Data Assimilation Using Sequential Monte Carlo Methods in Wildfire Spread Simulation

Transcription

1 Daa Assimilaion Using Sequenial Mone Carlo Mehods in Wildfire Spread Simulaion HAIDOG XUE, FEG GU, XIAOLI HU, Georgia Sae Universiy Georgia Sae Universiy Georgia Sae Universiy Assimilaing real ime sensor daa ino large-scale spaial-emporal simulaions, such as simulaions of wildfires, is a promising echnique for improving simulaion resuls. This asks for advanced daa assimilaion mehods ha can work wih he complex srucures and non-linear behaviors associaed wih he simulaion models. This paper presens a daa assimilaion framework using Sequenial Mone Carlo (SMC) mehods for wildfire spread simulaions. The models and algorihms of he framework are described, and experimenal resuls are provided. This work demonsraes he feasibiliy of applying SMC mehods o daa assimilaion of wildfire spread simulaions. The developed framework can poenially be generalized o oher applicaion areas where sophisicaed simulaion models are used. Caegories and Subjec Descripors: I.6.3 [Simulaion and Modeling]: Applicaions; I.6.8 [Simulaion and Modeling]: Types of Simulaion Discree Even; G.3 [Probabiliy and Saisics]: Probabilisic algorihms (including Mone Carlo) General Terms: Algorihm, Theory Addiional Key Words and Phrases: Daa assimilaion, Sequenial Mone Carlo mehods, Wildfire, DEVS 1 ITRODUCTIO Wildfires have significan impacs on boh he ecosysems and human sociey. The effecs on ecological sysems include burning local plans, reducing species diversiy due o he emission of carbon dioxide, desroying organic nuriens o cause flash floods, and leading o climae changes by releasing carbon ino amosphere [Keeley 1995; Lindsey 2008; Running 2008]. Wildfires also cause massive losses of naural fores resources, endangered species, properies, and even human lives. I is esimaed ha more han 11,000 communiies close o federal land are subjec o hreas from wildfires in he US [Rey 2004]. In he 2007 wildfire season, over 85,500 fires across he enire US burned more han 9.3 million acres of land. I coss 1.8 billion dollars in an effor o figh wildfires and a poenial 2.5 billion dollars in insured loss in California alone [Grossi 2007]. To effecively manage wildfires, simulaion models are used o sudy and predic wildfire spread. Over he years, several major wildfire spread simulaion models have been developed, including FARSITE [Finney 1998], BehavePlus [Andrews e al. 2005], DEVS-FIRE [aimo e al. 2008; Hu e al. 2011], and HFire [Morais 2001]. The accuracy of wildfire spread simulaions depends on many facors, including he GIS daa, he fuel daa, he weaher daa, and high fideliy wildfire behavior models. Unforunaely, due o he dynamic and complex naure of wildfire, i is impracical o obain all hese daa wih no error. For example, he weaher daa used in he simulaion is ypically obained from local weaher saions in a imebased manner (e.g., every 10 minues). Before he nex daa arrives, he weaher is considered as unchanged in he simulaion model. This is differen from he realiy where he real weaher consanly changes (e.g., due o he ineracions beween This research was suppored in par by grans CS and CS from he aional Science Foundaion. Auhors addresses: H. Xue, Deparmen of Compuer Science, Georgia Sae Universiy; F. Gu, Deparmen of Compuer Science, Georgia Sae Universiy; X. Hu, Deparmen of Compuer Science, Georgia Sae Universiy.

2 wildfires and weahers). The GIS daa and fuel daa also have errors and are consrained by heir spaial resoluions. Besides daa errors, he wildfire behavior model also inroduces errors because of is compuaional absracion. Wih hese errors, he predicions from he simulaion model will be differen from wha i is in a real wildfire. Wihou assimilaing daa from he real wildfire and dynamically adjusing he simulaion, he difference beween he simulaion and he real wildfire is likely o coninuously grow. To improve he accuracy of wildfire spread simulaions, daa assimilaion mehods ha assimilae sensor daa from real wildfires are needed. Daa assimilaion is an analysis echnique, in which he observed daa is assimilaed ino he model o produce a ime sequence of esimaed sysem saes [Bouier and Courier 1999]. While daa assimilaion has been widely used in areas such as amospheric, climae, and ocean modeling [Daley 1991; Kalnay 2003; akamura e al. 2006], i is no well developed in wildfire simulaions. Assimilaing daa in large-scale wildfire simulaions is a challenging ask. Because of he complexiy of he wildfire simulaion model, he number of possible sae variables and model parameers is exremely large, and many of hem are spaially dependen. Anoher noeworhy complexiy is associaed wih he non-saionary, non-linear, non-gaussian behavior of wildfire [Mandel e al. 2008], which makes i ineffecive o use convenional inference echniques such as Kalman filer and is varians (e.g., ensemble Kalman filer [Evensen 2003]). Furhermore, a wildfire simulaion model usually has complex model srucure and sae updae mechanism. For example, he DEVS-FIRE model used in his paper is a Discree Even Sysem Specificaion (DEVS)-based cellular space model wih discree even simulaion mechanism. I is hen difficul o apply convenional inference echniques ha rely on equaion-based model represenaions. Moivaed by hese challenges, we selec Sequenial Mone Carlo mehods o suppor he daa assimilaion of wildfire simulaions in his paper. Sequenial Mone Carlo (SMC) mehods, also called Paricle Filers, are a se of sample-based mehods ha use Bayesian inference and sochasic sampling echniques o recursively esimae he sae of dynamic sysems from some given observaions [Douce e al. 2001; Schön 2006]. In pracical applicaions of SMC mehods, he probabiliy densiy funcion is represened by a se of samples, each of which is a paricle associaed wih a weigh. Many mehods have been developed o generae samples, including perfec sampling, sequenial imporance sampling and resampling, accepance-rejecion resampling, and Meropolis-Hasings independence sampling. SMC mehods are able o approximae arbirary probabiliy densiies and have lile or no assumpion abou he properies of he sysem model. This makes hem effecive mehods for supporing daa assimilaion of wildfires ha have complex, nonlinear, non-gaussian, unseady behaviors [Mandel e al. 2008]. Meanwhile, SMC mehods are able o recursively adjus heir esimaions of sysem saes when new observaion daa become available. This feaure is suied for daa assimilaion where new sensor daa sequenially arrive, and he simulaion sysem needs o be coninuously updaed. This paper presens a daa assimilaion mehod for wildfire spread simulaion based on SMC mehods. The developed mehod dynamically assimilaes ground emperaure sensor daa of a wildfire o improve simulaion resuls. The wildfire spread simulaion model used in his work is a discree even model called DEVS- FIRE [aimo e al. 2008; Hu e al. 2011]. Applying SMC mehods o daa assimilaion for he DEVS-FIRE model needs o develop several key componens: formulaing he discree even simulaion model as a sae space model, defining he observaion model ha compues sensor daa of ground emperaure sensors from he wildfire simulaion, and developing he sampling, resampling, and weigh updaing

