A wildland fire model with data assimilation

Jan Mandel ∗,a,b, Lynn S. Bennethum a, Jonathan D. Beezley a,Janice L. Coen b, Craig C. Douglas c,d, Minjeong Kim a, and

Anthony Vodacek e

aCenter of Computational Mathematics and Department of Mathematical Sciences,

University of Colorado at Denver and Health Sciences Center, Denver, CO

bMesoscale and Microscale Meteorology Division,

National Center for Atmospheric Research, Boulder, CO

cDepartment of Computer Science, University of Kentucky, Lexington, KY

dDepartment of Computer Science, Yale University, New Haven, CT

eCenter for Imaging Science, Rochester Institute of Technology, Rochester, NY

Abstract

A wildfire model is formulated based on balance equations for energy and fuel, wherethe fuel loss due to combustion corresponds to the fuel reaction rate. The resultingcoupled partial differential equations have coefficients that can be approximatedfrom prior measurements of wildfires. An Ensemble Kalman Filter technique is thenused to assimilate temperatures measured at selected points into running wildfiresimulations. The assimilation technique is able to modify the simulations to trackthe measurements correctly even if the simulations were started with an erroneousignition location that is quite far away from the correct one.

Key words: wildfire; combustion; ensemble Kalman filter; parameteridentification; data assimilation; reaction-diffusion equations; partial differentialequations; sensors

1 Introduction

Modeling forest fires is a multi-scale multi-physics problem. One can try toaccount for the many physical processes involved at the appropriate scales,but at the cost of speed. Simplifying appropriate physical processes is one

∗ Corresponding author. Campus Box 170, Denver CO 80217-3364, USA

Preprint submitted to Elsevier 27 March 2007

way to obtain a faster-running model. In this paper we also propose usinga data assimilation technique to incorporate data in real-time. The purposeof this paper is a demonstration-of-concept: we take a very simple model,develop a data assimilation technique, and show how, even with this verysimple model, realistic results can be obtained even with significant errors inthe initial conditions (location of the fire). This is the first step of a longer-termgoal in which a more realistic model will be used.

Data assimilation is a technique used to incorporate data into a running modelusing sequential statistical estimation. Data assimilation is made necessary bythe facts that no model is perfect, the available data is spread over timeand space, and it is burdened with errors. The model solution is just oneout of infinitely many possible realizations (Evensen, 2007). Data assimilationmethods have achieved good success in fields like oil and gas pipeline models(Emara-Shabaik et al., 2002) and atmospheric, climate, and ocean modeling(Kalnay, 2003), and they are a part of virtually any navigation system,from steering the Apollo moon spaceships in the 60’s to GPS and operatingunmanned drones or rovers in hostile conditions like Afghanistan or Marstoday. Data assimilation can also dynamically steer the measurement process,by suggesting locations where the collection of data would result in the bestreduction of uncertainty in the forecast (Kalnay, 2003).

A new paradigm in modeling beyond current techniques in data assimilationis to use Dynamic Data-Driven Application System (DDDAS) techniques(Darema, 2004). Data assimilation is just one of the techniques fromthe DDDAS toolbox, which entails the ability to dynamically incorporateadditional data into an executing application, and in reverse, the ability ofan application to dynamically steer the measurement process. Other DDDAStechniques include deterministic methods such as time rollback, checkpointing,data flow computations, and optimization. One aspect of DDDAS is using dataassimilation and measurement steering techniques from weather forecasting inother fields. In a DDDAS, simulations and measurements become a symbioticfeedback control system. Such capabilities promise more accurate analysis andprediction, more precise controls, and more reliable outcomes.

Our ultimate objective is to build a real-time coupled atmospheric-wildlandfire modeling system based on DDDAS techniques that is steered dynamicallyby data, where data includes atmospheric, fire, fuel, terrain, and other datathat influence the growth of fires (Mandel et al., 2004b,a; Douglas et al., 2006).This work describes one stage of our investigation, that is, to develop andvalidate techniques to ingest fire data that might originate from in situ andremote sensors into a newly developed fire model. The purpose of this paperis to combine a data assimilation method with a partial differential equation(PDE) based model for wildland fire that achieves good physical behaviorand can run faster than real time. The model in this paper does not yet

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include coupling with the atmosphere, though it is known that such couplingis essential for the wildland fire behavior (Clark et al., 1996). Coupling the firemodel with atmospheric dynamics as well as with data assimilation is currentlyunder development. Models using explicit, detailed combustion physics are notfeasible for prediction, since they require a large number of chemical reactionsand species and extremely high resolution (grid cells << 1m) fluid dynamics(Zhou and Mahalingam, 2001). The actual interaction between the atmosphereand the fire and vegetation is much more sophisticated, involving turbulencein the vegetation layer and its consequences on heat transfer and combustion(Baines, 1990). The example provided in this paper is for 250 × 250 cells of2m size each. A model like FIRETEC (Linn, 1997; Linn et al., 2002) coulddo the same, but including the full interaction between fire, vegetation, andthe atmosphere, would come at a much higher computational cost. FIRETECis an example of a physically-based model that simplifies parts of the physics(coarser description than in Zhou and Mahalingam (2001)), but includes theessence of atmosphere-fire interaction. Future developments of the numericsand parallelization of our model are expected to be able to handle realisticfires of the size of several km, coupling with the atmosphere, and assimilationof real-time data.

An important point is that our paradigm attempts to strike a balance betweenmodel complexity and fast execution. Thus, the present model is based onjust two PDEs in two horizontal spatial dimensions: prognostic (predictive)equations for heat and fuel. We use a single semi-empirical reaction rate toachieve the desired combustion model results. In other words, we solve the setof equations known as the reaction-convection-diffusion problem using reactionrates based on the Arrhenius equation, which relates the rate at which achemical reaction proceeds to the temperature.

This is the simplest combustion model and it is known to produce solutionswith traveling combustion waves, that is, a propagating area of localizedcombustion made up of the preheated area ahead of the fire, the combustionzone, and the post-frontal burning region. One reason for considering a PDE-based model is that even simple reaction-diffusion equations are capable ofthe complex nonlinear, unsteady behavior such as pulsation and bifurcationthat is seen in reality but cannot be reproduced by empirical models.

The characteristics of the combustion wave (maximal temperature, width ofthe burning region as defined by the leading and trailing edge, and the speed ofpropagation) are used to calibrate the parameters of the model. We note thatphysical behavior can be achieved by a very simple model that can reproducerealistic fire behavior very quickly on today’s computers. The PDE-basedmodels in this paper are not necessarily original, cf., e.g., Weber et al. (1997),and PDE coefficients have been determined empirically from measured time-temperature curves (Balbi et al., 1999; Giroud et al., 1998; Morandini et al.,

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2001), though not in a reaction-diffusion PDE model like the model here. Weprovide a new systematic procedure for the calibration of the PDE modelon real wildland fire data based on separation of nondimensional propertiesand solution scales. The general calibration problem is an interesting optimalcontrol and stochastic parameter estimation problem in its own right that willbe studied in detail in future works.

