On the merits of sparse surrogates for global sensitivity analysis of multi-scale nonlinear problems: application to turbulence and fire-spotting model in wildland fire simulators. (English) Zbl 1464.62359

Summary: Many nonlinear phenomena, whose numerical simulation is not straightforward, depend on a set of parameters in a way which is not easy to predict beforehand. Wildland fires in presence of strong winds fall into this category, also due to the occurrence of firespotting. We present a global sensitivity analysis of a new sub-model for turbulence and fire-spotting included in a wildfire spread model based on a stochastic representation of the fireline. To limit the number of model evaluations, fast surrogate models based on generalized Polynomial Chaos (gPC) and Gaussian Process are used to identify the key parameters affecting topology and size of burnt area. This study investigates the application of these surrogates to compute Sobol’ sensitivity indices in an idealized test case. The performances of the surrogates for varying size and type of training sets as well as for varying parameterization and choice of algorithms have been compared. In particular, different types of truncation and projection strategies are tested for gPC surrogates. The best performance was achieved using a gPC strategy based on a sparse least-angle regression (LAR) and a low-discrepancy Halton’s sequence. Still, the LAR-based gPC surrogate tends to filter out the information coming from parameters with large length-scale, which is not the case of the cleaning-based gPC surrogate. The wind is known to drive the fire propagation. The results show that it is a more general leading factor that governs the generation of secondary fires. Using a sparse surrogate is thus a promising strategy to analyze new models and its dependency on input parameters in wildfire applications.


62K99 Design of statistical experiments
62P12 Applications of statistics to environmental and related topics
35F20 Nonlinear first-order PDEs
35F31 Initial-boundary value problems for nonlinear first-order PDEs
60G15 Gaussian processes
Full Text: DOI


[1] Viegas, D., Forest fire propagation, Philosoph Trans, 356, 2907-2928 (1998)
[2] Linn, R.; Reisner, J.; Colman, J.; Winterkamp, J., Studying wildfire behavior using FIRETEC, Int J Wildland Fire, 11, 233-246 (2002)
[3] Mell, W.; Jenkins, M.; Gould, J.; Cheney, P., A physics-based approach to modelling grassland fires, Int J Wildland Fire, 16, 1-22 (2007)
[4] Strada, S.; Mari, C.; Filippi, J. B.; Bosseur, F., Wildfire and the atmosphere: modelling the chemical and dynamic interactions at the regional scale, Atmos Environ, 51, 234-249 (2012)
[5] Finney, M.; Cohen, J.; McAllister, S.; Matt Jolly, W., On the need for a theory of wildland fire spread, Int J Wildland Fire, 22, 1, 25-36 (2013)
[6] McAllister, S.; Finney, M., Convection ignition of live forest fuels, Fire Safety Sci, 11, 1312-1325 (2014)
[7] Clark, T. L.; Jenkins, M. A.; Coen, J.; Packham, D., A coupled atmospheric-fire model: convective feedback on fire-line dynamics, J Appl Meteor, 35, 875-901 (1996)
[8] Potter, B. E., A dynamics based view of atmosphere-fire interactions, Int J Wildland Fire, 11, 247-255 (2002)
[9] Potter, B. E., Atmospheric interactions with wildland fire behaviour - i. basic surface interactions, vertical profiles and synoptic structures, Int J Wildland Fire, 21, 779-801 (2012)
[10] Potter, B. E., Atmospheric interactions with wildland fire behaviour - ii. plume and vortex dynamics, Int J Wildland Fire, 21, 802-817 (2012)
[11] Mandel, J.; Beezley, J. D.; Kochanski, A. K., Coupled atmosphere-wildland fire modeling with WRF 3.3 and SFIRE 2011, Geosci Model Dev, 4, 591-610 (2011)
