Bayesian Inference for front-tracking problems - 2013 IPDO conference

1. Applications of particle filters in moving frontier problems: Wildfire spread forecasting W. Da Silva, M. Rochoux, H. Orlande, M. Colaço, O. Fudym, M. El Hafi, B. Cuenot & S. Ricci © Pauline Crombe/e © Domingo Viegas

2. 2 INTRODUCTION Wildfire modeling challenges 2 Uncertain)es in large-‐scale fire spread predic)ons due to ➔ large range of length scales (pyrolysis vegetaFon scales to plume dynamic scales) ➔ unknown boundary and iniFal condiFons • non-‐homogeneous and poorly defined vegetal fuels • atmospheric external forcing ➔ difficult validaFon (lab-‐scale and field-‐scale experiments) cm m km • Cost-‐effecFve • Front-‐tracking simulator • Empirical model of the fire front spread-‐rate OperaFonally-‐oriented front-‐tracking simulator © ANR-‐IDEA

3. 3 INTRODUCTION Regional-‐scale wildﬁre spread modeling 3 ➔ ParameterizaFon of the rate of spread (ROS) as a funcFon of the local condiFons: Weather •  Wind velocity and direcFon •  Air temperature and humidity •  Rainfall Terrain •  Terrain slope Vegetal fuel •  Moisture content •  Depth of the vegetal layer •  Packing raFo •  Fuel parFcles (density, size, …) Front topology © Cheney (CSIRO) ➔ ISSUE -‐ Need to quanFfy and reduce uncertainFes in • Model formulaFon • Input model parameters • External forcing R R(x, y, t) = f(uw, αsl, Mf , δf , βf , Σf ...) FOCUS © ANR-‐IDEA

4. INTRODUCTION Why parFcle filters for tracking wildfire spread? 4 ➔ In principle, parFcle filters can handle the non-‐lineariFes present in a physical system (in a more formal way than the Kalman filter and its extensions). • Time-‐varying wind • Highly heterogeneous vegetal fuel properFes that change over Fme Normalizing constant © M. Finney (2011) Skewed fire size distribuFon

5. 5 OUTLINE Wildfire spread forecasFng using parFcle filters 5 © Horus (SDIS 66) ①  Regional-‐scale wildfire spread simulaFon capability ②  ParFcle filter algorithms ③  ApplicaFon to a controlled burning experiment

6. Focus: surface fire spread ➔ Build a simplified model that gives the Fme-‐evoluFon of the flame front locaFon • Front-‐tracking strategy • 2-‐D propagaFon within the vegetal fuel bed (li/er) PART. 1 PART. 2 PART. 3 InformaFon at regional-‐scales: model 6 ➔ Level-‐set-‐based front propagaFon solver • 2-‐D variable: reacFon progress variable c • Flame front marker: isoline c = 0.5 ∂c ∂t = R|∇c| FIREFLY: c = 1 c = 0 ➔ Issue: How to accurately describe uncertainFes in input parameters of the rate of spread R?

7. PART. 1 PART. 2 PART. 3 InformaFon at regional-‐scales: data 7 ➔ GeolocaFon of acFve fire areas • Middle InfraRed (MIR) camera aboard • FRP (Fire RadiaFve Power) measurements sensiFve to acFve fire areas Airborne-‐based thermal infrared imaging ➔ Requirements for inverse problems • High-‐spaFal resoluFon imagery (< 30 m) • Short revisit period X (m) Y(m) 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 © Ronan Paugam (King’s College) Assume iso-‐ temperature for fire igniFon (600K) Temperature field [K] ReconstrucFon of fire front posiFon © D. Viegas

8. PART. 1 PART. 2 PART. 3 Inverse problem strategy 8 Why? 1-‐ Uncertainty on inputs Uncertainty on outputs 2-‐ Find best esFmate of control variables given available observaFons ➔ Which input model parameters are criFcal to control? • SensiFvity analysis of Rothermel spread-‐rate model • IllustraFon of the non-‐lineariFes present in the wildfire spread model R(x, y, t) = f(uw, Mf , δf , βf , Σf , ...) Mf [!] ![m/s] 0 0.05 0.1 0.15 0.2 0.25 0.3 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 R [m/s] Mf [-] Wind-‐aided fire spread (1 m/s) Short grass Long grass Timber li/er Control parameters Simulated fronts Firefly simulator ¤  level-‐set simulator ¤  moisture content Mf ¤  fuel parFcle surface/volume Σf ¤  wind speed uw

