Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | ) 2002 ( Open Access Fujioka Discussions Sciences Natural Hazards Hazards Natural ). Basically it is clear from and Earth System System and Earth 2013 ( 1 Filippi et al. 3220 3219 , and B. Nader 2,3 erent front velocity models. The results are compared with several ff , V. Mallet 1 erent sizes, which come with the limited information available in an operational ff Nevertheless, the problem of evaluating model performance must be tackled as it This discussion paper is/has beenSystem under Sciences review (NHESS). for Please the refer journal to Natural the Hazards corresponding and final Earth paper in NHESS if available. CNRS SPE, Università di ,INRIA, BP BP 52, 105, 20250 78 Corte, 153 CEREA Le (joint Chesnay laboratory cedex, France École des Ponts ParisTech & EDF R&D, université Paris Est), establish model’s potential errors andformance is credibility. The to be first able stepscoring to to methods evaluate evaluate single have model simulation been results per- the against subject observations, of such several studies, initiated by community. 1 Introduction Model evaluation requires comparing predicted to observed values and is critical to carried out with 4 di error scoring methods applied toin a each fully of automated the manner, with 320of simulations simulations. the ran All as models first tasks or guesses are cases.including with performed This no some tuning approach of for leads the any a badRegardless wide simulation the range results of quality to simulation bebased of performance, expected on the in their an input performance operationalempirical data, and context. ones. it that Data is the and most found source complex that code models the used outperform models for the this can more paper be are ranked freely available to the This paper presents the evaluationof of several observed fire fires. propagation models The usingof observation a di base large set is composedcontext of (burned surface 80 and Mediterranean approximative ignition fire point). cases Simulations for all cases are Abstract Nat. Hazards Earth Syst. Sci.www.nat-hazards-earth-syst-sci-discuss.net/2/3219/2014/ Discuss., 2, 3219–3249, 2014 doi:10.5194/nhessd-2-3219-2014 © Author(s) 2014. CC Attribution 3.0 License. is important to knowtinue if the a process parametrisation of or enhancing a models, new codes formulation and is data. superior, and From an to operational con- point of and recently compiled andall extended these studies in that aas single it value cannot only be givesa representative human of limited eye a insights provides model a on performance, better all understanding of aspects what of was good performance, and while what went the wrong. analysis of 1 2 3 Marne-la-Vallée, France Received: 16 April 2014 – Accepted: 17Correspondence April to: 2014 J.-B. – Filippi Published: (fi[email protected]) 8 MayPublished 2014 by Copernicus Publications on behalf of the European Geosciences Union. Evaluation of forest fire modelsobservation on database a large J.-B. Filippi 5 20 25 15 10 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | , , i e T erent ff ), a spa- a T , height l, d is taken here in the normal ρ ) and is based n V · v 1952 = ( and mass exchange s defined as the fraction s W m and temperature a ) forms the basis of the United ρ 1972 ) carried out such comparison with , density of live/dead , Byram and Fons σ ). A major step in such model evalua- , stoichiometry h 2013 ∆ ( , air density 2013 ( v 3222 3221 and moisture content erent scoring methods, which fulfills the goal of Rothermel v ff  ). As this study focuses on the evaluation of large b , . This “rule of thumb model” is sometimes used by s W 2009a 0.03 , , emissivity v = Cruz and Alexander S ). Each model prognoses the fire front velocity V , pointing toward the unburned fuel. ˙ , combustion enthalpy Alexander and Cruz σ n v , p Sullivan c ). It builds on earlier works from erent models are presented in the first section along with the simulation ff 1986 , Depending on their complexity, the models can take into account the terrain slope, The four di Andrews locity everywhere on the fireas front, wind the wind velocity, normal and firefighters, to with the caution front because ofof being, its hopefully, lack the of lowest reliability. reference It in will terms serve of2.2 here performance. the purpose Rothermel model The quasi-empirical Rothermel model ( 2.1 Three-percent model The first and most simpleing model makes at the three strong percents assumption of thatof the the the fire wind vegetation is velocity, changes propagat- as or long the as terrain there slope. is In fuel practice, available, in regardless order to compute the ve- the atmospheric properties (wind velocity tial characterization of thesurface to fuels volume (mass ratio loading of water over total weight)calorific and capacity the fuel combustionrate properties due (ignition to temperature pyrolysis direction to the front fire-front velocity. A velocityburned areas. is A obviously fire-frontsame not solver for enough code all and to models inputselection and obtain data is described fire unfortunately are in not needed. progression the exhaustiveon These of next and representing all two section. some existing are Note kind formulations, that the of but evolution the rather in focuses proposed the model model types. scale fire simulation, our selected definition of “fire model” is the formulation of the as noted recently by view, it also appears that models are used with a clear lack of systematic evaluation States National Fire Danger Rating( System and fire behavior prediction tool BEHAVE 2 Models description Fire propagation modeling canor refer even to datasets ( a vast family of codes, formulations, systems method used to compute thetion front method, propagation. data The preprocessing second section anddiscussed details numerical in the set-up. the evalua- The last results section are along presented with focuses and on specific cases. is the Mediterranean islanda of variety Corsica of where configurations. numerousmodels The wildfires and test occur a consists every large incomparison year number running scores in of simulations between simulated observations, with and and fourrun observed di compiling in fires. a the These fully simulations results automatedjustment must in manner, of be without the fuel, observation wind form or biases of ignitionare introduced location the by in distributions manual order of ad- to scores, enhance usingranking results. di the The models overall results according to their specific use. area. While evaluating theis absolute proposed model here to performancebe evaluate is specific linked yet model to out performance. a This typical ofresponds specific model the to evaluation usage question, the will and it plausiblewith to “first no a direct territory. guess” observation The case of typical where the usage wind just proposed or cor- an fuel ignition moisture near location the is fire. The known selected area tion is to compareexists observed in rate of the spread literature. the (ROS) clear with and simulated reassuring ROS, conclusionmay as that well provide much empirical a data and good semi-empirical ROSels built approximation. models to Our simulate study focuses the on overall the two-dimensional use fire of spread these mod- and its corresponding burned 5 5 15 20 10 25 25 15 20 10 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | ) such as moisture cient on all fuel types. Rothermel and Ander- ffi 1972 ( ) . (2) 3 ) χ ). This model is widely used toward the unburned fuel. The the water evaporation enthalpy). γ m/M w 1966 1) were used. h , Rothermel 0.1)) . (3) ∆ = + 2.59( ). It highly relies on a set of parameters s β η − ( v 2 ) S ] is the contribution of the vegetation under- 1971 χ ( w . (4) McArthur q 3224 3223 h ∆ 0.54 v m/M ) , (11) , (1) S )) , (9) m γ × 0.681 , )) + op ); . (8) v ) λ ) like Rothermel can be classified as a quasi-physical + ( a 1.5 5.11( v Frandsen T /S ˙ S σf β/β + − 0.02526 i s h ) 2009 − T χ , ) ( ∆ 138 W ) v 0 P (1 , − χ p A φ 0.0594 c 0.55 of the combustion energy is released as radiation because the + v exp(0.792 [ m/M + ) + 1 S 0 exp( σ a V − χ exp( 2 T . (6) ) m . (5) 7.27) . (10) φ , 0.3) and mineral damping ( v A is the Boltzmann constant and 2 2.59( e/ S )) − + Balbi et al. m = (495 4 i , α 0.133 B − / op χ (1 σ 0.1 − v 0.8189 , 0.3 ξ 1.116 (1 BT (1 M − r v S − v e I s + , × S  i β β/β 0.2595 ( . (7) T ) and Australian experiments ( Hη , = l 1.5 v + v 0 ∆ 250 0 S Max  V d /ρ (4.774 ( σ 3.338 R d 0 7.47exp( 5.275 / 0 = ρ 1966 ( ( V ρ 1 = R (192 , = = = This quasi-physical model uses also a number of fitted parameters (in US Customary = = = = = V P 0 0 = Max op r 2.3 Balbi model The Balbi model ( R The reaction rate is given as