.slidesharecdn.com/rochouxuncecomp15slides-150902075419-lva1-app6891/85/Reduced-cost-ensemble-Kalman-filter-for-front-tracking-problems-7-320.jpg){kind=tracking}
![WILDFIRE SPREAD APPLICATION ●●●
Synthetic experiment!
• Estimation of a uniform proportionality coefficient P in the ROS formulation
• True parameter at the tail of the Gaussian distribution associated with the forecast estimates
• Reduced-cost approach:
• 5 model integrations to build the surrogate model
• 1000 members in the ensemble
• Observation error STD = 2 m
!
75 85 95 105 115 125
75
85
95
105
115
125
x [m]
y[m]
m
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
75
85
95
105
115
125
P [1/s]x−coordinate[m]
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
75
85
95
105
115
125
P [1/s]
y−coordinate[m]
forecast true
trueforecast
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
75
85
95
105
115
125
P [1/s]
x−coordinate[m]
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
75
85
95
105
115
125
y−coordinate[m]
analysis
analysis true
true
- forecast
- analysis
+ observations
quadrature points
▾ Response surface for the x-coordinate front marker m
◀ Fire front positions at time 50 s (analysis time)
13 !Rochoux et al. – UNCECOMP 2015 – MS-10!](https://image.slidesharecdn.com/rochouxuncecomp15slides-150902075419-lva1-app6891/85/Reduced-cost-ensemble-Kalman-filter-for-front-tracking-problems-13-320.jpg){kind=tracking}
![WILDFIRE SPREAD APPLICATION ●●●
Controlled fire experiment!
• Reduced scale fire experiment (4 m x 4 m) over quasi-homogeneous short grass
!
1min32s
50s
1min46s
1min04s
1min18s
Wind 1m/s
0 0.5 1 1.5 2 2.5 3 3.5
0
0.5
1
1.5
2
x [m]
y [m]
• Mid-Infrared imaging
• Quasi-homogeneous short
grass (22% moisture content)
• Mean wind speed: 1m/s in
northwestern direction
• Mean ROS = 2 cm/s
• Max. ROS = 5 cm/s
ANALYSIS
TIME
FORECAST
TIME
14 !Rochoux et al. – UNCECOMP 2015 – MS-10!](https://image.slidesharecdn.com/rochouxuncecomp15slides-150902075419-lva1-app6891/85/Reduced-cost-ensemble-Kalman-filter-for-front-tracking-problems-14-320.jpg){kind=tracking}
![WILDFIRE SPREAD APPLICATION ●●●
Controlled fire experiment!
• Reduced scale fire experiment (4 m x 4 m) over quasi-homogeneous short grass
• Estimation of 2 biomass fuel parameters: moisture content (Mv), geometrical parameter (Σv)
• Reduced-cost approach:
• 25 model integrations to build the surrogate model
• 1000 members in the ensemble
• Observation error STD = 5 cm
!
15 11500
Σv [1/m]Mv [%]
Σv [1/m]Mv [%]
15 11500
x-coordinate[m]y-coordinate[m]
13.8 2234513.8 22345
13.8 22345
Σv [1/m]Mv [%]
x-coordinate[m]y-coordinate[m]
0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2
0
0.5
1
1.5
2
x [m]
y[m]
m
▾ Response surface for the x-coordinate front marker m
◀ Fire front positions at time 1min18 s
quadrature
points
forecast!
analysis!
- Forecast (PC-EnKF)
- Analysis (PC-EnKF)
□ Analysis (standard EnKF)
+ observations
ANALYSIS TIME
15 !Rochoux et al. – UNCECOMP 2015 – MS-10!](https://image.slidesharecdn.com/rochouxuncecomp15slides-150902075419-lva1-app6891/85/Reduced-cost-ensemble-Kalman-filter-for-front-tracking-problems-15-320.jpg){kind=tracking}
![WILDFIRE SPREAD APPLICATION ●●●
Controlled fire experiment!
• Reduced scale fire experiment (4 m x 4 m) over quasi-homogeneous short grass
• Estimation of 2 biomass fuel parameters: moisture content (Mv), geometrical parameter (Σv)
• Reduced-cost approach:
• 25 model integrations to build the surrogate model
• 1000 members in the ensemble
• Observation error STD = 5 cm
!