3 mehods for he applicaion of DEVS-FIRE simulaion. We presen he models and algorihms for each of hese componens and describe how hey work ogeher o suppor daa assimilaion. The conribuions of his paper are wo folds. Firs, his work develops a daa assimilaion framework using SMC mehods for wildfire spread simulaions. I is one of he firs effors in assimilaing real ime daa in wildfire spread simulaions. Second, his work demonsraes he feasibiliy of applying SMC mehods o daa assimilaion for complex discree even simulaions. The developed framework can poenially be generalized o oher applicaion areas, where sophisicaed simulaion models are used, and real ime daa are available. The res of his paper is organized as follows. Secion 2 inroduces he relaed work in daa assimilaion, SMC mehods, and wildfire simulaions. Secion 3 reviews he basic conceps of SMC mehods. Secion 4 highlighs how he DEVS-FIRE wildfire spread simulaion model works. Secion 5 presens he key componens of he daa assimilaion framework using SMC mehods. Secion 6 presens experimenal resuls of applying he developed daa assimilaion o DEVS-FIRE simulaion. Secion 7 discusses several perinen issues of his work and Secion 8 draws he conclusions and poins ou fuure work. 2 RELATED WORK Daa assimilaion is used in many differen fields, such as geosciences, weaher forecasing, hydrology, and oher environmenal sysems. The purpose of daa assimilaion is o use observaion informaion o improve sae esimaion of a sysem under sudy. I ries o find he soluions by minimizing he errors beween he real sysem and he models. The opimal ineroperaion analysis echniques are widely used in daa assimilaion, such as hree-dimensional variaional analysis (3D-VAR) and four-dimensional variaional assimilaion (4D-VAR). Alhough boh of hem minimize he cos funcion o obain opimal esimaions, 4D-VAR incorporaes a predicion model o compare he model sae and he observaions a differen ime seps. The work of Bei e al. [2008] proposed a daa assimilaion sysem o improve ozone simulaions in Mexico Ciy basin using 3D-VAR ha generaes he opimal esimae of he rue amospheric sae during he analysis ime. Wilkin e al. [2008] used saellie remoely sensed observaions and applied 4D-VAR o a regional ocean modeling sysem o produce an opimal esimaion of he real ocean sae. Kalman filer [Kailah e al. 2000] is an analysis echnique ha esimaes he sae of a dynamic sysem wih observaions represened by a linear sae space model. For applicaions wih non-linear behaviors, he classic Kalman filer needs o be exended. Anoniou e al. [2007] sudied hree exensions of he Kalman filer including exended Kalman filer, limied exended Kalman filer, and unscened Kalman filer, o find he soluions o non-linear discree-ime sae space models. SMC mehods approximae he sae of dynamic sysems using paricles and associaed weighs. SMC mehods find applicaions in many problem domains, including signal processing, wireless communicaion, arge racking, speech recogniion, compuer vision, mobile robo localizaion, and DA sequence analysis (see an overview of hese applicaions in Chen [2004]). Gusafsson e al. [2002] developed a framework and several algorihms for he problems of posiioning, navigaion, and racking using SMC mehods. Fox e al. [2001] used SMC mehods o solve robo localizaion, an imporan problem for mobile robos. In Mihaylova e al. [2007], he auhors proposed Mone Carlo echniques for mobiliy racking in wireless communicaion neworks in erms of he signal srengh, by which he mobile saion s posiion and speed can be correcly esimaed. In Azzabou e al. [2005], SMC mehods were used in image processing o improve he image qualiy. Oher applicaions of SMC mehods include biology and chemisry. Zhang e al. [2003]

4 provided an applicaion of SMC mehods in biology, in which populaions of compac long chain polymers were creaed by he Mone Carlo mehods o sudy he relaionships beween packing densiy and chain lengh. The work of Chen e al. [2008] se up a probabilisic framework for he dynamic daa recificaion, which provided a basis for process faul diagnosis. Simulaions of wildfire can be caegorized ino wo approaches, he physical approach and he empirical approach. The physical approach considers fire spread as hea ransfer beween burning and unburned fuel using parial differenial equaions o solve for prediced fire spread (see [Douglas e al. 2006; Linn e al. 2002; Pasor e al. 2003; Weber 1991]). The empirical approach relies on saisical correlaion beween variables known o influence fire spread wih field observaions of raes of spread. A widely used empirical fire behavior model is Rohermel s model [Rohermel 1972]. Several major wildfire simulaion sysems have been developed o dae, including FARSITE [Finney 1998], BehavePlus [Andrews e al. 2005], and HFire [Morais 2001]. These sysems use Rohermel s fire behavior model o compue he rae of fire spread, and deermine he fire size according o an ellipical shape. They are raser-based spaially explici models, and use a discree ime approach for simulaing he wildfire growh. More recenly, Trunfio e al. [2011] develops a new algorihm for simulaing wildfire spread hrough cellular auomaa and also uses Rohermel s fire behavior model. Many complex simulaion models allow he wildfire o feed back upon he amosphere. FIRETEC, a wildfire behavior model developed a Los Alamos aional Laboraory, explored he ineracions beween he fire model and wind condiions [Linn e al. 2002]. Based on Rohermel s fire behavior model, we developed a cellular auomaa based discree even wildfire simulaion model called DEVS-FIRE [aimo e al. 2008; Hu e al. 2012], which is used in his paper. The deails of DEVS-FIRE are described in Secion 4. While much work has been conduced in developing wildfire simulaion models, less research exiss in wildfire daa assimilaion. In he limied work ha we are aware of, Bradley [2007] proposed an approach o esimae fores fires based on SMC mehods from video images. In his work, a blurring funcion was used o add uncerainy o he images, and he informaion from miniaure air vehicles was used as he measuremen daa o esimae fire poses. o wildfire spread simulaion model is used in his work. Anoher group of work [Mandel e al. 2004; Mandel e al. 2009; Douglas e al. 2006; Coen e al. 2007] used daa assimilaion o invesigae fire behaviors of wildfire spread models. Their work was buil on a reacion-diffusionconvecion parial differenial equaion model and used ensemble Kalman filer as he esimaion mehod. In heir work, he real ime daa was assimilaed ino he fire model o esimae he emperaure of each cell. Our previous work [Gu and Hu 2008; Gu e al. 2009; Yan e al. 2009] explored applicaions of SMC mehods o sae esimaion in wildfire spread simulaion using a discree even simulaion model. Preliminary resuls showed ha SMC mehods were promising echniques for supporing daa assimilaion in wildfire spread simulaions. This paper exends he previous work and presens he overall framework and associaed models and algorihms of daa assimilaion using he discree even wildfire spread simulaion model of DEVS-FIRE. 3 OVERVIEW OF SMC METHODS SMC mehods are sample-based mehods ha use Bayesian inference and sochasic sampling echniques o recursively esimae he sae of dynamic sysems from some given observaions. A dynamic sysem is formulaed as a discree dynamic saespace model, which is composed of he sysem ransiion model of equaion (1) and he measuremen model of equaion (2) [Jazwinski 1970]. In he equaions, is he