[12] Filippi, J. B.; Pialat, X.; Clements, C. B., Assessment of forefire/meso-NH for wildland fire/atmosphere coupled simulation of the fireflux experiment, Proc Combust Inst, 34, 2633-2640 (2013)
[13] Viegas, D.; Pita, L., Fire spread in canyons, Int J Wildland Fire, 13, 274, 1-22 (2004)
[14] Viegas, D.; Simeoni, A., Eruptive behaviour of forest fires, Fire Technol, 47, 2, 303-320 (2010)
[15] Nijhuis, M., Forest fires: burnt out, Nature, 489, 352-354 (2012)
[16] Cruz, M.; Sullivan, A.; Gould, J.; Sims, N.; Bannister, A.; Hollis, J., Anatomy of a catastrophic wildfire: the black saturday kilmore east fire in victoria, australia, Forest Ecol Manag, 284, 269-285 (2012)
[17] Paugam, R.; Wooster, M.; Freitas, S.; Val Martin, M., A review of approaches to estimate wildfire plume injection height within large-scale atmospheric chemical transport models, Atmos Chem Phys, 16, 2, 907-925 (2016)
[18] Koo, E.; Pagni, P. J.; Weise, D. R.; Woycheese, J. P., Firebrands and spotting ignition in large-scale fires, Int J Wildland Fire, 19, 7, 818-843 (2010)
[19] Finney, M., FARSITE: Fire Area Simulator - Model Development and Evaluation, Tech. Rep (1998), US Department of Agriculture, Forest Service, Rocky Mountain Research Station
[20] Filippi, J.-B.; Bosseur, F.; Mari, C.; Lac, C.; Le Moigne, P.; Cuenot, B., Coupled atmosphere-wildland fire modelling, J Adv Model Earth Sy, 1, 4, Quarter4 (2009)
[21] Tymstra, C.; Bryce, R. W.; Wotton, B. M.; Taylor, S. W.; Armitage, O. B., Development and structure of Prometheus: the Canadian Wildland fire growth simulation model, Tech. Rep (2010), Natural Resources Canada, Canadian Forest Service, Northern Forestry Centre, Edmonton, Alberta, Information Report NOR-X-417
[22] Chong, D.; Tolhurst, K. G.; Duff, T. J.; Cirulis, B., Sensitivity analysis of PHOENIX RapidFire, Tech. Rep (2013), Bushfire CRC, University of Melbourne
[23] Lautenberger, C., Wildland fire modeling with an eulerian level set method and automated calibration, Fire Safety J, 62, Part C, 289-298 (2013)
[24] Sullivan, A., -2007. 2: Empirical and quasi-empirical modelsWildland surface fire spread modeling, 1990- 2007. 2: empirical and quasi-empirical models, Int J Wildland Fire, 18, 369-386 (2009)
[25] Gollner, M.; Trouvé, A.; Altintas, I.; Block, J.; De Callafon, R.; Clements, C., Towards data-driven operational wildfire spread modeling - report of the nsf-funded wifire workshop, Tech. Rep (2015), University of Maryland
[26] Cruz, M. G.; Alexander, M. E., Limitations of the accuracy of model predictions of wildland fire behaviour: a state-of-the-knowledge overview, Forest Chron, 89, 3, 372-383 (2013)
[27] Cruz, M. G.; Alexander, M. E.; Sullivan, A. L.; Gould, J. S.; Kilinc, M., Assessing improvements in models used to operationally predict wildland fire rate of spread, Environ Modell Softw, 105, 54-63 (2018)
[28] Jimenez, E.; Hussaini, M.; Goodrick, S., Quantifying parametric uncertainty in the rothermel model, Int J Wildland Fire, 17, 638-649 (2008)
[29] Liu, Y.; Hussaini, M. Y.; Oktenb, G., Global sensitivity analysis for the rothermel model based on high-dimensional model representation, Can J For Res, 45, 11, 1474-1479 (2015)
[30] Denham, M.; Wendt, K.; Bianchini, G.; Cortés, A.; Margalef, T., Dynamic data-driven genetic algorithm for forest fire spread prediction, J Comput Sci-Neth, 3, 398-404 (2012)
[31] Rochoux, M. C.; Delmotte, B.; Cuenot, B.; Ricci, S.; Trouvé, A., Regional-scale simulations of wildland fire spread informed by real-time flame front observations, P Combust Inst, 34, 2641-2647 (2013)
[32] Rochoux, M. C.; Ricci, S.; Lucor, B.; Cuenot, D.; Trouvé, A., Towards predictive data-driven simulations of wildfire spread - Part 1: reduced-cost ensemble kalman filter based on a polynomial chaos surrogate model for parameter estimation, Nat Hazard Earth Syst Sci, 14, 11, 2951-2973 (2014)