9. PART. 1 PART. 2 PART. 3 Inverse problem strategy 9 Why? 1-‐ Uncertainty on inputs Uncertainty on outputs 2-‐ Find best esFmate of control variables given available observaFons. ➔ How to compare simulated fire front posiFons and observaFons? Discrete Fme-‐evolving fire front posiFons Uncertainty range for each front posiFon x y Fme Control parameters Simulated fronts Firefly simulator ¤  level-‐set simulator ¤  moisture content Mf ¤  fuel parFcle surface/volume Σf ¤  wind speed uw ObservaFons prior distribuFon likelihood DistribuFons for modeling and observaFon errors ¤  selecFon of the front at the assimilaFon Fme xk zk hkObserva)on model

10. PART. 1 PART. 2 PART. 3 Inverse problem strategy 10 Why? 1-‐ Uncertainty on inputs Uncertainty on outputs 2-‐ Find best esFmate of control variables given available observaFons. Control parameters Simulated fronts Fireﬂy simulator ¤  level-‐set simulator ¤  moisture content Mf ¤  fuel parFcle surface/volume Σf ¤  wind speed uw ObservaFons prior distribuFon Bayesian ﬁltering Data-‐driven feedback Simulated front Observed front (xf , yf )1 (xf , yf )p (xf , yf )j (xo f , yo f )j (xo f , yo f )1 (xo f , yo f )p Posterior distance Extended state es)ma)on

11. PART. 1 PART. 2 PART. 3 Inverse problem strategy 11 ➔ Bayesian ﬁltering in 2 steps: • PredicFon of the physical model • Update of the control parameters based on Bayes’ theorem πposterior(xk) = π(xk|zk) = πprior(xk)π(zk|xk) π(zk) Likelihood (measurement model including uncertainFes) Normalizing constant πprior(xk) = π(xk|xk−1) ➔ ISSUE: How to describe the prior model? • Is represented as a transiFon probability density from Fme (k-‐1) to Fme k • Includes a random walk model for the parameter evoluFon ➔ SOLUTION: ParFcle ﬁlters to obtain the posterior • Monte-‐Carlo technique: representaFon of the posterior by a set of random samples (parFcles) with associated weights

12. reality model predicFon diagnosis measurements analysis PART. 1 PART. 2 PART. 3 Inverse problem strategy 12 ➔ Bayesian ﬁltering in 2 steps: • PredicFon of the physical model • Update of the control parameters based on Bayes’ theorem πposterior(xk) = π(xk|zk) = πprior(xk)π(zk|xk) π(zk) Likelihood (measurement model including uncertainFes) predicFon update predicFon ➔ SequenFal esFmaFon Normalizing constant

13. PART. 1 PART. 2 PART. 3 Inverse problem strategy 13 ➔ Sampling Importance Resampling (SIR) algorithm 1 i N parFcles • Ref. RisFc et al. (2004), Beyond the Kalman ﬁlter 1) PredicFon π(xk|xi k−1) 2) Likelihood 4) Resampling (avoid parFcles with negligible weight) 3) Update π(xk|zk)(xi k, wi k) (xi∗ k , 1/N) • LimitaFon in the parallelizaFon • Loss of diversity (sample impoverishment) ISSUES wi k = π(zk|xi k)

14. PART. 1 PART. 2 PART. 3 Inverse problem strategy 14 ➔ New algorithm: Auxiliary Sampling Importance Resampling (ASIR) 1 i N parFcles • Ref. W. Da Silva et al., ApplicaFon to one-‐dimensional solidiﬁcaFon problem, COBEM 2011 • Key idea: improve the prior informaFon based on some point esFmate μi k using an auxiliary set of parFcles 1) PredicFon π(xk|xi k−1) 2) Likelihood 4) Resampling (avoid parFcles with negligible weight) 3) Update π(xk|zk)(xi k, wi k) wi k = π(zk|µi k) wi k−1 wi k = π(zk|xi k) (xi∗ k , wi∗ k ) • more realisFc parFcles • less sensiFve to outliers than SIR ADDED-‐VALUES FOR ASIR