5 ime sep, s and m are he sae variable and he measuremen variable respecively, he funcion f defines he evoluion of he sae variable and he funcion g defines he mapping from he sae variable o he measuremen variable where γ and ω are wo independen random variables represening he sae noise and he measuremen noise. s+ 1 = f ( s, ) + γ. (1) m = g s, ) + ω. (2) ( For sae esimaion, one needs o seek esimaes of s based on he se of all measuremens m1: = { mi, i = 1,2,..., }. In Bayesian filering, boh he sae and he measuremen variables are sochasic variables, and he poserior densiy p ( s m 1 : ) is recursively obained. Assuming he poserior probabiliy densiy p ( s 1 m1: 1 ) a ime sep -1 is available, he prior probabiliy densiy p ( s m 1: 1) of he sae a ime sep can be calculaed using equaion (3), where p ( s s 1) is he sysem ransiion densiy ha is consruced from he sysem ransiion model as shown in equaion (1). If he measuremen a ime sep is available, one can compue he poserior probabiliy densiy funcion according o Bayes heorem as shown in equaion (4). In equaion (4), p ( m s ) is obained from he measuremen model as shown in equaion (2), and p ( m m 1: 1) is a normalizing consan compued by equaion (5) according o Bayes heorem and Markov propery. Equaion (3) and equaion (4) form he foundaion o recursively predic he prior probabiliy densiy funcion and updae i o he needed poserior probabiliy densiy funcion of he curren sae. p s m ) p( s s ) p( s m ) ds. (3) ( 1: 1 = 1 1 1: 1 1 p p( m s ) p( s m1: 1) p ( s m1: ) =. (4) p( m m ) 1: 1 ( m m1: 1 1: 1 ) = p( m s ) p( s m ) ds. (5) Because i is difficul o solve he mulidimensional inegrals as shown in equaions (3)-(5), many approximaion algorihms are proposed, among which are SMC mehods described below. SMC mehods approximae he poserior probabiliy densiy funcion p ( s m 1 : ) by a se of paricles (samples) and heir corresponding weighs as in equaion (6), (i) (i) where s and w are he paricle i a ime sep and is normalized weigh, and δ(x) is he Dirac dela funcion. An imporan algorihm in SMC mehods is sequenial imporance sampling (SIS). In SIS, weighs are updaed using equaion (7), where q( s s, m ) is a proposal densiy ha can be easily sampled. 1 p s m1 : ) w δ( s s ). i = 1 ( (6)

6 w p( m s ) p( s s 1) w 1. (7) q( s s, m ) In pracical usage, a common choice of he proposal densiy is he sysem ransiion densiy, ha is, q( s s 1, m ) = p( s s 1). Wih his choice of he proposal densiy, equaion (7) is reduced o: w 1 1 w p( m s ). (8) oe ha, in our work, we have aken his approach by choosing he proposal densiy o be he sysem ransiion densiy. More deails will be given in Secion 5. The sequenial imporance sampling has he limiaion ha he enire process relies on he iniially generaed samples. To improve he algorihm, a resampling sep is added by using replicaed paricles in proporion o heir weighs for fuure use. This gives rise o he sequenial imporance sampling wih resampling (SISR), which forms he basic srucure of SMC mehods. Wih SMC mehods, i has been shown ha a large number of paricles are able o converge o he rue poserior densiy even in non-gaussian, non-linear dynamic sysems [Crisan 2001]. For sysems wih srongly non-linear behaviors, SMC mehods are more effecive han he widely used Kalman filer and is various exensions. More deails abou he algorihm can be found in Gordon e al. [1993]. To summarize, a basic SMC algorihm ha implemens he SISR procedure goes hrough muliple ieraions. In each ieraion, he algorihm receives a sample (paricle) se S 1 and an observaion m. S 1 represens he previous belief of he sysem sae where S 1 = n is he number of samples. In he imporance sampling p 1 S sep, each sample s 1 is used o predic he nex sae. This is done by sampling from he proposal densiy q( s s 1, m ). The imporance weigh of each paricle is hen updaed using equaion (7) and normalized. In he resampling sep, n offspring samples are drawn wih a probabiliy proporional o he normalized p sample weighs. These samples represen he poserior belief of he sysem sae and are used for he nex ieraion. To apply his algorihm o daa assimilaion for wildfire spread simulaion, we need o formulae he problem accordingly and develop associaed models and echniques following he algorihmic srucure of SMC mehods. 4 WILDFIRE SPREAD SIMULATIO MODEL OF DEVS-FIRE The DEVS-FIRE model is an inegraed wildfire spread and suppression simulaion model buil on Discree Even Sysem Specificaion (DEVS) formalism [Zeigler e al. 2000]. The srucure of DEVS-FIRE is shown in Fig. 1. Fire spread model is he core of DEVS-FIRE, which is modeled as a cellular space model conaining individual cells coupled ogeher. The cellular space model reads errain (slope and aspec), fuel, and weaher daa. I uses Rohermel s fire behavior model (he Behave model) o compue he speed and direcion of fire spread. DEVS-FIRE also suppors fire suppression simulaion. This can be achieved by adding ineracions beween he fire spread model and he firefighing model. More deails abou fire suppression simulaion can be found in Hu and aimo [2009] and are omied in his paper. Below we focus on he fire spread model of DEVS-FIRE.

7 Fig. 1. Srucure of DEVS-FIRE model. In DEVS-FIRE, he fire area is modeled as a wo-dimensional cell space, which is divided ino recangular cells whose dimensions depend on he resoluion of he GIS fuel and errain daa. Each cell has an ID (x, y) denoing is locaion in he cell space. Cells are coupled wih heir neighbors according o he Moore neighborhood (excep for he boundary cells), in which a cenral cell has eigh surrounding cells. For an individual cell, is fuel and errain are assumed o be uniform wihin he cell. All cells are coupled o a weaher model o receive weaher daa (wind speed and wind direcion) ha may change over ime. In DEVS-FIRE, fire spread simulaion is modeled as a propagaion process as burning cells ignie heir unburned neighbors. A cell, once ignied, calculaes is rae of spread and spread direcion using Rohermel s fire behavior model based on is fuel, slope, aspec, and weaher daa. The rae (and direcion) of spread is hen decomposed ino eigh direcions corresponding o is eigh neighbor cells based on an ellipical shape as illusraed in Fig. 2. The shape of his ellipse is compued by he mid flame wind speed and he fire spread rae (see [Finney 1998]). oe ha in Fig. 2, we assume he maximum rae of spread is in he souh direcion. Fig. 2. Fire spread decomposiion schema of DEVS-FIRE. In DEVS-FIRE, all cells are iniially se o he unburned (passive) sae. If a cell receives an igniion message and is fireline inensiy is larger han is burning hreshold, is sae changes o burning. A burning cell changes o he burned sae afer is maximum burn ime delay expires. The maximum burn ime delay of a cell depends on he size of he cell, and he fire spread speed compued from Rohermel s model. Based on his implemenaion, we can define he wildfire sae as a dimension vecor, where,