[33] Artes, T.; Cencerrado, A.; Cortes, A.; Margalef, T.; Rodriguez-Aseretto, D.; Petroliagkis, T., Towards a dynamic data driven wildfire behavior prediction system at european level, Procedia Comput Sci, 29, 1216-1226 (2014)
[34] Rochoux, M. C.; Emery, C.; Ricci, S.; Cuenot, B.; Trouvé, A., Towards predictive data-driven simulations of wildfire spread – part ii: ensemble kalman filter for the state estimation of a front-tracking simulator of wildfire spread, Nat Hazard Earth Syst Sci, 15, 8, 1721-1739 (2015)
[35] Zhang, C.; Rochoux, M. C.; Tang, W.; Gollner, M.; Filippi, J. B.; Trouvé, A., Evaluation of a data-driven wildland fire spread forecast model with spatially-distributed parameter estimation in simulations of the fireflux i field-scale experiment. Fire Safety Science: Proceedings of the 12th International Symposium, Fire Safety J, 91, 758-767 (2017)
[36] Rochoux, M. C.; Collin, A.; Zhang, C.; Trouvé, A.; Lucor, D.; Moireau, P., Front shape similarity measure for shape-oriented sensitivity analysis and data assimilation for eikonal equation, ESAIM, 63, 215-236 (2018)
[37] Manzello, S. L.; Cleary, T. G.; Shields, J. R.; Maranghides, A.; Mell, W.; Yang, J. C., Experimental investigation of firebrands: generation and ignition of fuel beds, Fire Safety J, 43, 3, 226-233 (2008)
[38] Sardoy, N.; Consalvi, J.; Kaiss, A.; Fernandez-Pello, A.; Porterie, B., Numerical study of ground-level distribution of firebrands generated by line fires, Combust Flame, 154, 3, 478-488 (2008)
[39] Kortas, S.; Mindykowski, P.; Consalvi, J. L.; Mhiri, H.; Porterie, B., Experimental validation of a numerical model for the transport of firebrands, Fire Safety J, 44, 1095-1102 (2009)
[40] Perryman, H. A., A mathematical model of spot fires and their management implications (2009), Humboldt State University: Humboldt State University Arcata, CA, Master’s thesis
[41] Perryman, H. A.; Dugaw, C. J.; Varner, J. M.; Johnson, D. L., A cellular automata model to link surface fires to firebrand lift-off and dispersal, Int J Wildland Fire, 22, 428-439 (2013)
[42] Tohidi, A.; Kaye, N.; Bridges, W., Statistical description of firebrand size and shape distribution from coniferous trees for use in metropolis monte carlo simulations of firebrand flight distance, Fire Safety J, 77, 21-35 (2015)
[43] Tohidi, A., Experimental and numerical modeling of wildfire spread via fire spotting (2016), Clemson University, South Carolina, USA, Ph.D. thesis
[44] Tohidi, A.; Kaye, N. B., Stochastic modeling of firebrand shower scenarios. Fire Safety Science: Proceedings of the 12th International Symposium, Fire Safety J, 91, 91-102 (2017)
[45] Pagnini, G.; Massidda, L., Modelling turbulence effects in wildland fire propagation by the randomized level-set method, Tech. Rep, 2012/PM12a (2012), CRS4
[46] Pagnini, G., Fire spotting effects in wildland fire propagation, (Casas, F.; Martínez, V., Advances in Differential Equations and Applications. Advances in Differential Equations and Applications, SEMA SIMAI Springer Series, 4 (2014), Springer International Publishing Switzerland), 203-216
[47] Pagnini, G.; Mentrelli, A., Modelling wildland fire propagation by tracking random fronts, Nat Hazards Earth Syst Sci, 14, 2249-2263 (2014)
[48] Kaur, I.; Mentrelli, A.; Bosseur, F.; Filippi, J.-B.; Pagnini, G., Turbulence and fire-spotting effects into wild-land fire simulators, Commun Nonlinear Sci Numer Simul, 39, 300-320 (2016)
[49] Saltelli, A.; Ratto, M.; Andres, T.; Campolongo, F.; Cariboni, J.; Gatelli, D., Global sensitivity analysis. the primer (2007), John Wiley & Sons, Ltd: John Wiley & Sons, Ltd Chichester, UK
[50] Storlie, C.; Swiler, L.; Helton, J.; Sallaberry, C., Implementation and evaluation of nonparametric regression procedures for sensitivity analysis of computationally demanding models, Reliab Eng Syst Safe, 94, 11, 1735-1763 (2009)