We then proceed to data assimilation to modify the state of the running modelfrom data. A version of the ensemble Kalman filter (EnKF) is used for thedata assimilation, that is, to inject observations into the model and modify itsstate while it is running. This work appears to be the first wildland fire modelwith data assimilation.

Future extensions of this work include coupling with a numerical weatherprediction model, modeling of water content in the fuel, multiple fuellayers, separate treatment of the gas phase (i.e. pyrolysis), crown fire,modeling spotting by stochastic differential equations, preheating by long-range radiation, and contemporary numerical methods such as finite elementsand level set methods. The exact number of data points (in space and time)that are necessary for recovering good predictions when the numerical solutiondiverges from reality depends on the particular data assimilation method used.For the method in this paper, the correction in the location of the firelineshould not be larger than the width of the reaction zone. This is similar to thesituation for data assimilation into hurricane models, where the correction inthe location of the vortex should not be larger than the vortex size (Chen andSnyder, 2006). Advanced data assimilation methods that allow sparser dataand larger correction are the subject of further research. While real time dataare routinely available for weather forecasting systems, in a wildland fire thedata collection is less straightforward. Available data include multi-spectruminfrared airborne photographs, processed to recover the fire region and tosome extent the temperature, and radioed data streams from hardened sensorsput in the fire path (Kremens et al., 2003; Ononye et al., 2005, to appear).For overviews of the whole project including computer science aspects, datacollection, and visualization, see Mandel et al. (2004b, 2005, 2007) and Douglaset al. (2006).

The remainder of the paper is organized as follows. In Section 2, we stateour PDE-based fire model. Then in Section 3, we describe the relation of ourmodel to other models in the literature. In Section 4, the model is derivedfrom physical principles in more detail. In Section 5, we develop a methodto determine the coefficients of the PDE model using wildland fire data. InSection 6, we describe the ensemble Kalman filter techniques for the dataassimilation. In Section 7, we test the PDE and ensemble Kalman filtermethods on a two-dimensional representation of a wildland fire and calibratethe models against real data. Finally, Section 8 contains our conclusions.

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2 Formulation of the model

We consider the model of fire in a layer just above the ground, given by thefollowing equations derived from the conservation of energy, balance of fuelsupply, and the fuel reaction rate,

dT

dt= ∇ · (k∇T ) + −→v · ∇T + A

(Se−B/(T−Ta) − C (T − Ta)

), (1)

dS

dt= −CSSe−B/(T−Ta), T > Ta, (2)

with the initial values

S (tinit) = 1 and T (tinit) = Tinit. (3)

In (1-3),

T (K) is the temperature of the fire layer,S ∈ [0, 1] is the fuel supply mass fraction (the relative amount of fuelremaining),

k (m2s−1) is the diffusion coefficient,A (Ks−1) is the temperature rise per second at the maximum burning ratewith full initial fuel load and no cooling present,

B (K) is the proportionality coefficient in the modified Arrhenius law,C (K−1) is the scaled coefficient of the heat transfer to the environment,CS (s−1) is the fuel relative disappearance rate,Ta (K) is the ambient temperature,−→v (ms−1) is the wind speed given by atmospheric data or model.

The diffusion term ∇ · (k∇T ) models short-range heat transfer by radiationin a semi-permeable medium, −→v · ∇T models heat advected by the wind,Se−B/(T−T0) is the rate fuel is consumed due to burning, and AC (T − Ta)models the convective heat lost to the atmosphere. The reaction ratee−B/(T−Ta) is obtained by modifying the reaction rate e−B/T from the Arrheniuslaw by an offset to force zero reaction at ambient temperature, with theresulting reaction rate smoothly dependent on temperature.

A more detailed derivation of the model from physical principles is contained inSection 4. Calibration of the coefficients from physically observable quantitieswill be described in Section 5.

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3 Relation to other models

For recent surveys of wildland fire models and their history, see Morvan et al.(2002), Pastor et al. (2003), and Seron et al. (2005).

3.1 Models based on diffusion-reaction PDEs

It is known that systems of the form (1-2) admit traveling wave solutions. Thetemperature in the traveling wave has a sharp leading edge, followed by anexponentially decaying cool-down trailing edge. This was observed numericallybut we were not able to find a rigorous proof in the literature in exactly thiscase, though proofs for some related systems exist.

For a related system with fuel diffusion, the existence and speed of travelingwaves were obtained by asymptotic methods already in classical work,summarized in the monograph by Zeldovich et al. (1985). For the system(1-2), Weber et al. (1997) obtained approximate combustion wave speed byheuristic asymptotic methods under the assumption that no heat is lost andambient temperature is absolute zero, which is equivalent to our setting C = 0.Models that do not guarantee zero combustion at ambient temperature sufferfrom the “cold boundary difficulty”: by the time a combustion wave gets toa given location, the fuel at that location is depleted by the ongoing reactionat ambient temperature. So, no perpetual traveling combustion waves canexist, and there are only “pseudo-waves” that travel only for a finite time(Berestycki et al., 1991; Mercer et al., 1996; Zeldovich et al., 1985). Camposet al. (2004) have derived the speed of traveling waves in a simplified modelwith the reaction started by ignition at a given temperature, followed by anassumed temperature-indepentent reaction rate that is only proportional tothe fuel remaining. This is similar to the model by Balbi et al. (1999).

Equation (1) without fuel depletion (i.e., with constant S) and without wind(i.e., −→v = 0) is a special case of the nonlinear reaction-diffusion equation,

dT

dt= ∇ · (∇T ) + f (T ) . (4)

Reaction-diffusion equations of the form of equation (4) are known to possesstraveling wave solutions, which switch between values close to stationary statesgiven by f (T ) = 0 (Gilding and Kersner, 2004; Infeld and Rowlands, 2000).The simplest model problem is Fisher’s equation with f (T ) = T (1 − T ),for which the existence of a traveling wave solution and a formula for itsspeed were proven by Kolmogoroff et al. (1937). For an analytical study ofthe evolution of waves to a traveling waveform, see Sherratt (1998), and for

6

a numerical study, see Gazdag and Canosa (1974) and Zhao and Wei (2003).Robinson (2001, Ch. 8 and 11) gives proofs of the existence of solution andattractors for reaction-diffusion equations, but does not mention travelingwaves. Asensio and Ferragut (2002) proved the existence of a solution, but nottraveling waves, for the reaction-diffusion equation (1) except with a nonlineardiffusion term ∇ · (kT 3∇T ) and again without considering fuel depletion.