0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2
0
0.5
1
1.5
2
x [m]
y[m]
m
ANALYSIS TIME
0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2
0
0.5
1
1.5
2
x [m]
y[m]
- Forecast (PC-EnKF)
- Analysis (PC-EnKF)
□ Analysis (standard EnKF)
+ observations
FORECAST TIME
Good behavior
of the PC
surrogate model!
16 !Rochoux et al. – UNCECOMP 2015 – MS-10!](https://image.slidesharecdn.com/rochouxuncecomp15slides-150902075419-lva1-app6891/85/Reduced-cost-ensemble-Kalman-filter-for-front-tracking-problems-16-320.jpg){kind=tracking}
![Cxy = Pt
f
Gt
T
xt
a,(k)
= xt
f,(k)
+Cxy (Cyy + R)−1
(yt
o
+ξo,(k)
− yt
f,(k)
)
Prior
parameters
Prior
fire fronts
Posterior
parameters
xt
f,(1)
xt
f,(2)
xt
f,(Ne )
yt
f,(1)
yt
f,(2)
yt
f,(Ne )
Covariance matrices
EnKF
update
Cyy = GtPt
f
Gt
T
xt
a,(1)
xt
a,(2)
xt
a,(Ne )
yt
o
+ξo,(1)
Ke
t
Posterior
fire fronts
yt
a,(1)
yt
a,(2)
yt
a,(Ne )
EnKF
prediction
FORECAST ANALYSIS
EnKF
prediction
yt
o
+ξo,(Ne )
yt
o
+ξo,(2)
Gt Gt
ALGORITHM ●●●
Standard EnKF!
➙ Key idea: 3D-Var approach with stochastically-based estimation of the error
covariance matrices over the assimilation cycle [t-1, t]
!
• Local error (over one assimilation cycle)
!
• Global error (over all assimilation cycles)
Specificities!!
• Random walk model for
parameter evolution
• Data randomization
➙ Burgers et al. 1998
• Limitations
➙ Slow convergence
rate (large number of
members)
➙ Sampling errors (Li
2008) – local & global
20 !Rochoux et al. – UNCECOMP 2015 – MS-10!](https://image.slidesharecdn.com/rochouxuncecomp15slides-150902075419-lva1-app6891/85/Reduced-cost-ensemble-Kalman-filter-for-front-tracking-problems-20-320.jpg){kind=tracking}
Reduced-cost ensemble Kalman filter for front-tracking problems
- 1. Reduced-cost ensemble Kalman filter for parameter estimation! Application to front-tracking problems! Mélanie Rochoux! in collaboration with S.Ricci, D. Lucor, B. Cuenot & A. Trouvé! * melanie.rochoux@cerfacs.fr! MS-10 Reduced-order models for stochastic inverse problems – U626
- 2. INTRODUCTION ●●●● Data assimilation: why? how?! 2 !Rochoux et al. – UNCECOMP 2015 – MS-10! ➙ Key idea: “optimal combination of observations and forward model”! Determine best estimate of a dynamical system given Weather forecast! Atm. chemistry! Hydrology! Biomechanics! - Sparse and imperfect - Relation between observations and model outputs Observations Numerical model Model formulation Model parameters Initial condition Forcing data Mathematical technique based on estimation theory • The “true state” is unknown and should be estimated • Measurements and models are imperfect • The estimate should be an optimal combination of both measurements and models ➙ error minimization problem Ex. applications
- 3. INTRODUCTION ●●●● Data assimilation: why? how?! ➙ Key idea: “optimal combination of observations and forward model”! Ensemble Kalman filter (EnKF) • Forecast step ➙ uncertainty propagation - Explicit propagation of the error statistics - Nonlinear extension of the Kalman filter • Analysis step ➙ Kalman filter update equation ! reality model forecast Diagnostic! measurements analysis Time! Sequential approach = + K [ -‐ ] Distance to observations! G( ) Kalman gain matrix! Stochastic characterization Estimation of error covariance matrices Control variables! 3 !Rochoux et al. – UNCECOMP 2015 – MS-10!