R where A β The maximum reaction rate is The packing ratio is given by on a heat balance developed by V where going pyrolysis ( the flame and aonly “conductive” a part given within portion theflame is fuel viewed layer. as Assumption a isequation tilted governing also radiant the panel made propagation with that velocity an of angle the front reads An optimal packing ratio with β model. Its formulation isdiating based panel on in the thewind, assumption direction normal that terrain to the and the front fuelergy front. propagates configuration directed The as toward model the a the verifies absorbed ra- that unburned energy for fuel. equals a This specific the energy combustion is the en- sum of “radiant part” from φ The slope factor is φ with reaction intensity I A propagating flux ratio given by ξ The wind factor is of extinction ( units) that read: V obtained from wind tunnelsson experiments in artificial fuelalso beds in Mediterranean ( countries forally various requires purposes some adjustments (fire by riskSince experts and this in behavior), study order but is to usu- be createdperformed fully as here; e a hence blind the test default for values each given in model, these adjustments were not 5 5 25 10 20 15 20 25 15 10 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | v of i (17) cient n ffi 2 is also , such as / κ m and wind d α = f are always avail- d the marker index. i κ · is an evolution coe or its curvature µ λ ) . (16) ) uses a front tracking method γ ( + and curvature is the local front speed. The max- R i λ 2009 H v ) , γ ] , (15) w h ) , (12) cos ∆ γ − , and that all energy is absorbed within the ( depends on the slope angle m + γ . (13) 3226 3225 R γ + ) , (14) κ/τ n ) H γ sin Filippi et al. ) a , = T γ λ + ( V − with with a reasonable numerical cost, introducing in the . Because of these assumptions, front velocity is not i (1 ˙ σf 0 ˙ cos T is the burning duration given by the Andersen model σ γ κ ( V h ) v − , τ s ∆ p γ 0 + cos c χ and [ λ σ sin and + λ µλ 2(1 2 ) v + a / + ) S T , determines the resolution of the front propagation and should e/ 2 , (1 ) n 4 ) i · n γ ( i m allowed between two consecutive markers is called the perimeter κ δl , v n , BT ( · σ ). The flame tilt angle m v ) a , cos refers to the value of a variable at the next step, and d n  cos ( ρ λ i δl δl R ,  v 1) − + µτ d e + + 4 1969 , is the Heavyside function for positive reals and + ) n i λβ α + (1 ( n ) , · T ( i + e n , 2 defines the direction of propagation, and ( i = R x v t a radiation dumping ratio that depends on fuel packing. The second term re- − i tan )  = H = ( γ 1 d = ) = , 0  β 1) γ V λ 1) γ , , + + = The main disadvantage of the model is that it is now tight to a solver able to locally A major assumption of Balbi and Rothermel models is that the fire is always traveling n = κ V n ( i ( i 0 ( ( Anderson t x tan V with surface ratio is noted ( f where of the ratio between radiated energy and released combustion energy. The volume-to- The second term accounts for the propagation by radiation and reads needed to avoid over-crossingThe of advection scheme two is markers an and first-order Euler potential scheme inversion in of space: the normal. f that relies onmarker a discretization withimum Lagrangian distance markers. Theresolution. If outward two markers normal areis further carried away than in this order distance, to a keep re-mapping the of resolution the constant. front A filtering distance where the Spatial increment and moves the requirement for the stationary speed with and constantly diagnose process some additional numericaltracking code errors. ForeFire. The solver used for the2.5 study is Fire the propagation front method The fire propagation solver ForeFire ( model was rather simplereads: as it removed strong assumptions. The updatedV formulation 2.4 Balbi non-stationary model By using the front trackingable solver, as local front numerical depth diagnostics of the front. The introduction of these variables in the projected onto the normal to the front: at a stationary speedfuel that bed verifies for the computationdependent of on the localaccelerate fire or state (previous go intensity,terrain to front properties. curvature, extinction, These depth). and assumptions Itof it are cannot spread is required without only to knowingin compute explicitly dependent the a the on local priori BEHAVE the front tool. potentialor depth local Later rate extinction. versions fuel, A of wind more Rothermela and fundamental non-stationary added approach formulation. sub-models was for developed acceleration for the Balbi model with 5 5 20 15 10 25 10 15 20 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | , Forthofer ). Data within GDAL Development ( 1 m. ≈ δl http://www.ign.fr/ ogr2ogr erent cases. Once a simulation ) a repository of wildfire observa- ff 3228 3227 . erent from the real values. For the considered fires, ff 3.3 http://www.promethee.com/ ers precise burned surface for many wildfires, along with an informa- ff ) using a conformal projection. 201x , ). WindNinja inputs consist of the elevation field and the wind station data. The The elevation field originates from the IGN BDAltiWind at information 25 originates m from resolution. the nearest It of is 12 originally automatic 10 m-high weather sta- Wind fields is extrapolated with the WindNinja mass consistent code ( The simulation configuration defines the size of the domain, the date and the numer- The design of the study implied that the data was not tailored for any model or test ical set-up. The rest ofextent the input and data date is extracted using from data the files GDAL with the library same and domain its tool case. All of thedate. preprocessing We is are thus aware automaticallyseem that done totally this for unrealistic automatic the compared generationfuel ignition to will location state the generate observation. and can input In data beor particular, that significantly for wind might di any direction fire and thatmodels may are ignite not at observed anytime,provided or anywhere, in stored. the an This operational exact study context. inputs tries to the to simulation rely on the best available data The first step inthat order it to can launch beForeFire, a the processed simulation input by is data the toseveral is fields fire compile composed of propagation and of wind direction format solver. a and For configuration data speed, file, the in a an a fire land elevation use way simulation field, field code and one a or fuel field. 3.2 Data preprocessing A selection of 80 firestudy. cases For have each been fire, compiledconditions into the and an the required observation data database initialare required for then data by this run the is by selected preprocessedresult distributing propagation is the to model. available computation The for generate of simulations anthe the the observation/simulation scores initial di pair, introduced the in comparison Sect. is computed with usually fire breaks such as roads, i.e., in typical simulations 3 Evaluation method be smaller than the smallest space scale influencing the fire propagation, which are wind field is outputted atwhole the duration same of resolution the aschange simulation the direction (under elevation during the field the strong and fire). assumption is that used the for wind the does not tions of the Météo Francewind network, speed relatively and evenly direction distributed isstation. across taken the at nearest island. time The with data available from this2007 nearest Team available in DEM ASCII format. model uses. ignition points and date,lection which of reduced the the Corsica amountas island of adequate was relevant data made observations. and because homogeneity The(mostly in field se- fire expertise shrubs dynamics was and given available, theone Mediterranean as relatively hectare limited well maquis). to dataset Fire several hundreds sizes hectares, in in order the to be selection representative ranged of from all potential tions managed by the “Institutthis Géographique database National” o ( tion on the ignition datealways and be sometimes location. trusted, Nevertheless, so this each information cannot case was reviewed with a field expert in order to validate 3.1 Observation database The observation databaseranean is island composed of of Corsica 80the between Prométhée fires database 2003 ( that and 2006. all These occurred cases in were the extracted Mediter- from 5 5 25 15 20 10 20 15 10 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | ) (23) (24) in the 2011 stands , t ) for the ) for the o ( X ( o f o t 1981 T ( X , ),0)d Santoni et al. X    ( o X ) that takes into account is the simulation domain. e ))d P . − T X Ω . We assume that the ignition ) ( cient compare the final simu- Anderson et al.    o t X ). ): ffi |S| ( d o f o f t t