Seron et al. (2005) consider a two reaction model (solid and pyrolysis gas),and argue that the modeling of pyrolysis by a separate reaction is essential forcapturing realistic fire behavior. For a more complicated model of this typethat includes also, e.g., water vapor, see Grishin (1996).

Various aspects of special cases of equations (1-2) were studied in a numberof papers. Weber (1991b) uses a formal expansion in an Arrhenius reactionmodel to get the wave speed and a prediction of whether a small fire will orwill not spread. Mercer and Weber (1995) compute the ignition wave speedand extinction wave speed numerically. The speed and stability of combustionwaves were analyzed by asymptotic expansion by Gubernov et al. (2003). Anapproximation to the temperature reaction equation gives a size of the reactionzone and slope of the temperature curve. Norbury and Stuart (1988a,b) derivea nonlinear eigenvalue problem for a traveling wave in a different combustionproblem, with fuel reaction, solve it numerically by the shooting method,and study the existence and stability of the traveling wave solution. See alsoGubernov et al. (2003, 2004) for the gas case (i.e., also with fuel diffusioninstead of only temperature diffusion).

Enriched finite element methods for the linear diffusion-advection-reactionproblem were designed and error estimates provided by Asensio et al. (2004),Franca et al. (2005), Franca et al. (2006), and Codina (1998). However, inall those works, the reaction function f is replaced by a linear function, sothere are no traveling wave solutions, and fuel consumption is not considered.Mickens (2005) compares several time discretization techniques in the presenceof nonlinear reaction terms. Asensio and Ferragut (2000) and Ferragut andAsensio (2002) provide error estimates for mixed finite elements applied tothe single species combustion equation and more general reaction-diffusionproblems, but again without a fuel balance equation; cf., also Arbogast et al.(1996). Sembera and Benes (2001) proposed a nonlinear Galerkin methodfor reaction-diffusion problems, and proved convergence by a compactnessargument. Even simple nonlinear models exhibit bifurcations, which can beexamined by direct simulation (Theodoropoulos et al., 2000). Approximationof Fisher’s equation by finite elements were studied numerically by Roesslerand Hussner (1997) and Carey and Shen (1995), especially regarding thecorrect wave speed, but no error estimates were given. For finite differencesapplied to a reaction-diffusion equation, see Liao et al. (2006). Since thecommon feature of the solutions of reaction-diffusion equations is the

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development of a sharp wave, with the solution being almost constantelsewhere, interface tracking techniques such as level set methods (Sussmanet al., 1994; Sethian, 1999) are relevant here as well.

3.2 Fireline evolution, fire spread, and empirical models

The reaction zone in reaction-diffusion models is typically very thin, andresolving it correctly requires very fine meshes. Hence, a number of modelsconsider the evolution of the fireline instead. Combustion equations in the so-called reaction sheet limit or large activation energy asymptotics reduce to arepresentation of the reaction zone (here, the fireline) as an evolving internalinterface (Chen, 1992; Class et al., 2003; Dold et al., 2003; Fife, 1988; Lawet al., 1993; Rastigejev and Matalon, 2006), though this reduction does notseem to have been done for exactly the same equations as here. The asymptoticmodels typically compute the speed of the movement of the reaction interfacein the normal direction, often involving its curvature.

Fireline evolution models often postulate empirically observed properties ofthe fire, such as the fire spread rate in the normal direction, instead ofphysically based differential equations. Modeling of fireline evolution wasreviewed by Weber (1991a). Albini and Reinhardt (1997), Albini (1994), andFrandsen (1971) derived fire spread rates without using reaction kinetics.Richards (1995, 1999) studied evolution of the fireline as a curve. Fire spreadmodels based on radiation like Albini (1986) were tested against data byDupuy (2000). Dupuy and Larini (1999) introduced a convective term in aradiation based model in an attempt to better describe slope effects on rateof spread.

Far fewer examples exist where data was used to calibrate models. Albini(1994) and Albini and Reinhardt (1997) calibrated a mass loss model bycrib burning experiments. Rothermel (1972) formulated a model for surfacefire spread rate with a physically based core rate of spread in zero windon flat ground, calibrated to other wind speed and slopes using laboratorymeasurements. Balbi et al. (1999) used the energy balance equation coupledwith a model of ignition at a threshold temperature, followed by exponentialdecay. Measured temperature profiles were used to identify parameters of themodel. Simeoni et al. (2001) and Morandini et al. (2001) used laboratorydata to validate predictions made with different systems of PDEs. Wottonet al. (1999) used actual fire spread data and theoretical calculations totest the effect of fire front width on surface spread rates through radiativetransfer terms. Viegas (2005) postulates empirical rates of fire spread and ofthe wind created by the fire and identifies the coefficients from experiments.The feedback between the fire and the surrounding flow is then modeled by

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a simple one-dimensional differential equation, which is sufficient to explainconditions for the fire spread to stop or accelerate to a blowup.

3.3 Coupled fluid-fire models

Wildland fire models (either empirical, semi-empirical, or PDE-based) havebeen coupled to a fluid environment that may be (for small domains) acomputational fluid dynamics model (i.e. models the flow and thermodynamicsof air) or a numerical weather prediction model (a computational fluiddynamics model that also considers moist atmospheric processes, theformation of precipitation, and flow over topography).

The FIRETEC model (Linn, 1997; Linn et al., 2002) simulates wildlandfires by representing the average reaction rates and transport over a resolvedvolume, usually on the order of 1m3 in three dimensional space. This attemptsto resolve the effects of heat transfer processes without representing each indetail. The ambient environment is air, but the model omits weather processes.Larini et al. (1998) gave a multiphase, reactive, and radiative one-dimensionalmodel specialized for wildland fires. Dupuy and Morvan (2005) added detailedfluid modeling to the model from Larini et al. (1998) to study crown fires inair, again with no weather processes, in two-dimensions. Grishin and Shipulina(2002) presented a complicated model of surface and canopy fire based onfluid dynamics and chemical reaction equations where prognostic equations arecreated for gases that have been grouped into reactive combustible gases, non-reactive combustion products, and an oxidizer (O2). Methods of this type arestandard in combustion modeling; for a flame model with detailed chemistryand physics, see Ern et al. (1995).

An alternate approach was adopted by Clark et al. (1996, 2004), wherea semi-empirical fire spread model based upon the Rothermel (1972) firespread equation and a canopy fire model were coupled to a numerical weatherprediction model to model the interactions between wildland fires and theatmospheric environment. Here, weather processes ranging from synoptic toboundary layer scale are simulated with good fidelity, and the combustionprocesses are represented by the semi-empirical formula in order to capturethe sensible (temperature) and latent (water vapor) heat fluxes into theenvironment.