- 4. INTRODUCTION ●●●● Uncertainty quantification! ➙ Challenging idea: Use uncertainty quantification to overcome the slow convergence rate and sampling errors of the Monte Carlo-based EnKF! ! reality model forecast Diagnostic! measurements analysis Time! Sequential approach Npc X k=1 ˆck k( ) ● Basis functions 4 !Rochoux et al. – UNCECOMP 2015 – MS-10! = + K [ -‐ ] G( ) Control variables! Hybrid Ensemble Kalman filter (PC-EnKF) • Forecast step ➙ uncertainty propagation - Use of surrogate model to compute model trajectories - Polynomial Chaos (PC) expansion • Analysis step ➙ Kalman filter update equation !
- 5. INTRODUCTION ●●●● Parameter estimation! ➙ Objective: Improvement of the forecast performance • State estimation limitation ➙ no long persistence of the initial condition for a chaotic system • Parameter estimation ➙ accounting for the temporal variability in the errors Difficulties ➙ Possible nonlinear relationship between input parameters and model counterparts of the observations ➙ Existence of an evolution model for parameters? ! Forward model Parameters Initial condition Boundary conditions Comparison Model outputs Observations Ensemble Kalman filter Parameter estimation State estimation 5 !Rochoux et al. – UNCECOMP 2015 – MS-10!
- 6. INTRODUCTION ●●●● Outline! ! Reduced-cost ensemble Kalman filter for parameter estimation (PC-EnKF)! ! u Algorithm! u Application to wildfire spread forecasting! • Front-tracking problem • Synthetic case • Controlled fire experiment 6 !Rochoux et al. – UNCECOMP 2015 – MS-10!
- 7. ALGORITHM ●●● Standard EnKF! Cxy = Pt f Gt T xt a,(k) = xt f,(k) +Cxy (Cyy + R)−1 (yt o +ξo,(k) − yt f,(k) ) Prior parameters Prior fire fronts Posterior parameters xt f,(1) xt f,(2) xt f,(Ne ) yt f,(1) yt f,(2) yt f,(Ne ) Covariance matrices EnKF update Cyy = GtPt f Gt T xt a,(1) xt a,(2) xt a,(Ne ) yt o +ξo,(1) Ke t Posterior fire fronts yt a,(1) yt a,(2) yt a,(Ne ) EnKF prediction FORECAST ANALYSIS EnKF prediction yt o +ξo,(Ne ) yt o +ξo,(2) Gt Gt ➙ Key idea: 3D-Var approach with stochastically-based estimation of the error covariance matrices over the assimilation cycle [t-1, t] Specificities!! • Random walk model for parameter evolution • Data randomization ➙ Burgers et al. 1998 • Limitations ➙ Slow convergence rate (large number of members) ➙ Sampling errors (Li 2008) – local & global 7 !Rochoux et al. – UNCECOMP 2015 – MS-10!
- 8. ALGORITHM ●●● Hybrid PC-EnKF! ➙ Objective: Reduce computational cost of forward model integration • Integrating Polynomial Chaos (PC) into forecast • Control parameters projected onto a stochastic space spanned by orthogonal PC functions of independent Gaussian random variables Surrogate model! Model inputs! Model outputs! Random event! • Easy access to statistics (mean, covariance, ensemble sampling) Ensemble sampling! • Integrating Polynomial Chaos (PC) into observation • Use the same basis for the model and for the data space (not obvious since observations and model counterparts should remain uncorrelated, Evensen 2009) 8 !Rochoux et al. – UNCECOMP 2015 – MS-10!
- 9. ALGORITHM ●●● Coupling PC and EnKF approaches! EnKF prediction Surrogate model Surrogate model Forward modelHermite quadrature Simulated fire fronts Hermite polynomials Surrogate model Forecast ! distribution! ➀! Monte-Carlo sampling Predicted fire front positions Posterior estimate of parameters Updated fire front positions ➁ ➂ EnKF update ('q)q = 1, · · · , Npc k = 1, · · · , Ne k = 1, · · · , Ne EnKF prediction k = 1, · · · , Ne k = 1, · · · , Ne FIREFLY j = 1, · · · , (Nquad)n j = 1, · · · , (Nquad)n ⇣ x f,(j) t , !j ⌘ ⇣ y f,(j) t ⌘ pf (xt) yf t = Gpc,t(xf t) ⇣ x f,(k) t ⌘ ⇣ x a,(k) t ⌘ ⇣ y a,(k) t ⌘ ⇣ y f,(k) t ⌘ ➙ Non-intrusive approach: PC used to build a surrogate model of the observation operator 9 !Rochoux et al. – UNCECOMP 2015 – MS-10!