( T , ) − T f t S t f (

t ) ( ) for the simulation and ) the burned surfaces at time o f , (19) ) f t \S t max( t X ( ) ( ( , (18) | ( ) o f max( o ) ) o S f t \S t

u u | , (21) ) ( Z T ( S ) t t is denoted ) = , (22) o f o o ( (

u t u

| ( u t Z ). t S ) S o t ( ∩S ( . At the end of the observed fire )) ) u 3230 3229

f S f u t ∩ S ∪ S t t [ ( t |S ( + f ) ) ( t ∩ S S , u u Z + 2013 o f ) t t    X \S | t ( ( ( ), with an adjustment ( u [ ) ) t ∪ S o o a Ω | u o | f ( + ) 2 P t t S S | is denoted o ( Ω t |

u , |

) f t o ). We denote ),0)d d S Ω ( t u | X

S t o

X | = ( ) ( 2 S o o f ( t , (20) \ ( 2013 | = max( ( ) \S o e Ω − T t e

cient ( Filippi et al. Ω P )

o f P \S o

f ffi 1 t ) |S − t + − t ( ( + a o 1

| P S ) ) u

u max( t = t ∪ S ] ) ( ( o o f f ) t t f Z ( t | |S o Filippi et al. 2 ( ]0, | ∩ S

