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Reduced-cost ensemble Kalman filter
for parameter estimation!
Application to front-tracking problems!
Mélanie Rochoux!
in c...
INTRODUCTION ●●●●
Data assimilation: why? how?!
2 !Rochoux et al. – UNCECOMP 2015 – MS-10!
➙ Key idea: “optimal combinatio...
INTRODUCTION ●●●●
Data assimilation: why? how?!
➙ Key idea: “optimal combination of observations and forward model”!
Ensem...
INTRODUCTION ●●●●
Uncertainty quantification!
➙ Challenging idea: Use uncertainty quantification to overcome the slow
conver...
INTRODUCTION ●●●●
Parameter estimation!
➙ Objective: Improvement of the forecast performance
•  State estimation limitatio...
INTRODUCTION ●●●●
Outline!
!
Reduced-cost ensemble Kalman filter for parameter
estimation (PC-EnKF)!
!
u  Algorithm!
u  A...
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
paramet...
ALGORITHM ●●●
Hybrid PC-EnKF!
➙ Objective: Reduce computational cost of forward model integration
•  Integrating Polynomia...
ALGORITHM ●●●
Coupling PC and EnKF approaches!
EnKF prediction
Surrogate model
Surrogate model
Forward modelHermite quadra...
WILDFIRE SPREAD APPLICATION ●●●
Front-tracking problem!
Experimental grassland fire
(100m x 100m), N.S. Cheney,
Annaburroo ...
WILDFIRE SPREAD APPLICATION ●●●
Front-tracking problem!
•  2-D state variable: reaction progress variable c 

•  Front mar...
WILDFIRE SPREAD APPLICATION ●●●
Front-tracking problem!
•  2-D state variable: reaction progress variable c 

•  Front mar...
WILDFIRE SPREAD APPLICATION ●●●
Synthetic experiment!
•  Estimation of a uniform proportionality coefficient P in the ROS f...
WILDFIRE SPREAD APPLICATION ●●●
Controlled fire experiment!
•  Reduced scale fire experiment (4 m x 4 m) over quasi-homogene...
WILDFIRE SPREAD APPLICATION ●●●
Controlled fire experiment!
•  Reduced scale fire experiment (4 m x 4 m) over quasi-homogene...
WILDFIRE SPREAD APPLICATION ●●●
Controlled fire experiment!
•  Reduced scale fire experiment (4 m x 4 m) over quasi-homogene...
CONCLUSION
Key ideas!
Rochoux et al (2014), NHESS!
Rochoux et al (2012), CTR brief!
➙ Reduced-cost ensemble Kalman filter (...
CONCLUSION
Key ideas!
Rochoux et al (2014), NHESS!
Rochoux et al (2012), CTR brief!
➙ Reduced-cost ensemble Kalman filter (...
*	
  Melanie.Rochoux@cerfacs.fr
Thank you for your attention!
Any question?
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
Posteri...
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Reduced-cost ensemble Kalman filter for front-tracking problems

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In this talk, a reduced-cost ensemble Kalman filter (PC-EnKF) is implemented for the estimation of the model input parameters in the context of a front-tracking problem. The forecast step relies on a probabilistic sampling based on a Polynomial Chaos (PC) surrogate model. The performance of the hybrid PC-EnKF strategy is assessed for synthetic front-tracking test cases as well as in the context of wildfire spread, which features a front-like geometry and where the estimation targets are the unknown biomass fuel properties and the surface wind conditions. Results indicate that the hybrid PC-EnKF strategy features similar performance to the standard EnKF algorithm, without loss of accuracy but at a much reduced computational cost.

Reference published in NHESS (2014)
➞ Rochoux, M.C., Ricci, S., Lucor, D., Cuenot, B., and Trouvé, A. (2014) Towards predictive data-driven simulations of wildfire spread. Part I: Reduced-cost Ensemble Kalman Filter based on a Polynomial Chaos surrogate model for parameter estimation, Natural Hazards and Earth System Sciences, Special Issue: Numerical Wildland Combustion, from the flame to the atmosphere, vol. 14, pp. 2951-2973, doi: 10.5194/nhess-14-2951-2014, published.

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Reduced-cost ensemble Kalman filter for front-tracking problems

  1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 19. *  Melanie.Rochoux@cerfacs.fr Thank you for your attention! Any question?
  20. 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!

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