Pixel-Level Statistical Analyses of Prescribed Fire Spread
Miles Currie, Kevin Speer, Kevin Hiers, Joseph O'Brien, Scott Goodrick, Bryan Quaife
PPixel-Level Statistical Analyses of Prescribed Fire Spread
M. Currie a , K. Speer a,b , J. K. Hiers c , J.J. O’Brien d , S. Goodrick d , B. Quaife a,e, ∗ a Geophysical Fluid Dynamics Institute, Florida State University, Tallahassee, FL b Earth, Ocean and Atmospheric Sciences, Florida State University, Tallahassee, FL c Tall Timbers Research Station, Tallahassee, FL d U.S. Forest Service, Southern Research Station, Athens, GA e Department of Scientific Computing, Florida State University, 400 Dirac Science Library, Tallahassee, FL, 32306
Abstract
Wildland fire dynamics is a complex turbulent dimensional process. Cellular automata (CA) is an efficienttool to predict fire dynamics, but the main parameters of the method are challenging to estimate. Toovercome this challenge, we compute statistical distributions of the key parameters of a CA model usinginfrared images from controlled burns. Moreover, we apply this analysis to different spatial scales andcompare the experimental results to a simple statistical model. By performing this analysis and making thiscomparison, several capabilities and limitations of CA are revealed.
Keywords:
Cellular automata, Prescribed fire, Data analytics.
1. Introduction
Wildland fire dynamics involve both complex fuel geometries and three-dimensional turbulent flow.Measuring, modeling, and predicting the behavior of fire under realistic conditions is a major technical andcomputational challenge. The fundamental impact of turbulence in the surface boundary layer and in theself-generated convective flow leads to difficulties relating the smaller scale flow and heat transfer to largerscale fire spread. Understanding these dynamics is nevertheless essential for natural resource conservation,and for developing tools to protect life and property.Much effort has been put into developing efficient numerical simulations on both discrete systems, suchas cellular automata (CA) [1, 4, 9, 22, 25], and continuous systems, typically represented by a set of partialdifferential equations (PDEs) [11, 14, 18, 24], or using both in conjunction [6, 15]. PDE models incorporatethe complex physics involved in wildfire, such as the chemistry, radiation, and fluid dynamics, but thesemethods can be too computationally expensive to be used in a real-time framework. Alternatively, thediscrete, lattice-type approach, with individual elements or cells having defined states (for example unburnt ∗ Corresponding author.
Email address: [email protected] (B. Quaife)
Preprint submitted to Elsevier July 24, 2018 a r X i v : . [ n li n . C G ] D ec nd burnt) has the advantage of being computationally fast, but may be difficult to relate to the actualproperties of the fuel and other environmental conditions.The basic CA approach, with probabilities for changing state defined by some rules, is the simplest caseof a discretized model governing the behavior of fire propagation [21]. The possible states of a cell in a CAmodel for wildfires includes burnt, ignited, and combustible. Two of the most important parameters for thebasic CA model are the transition probabilities between these states, and the amount of time a cell remainsin each of the states (burn time). We compute spatial statistics of probabilities by analyzing infrared (IR)images from prescribed burns. The images were derived from an IR camera with a spatial resolution of 1cm and a time resolution of 1 Hz. These images have been successfully used for other studies such as O’Brienet al. [17] who analyzed the radiant power output. Related Work.