Z Ω    \S ) ) | S ) Ω f f u u |

t 1 t t t ( ( ( S − o o − + 1 S S 1

= = = = Sørensen similarity index and Jaccard similarity coe Fuel distribution data is taken in vector format (shape file) from the IGN IFEN (“Inven- We relied on the following scores, with e a ATA SA P and taire Forestier National”), with locally 10vegetation fuel in classes Corsica. corresponding Fuel to the parameters main are burning derived from The 4 models were run320 using simulations. the Each ForeFire simulation code,to is with carry set-up 80 out using cases the the it simulation. ignition resulted The point in simulation that a domain defines total is of where centered on the ignition point. In at the end of thetially fire (i.e., evaluate the the final time burnedscores surface), evolution may these of be scores the were found simulation designed in to dynamics. par- Further details3.4 on these Simulation set-up lated and observed areas.ulation Kappa agrees statistics with computes observationagreement the ( by chance. frequency The with arrivalinto time which account agreement sim- the and dynamics the of shape the agreement simulation. both Even take though there is only one observation where P 3.3 Comparison scores Our evaluation reliesproposed, on in scoring methods that were analyzed, and for two of them Jaccard Similarity Coe Kappa Statistics grass and pine forest types,is while used for the the Proterina shrubs parametrisationture and is ( maquis. set As as little moderate ishigh to known fire about high danger the water in fuel stress summer state, for the every day. fuel fuel model, mois- which corresponds to for observation), the observedThe arrival burned time surface at is someobservation. point Finally, the area of atime surface is 0. Sørensen Similarity Index simulation. The final simulation time is 5 5 15 10 20 15 10 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | f d cient and kappa ffi . The filtering distance 2 is in m A 4) m, if − 3232 3231 A depends on the fire size. If the final observed 10 δl max(1,log = δl , then A . δl cient, are very similar with worst score distributions for the 3-percent model. The ffi Based on all the distributions, a conclusion is that the non-stationary Balbi model In Fig. 1, the scores are sorted in ascending order, independently for each model. All simulations were carried out at most until the burned area equals the observed is taken into account, leads to very significant improvements, at least as high as the and the 3-percent modeltypes the of worst. fires It and means conditions.agreement, there The since is conclusion it is a may not lack consistent relevance soperformance improvement in clear this for suggests concerning context. all the The that large arrival the variance time needed in generation in the of models the large simulation ensemblesa process single of of fire simulations and a may weak single be data fire. quality, a Indeed, poor with simulationconsistently a is single likely gives to simulation the occur. for model. best It is results. worth The noting that second the switch model to a is non-stationary model, then where the the fire stationary depth Balbi tions are shown inthe Fig. models. 2 The with sorting anpaired ascending is in sorting therefore the based the plot. on sameat There the for each is mean all individual clearly score fire. models a There across portantly, and large is for their variability however most in scores an the fires, overall remain ascending performance the tendency of non-stationary and, the Balbi more models im- model shows the best performance the most reliable models,the but distribution. its We stationary point version outwhich performs is that better a all at very observed rough theditions, fires approximation high arrival to are end time the assumed actual agreement of to duration isas last of a it at the less strongly fire most relevant penalizes and, scoring 24 over-prediction. in h, method these than con- the other scores It is therefore impossiblemake to sure compare that the a performance model for consistently each shows individual better fire performance. and The to score distribu- additional improvements over Rothermel simulations.clearly The provides non-stationary the Balbi bestwith model overall the results distributions for ofmodels this shape with data-set. agreements the The distributions (Fig. samemodel of 1e). still ranking seems the It is to arrival is be found time the harder worst agreements to model. (Fig. The discriminate 1d). non-stationary the Balbi The model 3-percent is still among Rothermel models gives significantly better results. The Balbi model arguably brings classical scores, Sørensencoe similarity index, Jaccard similarity coe An important aspectare of presented. the At comparison first, isand let scoring to consider method. select The the andIt 80 distribution understand is scores of the important are the wayso to here 80 scores the sorted note scores number in that for ascendingtions in the each order (per the sorting model and model), abscissae is plotted. The corresponds but carried Fig. not to out 1 a to independently shows rank the for the in each same distributions model; the fire for list case all of in scoring sorted all methods. simula- 4 The distinct distributions evaluated for models. the and observation). The performancelyzed since of it each can model varyconclusions for a that lot, individual are depending supported case on by is the the quality hardly overall of performance. ana- the4.1 data. We essentially Distribution draw of the scores larger because the stopping criterioncannot is be checked more every 7 than min 7the (so min front the of velocity over-development fire is propagation). zero everywhere The (stopped simulation fire). can also stop earlier if 4 Results In this section, weAll investigate simulations are the concisely performance described of in the appendix, with models plotted on contours all (simulations fire simulations. of the fire.burned The area spatial is increment is set to 20 final burned area. In practice, on small fires, the area burned in the simulation may be north-south and east-west directions, the domain size is about four times the extension 5 5 25 15 20 10 25 20 15 10 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | . 2 cult to understand the reason ffi 3234 3233 erent light on a model. ff . 