15. 15 PART. 1 PART. 2 PART. 3 ApplicaFon to controlled burning experiment 15 Environmental condi)ons ➔ Reduced-‐scale ﬁre: 4m x 4m ➔ Homogeneous short grass vegetaFon •  Fuel bed depth: 8 cm •  Moisture content: 22% ➔ Mean rate of spread: 1-‐2 cm/s (max. 5 cm/s) ➔ ObservaFon: •  Error due to the resoluFon of the MIR camera •  Error esFmaFon: 5 cm (1% burning area) 2min14s 3min10s 2min42s 1min28s 1min46s ! Mean wind 1 m/s, 307° Time series of surface temperature ﬁeld (Ronan Paugam, King’s College of London) Time

16. 16 PART. 1 PART. 2 PART. 3 ApplicaFon to controlled burning experiment 16 3 control parameters ➔ Wind magnitude (ﬂuctuaFons between 0-‐2 m/s) ➔ Fuel moisture content (22%) ➔ Fuel parFcle surface/volume (11500 m-‐1) 2min14s 3min10s 2min42s 1min28s 1min46s ! Mean wind 1 m/s, 307° Time series of surface temperature ﬁeld (Ronan Paugam, King’s College of London) Time R(x, y, t) = f(uw, Mf , δf , βf , Σf , ...)

17. PART. 1 PART. 2 PART. 3 ApplicaFon to controlled burning experiment 17 ➔ Sequen)al es)ma)on: 5 successive esFmaFons of the control parameters SIR algorithm (N = 200) ASIR algorithm (N = 50) Results: •  Consistent results of the SIR and ASIR algorithms •  Good tracking of the observed ﬁre front.

18. PART. 1 PART. 2 PART. 3 ApplicaFon to controlled burning experiment 18 ➔ Sequen)al es)ma)on: 5 successive esFmaFons of the control parameters SIR algorithm (N = 200) ASIR algorithm (N = 50) Moisture content Fuel parFcle surface/ volume 99% Conﬁdence interval Mean value EKF result

19. PART. 1 PART. 2 PART. 3 ApplicaFon to controlled burning experiment 19 ➔ Sequen)al es)ma)on: 5 successive esFmaFons of the control parameters Wind magnitude (m/s) SIR algorithm (N = 200) ASIR algorithm (N = 50) Results: •  Same level accuracy reached by the SIR and ASIR algorithms •  ValidaFon against independent measurements of the wind velocity magnitude, even though the wind is subject to signiﬁcant ﬂuctuaFons In-‐situ measurements of the wind magnitude In-‐situ measurements of the wind magnitude

20. CONCLUSIONS ApplicaFons of parFcle filters to moving fronFer problems •  SIR and ASIR par)cle filters able to ➔ achieve mulF-‐parameter esFmaFon ➔ reduce fire modeling uncertainFes ➔ track fire front for a controlled burning experiment •  Valida)on of the ASIR algorithm: shown to be less computaFonally expensive than the SIR algorithm in a wide range of experiments [W. Da Silva et al., ApplicaFon to one-‐dimensional solidificaFon problem, COBEM 2011]

21. •  Comparison to Ensemble Kalman filter algorithm (CERFACS-‐University of Maryland, M. Rochoux’s PhD thesis) •  Applica)ons of ASIR par)cle filters to new fields of applica)ons (Wellington) ➔ temperature field predicFon of a mulF-‐layer composite pipeline ➔ reservoir history matching problem PERSPECTIVES ApplicaFons of parFcle filters to moving fronFer problems Parameter esFmaFon • CorrecFon on the model physics (dynamic learning) • Surrogate model of the fire spread simulator to limit computaFonal cost [Rochoux et al. (2012), CTR Summer Program] Polynomial Chaos

22. Thank you for your a/enFon!

23. Acknowledgments •  FAPERJ, CAPES and CNPq, Brazilian agencies and French Ministry of foreign affairs. •  Centre NaFonal pour la Recherche ScienFfique (CNRS). •  Project «11STIC06-‐I3PE-‐Inverse Problems in Physical Property EsFmaFon». •  Project «IDEA ANR-‐09-‐COSI-‐006-‐06, Wilfires: From PropagaFon to Atmospheric Emissions» •  Dept. of Geography, King’s College of London (MarFn Wooster and Ronan Paugam for the data of the controlled burning experiment).

Bayesian Inference for front-tracking problems - 2013 IPDO conference

Mélanie Rochoux

Bayesian Inference for front-tracking problems - 2013 IPDO conference