8 nc is he oal number of cells in he whole cell space, he second elemen in each uple indicaes he fireline inensiy. I is always 0 in he saes unburned and + burned; in burning sae, FI {0 } R. When he wildfire spreads, he fire sae evolves over ime and resuls in differen shapes due o he nonuniform fuel, errain, and weaher daa of he fire area: fire+ = DEVSFIRE( fire, θ, ), (9) where fire and fire + are he fire saes a ime and +, θ is a vecor conaining all he oher model inpus (informaion of errain, weaher and so forh), is he ime duraion. More echnical deails of he DEVS-FIRE model can be found in aimo e al. [2008] and Hu e al. [2012]. DEVS-FIRE is a deerminisic model, i.e., saring from he same fire sae and inpus, wih he same ime duraion, DEVS-FIRE always produces he same resul. 5 DATA ASSIMILATIO USIG SMC METHODS I DEVS-FIRE SIMULATIO To presen he daa assimilaion framework using SMC mehods based on he DEVS- FIRE model, we firs formulae he sysem ransiion model and measuremen model for wildfire simulaion. Based on hose models, we propose a sysem ransiion densiy by providing a sampling algorihm. We hen describe how o use he mulivariae Gaussian disribuion o consruc he measuremen densiy. Wih he densiies proposed, we presen he procedure of SMC mehods for assimilaing daa in DEVS-FIRE simulaion. 5.1 Sysem Model and Measuremen Model To apply SMC mehods for daa assimilaion, he sysem ransiion model of sae evoluion and he measuremen model ha maps sysem sae o measuremen daa need o be defined. Based on he DEVS-FIRE simulaion model, we formulae a nonlinear sae-space model as shown in equaion (10). fire+ 1 = SF( fire, ) + γ, m = MF( fire, ) + ω. (10) In equaions (10), fire and fire+ 1 are sysem sae variables of fire spread a ime sep and ime sep +1 respecively; SF is he sysem ransiion funcion: SF( fire, ) = DEVSFIRE ( fire, θ, ), (11) where is he ime duraion of a ime sep; γ is he sysem ransiion noise ha inroduces sochasic elemens o he sysem ransiion model; is he measuremen variable, MF( fire, ) is he measuremen funcion mapping fire saes o measuremens, and ω is he measuremen noise. In his work, as described in Secion 4, he sysem sae variable fire is a n c (he number of cells) dimension vecor conaining all he cell saes, and he measuremen variable m is a n s (he number of sensors) dimension vecor conaining emperaure values from he sensors deployed in he fire field. The sysem ransiion funcion is he DEVS-FIRE simulaion model. The slope, aspec, fuel, and weaher daa used in compuing fire spread behavior are considered as he parameer θ of he DEVS- m

9 FIRE model. We also assume he number of sensors and heir locaions are predefined in he measuremen funcion. The deail of he measuremen funcion is described in Secion 5.3. Given a fire sae a ime sep, hrough SF, a fire sae a ime sep +1 is obained; also, hrough MF, he corresponding measuremens can be calculaed. Equaions (10) define he sysem ransiion densiy p ( fire fire 1) and measuremen densiy p ( m fire ). To draw samples from hese wo densiies based on equaions (10), besides he ransiion funcion SF and measuremen funcion MF one also needs o know he disribuions of γ and ω. In his work, we define ω as a mulivariae Gaussian noise, so p( m fire ) is in he form of mulivariae Gaussian disribuion. For he fire sae noise γ, we focus on he fire perimeer shape and define γ as a graph noise over he fire shape. Focusing on he perimeer shape of he fire is due o he fac ha he spread of a wildfire is mainly influenced by he burning cells on he fire fron (we do no consider jumping fires or spoing fires where burning branches/leaves inside he fire perimeer are carried by winds and sar disan fires). Based on his idea, we propose a mehod o add graph noise for a given fire shape and define a sampling algorihm for drawing samples from p ( fire fire 1). 5.2 Sampling Algorihm of Sysem Transiion Densiy The goal of he sampling algorihm is o generae a fire sae sample for he nex ime sep given he curren fire sae based on he disribuion p ( fire fire 1). Given fire 1, in order o draw a sample from p ( fire fire 1), we firs apply he ransiion funcion as shown in equaion (11). Specifically, for each paricle we creae a DEVS- FIRE simulaion saring from he fire sae of ha paricle, and run he simulaion for one ime sep. The lengh of a ime sep is deermined by how ofen he sensor daa is colleced, which is every 20 minues in our experimens, so in equaion (11) we se o 20 minues. Le fire denoe he resul of SF ( fire 1, ). Given fire, we firs exrac he fire perimeer (also called fire fron) from fire, and add graph noise o his fire fron. We hen reconsruc a fire sae from he noised fire fron. We consider he reconsruced fire sae as a sample of p ( fire fire 1). In order o add graph noise o he fire fron, we divide i ino muliple segmens (he number of segmens is denoed as C1), each of which consiss of an equal number of burning cells. For each segmen, we hen inroduce a noise denoed as di ha defines he change (in number of cells) inside or ouside a cell along he direcion from he igniion poin o his cell. Differen segmens may have differen noise, bu all cells in he same segmen share he same noise. ex, for each cell in a segmen, i is moved o a new posiion ha is di cells away. Afer reconnecing all he moved cells, a new fire fron, referred o as he noised fire fron, is formed. We hen se all he cells inside he noised fire fron o burned, and obain a noised fire sae. This noised fire sae is considered as a sample of p ( fire fire 1). The algorihm of drawing samples from p ( fire fire 1) is given below. To illusrae he effec of his algorihm, Fig. 3 shows he resuls of 4 differen runs of he algorihm for a given fire sae. In each run, we compare he original fire fron of fire wih a noised fire fron.