1 ). The dynamic-aware scores (arrival time agreement and shape erences between the models simulations. When we only consider 4.1 ff : the case (21 August 2004, N3 in appendix) and the Santo Pietro di Tenda 1 In Fig. 4, we plotted the case for which the Balbi model attained its best shape The Sørensen, Jaccard and kappa scores identify the same “best” cases. It is con- observed, so the locationwe was computed both burned, the just burn like probability the and the model detection “predicted”. skill. Consequently, See Table be of high importance.to One evaluate way its to burn assesslocation, probability. the we When compute reliability the the of modelare frequency a interested predicts with model in that which the in this a frequency thisburn. with actually fire We which context happens. will refer a is Conversely, to model burn we this correctlyif a indicator predicts the that as given model a the location burns detection will skill. verybe The large burned detection in regions: skill the any simulation may location aswill be that well. perfect be is On actually the perfect contrary, burned if the in first the reality indicator (burn will fire probability) stops right after it started: at the ignition point, the fire was contributed to the shapedynamics agreement. in This the evaluation suggests can that cast taking di into4.3 account the front Forecast reliability In an operational context, the ability of a model to predict that a location will burn can the other models questions themight reliability of be the erroneous input and data.possible It compensating that is the for likely Balbi that deficiencies the modelthat in input would exposing data the under-perform the Balbi with actual the model.models prediction exact and It data. the skill is exploration This of of fairly suggests the fire full simulation probability requires distribution ofagreement. the the The use uncertain visual of data. agreement several as with the in observed the contour previous does fire not case. seem as Hence good the dynamics of the fire must have significantly than the observed one, while theover-burning Rothermel at model the and head the of 3-percent thecan model fronts. show So, still even large be when the large wind di the direction Balbi is simulations correct, for there thisprediction fire case, skill we might once conclude proper that a input model data can reach is good provided. Nevertheless the performance of The overall shape of the final simulated contour by Balbi model has similar aspect ratio cases are shown in Table 4.2 A look at the individualFor simulations many simulations, themance. input When data the is datawhich is really is probably poor, reliable, detected hence the by afound models the the dramatically may simulation scoring low perform with methods. reasonably perfor- the well, For highest performance each across scoring the method 80 fires. and These model, selected we to rank models. In order toof conclude the on Rothermel the Rothermel model rank,The would a 3-percent be renewed model implementation required, is asis clearly well needed the as to worst an attain of extendedinput the state-of-the-art database data four. of models may This be fires. performance, suggests of evendata poor that in quality quality. more Overall, a (which it physics context is weranked, where interesting might hence to the expect even note a in that, poor despite operational database the forecasts), can poor help the objective models model are development. clearly in the numerical modelby is removing a assumptions key to aspectThe the of Rothermel a model model is simulation can rankedcause and enhance third making we its in it used this versatility more study the andbeen “physical” but original developed performance. we in should formulation the mitigate of context this Rothermel, of rank Corsican be- and fires. the The Balbi study formulation is more has about the method changes due to the fire spread rate. This suggests that the representation of the fire full dynamics. Let us considerble more closely two fires which(1 appear July several 2003, times in T0 Ta- clear in that appendix). In the Fig. Balbi 3, model we shows plotted a the much former better case. performance In than this the case, it other is models. sistent with the strong similarities founding between methods the (Sect. scores distributions for theseagreement) scor- select a varietyfor of these fire selections cases. from It the is final more contours, di since these scores take into account the 5 5 25 15 20 10 25 15 20 10 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | . . The results are displayed in Figs. 5–7. 3 3236 3235 ). With models results being so variable, a promising http://anridea.univ-corse.fr/ 2011 ( This research is supported by the Agence Nationale de la Recherche, Finney et al. erent. The vegetation cover and the associated fuel load were not tuned by ff Considering the high uncertainties in both models and the input data, the simulation We evaluated the skill of the models to forecast that a certain location will be burned. Despite the crude application setting, we were able to rank these firespread models. Overall, the performance may not be good enough for a model to be reliable in The burn probability and detection skill confirm previous results. The non-stationary context is barely deterministic.developed There by is a need for probabilistic approaches such as We also evaluated whether the locationsreality. In burned both in a cases, simulation there is isthe likely a best to wide model be spread may burned not in in before the high we models enough conclude skills. for that In a further addition, reliableinput the work use data, skill on in especially of the operational for physical context. the There- models windgains is direction, in are still obviously performance needed needed. and may For improved properly instance, be largest forecast obtained wind from direction. local micro-meteorological forecasts that pect, just like thewithout rate tuning) of was spread. rankedtaking The third. 