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4 Derivation of the model

We consider fire in a ground layer of some unspecified finite small thicknessh. The fire layer consists of the fuel and air just above the fuel. Allmodeled quantities are treated as two-dimensional, homogenized in the verticaldirection over the ground layer. We will not attempt to derive equationsand substitute coefficients from material properties because of the degree ofsimplification and uncertainty present in the homogenization. Instead, physicallaws will be used to derive the form of the equations and the coefficients willbe identified later from the dynamical behavior of the solution. We first derivethe system of PDEs based on conservation of energy and fuel reaction rate inSection 4.1, and then discuss the choice of the reaction term in Section 4.2.

4.1 Heat and fuel supply balance equations

The chemical reactions are a heat source. Heat transfer is due to radiation andconvection to the atmosphere. The short-range heat transfer due to radiationand turbulence is modeled by diffusion; the two-dimensional heat flux througha segment per length unit then is

−→q r = −k1∇T (Wm−1). (5)

The constant k1 (WK−1) will be identified later.

Heat is generated by the chemical reaction of burning. We model the burningas a reaction in which the rate depends on temperature only, so the reactionrate is CSr (T ), where CS is a coefficient of proportionality (1/s), and r isdimensionless. Let F > 0 (kg m−2) be the concentration of fuel remaining;then the rate at which the fuel is lost is proportional to the rate of reactionand the amount of fuel available,

dF

dt= −FCSr (T ) , (6)

and the heat generated per unit surface area is then proportional to the fuellost,

qg = A1FCSr (T ) , (Wm−2) (7)

where A1 (J kg−1) is the heat released per unit mass of fuel.

Heat per unit area lost due to natural convection to the atmosphere due tobuoyancy is given by Newton’s law of cooling,

qc = Ca(T − Ta),(Wm−2

)(8)

10

where Ta is the ambient temperature (K) and Ca (Wm−2K−1) is the heattransfer coefficient. In this model, it is assumed that the convective heattransfer is dominant, and so the effect of radiation into the atmosphere isincluded in (8).

From the divergence theorem, we now obtain the conservation of energy inthe fire layer as

hρcpDT

Dt= ∇ · −→q r + qg − qc, (9)

whereDT

Dt=

dT

dt+ −→v · ∇T (10)

is the material time derivative of the temperature, dT/dt is the Eulerian (orspatial) time derivative of temperature, −→v is the (homogenized) velocity ofthe air, ρ is the homogenized surface density of the fire layer (kg m−3), andcp is the homogenized specific heat of the fire layer (J kg−1K−1). Again, noneof the coefficients h, ρ, or cp can be assumed to be actually known. The unitsof the product hρcp are JK−1m−2.

The velocity vector −→v is obtained from the state of the atmosphere as data, orin future work by coupling with an atmospheric model. In the present model,the velocity vector also incorporates the effect of slope: adding to the wind asmall multiple of the surface gradient somewhat simulates the effect that firespreads more readily uphill. In addition, since the speed of the air is zero atthe ground level if as usual no-slip conditions are assumed, the homogenizedspeed through the fire layer should be approximated by scaling the given windvelocity by a constant less than one.

We now write the equations in a form suitable for identification of thecoefficients (Sec. 5). The goal is to obtain a system of equations with aminimal number of coefficients in as simple form as possible. In addition,we wish to relate the coefficients to the behavior of the solution for certainparticular coefficient values, rather than to material and physical propertiesof the medium, which are in general unknown. We introduce the mass fractionof fuel by

S =F

F0

,

where F0 is the initial fuel quantity. Substituting in the appropriate valuesfor the heat sources and fluxes, (5), (7), and (8), into (10) and (9), and somesimple algebra, we obtain the energy balance and fuel reaction rate equations

dT

dt= ∇ · (k∇T ) + −→v · ∇T + A (Sr (T ) − C0 (T − Ta)) , (11)

dS

dt= −CSSr (T ) , (12)

11

with

k = k1/(hρcp), A = A1CS/(hcpρ), C0 = Ca/A1.

Alternatively, we could have taken disappearance of fuel on the left hand sideof the heat balance equation (9) (fuel that has burned does not need to beheated), which would lead to an equation of the form

(1 + C1S)dT

dt= ∇ · (k∇T ) + −→v · ∇T + A (Sr (T ) − C0 (T − Ta))

instead of (11). We have chosen not to do that because our aim is to workwith the simplest possible model, whose coefficients can be identified. The fueldisappearance in the heat equation affects the temperature profile significantlyonly in the reaction zone, which is the highest part of the temperature curve.Before the ignition, there is a full fuel load, S = 1, and after a fairly shortreaction time, well before most of the cooling takes place, the remaining fuelsettles to some residual value which then remains constant. The effect of thedecreased heat capacity of the remaining fuel is then absorbed in the coolingcoefficient AC0.

4.2 Reaction rate

The Arrhenius reaction rate from physical chemistry is given by

r (T ) = e−B/T , (13)

where the coefficient B has units K. This equation is valid only for gas fuelpremixed with a sufficient supply of oxygen. This approximation ignores fuelsurface effects but it is widely used nonetheless. One consequence of (13) isthat the reaction has a nonzero rate at any temperature above absolute zero.Because the time scale for burning is much smaller than the oxidation rate atambient temperature, we modify (13) so that no oxidation occurs below somefixed temperature, T0 (see Section 5), and take instead

r (T ) =

e−B/(T−T0), T > T0,

0, T ≤ T0.(14)

Note that the fuel consumption rate is a smooth function of T , which isfavorable for a numerical solution, unlike in (Asensio and Ferragut, 2002),where a cutoff function was used. Consider first the hypothetical case in whichT is constant in space, so that only heat due to the reaction (burning) andnatural convection contribute non-zero terms in the heat equation, (1), andessentially the full initial fuel supply F0 is present at all times (the rate of fuel

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consumption is negligible, CS ≈ 0, so S ≈ 1):

dT

dt= A

(e−B/(T−T0) − C (T − Ta)

). (15)

Constant values of temperature which are solutions to (15) are calledequilibrium points, and at these points the heat produced by the reactionequals the heat lost to the environment,

f(T ) = e−B/(T−T0) − C (T − Ta) = 0. (16)