10 Algorihm 1. Sysem Transiion Densiy Sampling Inpu: The fire sae a ime sep -1 ( fire 1 ), segmen denominaor ( C 1 ), noise denominaor ( C 2 ), and noise variance ( C 3 ). Oupu: A sample of p ( fire fire 1). 1. fire = SF ( fire 1, 1) ; 2. Scan he fire fron of fire ; 3. Divide he fire fron ino C 1 consecuive segmens, denoed as 4. Generae noise d 1, d 2,..., dc for all he segmens where 1 SEG, SEG2,..., SEG ; 1 C lengh of SEGi di ~ Gaussian (, C3) ; C2 5. For every burning cell c in segmen SEG i, move i o d i cells away along he direcion from he igniion poin o he cell; 6. Connec all he segmens according o he segmen order o form a closed shape (referred o as he noised fire fron), and se all he cells on he noised fire fron o burning; 7. Se all he cells inside he noised fire fron o burned and all he cells ouside he noised fire fron o unburned; 8. Reurn he fire sae. 1 (a) (b) (c) (d) Fig. 3. oised fire frons generaed from Algorihm 1. In each of hem, he original fire fron of fire is shown in blue and black; he noised fire fron is shown in red and black. (The black cells are on boh he original fire fron and he noised fire fron.) The sampling algorihm represens he sysem ransiion densiy and plays criical roles in he SMC mehod. In his algorihm, he purpose of he graph noise mehod is o model he simulaion error, and hen generae a new and realisic noised fire fron from an exising fire fron. The effeciveness of his mehod is

11 suppored by several feaures ha we build ino he mehod. Firs, he cells of he same segmen have he same noise level and differen segmens can have differen noise levels. This is based on he fac ha, in wildfire spread simulaions, cells in nearby regions of a fire fron end o have similar disance errors. Second, guaraneed by sep 4, larger noise comes wih lower probabiliy. In his way, wih a high probabiliy, he belief of curren paricles is inheried by he successor paricles in he nex sep. Boh of hese feaures can be seen from Fig. 3. Finally, small fires have smaller noise, and he noise level is conrolled by C1, C2 and C3. The fire shape error on he fire fron could be caused by imprecise fuel daa, errain daa, weaher condiion, fire model error and oher uncerain elemens affecing fire spread. The assumpion here is ha he effec of hese imprecise daa and errors can be modeled by he fire fron graph noise as presened in Algorihm 1, where he mean of noise is affeced by he lengh of fire perimeer. This assumpion is no always rue, bu when he updaing inerval is relaively shor and he underlying errors are no large, he proposed graph noise can be a reasonable model of hose errors even if wih nonhomogeneous fuel and complex errain. However, when his assumpion is violaed, in order o achieve an effecive he wildfire spread SMC based daa assimilaion mehod, more advanced sampling algorihms need o be developed, which are considered as fuure work. 5.3 Measuremen Densiy Afer obaining a sample from p ( fire fire 1), is imporance weigh can be calculaed using equaion (7). As menioned before, in our work he proposal densiy is chosen o be he sysem ransiion densiy, and hus equaion (7) is reduced o equaion (8), where p ( m s ) = p( m fire ) in our work. Since he measuremen m is a emperaure vecor, like in many Kalman Filer applicaions, we define p ( m fire ) as a mulivariae Gaussian disribuion, i.e.: 1 1 exp( ( m MF( fire, ))' Σ ( m MF( fire, ))) p ( m ) = 2 fire, (12) n / 2 1/ 2 (2π) s Σ where MF is he measuremen funcion, Σ is he covariance marix. We assume sensor daa are independen wih each oher, so we se Σ as a diagonal marix. The measuremen funcion MF maps a fire sae o a measuremen vecor n s (emperaure daa of deployed sensors): c n MF( fire, ) : FIRE R, where FIRE is defined in Secion 4, n c is he number of cells, and ns is he number of sensors. In DEVS-FIRE, he wildfire field is represened by a discreized cellular space. Wihin his conex, he measuremen funcion includes wo main aspecs: he deploymen schema of ground emperaure sensors and he funcion of compuing he emperaure daa for a specific sensor from a given fire sae. The deploymen schema defines how he sensors are deployed in he wildfire field. Examples of deploymen schema include regular deploymen, e.g., one sensor every 10 cells or every 20 cells, random deploymen where sensors are randomly disribued in he cell space, and fire-direced deploymen where more sensors are deployed around he acive fire regions. These deploymen schemas (and he oal number of sensors) resul in differen locaions of he sensors. The locaion informaion is used in compuing he emperaure daa of he sensors. In his work, he deploymen schema is considered as predefined, and sensors do no move over ime.

12 For a specific sensor and a given fire sae, he funcion of compuing he sensor s emperaure is defined in equaion (13). d T = T e σ + T. c 2 2 a (13) In equaion (13), T is he emperaure of a sensor; Tc ( C) refers o he emperaure rise above ambien emperaure of he closes burning cell on he fire fron; Ta denoes he ambien emperaure ( C); d denoes he disance from he sensor o he closes burning cell; σ is a consan and is se o 50 (m) in our work. This formula is based on he work of Mandel e al. [Mandel e al. 2008]. To compue Tc for a burning cell, equaion (14) is used [Van Wagner 1973; Van Wagner 1975]. 2 T c = FI / h. (14) In equaion (14), FI is he fireline inensiy of he burning cell (kw m -1 ), and h is he heigh above ground (m). In he DEVS-FIRE simulaion, he fireline inensiy of a burning cell is compued from Rohermel s model a runime. For ground emperaure sensors, he heigh h would be heir insallaion heighs (see an example of ground emperaure sensor in [Kremens e al. 2003]). Fig. 4 illusraes how he measuremen funcion works in compuing he emperaure daa of ground emperaure sensors. I displays a simplified fire sae, where 8 cells are burning (displayed in red) in a 9 9 cell space wih each cell s resoluion being 15 (m). The emperaure sensors are regularly deployed in he cell space wih one emperaure sensor every hree cells (in boh horizonal and verical direcions). In Fig. 4, he cells where sensors are deployed are displayed in gray. To illusrae how he emperaure daa are calculaed, below we assume all burning cells Tc (see equaion (13)) is 376 C and he ambien emperaure is 27 C. From d 2 / 2σ 2 equaion (13), we have T = 376e + 27 (σ = 50), where d is he disance from a sensor o is closes burning cell on he fire fron. Based on his formula, we can obain he emperaure daa of all he sensors, which are {149, 267, 267, 267, 386, 386, 267, 386, 371} ( C) indexed from lef o righ and from op o boom. oe ha in his example, he closes disances o he fire fron for hese sensors are {75, 45, 45, 45, 15, 15, 45, 15, 21} (m). Burning cell Cell wih sensor Fig. 4. An example of he measuremen model.