- 10. WILDFIRE SPREAD APPLICATION ●●● Front-tracking problem! Experimental grassland fire (100m x 100m), N.S. Cheney, Annaburroo site (Australia)! ➙ Wildfires feature a front-like geometry at regional scales! FRONT! • Scales ranging from meters up to several kilometers • Thin flame zone propagating normal to itself towards unburnt vegetation • Local propagation speed of the front called “rate of spread” (ROS) 10 !Rochoux et al. – UNCECOMP 2015 – MS-10!
- 11. WILDFIRE SPREAD APPLICATION ●●● Front-tracking problem! • 2-D state variable: reaction progress variable c • Front marker: contour line c = 0.5 • Submodel for the local ROS along the normal direction to the front • Semi-empirical formulation (Rothermel) • Function of the local environmental conditions ➙ Level-set-based front propagation simulator ROS = f(uw, ↵sl, Mv, v, ⌃v, ...) Simulated front c= 0.5 ! (x1 , y1 ) (x2 , y2 ) (x3 , y3 ) (x4 , y4 ) @c @t = ROS |rc| 11 !Rochoux et al. – UNCECOMP 2015 – MS-10!
- 12. WILDFIRE SPREAD APPLICATION ●●● Front-tracking problem! • 2-D state variable: reaction progress variable c • Front marker: contour line c = 0.5 • Submodel for the local ROS along the normal direction to the front • Semi-empirical formulation (Rothermel) • Function of the local environmental conditions ➙ Level-set-based front propagation simulator ROS = f(uw, ↵sl, Mv, v, ⌃v, ...) Simulated front c= 0.5 ! (x1 , y1 ) (x2 , y2 ) (x3 , y3 ) (x4 , y4 ) @c @t = ROS |rc|➙ Observation represented as a discretized fire front!! raw data: infrared imagery 12 !Rochoux et al. – UNCECOMP 2015 – MS-10!
- 13. WILDFIRE SPREAD APPLICATION ●●● Synthetic experiment! • Estimation of a uniform proportionality coefficient P in the ROS formulation • True parameter at the tail of the Gaussian distribution associated with the forecast estimates • Reduced-cost approach: • 5 model integrations to build the surrogate model • 1000 members in the ensemble • Observation error STD = 2 m ! 75 85 95 105 115 125 75 85 95 105 115 125 x [m] y[m] m 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 75 85 95 105 115 125 P [1/s]x−coordinate[m] 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 75 85 95 105 115 125 P [1/s] y−coordinate[m] forecast true trueforecast 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 75 85 95 105 115 125 P [1/s] x−coordinate[m] 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 75 85 95 105 115 125 y−coordinate[m] analysis analysis true true - forecast - analysis + observations quadrature points ▾ Response surface for the x-coordinate front marker m ◀ Fire front positions at time 50 s (analysis time) 13 !Rochoux et al. – UNCECOMP 2015 – MS-10!
- 14. WILDFIRE SPREAD APPLICATION ●●● Controlled fire experiment! • Reduced scale fire experiment (4 m x 4 m) over quasi-homogeneous short grass ! 1min32s 50s 1min46s 1min04s 1min18s Wind 1m/s 0 0.5 1 1.5 2 2.5 3 3.5 0 0.5 1 1.5 2 x [m] y [m] • Mid-Infrared imaging • Quasi-homogeneous short grass (22% moisture content) • Mean wind speed: 1m/s in northwestern direction • Mean ROS = 2 cm/s • Max. ROS = 5 cm/s ANALYSIS TIME FORECAST TIME 14 !Rochoux et al. – UNCECOMP 2015 – MS-10!
- 15. WILDFIRE SPREAD APPLICATION ●●● Controlled fire experiment! • Reduced scale fire experiment (4 m x 4 m) over quasi-homogeneous short grass • Estimation of 2 biomass fuel parameters: moisture content (Mv), geometrical parameter (Σv) • Reduced-cost approach: • 25 model integrations to build the surrogate model • 1000 members in the ensemble • Observation error STD = 5 cm ! 15 11500 Σv [1/m]Mv [%] Σv [1/m]Mv [%] 15 11500 x-coordinate[m]y-coordinate[m] 13.8 2234513.8 22345 13.8 22345 Σv [1/m]Mv [%] x-coordinate[m]y-coordinate[m] 0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 0 0.5 1 1.5 2 x [m] y[m] m ▾ Response surface for the x-coordinate front marker m ◀ Fire front positions at time 1min18 s quadrature points forecast! analysis! - Forecast (PC-EnKF) - Analysis (PC-EnKF) □ Analysis (standard EnKF) + observations ANALYSIS TIME 15 !Rochoux et al. – UNCECOMP 2015 – MS-10!