Equation (16) has at most three roots (Frank-Kamenetskii, 1955), see forexample, Figure 1. The first zero, denoted as Tp, called the lower temperatureregime by Frank-Kamenetskii (1955), is a stable equilibrium temperature - ifthe temperature goes below this temperature then the heat generated from thereaction dominates and the temperature rises. If the temperature goes abovethis temperature then convective cooling dominates and the temperaturedecreases. If T0 < Ta, then this point is typically just above the ambienttemperature, Ta, since some reaction is present even at ambient temperature.The middle zero, Ti, is an unstable equilibrium point. If the temperature goesbelow Ti then convective cooling dominates and the temperature decreases.Above Ti the heat due to chemical reactions dominate and the temperatureincreases. We refer to Ti as the auto-ignition temperature, the temperatureabove which the reaction is self-sustaining (Quintiere, 1998). The stableequilibrium at a high temperature, Tc, is the maximum stable combustiontemperature, assuming replenishing of the supply of fuel and oxygen. Thetemperature Tc is called the high temperature regime by Frank-Kamenetskii(1955). The stability properties of the equilibrium points are also clear from thegraph of the potential U(T ), defined by U ′(T ) = f(T ); the stable equilibriumpoints are local minima of the potential, while the autoignition temperatureis a local maximum and thus an unstable equilibrium (Fig. 2).

While the coefficients B and C/F0 in (16) are generally unknown, they canbe found from the equilibrium temperature points. Suppose that two roots Ti

and Tc of f(T ) from (16) are given such that T0 ≤ Ta < Ti < Tc. Then simplealgebra gives

B =ln

(Ti−Ta

Tc−Ta

)

1Tc−T0

− 1Ti−T0

, C =e−B/(Ti−T0)

Ti − Ta

. (17)

It should be noted that the coefficients B and C from (17) result in threeequilibrium points only when Ti is significantly higher than Ta; for Ti tooclose to Ta the resulting energy balance equation f(T ) = 0 has only two roots.This, however, does not occur for the values of Ta, Ti, and Tc of interest.

First consider the solution of (11-12) and the reaction rate (14) with T0 = 0.

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400 600 800 1000 1200−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Temperature(K)

f(T

)

Ta=300 T

c=1200 T

i=670

.

Fig. 1. Sample reaction heat balance function f(T ) from equation (16)

200 400 600 800 1000 1200 1400−60

−50

−40

−30

−20

−10

0

10

Ta=300 T

c=1200 T

i=670

Temperature(K)

Pot

entia

l U

Fig. 2. Reaction heat balance potential U

Then the reaction is the Arrhenius rate known from chemistry, and there isa nonzero reaction rate at T = Ta. This results in fuel loss everywhere, and,in our computational experiments, no traveling combustion wave developed(Fig. 3) since, after a relatively short time, there was not enough fuel tosustain combustion. This phenomenon is known as the cold boundary effectin combustion literature (Weber et al., 1997): a traveling combustion wavesolution does not exist, and there can only be pseudo-waves that propagatefor a limited time (Berestycki et al., 1991; Mercer et al., 1996) and thenvanish. Since for the values of B and C we obtain from realistic Ti and Tc

the fuel disappears rather quickly, we force the reaction rate, r(T ), to be zeroat ambient temperature by choosing the offset T0 = Ta. Using the offset by Ta

is essentially the same as assuming that the ambient temperature is absolutezero as commonly done in combustion literature (Weber et al., 1997). In thiscase, a propagating combustion wave develops (Fig. 4, 5). Therefore, from nowon, we use T0 = Ta. This concludes the derivation of the model (1-3).

14

0

500

1000

020

4060

80

0

0.5

1

x(m)

Fuel supply mass fraction

time(seconds)

Mas

s fr

actio

n

0

500

1000

020

4060

80

0

1000

2000

3000

x(m)

Propagating combustion wave

time(seconds)

Tem

pera

ture

(K)

Fig. 3. Solution with the Arrhenius reaction rate. Due to nonzero reactionrate at ambient temperature, fuel starts disappearing and thus a propagatingcombustion wave does not develop. The coefficients are k = 2.1360×10−1m2s−1K−3,A = 1.8793 × 102Ks−1, B = 5.5849 × 102K, C = 4.8372 × 10−5K−1, andCS = 1.6250 × 10−1s−1. Tc = 1200K and Ti = 670K were used.

5 Identification of coefficients

We wish to use observed behavior of the fire rather than physical materialproperties to identify the coefficients. It is not simple to obtain reasonablebehavior of the solution from substituting physical coefficients into theequations. Further, as explained in Section 4, because of a number ofsimplifying assumption employed and because of the homogenization ofcoefficients over a fire layer of unspecified thickness, it is not quite clear whatthe material properties should be anyway.

We first use basic reaction dynamics and a reduced model to find roughapproximate values of the coefficients that produce a reasonable solution.Then we transform the equations to a nondimensional form, which allows usto separate the coefficients into those that determine the qualitative behaviorof the solution and those that determine the scales. We propose to use theapproximate coefficients obtained from the reduced models as initial values

15

0

500

1000

0

1000

20000

0.2

0.4

0.6

0.8

x(m)

Reaction rate

time(seconds)

Rea

ctio

n

0

500

1000

0

1000

20000

0.5

1

x(m)

Fuel supply mass fraction

time(seconds)

Mas

s fr

actio

n

0

500

1000

0

1000

20000

500

1000

1500

2000

x(m)

Propagating combustion wave

time(seconds)

Tem

pera

ture

(K)

Fig. 4. Solution with Arrhenius reaction rate modified by temperature offset Ta toforce zero reaction rate at ambient temperature. A propagating combustion wavedevelops. The coefficients are k = 2.1360× 10−1m2s−1K−3, A = 1.8793× 102Ks−1,B = 5.5849×102K, C = 4.8372×10−5K−1, and CS = 1.6250×10−1s−1. Tc = 1200K

and Ti = 670K were used.

for identification of the coefficients by the nondimensionalization method tomatch observed temperature profiles.

16

5.1 Reaction and cooling coefficients

The coefficients B and C in the modified reaction form, (16), can bedetermined from reasonable values of Tc and Ti by (17). We want to determinethe remaining coefficient, A. This can be done from the characteristic coolingtime. Consider the trailing edge of a traveling combustion wave, after all ormost of the fuel has been depleted, temperature drops, and heat generated bythe reaction and diffusion drop to an insignificant level. From that point on,the temperature satisfies (approximately)

dT

dt= −AC(T − Ta).

Thus, at the trailing edge, given T at some time t0, we have

T (t) = Ta + (T (t0) − Ta) e−AC(t−t0)

and we can define the characteristic cooling time, tc, to be the time which thefire layer takes to cool by a factor of 1/e, i.e.