13 To illusrae how deploymen schemas affec emperaure daa colleced from sensors, Fig. 5 visualizes sensors emperaure daa for a given fire sae wih hree differen deploymen schemas. Fig. 5(a) is he real emperaure map of a wildfire in a cell space, displayed in colors varying smoohly from black hrough red, and yellow o whie. This is compued by assuming each cell has a deployed emperaure sensor. In realiy, much smaller number of sensors is used. Fig. 5(b), Fig. 5(c), and Fig. 5(d) show he emperaure daa for hree differen deploymen schemas, in which he sensors emperaure daa are displayed by he shades of red (he darker he red, he higher he emperaure is). In Fig. 5(b), sensors are regularly deployed wih one sensor every 20 cells. Overall 100 sensors are deployed in he cell space. Fig. 5(c) and Fig. 5(d) use he same number of sensors. However, in hese wo cases he sensors are randomly deployed (referred o as random deploymen 1 and random deploymen 2, respecively). (a) (b) (c) (d) Fig. 5. An example of he measuremen model: (a) Real emperaure map; (b) Temperaure daa wih one sensor per 20 cells; (c) Temperaure daa wih 100 sensors randomly deployed in random deploymen 1; (d) Temperaure daa 2 wih 100 sensors randomly deployed in random deploymen Daa Assimilaion Using SMC Mehods In his secion, we presen he SMC mehod for assimilaing emperaure sensor daa ino DEVS-FIRE wildfire spread simulaions. The SMC mehod used in his work

14 implemens he sequenial imporance sampling wih resampling (SISR) principle described in Secion 3. Fig. 6 shows he srucure of he SMC mehod and he procedure of he daa assimilaion algorihm. In he figure, he recangle boxes represen he major componens in one sep of he algorihm, and he circles and rounded recangles represen he daa/variables. The daa assimilaion algorihm runs in a sepwise fashion. A ime sep, he se of sysem sae variables (paricles), i.e., he fire saes, from ime sep -1 (denoed as Fire-1 in Fig. 6) are fed ino he sysem ransiion model. This model performs Algorihm 1 and produces a sample for each paricle in Fire-1 based on as described in Secion 5.2. The resuling fire sae se is denoed as Fire'. To compue he imporance weighs of he paricles, for each fire sae in Fire', is emperaure vecor is compued according o he measuremen funcion as described in Secion 5.3. The se of emperaure vecors for all fire saes in Fire' are denoed as M'. Then considering each emperaure vecor in M' as he mean vecor, he probabiliy densiy value of he real emperaure vecor m (he emperaures colleced from real ime sensors) is calculaed based on he mulivariae Gaussian disribuion shown in equaion (12). This densiy value is used o updae he imporance weigh of he corresponding paricle. Afer normalizing he weighs of all paricles, a resampling algorihm is applied o generae Fire, and i is he inpu for he nex sep. We noe ha in Fig. 6 he ime sep -1,, and +1 are used o indicae he sepwise naure of he algorihm. The acual ime inerval beween wo consecuive seps is usually defined by how ofen he sensor daa is colleced, for example, in every 20 minues. Fig. 6. Daa assimilaion based on SMC mehods. The algorihm of he SMC mehod ha implemens he above procedure is given in Algorihm 2. In his algorihm, he se of fire saes is represened by a se of paricles. The algorihm sars by iniializing paricles represening he iniial fire saes when he fire is ignied (we sar he daa assimilaion from when he fire is ignied). Each paricle s weigh is iniialized o 1/. Then he algorihm goes hrough muliple ieraions, each of which includes sampling, weigh updaing, and resampling sages. A he sampling sage, all he paricles go hrough Algorihm 1, so each paricle is replaced wih a sampled fire sae. A he weigh updaing sage, he weighs of he sampled fire saes are updaed using equaion (8). These weighs are hen normalized. Finally, a resampling algorihm (Algorihm 3 given laer) selecs he paricles based on heir normalized weighs o form a new se of paricles. These paricles are assigned a new weigh of 1/ and are used in he nex ieraion of he algorihm.

15 Algorihm 2. SMC Mehod in Wildfire Simulaion for One Time Sep Inpu: The fire saes and he corresponding imporance weighs a ime sep -1 ( { fire 1 } i= 1, { w 1 } i= 1 ), and he measuremen a ime sep ( m ). Oupu: The fire saes and he corresponding imporance weighs a ime sep ({ fire } i= 1, { w } i= 1 ). 1. Sampling For each fire sae in Algorihm Weigh updaing { fire 1 } i = 1, draw a sample ( ) fire' i from p( fire fire 1) using a. For each fire sae in { fire' } i= 1, updae he weigh: w ' = w p( m fire' ) ; (i) (i) w' = i (i) = 1w' b. Calculae he normalized weighs: w ''. 3. Resampling a. Draw paricles from { fire' } i= 1 and { w'' } i = 1 : { } i= = Algorihm 3( fire 1 ' i w 1 { fire } = 1, { '' } i= ); ( ) b. Se he weighs: w i = 1/, i = 1,2,...,. 1 We use he mulinomial resampling o implemen our resampling algorihm. The (i) algorihm is described below as Algorihm 3, where w is he normalized imporance weigh of he i-h paricle a ime sep, and is he oal number of paricles. Firsly, he cumulaive sums of he normalized weighs of paricles ~ ( 1) ~ (2) (,,..., ~,..., ~ ( ) i q q q q ) are compued, where = i j q ~ ( ) ( ) w. Then random numbers = 1 uk k 1 uk} k 1 (denoed as { } = ) are generaed beween 0 and 1. Finally, we coun he number of j ~ ( i 1) elemens in { = ha fall ino he inerval of q and how many copies of he i-h paricle will be seleced. ~ q. This number decides Algorihm 3: Mulinomial Resampling Inpu: The fire saes and he corresponding imporance weighs a ime sep ({ fire } i= 1, { w } i= 1 ). Oupu: Resampled fire saes a ime sep ({ fire' } i= 1 ). 1. Compue he cumulaive sums of he normalized weighs of paricles ( ~ (1) q,,, ~ ( ) i q ), where = i j q ~ ( ) ( ) w ; j = 1 2. Generae ordered random numbers { uk } k= 1, where u k (0,1] ; 3. Generae n i copies of (i) fire, where i 4. Reurn he new generaed fire saes as { fire' } i= 1. n is he number of ~ ( i 1) (, ~ uk q q ] ; ~ (2) q,, ~ q