3 Finally, Rothermel percents we of model, observed the (in windthat that velocity, current its is the firespread clearly first most models not may original empirical a benefitpoor good model, form from data option. a and quality. Overall, better it physical is description, guessed even with performance. This suggests that the numerical treatment of the fire front is a key as- performance (compared to theof other models) the was 80 consistently cases. observed The across stationary most version of Balbi model showed a significantly lower of 80 fires whosea final purely burned automated surfacesoperational manner, were context. using observed. The only meteorological Westation, values poor simulated even were data-set these though taken fires that the aticantly actual in the may di wind closest be direction observation available atany in the means exact this for fire any location cases or may models. be signif- The non-stationary version of the Balbi model gave overall the best results. Its higher 5 Conclusions The objective ofin this a paper realistic was operational to context with evaluate limited firespread information. models We and considered their a database relevance percent model with a detectionof skill the of non-stationary 11 Balbi % model. is of very low valueoperational compared applications. to the The 42 performancefurther % spread development on among the theNonetheless, it model models is has suggests clear from the that In the particular, potential simulation an to results erroneous that strongly windmeteorological the forecasts direction improve input alone spoiled the data would so plays be results. many a likely simulations key to that role. dramatically good improve performance. local Balbi is still the bestmodel. model, When followed by the the non-stationary (stationary) Balbireal model Balbi predicts model fire that and burned a the it location Rothermel in willnon-stationary 29 be Balbi % burned, model of the would all have predictedthat cases. it When the in the performance 42 real % varies of fire a all reaches cases. lot a It given from is location, noteworthy one the model to another. It is obvious that the 3- project ANR-09-COSI-006, IDEA Acknowledgements. This section showsInformation the about the simulation fires is results collected in of Table the 4 models, for all considered fires. Appendix A Simulations for all fires direction may be the useSuch of developments could ensembles be of used models,ensemble-based for forecasting. together uncertainty with quantification, perturbed risk input assessment data. and 5 5 25 20 15 10 25 15 20 10 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | , 3224 3221 3221 , 2009. 3229 3230 , 3222 , 2009b. 3229 , 2009a. , 3228 3220 3223 10.1177/0037549709343117 , 2013. , 2013. , 1971. 10.1071/WF06142 10.1071/WF06143 3222 www.gdal.org 3223 3221 3221 3238 3237 3236 10.1016/j.combustflame.2009.07.010 10.1071/WF12202 3223 3223 , , 2013. , 2013. 3229 3228 , 2011. 3222 , 2011. 3220 10.1016/S0010-2180(71)80005-6 3225 3226 10.1155/2011/613424 10.1016/j.envsoft.2012.11.001 10.1016/j.envsoft.2013.04.004 10.1007/s10666-010-9241-3 physical models, Int. J. Wildland Fire, 18, 349, doi: USDA Forest Service, 1972. Studies anddoi: Fuel Models for Landscape-Scale Fire Modeling, J. Combust.,empirical models, 613424, Int. J. Wildland Fire, 18, 369, doi: Forestry and Timber Bureau, 1966. Source Geospatial Foundation, available at: Forest Service research paper INT,est Intermountain Service, Forest US & Dept. Range of Experiment Agriculture, Station, 1966. For- Fire, 11, 193–203, 2002. models getting aheaddoi: of their evaluation again?, Environ. Modell. Softw., 41, 65–71, Forest Service, Division of Fire Research, 1952. 1969. models for estimating fireForest Service, behavior, Intermountain General Forest technical and report Range Experiment INT, Station, US 1981. Dept. of Agriculture, Flame, 16, 9–16, doi: mountain Forest and Range Experiment Station, Forest Service, US Dept. of Agriculture, Tech. rep., USDA Forest Service, Intermountain Research Station, 1986. Combust. Flame, 156, 2217–2230, doi: doi: of a physical fire-spread model, Simulation, 86, 629–646, doi: A method fordoi: ensemble wildland fire simulation, Environ.Colorado Model. State University, Assess., 2007. 16, 153–167, of surface and crown fire rates2009. of spread, Environ.simulations, Modell. Int. J. Wildland Softw., Fire, doi: 47, 16–28, Alexander, M. E. and Cruz, M. G.: Are the applications of wildland fire behaviour References Sullivan, A. L.: Wildland surface fire spread modelling, 1990–2007, 1: Physical and quasi- Sullivan, A. L.: Wildland surface fire spread modelling, 1990–2007, 2: Empirical and quasi- Santoni, P., Filippi, J. B., Balbi, J.-H., and Bosseur, F.: Wildland Fire Behaviour Case Rothermel, R.: A mahematical model for predicting fire spread in wildland fuels, Tech. rep. McArthur, A.: Weather and Grassland Fire Behaviour, Leaflet, Forest ResearchRothermel, R. Institute, and Anderson, H.: Fire spread characteristics determined in the laboratory, US GDAL Development Team: GDAL – Geospatial Data Abstraction Library, Version 1.10.0, Open Fujioka, F. M.: A new method for the analysis of fire spread modeling errors, Int. J. Wildland Byram, G. and Fons, W.: Thermal conductivity of some common forest fuels, Tech. rep., USDA Frandsen, W. H.: Fire spread through porous fuels from the conservation of energy, Combust. Anderson, H., Forest, I., and Range Experiment Station (Ogden, U.): Aids to determining fuel Anderson, H.: Heat transfer and fire spread, USDA Forest Service research paper INT, Inter- Andrews, P.L.: BEHAVE: fire behavior prediction and fuel modeling system – BURNBalbi, subsystem, J., Morandini, F., Silvani, X., Filippi, J., and Rinieri, F.: A physical model for wildland fires, Cruz, M. G. and Alexander,Filippi, J.-B., M. Morandini, F., Balbi, E.: J. H., and Uncertainty Hill, D. R.: associated Discrete event front-tracking with simulation model predictions Finney, M., Grenfell, I., McHugh, C., Seli, R.,Forthofer, Trethewey, D., J.: Stratton, Modeling R., Wind in and Complex Brittain, Terrain S.: for Use in Fire Spread Prediction, Thesis, Filippi, J.-B., Mallet, V., and Nader, B.: Representation and evaluation of wildfire propagation 5 5 15 10 15 30 20 25 10 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | . 3 3240 3239 Burn probability and detection skill for the 4 models, all fires included. The burn prob- The best simulations, according to the scoring method and the fire model. stands “ATA” SørensenJaccard N3 – Oletta (21 AugKappa 2004) N3 –ATA Oletta (21 Aug 2004)Shape N3 – Oletta (21 Aug 2004)Score N3 B0 – – Oletta Santo (21 Pietro AugSørensen 2004) di Tenda (1 Jul 2003) Original N3 E0Jaccard – Rothermel – Oletta Sisco (21 (24 Aug B0 JulKappa 2004) 2003) – Santo Pietro N3 E0 di – –ATA Tenda Oletta Sisco (1 (21 (24 Jul Aug Jul 2003) 2004) 2003)Shape N3 E0 – – Oletta Sisco (21 (24 Aug Jul 2004) 2003) H0 E0 – J3 – Corbara – Sisco (1 Olmeta (24 Jul di Jul 2003) Tuda 2003) (21 Aug 2004) B1 – (30 Jun 3-percent 2005) B1 L0 – – Calenzana (30 (22 Jun Jul 2005) 2006) B1 – Calenzana (30 Jun 2005) S1 – Ersa (17 Oct 2003) Score Non-stationary Balbi Stationary Balbi IndicatorBurn probabilityDetection skill Non-stationary Balbi Stationary Balbi 0.29 Original Rothermel 0.42 3-percent 0.23 0.35 0.21 0.26 0.12 0.11 Table 2. ability is the frequencyin with reality. The which detection the skillalso model is burned correctly in the the predicts frequency simulation. with that which a a location burned will location be (in observations) burned is Table 1. for “arrival time agreement”.found “Shape” in refers Figs. to 5–7, and “shape further agreement”. information The about final the contours fires may is available be in Table Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | . WD is the wind direction in degrees, defined 1 − 3242 3241 #A0 NameA1 AulleneA2 OlettaA3 Olmi CappellaB0 PietracorbaraB1 Santo Pietro di TendaB2 Calenzana 1 Date JulB3 29 Calenzana 2003 7 Aug 2003 Aug 2003C0 13 Solaro Aug 2004 192 ha 9C1 Aug 1310 ha 990 2003 ha 1082 haC2 Murzo 15 3 19 53 9 Size haC3 30 Jun 2005 210D0 19 150 260 Jul 2005 60 1418 m 60 ha 3D1 136 139 WS m m 6D2 17 ha 0.12 WD 133 0.53 0.47 m 21 13 275 Jun JanD3 2003 2004 Res. 0.15 2 Aug 0.63 0.88 0.8 320 27 2003 mE0 Aleria 5 49 28 ha ha 0.46 9 0.95 BJ 214 0.7 m AugE1 Sisco 2003 0.63 0.29 95 ha 0.42 220E2 0.69 5 2 9 180 Jul ha 0.86 BA 2007 13E3 Aleria m 1 0.85 Jan 7 2009 0.65 26F0 2 May Prunelli 130 129 Aug 2008 5 di BS 0.77 2003 0.26 Fiumorbo 56F1 ha Olmeta 23 13 2 57 m 40 di m ha ha TudaF2 0.75 782 4 Calenzana 210 ha Jul 2003 9 0.15 45 Aug 0.29 5 4 mF3 0.65 2003 Sep 52 m 2004 1 7 5G0 24 Jul 0.58 0.71 0.34 2003 26 33 ha ha 0.031 140 21 62G1 Porto ha Aug Vecchio 0.3 0 2004 0.6 20 110 0.76 30 0.71 20 m 441G2 Afa ha Jun 2004 80 ha 5 4 5 0.28 m 0.82 5 103G3 Loreto 19 m 23 di m Jul 0.13 Casinca 2005 1 80 ha CalenzanaH0 0.42 7 270 140 0.084 0.42 30 0.62H1 21 Corbara 75 ha 11 Aug 2006 276 Apr 18 16 0.71 m 4 m 2006 0.61 21 0.41 20H2 Vero Aug Apr 0.78 24 2004 240 2005 m 79 3 m 0.89 ha 0.22 81H3 Canale 0.53 ha 8 0.38 0.7 di 26 28 m Verde ha 90 41 ha 0.29I0 0.53 0.8 0.74 0.4 17 28I1 250 4 3 m Oct 0.71 2008 7 0.99 0.75 22 7 Jul 0.84 I2 38 m 2003 0.57 Pruno 4 0.79 0.17 59 Jan ha 0.77 240 2003 210 1I3 Jul 260 2003 260 0.28 242 ha 0.84 26 5J0 m 0.79 m 19 Sartene m 22 34 m 7 haJ1 0.4 0.66 Calenzana 15 5 10 Jul 0.15 ha 2003 0.19J2 0.077 0.82 0.21 11 Feb 210 3 12 2005 0.69 AugJ3 0.54 220 2003 535 Canari ha 0.65 8 25 0.67 m 0.38 12K0 0.39 ha 61 Olmeta 23 m 0.48 ha di 240 0.61 Tuda 3 Calenzana 20 0.1 210 Apr 14 2005 m 0.16 1 5 8 Jan 4 12 Aug 14 2006 m 2003 210 Jul 116 2008 ha 0.55 0.25 0.7 24 96 Aug m 120 0.28 21 2008 69 0.23 Aug 16 130 ha ha 2004 0.25 7 ha 0.73 0.66 13 m 2.8 16 ha 0.11 m 0.81 18 ha 0.77 1 10 Jul 31 2003 12 2 Jul 260 0.88 2005 0.21 0.6 0.21 4 270 37 m 6 260 0.74 91 275 94 ha 1 0.64 0.66 ha m 37 m 210 0.25 0.41 12 0.56 m 320 5 m 8 7 0.13 0.13 0.67 12 m 0.17 0.65 0.64 0.62 0.5 240 0.66 239 0.6 0.29 0.35 50 57 m m 0.83 0.37 0.76 0.89 0.13 0.31 0.71 0.78 0.66 0.61 0.79 #K1 NameK2 CalenzanaK3 CalenzanaL0 PianaL1 AjaccioL2 BastelicacciaL3 Propriano 15M0 Jul Oletta 2004 Date 30 SoveriaM1 Aug 2006 102 haM2 Sisco 143 haM3 Coti 13 Chiavari Jul 4 2008 22N0 7 Aug 2008 22 Jul Size 2006N1 0.3 Santa ha 2.2 Maria 260 ha 19 Poggio Jul 2006 230N2 13 32 ha m 4 26N3 WS m 3 0.5 ha 2 0.049 29O0 WD 3 Jul Oletta Aug 2003 6 Aug 4 2003 2003 May 0.17 2003 300 0.78O1 280 Propriano Res. 4 56 ha 2 14 m 0.33 58 0.65O2 6 Aug Calenzana 185 ha m 1126 190 2003 ha ha BJ 0.49 O3 210 Calvi 15 382 7 m ha 5 12 7 Aug 0.27 2003P0 1 San 1 0.086 1 m Giovanni Jul di 2003 BA Moriani 0.55 1P1 201 0.59 Luri ha 280 0.69 350 110 4 321 AugP2 0.052 0.21 Saint 2003 BS 168 0.9 0.46 m Florent 26 517 m ha 46 6 258 m 14 29P3 Jan m Peri 2004 27 0.41 Aug 0.56 10 21 2008 0.84 42 ha m 25 AugQ0 0.17 7 Jul 2004 Poggio 0.31 d’Oletta 2004 59 240 ha 0.55 0.39 0.25 0.77 haQ1 Borgo 186 ha 0.24 0.56 51 m 0.83 5 465 0.8 ha 239 0.77Q2 Sisco 3 2 0.41 0.65 13 0.62 80 0.88 Q3 8 Calenzana m 0.34 Jul 6 2009 5 340 Aug 0.31 2003 220R0 Bonifacio 260 72 11 0.56 0.74 10 Sep m 2004 2R1 Propriano m 44 200 m 78 25 ha ha 0.54 13 31 Jan m 53R2 0.78 2004 Tolla ha 0.57 72 m 0.67R3 0.84 0.4 Santa 50 ha Lucia 0.23 8 3 23 di 0.78 Jul Mercurio 0.48 2009S0 4 Alata 0.81 20 0.84 6 0.65 Jul 6 0.83S1 Sep 2003 Sep 0.9 2 210 0.77 2004 284 2004 749 ha 14 0.83 140 0.54 S2 Aug Ersa 2003 36 0.75 17 28 m m 40 ha Jul 19 27 2003 26 ha ha 2S3 m 270 Manso 357 19 ha Jul 2006 0.44 0.21T0 26 Olmeta m 477 di ha 0.38 1 Tuda 280 7 10 4T1 Santo 3.5 ha 0.67 0.84 Pietro di 0.36 152 Tenda 8 0.82 mT2 9 Calenzana 20 Aug 0.81 0.43 20 2003 0 238 0.51 0.066T3 4 0.71 1 19 43 280 31 Jul m m 17 0.76 21 942 m 2003 Aug ha 0.66 Jul 2003 2008 20 4 m 104 210 m Aug 0.59 2003 115 0.05 0.074 ha 0.24 2 7 0.12 0.11 110 m ha ha 17 0.31 0.79 8 Oct 13 Jul 2003 ha 0.59 2004 4 0.67 6 0.23 0.6 325 7 0.68 0.16 156 0.14 0.56 6 ha Sep 2004 146 m 1 0.67 140 11 0.58 ha 230 1 Jul 0.56 239 3 2003 0.4 33 m 28 ha 2 1 m Aug 0.28 37 2005 260 m 7 90 0.42 43 0.73 9 ha m 10 13 0.55 ha 0.1 0.73 230 69 0.57 m 40 0.77 5 0.16 8 0.63 m 5 0.7 0.24 0.83 20 m 0.69 265 0.13 0.083 1 40 0.52 21 0.14 m 0.62 12 m 0.32 0.57 0.79 0.27 0.37 0.28 0.76 0.7 0.7 0.56 Continued. Information about all the 80 fires shown in Figs. 5–7. The size is the final burned area cient (BJ), arrival time agreement (BA) and shape agreement (BS). ffi Table 3. Table 3. in hectares. WS stands forclockwise, “wind with speed”, 0 in ms correspondingcolumn a “Res”. westerly We wind finally (90 putcoe the southerly). best The scores resolution (among in the meters 4 models) is found in for Jaccard similarity Discussion PaperDiscussion | Discussion Paper Paper | Discussion| Discussion Paper Paper | | Discussion Discussion Paper Paper | | Discussion PaperDiscussion | Paper |Discussion Discussion Paper Paper | | Discussion Discussion Paper Paper | | Discussion Discussion Paper | | Discussion Paper | 80 70 70 60 60 50 50 40 40 30 30 80 (c) Kappa 70 20 (c) Kappa 20 70 Balbi non stationary Balbi stationary Original Rothermel 3-percent 10 60 10 60 0 0 50 50