T (t0 + tc) − Ta =1

e(T (t0) − Ta) ,

which gives ACtc = 1, or

A =1

Ctc. (18)

5.2 Scales and non-dimensional coefficients

We now write the model in terms of nondimensional variables, which controlthe qualitative behavior of the system (1-2). Again, we do not consider thewind here yet, and so

dT

dt= ∇ (k∇T ) + A

(Se−

B

T−Ta − C (T − Ta)), (19)

dS

dt= −SCSe−

B

T−Ta , T > Ta. (20)

The substitution

T =T − Ta

Bx =

x

k1/2B1/2A−1/2t =

tA

B.

17

transforms (19-20) into a non-dimensional form

dT

dt= ∇ ·

(∇T

)+ Se−1/T − λT , (21)

dS

dt= −βSe−1/T , T > 0, (22)

with two dimensionless coefficients

λ = CB, β =BCS

A. (23)

Therefore, the qualitative behavior of the solution is determined only by thenondimensional coefficients λ and β, which can be varied independently.

The nondimensional form (21-22) suggests a strategy for identification of thecoefficients k, A, B, C, CS: first match nondimensional properties of thetraveling combustion wave, such as the ratio of the width of the leading edgeand the trailing edge and the fuel fraction remaining after the combustionwave by varying λ and β. The width of the wave can be measured e.g.as the distance of the points where the temperature equals 50% of themaximum. The nondimensional traveling wave solution T (t, x), S(t, x) hassome (nondimensional) maximal temperature Tmax, width w, and speed v,while the data (a measured temperature profile) has maximal temperatureTmax, the width w, and the speed v of the traveling wave. This determines thescales

T1 =Tmax

Tmax

, x1 =w

w, t1 =

vw

vw.

By the substitution

T =T − Ta

T1

x =x

x1

t =t

t1,

the system (19-20) with the coefficients

A = T1/t1, B = T1, C = λ/T1, CS = β/t1, k = x21/

(T 3

1 t1), (24)

admits the scaled solution

T (t, x) = T1T(

t

t1,

x

x1

)+ Ta, S(t, x) = S

(t

t1,

x

x1

),

which has the desired nondimensional properties as well as the correct maximaltemperature, the width, and the speed of a traveling combustion wave.

A dimensionless system similar to (21-22) was studied by Weber et al. (1997)in the case λ = 0, i.e. combustion insulated against heat loss. Weber et al.(1997) have determined the speed of the traveling wave as a function of β

18

numerically and by an asymptotic expansion, and they have observed that atraveling combustion wave exists only for small values of β; with increasing β,the solution is periodic, then the period doubles, and eventually the solutionbecomes chaotic. We have observed that increasing β has a similar effect forthe equations (21-22) when λ > 0. Also, we have observed that a sustainedcombustion wave is possible only when λ is small enough. A systematic studyof the properties of (21-22) for various values of λ and β will be done elsewhere.For a dimensionless system similar to ours, but without the temperature offsetto force zero reaction at ambient temperature, see also (Asensio and Ferragut,2002).

6 Data Assimilation

The goal in data assimilation on a fire model involves determining whetheror not a filter can effectively track the location of the fireline given datain the form of temperature and remaining fuel at sample points inside ofthe domain. Such an application is particularly troublesome for EnKFs. Thestandard method for generating an initial ensemble is not sufficient for thisscenario. Namely, taking an initial guess at the model state (temperature andfuel) and adding to it a smooth random field. Here, if the data indicates thatthe fireline has shifted away from that of the ensemble, then the KalmanFilter will generally ignore the data entirely due to the extraordinarily smalldata likelihood. Clearly, such an initial ensemble does not properly representthe prior uncertainty in the location of the ignition region, only that of thetemperature of ignition. In order to represent this uncertainty as well, we havealso perturbed the state variables by a spatial shift (Johns and Mandel, 2005).However, this approach leads to further potential problems for EnKF. Due tothe relatively sharp temperature profile of the fireline, the temperature ateach grid point will tend to be close to that of the stable ambient or burningtemperatures. A similar situation occurs with the fuel near the fireline as well.This is indicative of a strongly bimodal or non-Gaussian prior distribution.Despite this violation of the underlying assumptions of EnKF, we have foundthat it is possible to track large changes in the fireline, as shown in thenumerical results section.

The Ensemble Kalman Filter (EnKF) is a Monte-Carlo implementation of theBayesian update problem: Given a probability distribution of the modeledsystem (the prior, often called the ‘forecast’ in geophysical sciences) anddata likelihood, the Bayes theorem is used to to obtain the probabilitydistribution with the data likelihood taken into account (the posterior or the’analysis’). The Bayesian update is combined with advancing the model intime, with the data incorporated from time to time. The original Kalman Filter(Kalman, 1960) relies on the assumption that the probability distributions

19

are Gaussian, and provided algebraic formulas for the change of the meanand covariance by the Bayesian update, and a formula for advancing thecovariance matrix in time provided the system is linear. However, this isnot possible computationally for high dimensional systems. For this reasons,EnKFs were developed by Evensen (1994); Houtekamer and Mitchell (1998).EnKFs represent the distribution of the system state using a random sample,called an ensemble, and replace the covariance matrix by the sample covarianceof the ensemble. One advantage of EnKFs is that advancing the probabilitydistribution in time is achieved by simply advancing each member of theensemble. EnKFs, however, still rely on the Gaussian assumption, thoughthey are of course used in practice for nonlinear problems, where the Gaussianassumption is not satisfied. Related filters attempting to relax the Gaussianassumption in EnKF include Anderson and Anderson (1999); Bengtsson et al.(2003); Mandel and Beezley (2006); van Leeuwen (2003).

We use the EnKF following Burgers et al. (1998); Evensen (2003), withonly some minor differences. This filter involves randomization of data. Forfilters without randomization of data, see Anderson (1999); Evensen (2004);Tippett et al. (2003). The data assimilation uses a collection of independentsimulations, called an ensemble. The ensemble filter consists of

(1) generating an initial ensemble by random perturbations,(2) advancing each ensemble member in time until the time of the data, which

gives the so-called forecast ensemble

(3) modifying the ensemble members by injecting the data (the analysis

step), which results in the so-called analysis ensemble

(4) continuing with step 2 to advance the ensemble in time again.