16 6 EXPERIMETS 6.1 Experimenal Design We use he idenical-win experimen, which is widely used in daa assimilaion research, o evaluae he daa assimilaion framework developed in his paper. The purpose of idenical-win experimens is o sudy he assimilaion in ideal siuaions and evaluae he proximiy of he predicion o he rue saes in a conrolled manner. In he idenical-win experimen, a simulaion is firs run, and he corresponding daa are recorded. This simulaion resul is considered as rue ; herefore, he observaion daa obained here are regarded as he real observaion daa (because hey come from he rue model). Consequenly, we esimae he sysem saes from he observaion daa using SMC mehods, and hen check wheher hese esimaed resuls are close o he rue simulaion resul. In his secion, we use hree erms: real fire, filered fire, and simulaed fire, o help us presen he experimenal resuls. A real fire is he simulaed fire spread from which he real observaion daa are obained. A simulaed fire is he simulaion resul based on some erroneous daa ( erroneous in he sense ha he daa are differen from hose used in he real fire), for example, imprecise weaher daa. This is o represen he fac ha wildfire simulaions usually rely on imperfec daa as compared o real wildfires. Finally, a filered fire is he daa assimilaion enhanced simulaion resul based on he same erroneous daa as in he simulaed fire. In our experimen, in every sep of he algorihm we choose he paricle wih he larges imporance weigh before resampling as he filered fire. The goal of our experimens is o show ha a filered fire gives more accurae simulaion resuls by assimilaing observaion daa from he real fire even if i uses he erroneous daa as in he simulaed fire. The differences beween a real fire and a simulaed fire are due o he imprecise daa such as wind speed, wind direcion, GIS daa, and fuel model, used in he simulaion. In our experimens, we choose o use imprecise wind condiions (wind speed and wind direcion) as he erroneous daa. Table 1 shows he configuraions of four ses of experimens. The real wind speed and direcion are 8 (mph) and 180 (degrees) wih random variances added every 10 minues. The variances for he wind speeds are in he range of 2 o 2 (mph) (denoed as 8±2 in he able), and he variances for he wind direcion are in he range of -20 o 20 (degrees) (denoed as 180±20 in he able). Our firs wo experimen cases inroduce errors o he wind speeds and make he wind direcions o be exacly he same as he real wind direcion. In case 1 he wind speed is randomly generaed based on 6 (mph) wih variances added in he range of 2 o 2 (mph). In case 2, he wind speed is randomly generaed based on 10 (mph) wih variances added in he range of 2 o 2 (mph). Our nex wo experimen cases inroduce errors o he wind direcions only: case 3 uses wind direcion of 160 (degrees) wih added variances in he range of ±30 (degrees); case 4 uses wind direcion of 200 (degrees) wih added variances in he range of ±30 (degrees). For wind direcions, he degrees indicae he angle beween he norh direcion clockwise o he direcion from where he wind comes. Case Table 1: Wind Daa Used in Experimens Error daa Speed (mph) Direcion (degrees) Speed (mph) 1 6±2 o error 2 10± ±30 o error 4 200±30 Real daa Direcion (degrees) 8±2 180±20

17 The sensor deploymen informaion ha includes he oal number of sensors and heir locaions is a criical facor o consider in experimens. In general, he more sensors deployed in he fire area, he more informaion he sensor daa conain. For all he experimen cases in Table 1, we employ a regular sensor deploymen schema where he sensors are regularly deployed wih one sensor every 10 cells. Besides ha, o compare he impacs of differen sensor deploymen schemas, we perform he proposed daa assimilaion on case 1 and employ wo addiional regular deploymen schemas including one sensor per 20 cells, one sensor per 40 cells, and wo differen random deploymen schemas wih 100 sensors randomly deployed. Limied by space, he resuls of he sensor deploymen impac comparison are no included in his paper, bu can be found in he online appendix. All simulaions use he real-world GIS daa and fuel daa. The cell space dimension is and he cell size is 15 (m). The GIS daa are airborne LiDAR (Ligh Deecion and Ranging) [Wagner e al. 2004] raser-based errain daa. The fuel daa was obained by classifying a mulispecral QuickBird (DigialGlobal) image [Mulu e al. 2008]. Those daa were acquired from Hunsville area, Texas, during he leaf-off season in March 2004 by M7 Visual Inelligence of Houson, Texas. The igniion poin is se o he cener poin of he cell space for all of he simulaions. The observaion daa (ground emperaure sensor daa) from he real fire are colleced every 20 minues. We use 50 paricles in all SMC experimens and se C1 o 6, C2 o 20, and C3 o 1in Algorihm Experimenal Resuls Wind speed. This se of experimens (case 1 and case 2) es he daa assimilaion resuls when he wind speed daa have errors. Fig. 7(a) displays he real fire afer 8 seps (20 minues each sep) of he simulaion. This real fire is used o generae he observaion daa for all our experimens. Fig. 7(b) and Fig. 7(c) show he simulaed fires for case 1 and case 2 respecively for he same simulaion duraion (160 minues). oe ha all hese fires are simulaed from DEVS-FIRE using heir corresponding daa shown in Table 1. In he figures, he burning cells and he burned cells are displayed in red and black respecively. The oher colors show differen fuel ypes of cells. From he figures, we know ha he real fire and he simulaed fires have large deviaions due o he errors of he wind speeds. In case 1, he real fire spreads faser han he simulaed fire because he real wind speeds (8±2 mph) are larger han he erroneous wind speeds (6±2 mph). In case 2, he real fire grows slower han he simulaed fire since he real wind speeds (8±2 mph) are smaller han he erroneous wind speeds (10±2 mph). By assimilaing he observaion daa, he filered fires are obained. Fig. 8 shows he resuls afer 8 seps for case 1. Fig. 8(a) shows he filered fire; Fig. 8(b) compares he real fire fron (displayed in green) wih he filered fire fron (displayed in blue. For comparison purpose, we also show he simulaed fire fron (displayed in red) from Fig. 7(b) in he figure. Fig. 8(c) shows he cells ha have mismached saes beween he real fire and he filered fire. From Fig. 8(b), i is observed ha he fire fron of he filered fire is much closer o he one of he real fire han ha of he simulaed fire. Specifically, due o he erroneous smaller wind speeds, he simulaed fire is significanly smaller han he real fire a he head of he fire (he op par of he fire shape as shown in Fig. 8). Using daa assimilaion, he filered fire overcomes his problem, and maches he real fire fron wih smaller difference. Fig. 8(c) confirms his and shows ha he mismached cells of he filered fire are roughly evenly disribued along he real fire fron.

18 (a) (b) (c) Fig. 7. Real fire and simulaed fires for case 1 and case 2. (a) Real fire afer 160 minues (average wind speed is 8 mph; average wind direcion is 180 degrees); (b) Simulaed fire for case 1 afer 160 minues (average wind speed is 6 mph; average wind direcion is 180 degrees); (c) Simulaed fire for case 2 afer 160 minues (average wind speed is 10 mph; average wind direcion is 180 degrees). (a) (b) (c) Fig. 8. Comparisons of he simulaed fire and he filered fire for case 1. (a) Filered fire; (b) Fire frons of he real fire (displayed in green), simulaed fire (displayed in red) and filered fire (displayed in blue); (c) Mismached cells (displayed in red) beween he real fire and he filered fire. Fig. 9 shows he resuls afer 8 seps for case 2 wih similar display arrangemen as in Fig. 8. As shown in Fig 9(b), because in case 2 he wind speeds are generally larger han he real wind speeds, he simulaed fire grows much larger han he real fire does. Fig. 9(b) and 9(c) show ha, afer assimilaing sensor daa by he SMC mehod, simulaion resuls are significanly improved since he difference beween he real fire and he filered fire is much smaller han he one beween he real fire and he simulaed fire. (a) (b) (c) Fig. 9. Comparisons of he simulaed fire and he filered fire for case 2. (a) Filered fire; (b) Fire frons of he real fire (displayed in green), simulaed fire (displayed in red) and filered fire (displayed in blue); (c) Mismached cells (displayed in red) beween he real fire and he filered fire.