CA codes have been used in the past to understand fire dynamics with the commonality thateach of these models have to choose rules for the propagation of the fire. The main choices are the probabilitythat a fire spreads from a burnt to an unburnt cell, which cells the fire can spread to, the amount of timethat a burning cell remains burning, and a fire-atmosphere coupling. While it is clear that the probabilitiesof spread are important, the burn time also plays a significant role, and brings with it a measure of fuelload. With increasing burn time comes increasing probability of the combustion of a neighboring area. Inaddition, the spatial scales: resolution and domain size, and effective time step of the CA model must bechosen.Achtemeier [1, 3] developed a CA model called “Rabbit Rules”. The model draws a parallel betweena propagating fire and a population of rabbits that are moving, reproducing, and dying. In addition,nonlocal effects are included by coupling the heated plume generated by the fire to either a model forthe atmosphere [2] or meteorological measurements. Dahl et al. [8] developed a coupled fire-atmosphereapproach using CA called a raster-based methodology. Their spread rate is computed from a number ofinput variables using the Rothermel model [19]. Both “Rabbit Rules” and raster-based models represent firedynamics as a complex diffusive process, and the characteristics of this diffusion depend on the nature ofthe probabilities used for driving the ignition and combustion of natural fuels, the burn times of these fuels,and the probability of new ignition sites through spotting. Almeida and Macau [5] presented a stochasticCA model that contains several parameters thought to be relevant to the fire spread. Their “D” is relatedto fuel concentration, “I” is probability of ignition, and “B” is the probability that a cell remains burningonce ignited. They make the argument that this persistence is also a combustion latency, or potential toignite a neighbor, in turn a function of the burn time.Trunfio et al. [23] showed that it is possible and overall more advantageous to utilize a cell-based firespread model that effectively does not limit the size or shape of a single cell. This mitigates the constraineddirections of travel and neighborhood size given by the inherent shapes and angles of a cell which is constant2n shape and size. With less computational cost, a grid of dynamic cell shapes is a significant improvementover methods with static cell shapes.When combined with PDE models, CA can be an extremely powerful tool for investigating fire spread. Anexample of this is diffusion-limited aggregation models, which Clarke et al. [6] compared to real experimentalburns and found that the pixels in the simulation capture the correct result roughly 80% of the time. Morerecently, groups have been actively pursuing more data-driven CA models which incorporate real fire datato improve accuracy [13, 16].Despite these advances, large numbers of parameters remain chosen by some combination of empiricalevidence or objective fitting procedure, typically based on on integral measures such as burn area ratherthan the detailed transitions from one cell to another. In this paper, we use robust experimental data fromrelatively small burn plots with light to moderate fuel characteristics to determine the detailed statisticalproperties of the fire spread under various conditions. The analysis provides several essential parametersthat are required in a CA code. In addition, we analyze the validity of local versus nonlocal transitionprobabilities at several different scales for these particular prescribed burns. By computing the parametersfor an empirical CA method, the method can be potentially applied to conditions outside the range of thedata [7].
Outline of the paper.
In Section 2, parameters of a CA method are extracted from experimental data.Section 3 describes a simple model for predicting fire spread and it is compared with experimental results.Section 4 analyzes the effect of scale by considering the CA parameters of aggregated cells. Finally, theresults are further discussed and concluding remarks are made in Section 5.
2. Extracting Cellular Automata Parameters
We present several basic statistics that are extracted at the pixel level from an IR time series of acontrolled burn. Namely, we determine the probabilities of a pixel igniting (assuming the pixel has thepotential of lighting on fire, i.e. necessary fuel) when its closest neighbors are burning in the previoustime step, thus quantifying the probability and direction of wildfire spread. We also compute burn timedistributions of the pixels.In all, 40 datasets comprising of time series of IR temperature of experimental prescribed burns areanalyzed. Hence, each dataset is a time series of IR images where each pixel is a temperature value at agiven 1cm ×
20 40 60 80 100 120 140 160Frame, time (s)02004006008001000 T e m p e r a t u r e ( C ) Spatial Temperature Statistics, 15fopen2 maxmeanmedianvariance
Figure 1: Basic temperature statistics of a single dataset ( ). The IR camera is calibrated to measure temperaturesabove 200 ° C. The black line indicates a temperature of 500 ° C that we use to distinguish between burning and non-burningpixels.
We start by computing basic spatial statistics of a time series of images: the mean, median, maximum,and standard deviation of the temperature values of each frame. To illustrate the typical behavior, in Figure 1we plot these quantities for the dataset , a prescribed fire from 2015 in a longleaf pine flatwoodsforest plot with no trees in the field of view. As the IR camera was calibrated to measure temperatures above200 ° C, the results are determined with this background rather than ambient temperature. For this particularfire, there is a sudden increase in the mean and maximum temperature near t = 20s which indicates that asignificant number of pixels have ignited. We use the maximum temperature achieved at this time, about500 ° C, to define a threshold between the burning and non-burning states. The choice is somewhat arbitraryin the sense that heterogeneous fuels imply a range of ignition temperatures; other choices are possible butlead to similar results.