0.1 0.2 0.1 0.5 0.7 0.3 0.8 0.0 0.6 0.4 0.1 0.2 0.1 0.5 0.7 0.3 0.8 0.0 0.6 0.4

Score 40 Score 40 30 30 80 70 20 20 70 Balbi non stationary Balbi stationary Original Rothermel 3-percent 10 60 10 60 (e) Shape agreement (e) Shape agreement 0 0 50 50

1.0 0.2 0.8 0.0 0.6 0.4

1.0 0.2 0.8 0.0 0.6 0.4 Score Score 40 40 17 15 30 30 80 3244 3243 70 20 20 70 Balbi non stationary Balbi stationary Original Rothermel 3-percent 10 60 10 60 (b) Jaccard similarity (b) Jaccard similarity 0 0 50 50

0.2 0.1 0.5 0.7 0.2 0.3 0.1 0.5 0.7 0.0 0.3 0.6 0.4

0.0 0.6 0.4 Score Score 40 40 30 30 80 70 20 20 70 Balbi non stationary Balbi stationary Original Rothermel 3-percent 10 10 60 60 0 0 50 50

(d) Arrival time agreement (d) Arrival time agreement 1.0 0.2 0.8 0.0 0.6

0.4 1.0 0.2 0.8 0.0 0.6 0.4 Score Score 40 40 30 30 20 20 Balbi non stationary Balbi stationary Original Rothermel 3-percent 10 10 The distribution of the scores for the 80 cases and for the 4 models. The sorting was The distribution of the scores for the 80 cases and for the 4 models. The sorting was 0 0

0.2 0.1 0.5 0.7 0.3 0.9 0.8 0.0 0.6 0.4

0.2 0.1 0.5 0.7 0.3 0.9 0.8 0.0 0.6 0.4 Score (a) Sørensen similarity index Score (a) Sørensen similarity index Fig. 2. carried out with respect tocan read the the mean four score scores acrossbounds for a the of given the models, fire lines. hence case. at The a gray area given is abscissa, delimited one by the upper and lower Fig. 1. carried out independently for each model, hence one cannot compare individuals scores. Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper |

Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | 4000 10000 3000 21 22 2000 3246 3245 5000 N3 - Oletta [186 ha] N3 - Oletta 1000 B0 - Santo Pietro di Tenda [1310 ha] 5000 1000 5000 3000 2000 6000 4000 15000 10000 20000 Case of Santo Pietro di Tenda fire (1 July 2003, T0 in appendix) as simulated by the Case of Oletta fire (31 August 2004, N3 in appendix) as simulated by the non-stationary Fig. 4. non-stationary Balbi model(black) (blue), and the the 3-percent stationary model Balbi (cyan). model (green), the Rothermel model Fig. 3. Balbi model (blue), the3-percent stationary model (cyan). Balbi model (green), the Rothermel model (black) and the Case of Santo Pietro di Tenda fire (2003-07-01, T0 in appendix) as simulated by the non- Case of Oletta fire (2004-08-31, N3 in appendix) as simulated by the non-stationary Balbi In figure4, we plotted the case for which the Balbi model attained its best shape agree- Fig. 4. stationary Balbi model (blue),the the 3-percent stationary Balbi model model (cyan). (green), the Rothermel model (black) and 4.3 Forecast reliability In an operationalbe context, the of ability high ofevaluate importance. its a One burn model way probability. When towe to the compute predict assess model the predicts that frequency the that with athe reliability which a frequency location this fire of with actually will which will a happens. a burn burn Conversely, modelindicator model a we correctly as can given are in predicts interested location, the that this in a detection locationlarge context skill. will regions: is burn. The any We to detection location refer toas that skill this well. is may On actually be the burned perfect contrary,right in the if after reality first it will the indicator started: be model (burn atjust burned probability) the burns will in like ignition very be point, the the the perfect simulation model firethe if “predicted”. was detection the Consequently, observed, fire skill. we so stops See computed the location table2. both was the burned, burn probability and ment. The visual agreementprevious with fire the case. observedthe Hence contour the shape does dynamics agreement. notevaluation of This can seem the suggests cast as fire different that good must light taking on as have into a in significantly model. account the contributed the to front dynamics in the Fig. 3. model (blue), the stationarymodel Balbi (cyan). model (green), the Rothermel model (black) and the 3-percent 5 10 5 Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | 3248 3247 3). The non-stationary Balbi model is in blue; the stationary 3). The non-stationary Balbi model is in blue; the stationary / / Simulation for all fires (2 Simulation for all fires (1 Balbi model is ingray green; area the is Rothermel the observed model final is burned in area. black; The the coordinates 3-percent are model meters. is in cyan. The Fig. 6. Balbi model is ingray green; area the is Rothermel the observed model final is burned in area. black; The the coordinates 3-percent are model meters. is in cyan. The Fig. 5. Discussion Paper | Discussion Paper | Discussion Paper | Discussion Paper | 3249 3). The non-stationary Balbi model is in blue; the stationary / Simulation for all fires (3 Balbi model is ingray green; area the is Rothermel the observed model final is burned in area. black; The the coordinates 3-percent are model meters. is in cyan. The Fig. 7.