[51] Lamboni, M.; Monod, H.; Makowski, D., Multivariate sensitivity analysis to measure global contribution of input factors in dynamic models, Reliab Eng Syst Safe, 96, 4, 450-459 (2011)
[52] Ruiz, J. J.; Pulido, M.; Miyoshi, T., Estimating model parameters with ensemble-based data assimilation: a review, J Meteorol Soc Japan Ser II, 91, 2, 79-99 (2013)
[53] Sobol, I., Sensitivity analysis for nonlinear mathematical models, Math Model Comput Exp, 1, 4, 407-414 (1993)
[54] Sudret, B., Global sensitivity analysis using polynomial chaos expansions, Reliab Eng Syst Safe, 93, 7, 964-979 (2008)
[55] Marrel, A.; Iooss, B.; Laurent, B.; Roustant, O., Calculations of sobol indices for the gaussian process metamodel, Reliab Eng Syst Safe, 94, 3, 742-751 (2009)
[56] Iooss, B.; Saltelli, A., Introduction to sensitivity analysis, Handbook of Uncertainty quantification, 1-20 (2016), Springer International Publishing
[57] Le Gratiet, L.; Marelli, S.; Sudret, B., Metamodel-based sensitivity analysis: polynomial chaos expansions and gaussian processes, Handbook of Uncertainty quantification, 1-37 (2017), Springer International Publishing
[58] Owen, N.; Challenor, P.; Menon, P. P.; Bennani, S., Comparison of surrogate-based uncertainty quantification methods for computationally expensive simulators, SIAM/ASA J Uncertain Quantif., 5, 1, 403-435 (2017)
[59] Ciriello, V.; Di Federico, V.; Riva, M.; Cadini, F.; De Sanctis, J.; Zio, E., Polynomial chaos expansion for global sensitivity analysis applied to a model of radionuclide migration in a randomly heterogeneous aquifer, Stoch Env Res Risk A, 27, 4, 945-954 (2013)
[60] Després, B.; Poette, G.; Lucor, D., Robust uncertainty propagation in systems of conservation laws with the entropy closure method. Springer International Publishing, 105-149 (2013)
[61] Dubreuil, S.; Berveiller, M.; Petitjean, F.; Salaün, M., Construction of bootstrap confidence intervals on sensitivity indices computed by polynomial chaos expansion, Reliab Eng Syst Safe, 121, 263-275 (2014)
[62] Xiu, D., Numerical methods for stochastic computations: a spectral method approach (2010), Princeton University Press: Princeton University Press Princeton, New Jersey, US
[63] De Lozzo, M.; Marrel, A., Sensitivity analysis with dependence and variance-based measures for spatio-temporal numerical simulators, Stoch Env Res Risk A, 31, 6, 1437-1453 (2017)
[64] Le Gratiet, L.; Cannamela, C.; Iooss, B., A bayesian approach for global sensitivity analysis of (multifidelity) computer codes, SIAM/ASA J Uncertain Quantif, 2, 1, 336-363 (2014)
[65] Marrel, A.; Perot, G.; Mottet, C., Development of a surrogate model and sensitivity analysis for spatio-temporal numerical simulators, Stoch Env Res Risk A, 29, 3, 959-974 (2015)
[66] Oakley, J.; O’Hagan, A., Probabilistic sensitivity analysis of complex models: a bayesian approach, J Roy Stat Soc B, 66, 3, 751-769 (2004)
[67] Rasmussen, C.; Williams, C., Gaussian processes for machine learning (2006), MIT Press: MIT Press Cambridge, MA, US
[68] Roy, P. T.; El Moçayd, N.; Ricci, S.; Jouhaud, J.-C.; Goutal, N.; De Lozzo, M., Comparison of polynomial chaos and gaussian process surrogates for uncertainty quantification and correlation estimation of spatially distributed open-channel steady flows, Stoch Env Res Risk A, 32, 6, 1723-1741 (2018)
[69] Schoebi, R.; Sudret, B.; Wiart, J., Polynomial-Chaos-based kriging, Int J Uncertain Quan, 5, 2, 171-193 (2015)
[70] Blatman, G., Adaptative sparse Polynomial Chaos expansions for uncertainty propagation and sensitivity analysis (2009), Université Blaise Pascal, Clermont-Ferrand, Ph.D. thesis
[71] Blatman, G.; Sudret, B., Adaptative sparse polynomial chaos expansion based on least angle regression, J Comput Phys, 230, 6, 2345-2367 (2011)