We now consider the analysis step in more detail. We have the forecastensemble

U f = [uf1 , . . . , u

fN ] = [uf

i ]

where each ui is vector size n, which contains the whole simulation state(in our case, the vector of T and S at mesh nodes). The data is given as ameasurement vector d and data error covariance matrix R size m by m. Thecorrespondence of the data and the simulation states is given by an observation

function h(u) that creates synthetic data, that is, what the data would havebeen if the simulation and the measurements were exact. We assume that his linear, h (u) = Hu, so the observation being assimilated is

Hu − d ∼ N (0, R)

for some matrix H. The observation function defines the data likelihood (theprobability density of d given the state u). Assuming the data error is normally

20

distributed, the data likelihood is

p (d|u) ∝ exp(−1

2(h (u) − d) R−1 (h (u) − d)

),

where ∝ means proportional. The forecast ensemble is considered a samplefrom so-called prior distribution p(u), and the EnKF strives to create ananalysis ensemble that is a sample from the so-called posterior distributionp (u|d), which is the probability distribution of u after the data has beeninjected. From Bayes theorem of probability theory, we have

p (u|d) ∝ p (d|u) p(u). (25)

cf., Evensen (2003, eq. (20)). If p ∼ N(uf , Qf ) (that is, the prior is normalwith mean uf and covariance matrix Qf ), then it is known that the posterioris also normal with mean

ua = uf + K(d − Huf

), (26)

where K is the Kalman gain matrix,

K = QfHT(HQfHT + R

)−1

. (27)

The EnKF proceeds by applying (26) to each ensemble member, with the datavector replaced by a randomly perturbed vector

dj = d + vj, vj ∼ N (0, R)

and the unknown covariance matrix Qf in (27) replaced by the samplecovariance of the forecast ensemble [uf

i ]. This gives the EnKF formulas,

uai = X + CHT

(HCHT + R

)−1

(di − Hufi ), (28)

C = [ckℓ] , ckℓ =1

N − 1

N∑

i=1

(uf

i,k − ufi,k

) (uf

i,ℓ − ufi,ℓ

)(29)

ufi,k =

1

N

N∑

i=1

ufi,k, (30)

where ufi,k is the entry k of uf

i . See Evensen (2003, eq. (20)) for details. Theonly difference between (28) and Evensen (2003, eq. (20)) is that we usethe covariance matrix R of the measurement error rather than the samplecovariance of the randomized data, so, because R is always positive definite,there is no difficulty with the inverse in (28), unlike Evensen (2003, eq. (20)).

21

6.1 EnKF implementation

When the data covariance R is diagonal, which will the case here, the EnKFformulas can be implemented in O (N3 + mN2 + nN2) operations, which issuitable both for large number n of the degrees of freedom and large number mof data points. Also, (28) can be implemented without forming the observationmatrix H explicitly, by only evaluating the observation function h. See Mandel(2006) for details.

The ensemble filter formulas are operations on full matrices, and theywere implemented in a distributed parallel environment using MPI andScaLAPACK. EnKF is naturally parallel: each ensemble member can beadvanced in time independently. The linear algebra in the Bayesian updatestep links the ensemble members together.

6.2 Regularization

EnKF produces the analysis ensemble in the span of the forecast ensemble.This results in nonphysical states especially if the states in the span are faraway from the data. For cheap numerical methods and a highly nonlinearproblem, EnKF can easily knock the state out of the stability region. In orderto ease this problem, we add an independent observation

ua −uf ∼ N(0, D),

where is the spatial gradient, computed by finite differences. This is easilyimplemented by running the EnKF formulas a second time. In practice, thismatrix, D, is of the form ρI, where ρ is a regularization parameter. Thistechnique prevents large, nonphysical gradients in the analysis ensemble. See(Johns and Mandel, 2005) for details.

7 Numerical results

7.1 Calibration of coefficients in one dimension

We have found initial values B = 5.5849 × 104, and C = 5.9739 × 10−4

from the values Ti = 670K and Tc = 1200K using (17), and then the valueA = 1.5217 × 101 from (18), using the value tc = 110s from Kremens et al.(2003). Not every initial condition gives rise to a traveling combustion wave(Mercer et al., 1996), so, inspired by Mercer and Weber (1997), we have used

22

0 200 400 600 800 1000300

400

500

600

700

800

900

1000

1100

1200

x(m)

Tem

pera

ture

(K)

Traveling combustion waves

Fig. 5. Temperature profile of traveling wave. The wave moved about 398m fromthe initial position in 2300s.

1.125 1.175 1.225 1.275 1.325

x 104

0

200

400

600

800

1000

time(seconds)

Tem

pera

ture

(C)

Fig. 6. Time-temperature profile (dotted line) measured in a grass wildland fire at afixed sensor location, digitized from Kremens et al. (2003), and a computed profile(solid line) from simulation.

an initial condition of the form T (x, t0) = Tce−

(x−x0)2

σ2 + Ta, where x0 is inthe center of the interval and σ = 10

√2m. This initial condition is smooth,

thus it does not excite possible numerical artifacts, it has numerically localsupport, and, for a modest σ, it provides ignition sufficient to develop into twosustained combustion waves traveling from the center. We have then foundempirically suitable values of k and CS that result in traveling combustionwaves, computed the nondimensional coefficients λ and β, adjusted them usingFig. 6 (dotted line), and scaled using (24) to match the maximal temperatureand the width of the wave in Fig. 6 (dotted line), and the speed of the travelingcombustion wave, 0.17m/s, from Kremens et al. (2003). There was a smallamount of wind in Kremens et al. (2003), however we have not consideredthe wind here. The resulting coefficients are k = 2.1360 × 10−1m2s−1K−3,

23

A = 1.8793 × 102Ks−1, B = 5.5849 × 102K, C = 4.8372 × 10−5K−1, andCS = 1.6250 × 10−1s−1. The corresponding nondimensional coefficients wereλ = 2.7000 × 10−2 and β = 0.4829. The computed traveling combustion wave(Fig. 4, 5; Fig. 6, solid line) is a reasonable match with the observation (Fig. 6,dotted line). The trailing edge of the computed temperature profile (Fig. 6,solid line) was not so well matched but this model is quite limited and othermatches reported in the literature are similar (Balbi et al., 1999). The realdata looks like the superposition of two exponential decay modes, possiblythe fast one from cooling and the long one from the heat stored in water inthe ground. This effect will be studied elsewhere.

We have also noted that when the ratio A/CS increases the temperature inthe traveling combustion wave increases, increasing the thermal diffusivitycoefficient k increases the width and speed of the combustion wave and thatthe maximum temperature in the traveling wave decreases if CS increases.Sufficiently small value of CS is needed for sustained combustion. We havenoted that the numerical solution by finite differences becomes unstable whenthe ratio k/h, where h is the mesh size, is too small.

7.2 Numerical results in two dimensions

We have implemented the fire model in two dimensions by central finitedifferences in space. The mesh size was 250 by 250 and the mesh step was2m. We have used the explicit Euler method with time step 1s. The initialconditions were given by the ambient temperature, Ta = 300K, everywhereexcept a square, 50m by 50m, ignition region which was ignited by elevatingthe temperature to 1200K. The mass fraction of the fuel was initialized to beone everywhere except for a 25m fuel break in the center of the domain. Then,at each grid point, the fuel was shifted by a random number between ±0.3.This is intended to simulate a natural uniform fuel supply and a road as a fuelbreak. The Neumann boundary conditions were specified on all boundarieswith no ambient wind across the domain.