- 16. WILDFIRE SPREAD APPLICATION ●●● Controlled fire experiment! • Reduced scale fire experiment (4 m x 4 m) over quasi-homogeneous short grass • Estimation of 2 biomass fuel parameters: moisture content (Mv), geometrical parameter (Σv) • Reduced-cost approach: • 25 model integrations to build the surrogate model • 1000 members in the ensemble • Observation error STD = 5 cm ! 0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 0 0.5 1 1.5 2 x [m] y[m] m ANALYSIS TIME 0 0.4 0.8 1.2 1.6 2 2.4 2.8 3.2 0 0.5 1 1.5 2 x [m] y[m] - Forecast (PC-EnKF) - Analysis (PC-EnKF) □ Analysis (standard EnKF) + observations FORECAST TIME Good behavior of the PC surrogate model! 16 !Rochoux et al. – UNCECOMP 2015 – MS-10!
- 17. CONCLUSION Key ideas! Rochoux et al (2014), NHESS! Rochoux et al (2012), CTR brief! ➙ Reduced-cost ensemble Kalman filter (PC-EnKF) for parameter estimation in front-tracking problems! • Stand-alone parameter estimation ➙ forecast improvement! • Prototype able to address multi-parameter sequential estimation at a reduced cost • Spatially-uniform and constant parameters over the time window • Application: Reduced-scale wildfire spread problem ➙ Need to extend the strategy at regional scales ➙ Need to combine parameter estimation and state estimation approaches to treat anisotropic uncertainties ! 17 !Rochoux et al. – UNCECOMP 2015 – MS-10!
- 18. CONCLUSION Key ideas! Rochoux et al (2014), NHESS! Rochoux et al (2012), CTR brief! ➙ Reduced-cost ensemble Kalman filter (PC-EnKF) for parameter estimation in front-tracking problems! • Stand-alone parameter estimation ➙ forecast improvement! • Prototype able to address multi-parameter sequential estimation at a reduced cost • Spatially-uniform and constant parameters over the time window • Application: Reduced-scale wildfire spread problem ➙ Need to extend the strategy at regional scales ➙ Need to combine parameter estimation and state estimation approaches to treat anisotropic uncertainties • Front-tracking problem ➙ dynamically-evolving observation operator over time! • Prototype able to track coherent features • Unusual application of the EnKF algorithm ➙ Need to test the sensitivity of the hybrid data assimilation algorithm to different representations of the front ! 18 !Rochoux et al. – UNCECOMP 2015 – MS-10!
- 19. * Melanie.Rochoux@cerfacs.fr Thank you for your attention! Any question?
- 20. Cxy = Pt f Gt T xt a,(k) = xt f,(k) +Cxy (Cyy + R)−1 (yt o +ξo,(k) − yt f,(k) ) Prior parameters Prior fire fronts Posterior parameters xt f,(1) xt f,(2) xt f,(Ne ) yt f,(1) yt f,(2) yt f,(Ne ) Covariance matrices EnKF update Cyy = GtPt f Gt T xt a,(1) xt a,(2) xt a,(Ne ) yt o +ξo,(1) Ke t Posterior fire fronts yt a,(1) yt a,(2) yt a,(Ne ) EnKF prediction FORECAST ANALYSIS EnKF prediction yt o +ξo,(Ne ) yt o +ξo,(2) Gt Gt ALGORITHM ●●● Standard EnKF! ➙ Key idea: 3D-Var approach with stochastically-based estimation of the error covariance matrices over the assimilation cycle [t-1, t] ! • Local error (over one assimilation cycle) ! • Global error (over all assimilation cycles) Specificities!! • Random walk model for parameter evolution • Data randomization ➙ Burgers et al. 1998 • Limitations ➙ Slow convergence rate (large number of members) ➙ Sampling errors (Li 2008) – local & global 20 !Rochoux et al. – UNCECOMP 2015 – MS-10!