19 To quaniaively show he daa assimilaion resuls, Fig. 10(a) and Fig. 10(b) show he fire perimeers and burned areas of he real fire, he simulaed fire, and he filered fire for case 1 from ime sep 1 o 8. Fig. 10(c) and Fig. 10(d) show he fire perimeers and burned areas of he real fire, he simulaed fire, and he filered fire for case 2 from ime sep 1 o 8. In boh cases, we carry ou 10 independen runs of he daa assimilaion experimens, and display he average of heir resuls in he figures. For case 1, afer he simulaion is finished, he sandard deviaions of he filered fires from he 10 runs are 0.36 km for perimeer and 1.26 hecares for burned area. For case 2, he sandard deviaions are 0.51 km for perimeer and 2.84 hecares for burned area. These figures show ha he differences beween he real fire s perimeers and he filered fires perimeers are smaller han hose for he simulaed fires. The same rend holds rue for he burned areas. Perimeer(km) Perimeer(km) Real Fire Simulaed Fire Filered Fire Time Sep (a) Real Fire Simulaed Fire Filered Fire Time Sep Time Sep (c) (d) Fig. 10. Perimeers and burned areas of he real fire, simulaed fires, and filered fires for case 1 and case 2. (a) Perimeers for case 1; (b) Burned areas for case 1; (c) Perimeers for case 2; (d) Burned areas for case 2. We also compue he symmeric se difference as anoher meric o measure he similariy beween wo fires. In mahemaics, he symmeric se difference of wo ses is he se of elemens in eiher se, bu no in boh. We use i o compare wo fire frons, which is he number of cells inside one of he fire fron shapes, bu no in boh. The smaller he symmeric se difference, he more similar he wo fire frons (he symmeric se difference of wo same fire frons is 0). Fig. 11 shows he symmeric se differences of he simulaed fire (compared o he real fire) and ha of he filered fire (compared o he real fire) in case 1 (Fig. 11(a)) and case 2 (Fig. 11(b)) from ime sep 1 o 8. In hese figures, he values of he filered fire are he averages of 10 independen runs. The horizonal axis represens he ime sep, and he verical axis represens he symmeric se difference value in erms of he number of cells. From he figures, i can be seen ha he symmeric se differences of he filered fires are smaller han hose of he simulaed fires afer sep 2. Wih he increase of ime sep, (i.e., when more sensor daa are assimilaed), he difference beween he simulaed fire and he filered fire becomes more and more noable. A sep 8, he symmeric se difference of he filered fire is less han half of he symmeric se difference of he Area(ha) Area(ha) Real fire Simulaed fire Filered fire (b) Real fire Simulaed fire Filered fire Time Sep

20 simulaed fire. This experimen demonsraes he effeciveness of he daa assimilaion mehod in wildfire spread simulaion when using imprecise wind speeds. umber of cells Simulaed fire Filered fire umber of cells Simulaed fire Filered fire Time Sep Time Sep (a) (b) Fig. 11. Symmeric se differences for case 1 and case 2. (a) Case 1; (b) Case Wind direcion. This se of experimens (case 3 and case 4) examines he daa assimilaion resuls when he wind direcion daa have errors. We run DEVS-FIRE o obain he simulaed fires using he wind direcion daa in Table 1. Fig. 12(a) and Fig. 12(b) show he fire growh of he wo simulaed fires for case 3 and case 4 afer 160 minues. (a) (b) Fig. 12. Simulaed fires for case 3 and case 4. (a) Simulaed fire for case 3 afer 160 minues (average wind speed is 8 mph; average wind direcion is 160 degrees); (b) Simulaed fire for case 4 afer 160 minues (average wind speed is 8 mph; average wind direcion is 200 degrees). Same as before, he emperaure sensor daa are dynamically assimilaed ino he simulaions. The daa assimilaion resuls are displayed in Fig. 13 for case 3, and Fig. 14 for case 4. As can be seen, because of he incorrec wind direcions, here are large differences beween he real fire and he simulaed fires. From Fig. 13(b) one can see ha he simulaed fire grows much faser han he real fire. This is because he fuel ypes along he wind direcion (norhwes) of he simulaed fire make i easier for he fire o spread fas. In Fig. 14(b), he simulaed fire is smaller han he real fire because he fuel ypes along he wind direcion (norheas) in case 4 are harder for he fire o spread. In boh cases, he differences beween he filered fires and he real fire are much smaller han hose of he simulaed fire, as can be seen from Fig. 13(b) and 14(b). However, from Fig 14(b), we noice ha he daa assimilaion in case 4 is no as effecive as in cases 1-3, alhough i sill gives improved resul compared o he simulaed fire. The reason behind his is ha, in case 4, he wind direcion in he filered fire is owards norheas, where i is hard for a fire o spread due o he fuel daa in ha direcion, whereas he wind direcion in he real fire is owards norh where i is easy o spread. This difference of wind direcion resuls in significanly

21 differen fire spreading behavior, which is also refleced by he large difference beween he real fire shape and he simulaed fire shape as shown in Fig. 14(b). In his siuaion, he developed daa assimilaion mehod does no work as effecively o keep rack of he real fire as in oher cases, where here are relaively smaller differences beween he real fire and he simulaed fires. This indicaes a fuure research ask o develop echniques o improve daa assimilaion resuls. For example, he curren mehod does no differeniae differen regions when modeling he sysem ransiion noise. A more advanced mehod may differeniae differen pars of he fire and add graph noise according o he fire spreading behavior (e.g., higher level of graph noise for faser spreading fire regions). (a) (b) (c) Fig. 13. Comparisons of he simulaed fire and he filered fire for case 3. (a) Filered fire; (b) Fire frons of he real fire (displayed in green), simulaed fire (displayed in red) and filered fire (displayed in blue); (c) Mismached cells (displayed in red) beween he real fire and he filered fire. (a) (b) (c) Fig. 14. Comparisons of he simulaed fire and he filered fire for case 4. (a) Filered fire; (b) Fire frons of he real fire (displayed in green), simulaed fire (displayed in red) and filered fire (displayed in blue); (c) Mismached cells beween he real fire and he filered fire. Fig. 15 displays he perimeers and burned areas of he real fires, he simulaed fires, and he corresponding filered fires for case 3 and case 4. In boh cases, we carry ou 10 independen runs of he daa assimilaion experimens and show he averaged resuls for he filered fires in he figures. In case 3, he sandard deviaions of he filered fires in he las sep are 0.36 km for perimeer and 1.30 hecares for burned area. In case 4, he sandard deviaions in he las sep are 0.67 km for perimeer and 3.73 hecares for burned area. In boh case 3 (Fig. 15(a) and Fig. 15(b)) and case 4 (Fig. 15(c) and Fig. 15(d)), compared o he ones of he simulaed fires he perimeers and burned areas of he filered fires are closer o hose of he real fire.