[72] Migliorati, G.; Nobile, F.; Von Schwerin, E.; Tempone, R., Approximation of quantities of interest in stochastic PDEs by the random discret L2 projection on polynomial spaces, SIAM J Sci Comput, 35, 3, A1440-A1460 (2013)
[73] Sethian, J., Level set methods and fast marching methods (1999), Cambridge University Press: Cambridge University Press Cambridge, UK
[74] Osher, S.; Fedkiw, R., Level set methods and dynamic implicit surfaces, 153 (2003), Applied Mathematical Sciences - Springer
[75] Mallet, V.; Keyes, D.; Fendell, F., Modeling wildland fire propagation with level set methods, Comput Math Appl, 57, 7, 1089-1101 (2009)
[76] Lautenberger, C., Mapping areas at elevated risk of large-scale structure loss using monte carlo simulation and wildland fire modeling. Fire Safety Science: Proceedings of the 12th International Symposium, Fire Safety J, 91, 768-775 (2017)
[77] Mentrelli, A.; Pagnini, G., Front propagation in anomalous diffusive media governed by time-fractional diffusion, J Comput Phys, 293, 427-441 (2015)
[78] Byram, G. M., Combustion of forest fuels, (Davis, K. P., Forest Fire: Control and Use (1959), McGraw Hill: McGraw Hill New York), 61-89
[79] Alexander, M. E., Calculating and interpreting forest fire intensities, Can J Bot, 60, 349-357 (1982)
[80] Sofiev, M.; Ermakova, T.; Vankevich, R., Evaluation of the smoke-injection height from wild-land fires using remote-sensing data, Atmos Chem Phys, 12, 4, 1995-2006 (2012)
[81] Xiu, D.; Karniadakis, G., The wiener – askey polynomial chaos for stochastic differential equations, SIAM J Sci Comput, 24, 2, 619-644 (2002)
[82] Jones, E.; Oliphant, T.; Peterson, P., SciPy: Open source scientific tools for Python (2001)
[83] Chu, K. T.; Prodanović, M., Level set method library (lsmlib), Tech. Rep. (2009)
[84] Bevins, C. D., Firelib: user manual and technical reference, Tech. Rep (1996), US Forest Service, Missoula Fire Sciences Laboratory, Fire Behavior Research Work Unit Systems for Environmental Management
[85] Baudin, M.; Dutfoy, A.; Iooss, B.; Popelin, A.-L., OpenTURNS: an Industrial software for uncertainty quantification in simulation. Springer International Publishing, 2001-2038 (2017)
[86] Efron, B.; Hastie, T.; Johnstone, I.; Tibshirani, R., Least angle regression, Ann Statist, 32, 2, 407-499 (2004)
[87] Wales, D. J.; Doye, J. P.K., Global optimization by basin-Hopping and the lowest energy structures of lennard-Jones clusters containing up to 110 atoms, J Phys Chem A, 101, 28, 5111-5116 (1997)
[88] Baudin, M.; Boumhaout, K.; Delage, T.; Iooss, B.; Martinez, J.-M., Numerical stability of Sobol’ indices estimation formula, 8th International Conference on Sensitivity Analysis of Model Output, Réunion Island (2016)
[89] Pedregosa, F.; Varoquaux, G.; Gramfort, A.; Michel, V.; Thirion, B.; Grisel, O., Scikit-learn: machine learning in python, J Mach Learn Res, 12, 2825-2830 (2012), arXiv:1201.0490
[90] Roy, P. T.; Ricci, S.; Dupuis, R.; Campet, R.; Jouhaud, J.-C.; Fournier, C., Batman: statistical analysis for expensive computer codes made easy, J Open Source Softw, 3, 21, 493 (2018)
[91] Taylor, S. W.; Woolford, D. G.; Dean, C. B.; Martell, D. L., Wildfire prediction to inform fire management: statistical science challenges, Stat Sci, 28, 586-615 (2013)
[92] San-Miguel-Ayanz, J.; Moreno, J. M.; Camia, A., Analysis of large fires in european mediterranean landscapes: lessons learned and perspectives, Forest Ecol Manage, 294, 11-22 (2013)
[93] Hernandez, C.; Keribin, C.; Drobinski, P.; Turquety, S., Statistical modelling of wildfire size and intensity: a step toward meteorological forecasting of summer extreme fire risk, Ann Geophys, 33, 1495-1506 (2015)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.