The initial ensemble was generated by perturbing the temperature profile ofwhat we call the comparison solution, T0, utilizing smooth random fields inthe following form:

u =d∑

n=1

vn

1 + n2αen, vn ∼ N (0, 1) , (31)

where α is the order of smoothness of the random field, and endn=1 is the

Fourier sine basis, ensuring that u is real valued (Evensen, 1994; Ruan andMcLaughlin, 1998). This process can be understood as a finite dimensional

24

version of sampling out of normal distribution on an infinite dimensional spaceof smooth functions, the Sobolev space Hα

0 (Ω) (Mandel and Beezley, 2006).For integer α, this is the space of functions with square integrable derivativesof order α and zero traces on the boundary.

A preliminary ensemble was generated by adding a smooth random field toeach state variable of the comparison solution. For example, the temperature ofkth ensemble member is given by Tk = T0 + cT uk, where the scalar cT controlsthe magnitude of this perturbation. Finally, the preliminary ensemble wasmoved spatially in both x and y directions by,

Tk(x, y) = Tk(x + cxuik(x, y), y + cyujk(x, y)).

Here, cx and cy control the magnitude of the shift in each coordinate, bilinear

interpolation is used to determine T on off-grid points, and the temperatureoutside of the computational domain is assumed to be at the ambienttemperature. The given simulation was run with the initialization parameterscT = 5 and cx = cy = 150. Fig. 7 shows the effect of these perturbations on asimple circular fire line on the center of the domain.

In each analysis cycle, the solution was advanced by 100s, and then the datawas injected. The data was created artificially by sampling the temperatureand fuel of one fixed solution, called the reference solution, every 10m (or 5grid points) across the domain. The data covariance matrix was taken to bediagonal with a variance 10 for each sample, and the regularization was usedwith regularization parameter ρ = 750. The reference solution was createdin the same manner as the ensemble with an ignition region located 100maway in one direction. This discrepancy is intended to demonstrate the powerof EnKF to attract the ensemble to the truth. After each analysis cycle, theensemble was further perturbed by 5% magnitude of the initial perturbationto assure sufficient ensemble spread for future assimilations.

Fig. 8 shows the reference and comparison solution 100s after initialization,at the end of the first cycle, and Fig. 9 shows the ensemble mean and varianceat the same time prior to performing an assimilation. Fig. 10 shows theensemble after applying the first assimilation. The analysis cycle was repeated10 times, with the results shown in Figs. 11 and 12. These figures showa remarkable agreement of the ensemble mean with the reference solution,even if the simulation ensemble was ignited intentionally far away from thereference ignition region. However, it should be noted that different runs ofthis stochastic algorithm produce different results. Sometimes the ensemble isattracted to the reference solution, and sometimes not, depending on if thereexists a good match to the data in the span of a fairly small ensemble.

25

0 100 200 300 400 5000

100

200

300

400

500

0 100 200 300 400 5000

100

200

300

400

500

(a.) (b.)

Fig. 7. Contour plots, 100K between contour lines, of (a.) the temperature profile ofa circular ignition region in the center of domain, and (b.) the same profile perturbedin magnitude with cT = 5, and spatially with cx = cy = 100.

0

20

40

0

20

40

500

1000

1500

Data

Tem

pera

ture

0

20

40

0

20

40

500

1000

1500

Comparison solution

Tem

pera

ture

(a.) (b.)

Fig. 8. Temperature profiles representing the (a.) data (reference solution, takenas the truth) and (b.) an unperturbed ensemble member comparison solution 100safter initialization.

8 Conclusion

A simple model based on two coupled PDEs can reproduce the time-temperature curve recorded as a wildfire burns over a sensor, which isa measurable feature of fire behavior. By separating the parameters thatdetermine the qualitative properties of the solution from the parameters thatdetermine the temperature, time, and space scales, we were able to identify theparameters of the model from actual wildfire observations. Assimilation of datainto a wildfire simulation poses a particular challenge, because the combustionregion is quite thin. We have shown that a version of the Ensemble Kalmanfilter is able to assimilate data into a wildfire simulation successfully. The

26

0

20

40

0

20

40

500

1000

1500

Ensemble mean

Tem

pera

ture

0

20

40

0

20

400

100

200

Ensemble variance

Tem

pera

ture

(a.) (b.)

Fig. 9. Pointwise prior ensemble mean (a.) and variance (b.) after advancing thesolution in time by 100s, before any data assimilations.

0

20

40

0

20

40

500

1000

1500

Ensemble mean

Tem

pera

ture

0

20

40

0

20

400

100

200

Ensemble varianceT

empe

ratu

re

(a.) (b.)

Fig. 10. Pointwise posterior ensemble mean (a.) and variance (b.) after advancingthe solution in time by 100s and performing a single data assimilation.

0

20

40

0

20

40

500

1000

1500

Ensemble mean

Tem

pera

ture

0

20

40

0

20

40

500

1000

1500

Data

Tem

pera

ture

(a.) (b.)

Fig. 11. After 10 analysis cycles with a 100s time update per cycle, (a.) the ensemblemean compared to (b.) the reference solution (the data).

27

0

20

40

0

20

40

500

1000

1500

Comparison solution

Tem

pera

ture

Fig. 12. Comparison solution advanced to 1000s. This is what the solution in Fig. 11(a.) would be without data assimilation.

filter uses penalization of nonphysical solution and perturbations by smoothrandom transformations of the spatial domain, in addition to standard smoothadditive perturbation.

9 Acknowledgments

This material is based upon work supported by the National ScienceFoundation (NSF) under grants CNS-0325314, CNS-0324989, CNS-0324876,CNS-0324910, EIA-0219627, ACI-0305466, OISE-0405349, CNS-0540178,DMS-0610039, and by a National Center for Atmospheric Research (NCAR)Faculty Fellowship. Computer time on IBM BG/L was provided in partby NSF MRI Grants CNS-0421498, CNS-0420873, CNS-0420985, NSFsponsorship of the NCAR, the University of Colorado, and a grant from theIBM Shared University Research (SUR) program. TeraGrid computer timewas provided by the NSF through TeraGrid resources at the National Centerfor Supercomputing Applications, the San Diego Supercomputing Center,the Texas Advanced Computing Center, and the Pittsburgh SupercomputingCenter. The authors would like to thank Bob Kremens for suggesting anexplanation for the behavior of the measured temperature in Fig. 6. Theauthors would like to thank an anonymous referee for useful comments thatcontributed to improving this paper.

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