With a value separating burning and non-burning states, we can compute the burn time of each pixel forthe same burn. Figure 2 shows the number of time steps that each pixel was above the 500 ° C threshold. Ourestimates of burn time are directly relevant to the fire spread rate since longer burning time can overcomelow probability of ignition and maintain fire spread. For instance, Almeida and Macau’s stochastic model [5]4 igure 2: The burn times for each pixel using a 500 ◦ C threshold. The colors indicate the number of time steps that each pixelburned. contains a universal parameter for the probability that a pixel burns (their “B”), which is a representationof the burn time, and this exerts an important control on fire spread in their model.In addition to spatial maps of pixel burn times, we present burn time distributions of the 40 datasets.These distributions can be organized into four general trends, and representative distributions from eachgroup are shown (Figure 3). The error is estimated with a ± ◦ C range on the defined burn threshold.Note that the maximum burn time in the bottom right distribution is about half the size of the other threedistributions. These distributions are described further Section 5.
Another key parameter of a CA model is the probability of spread. We compute probabilities of a pixeligniting when any number of its nearest eight neighbors are on fire. Spotting can be included by definingthe nearest neighbors to include additional layers of nearby pixels. In this work, however, we will coarsenthe grid by aggregating pixels (Section 4), and this has the effect of reducing the amount of spotting.To calculate the probability of a fire spreading between pixels, we first identify pixels that are newlyignited in each frame. For each of these pixels, we count the number of nearest neighbors that were onfire. This gives a statistic of the likelihood of a pixel igniting given the condition that some number of itsneighbors are burning. We can perform the same count for pixels that did not ignite when at least one of5
20 40 60 80 100Burn Time (s)02000400060008000100001200014000 P i x e l C o un t s Burn Time Distribution, sopen1 P i x e l C o un t s Burn Time Distribution, fsingle4 P i x e l C o un t s Burn Time Distribution, 15fopen2 P i x e l C o un t s Burn Time Distribution, fclump2
Figure 3: Four burn time distributions with 50 ◦ C error bars that are representatives of the 40 datasets.
Top left:
A burn timedistribution with exponential decay with an e -folding time of approximately four seconds. This is the most common distributionand can be found in twenty-four of our datasets. Top Right:
Burn time distribution with a characteristic single peak in thedistribution. This behavior is found in six of our datasets.
Bottom Left:
Burn time distribution with the characteristic ofleveling off for several seconds before decreasing roughly exponentially. This behavior is found in eight of our datasets, includingthe dataset in Figures 1 and 2.
Bottom Right:
Burn time distribution with the bi-modal characteristic of two distinctpeaks in the burn time distribution. Only one fire shows this characteristic. F r e q u e n c y ( I n s t a n c e s ) Neighbor Counts neighbors did NOT light pixelneighbors DID light pixel . . . . . P r o b a b ili t y o f I g n i t i o n Figure 4:
Left:
The number of instances where an unburnt pixel’s nearest neighbors did (orange) and did not (blue) causeignition.
Right:
The probability of a pixel igniting conditioned on the number of its neighbors being on fire. For example, iffour of an unburnt pixel’s neighbors are on fire, then it has approximately a 35% chance of igniting at the next time step. its neighbors was on fire. We plot these two values as a histogram in the left side of Figure 4. We excludethe case of newly ignited pixels where zero neighbors were on fire since this corresponds to spotting, aphenomenon that we are not investigating in this analysis. In the right plot of Figure 4, the histogram isused to compute the distribution of a pixel igniting.
3. A Simple Model for Fire Spread
A factor in fire spread that we have not explicitly treated in this analysis is wind. Wind can be modelledin a CA code by choosing probabilities of one pixel igniting another that are directionally dependent. Forexample, if there is no wind, then every burning neighbor of an unburnt pixel has the same probability ofigniting this pixel. However, if there is a strong eastward wind then the burning neighbors to the left arethe most likely to ignite an unburnt pixel.We propose a basic two parameter model to understand the effect of the wind on the data. The parametersare the following.1. M is the number of neighbors that have the ability of igniting an unburnt pixel.2. p is the probability that one of these neighbors ignites an unburnt pixel.The two extreme cases are M = 1, where only one neighbor can ignite a pixel (high wind case), and M = 8where any pixel can ignite a neighboring pixel (no wind case). We let X be the event of an unburnt pixeligniting at the next time step and Y be the number of neighbors of this unburnt pixel that are currentlyburning. For a fixed p and M , we are interested in computing p ( X | Y = N ) which is the probability thatan unburnt pixel ignites, given that it currently has N burning neighbors.7hese probabilities can be computed analytically. For example, if only one neighbor can ignite a pixel( M = 1), then p ( X | Y = N ) = N p. In the other extreme, if any neighbor can ignite a pixel ( M = 8), then p ( X | Y = N ) = (cid:0) − p ) + · · · + (1 − p ) N − (cid:1) p = 1 − (1 − p ) N − (1 − p ) p = 1 − (1 − p ) N . The expressions for these probabilities become unwieldy for other values of M , so we use a Monte-Carloapproach to compute the probabilities. For a fixed M , p , and N , 10 randomly selected configurations ofburning neighbors are chosen, and for each burning neighbor that is able to ignite the unburnt pixel, wedraw a random number to determine if the unburnt pixel is ignited. This method is verified with analyticexpressions for the probabilities with M = 2 and M = 3 (not reported in this paper).To find the best possible probability for each value of M , we compute the value of p that minimizes theroot mean square (RMS) for N = 1 , . . . , p at an equispaced partitioning of [0 , p that minimizes the RMS to choose lower and upper limits for a new search region. We iteratethis procedure until the first two digits of the RMS is unchanging. For each value of M , we report the bestprobability of spread p and the RMS value in Table 1. In Figure 5, we plot the experimental data and thebest model fits when M = 1 , ,
7. The other values of M are similar to M = 7. While the smallest RMSvalue is achieved when seven neighbors can ignite an unburnt pixel, this RMS value is comparable to othersimulations where fewer neighbors have the ability to ignite a pixel. In particular, even if only 3 pixels canignite an unburnt pixel, a RMS of almost 20% can be achieved for experimental data from complex dynamicswith a very simple CA model. Further discussion of this behavior is described in Section 5.
4. Pixel Aggregation
By only using the eight nearest neighbors as potential ignition sources for an unburnt pixel, we areignoring ignition due to spotting. Given the spatial resolution of 1cm of the IR images, it is unreasonableto assume that fire spreads to only its neighboring pixels at the 1 Hz speed of the camera. In reality, spottingoccurs due to radiative and convective transport of heat, or by transport of an ember. To investigate theeffect of spotting and to examine the fire dynamics at larger spatial scales, we aggregate groups of 3 × ×
5, and 10 ×
10 pixels. We use the maximum temperature of the pixels in the aggregate to assign atemperature to the aggregate. The maximum value is used instead of the mean since the mean would be8 p RMS1 0.6394 0.25632 0.3772 0.21393 0.2671 0.20164 0.2059 0.19525 0.1679 0.19286 0.1402 0.18977 0.1192 0.18858 0.1100 0.1891
Table 1: The best probability of spread p for each choice ofthe number M of neighbors that can ignite an unburnt pixel.Also reported is the root mean square error. . . . . . . P r o b a b ili t y o f I g n i t i o n M = 1 M = 2 M = 7Experimental Data Figure 5: The probabilities of ignition for a different numberof neighbors that can ignite an unburnt pixel (solid lines). Thecircled line is the experimental data from Figure 4. skewed by the 200 ° C threshold of the IR camera. We repeat the statistics reported in Figure 4 for thesenew aggregated pixels, and the results are in Figure 6.We also repeat our probabilistic model for the aggregated pixel data sets. As before, for each number M of pixels that can ignite an unburnt neighbor, we find the probability p that minimizes the RMS overthe number of burning neighbors N . For the 10 ×
10 aggregation, we do not include the N = 8 entry in theRMS as this value seems to be an outlier which is feasible since there are very few instances when a 10 ×
5. Discussion
Empirical data is needed to validate and inform new modelling tools in wildland fire [7]. Cellularautomata codes rely on probabilistic rules to determine fire dynamics, and we have used 40 IR datasets fromcontrolled burns to analyze two of the main parameters common in fire modeling using CA methods. Wehave focused on the burn times of a pixel and the probability of a unburnt pixel being ignited by its burningneighbors. We also analyzed the effect of the spatial resolution on these probabilities. Finally, we compareda simple model for fire spread to the experimental data at different aggregate resolutions.First, the burn times depend strongly on the fuel type, moisture, wind speed, and other factors [10],but we found that from these 40 datasets, the burn times could be binned into four distributions plotted inFigure 3. Most of the datasets’ burn times have a distribution with a small mean (only a few seconds) and9xponential decay (top left). This corresponds to a fast burning homogeneous fuel type. Another set of fireobservations are represented by a mean closer to 20 seconds with an exponential decay (top right). Thiscorresponds to a slower burning homogeneous fuel type. The third distribution (bottom left) has both aslow burning and fast burning fuels which indicates a heterogeneous fuel type. Finally, a single data set hasa distribution with a large variance and a shorter maximum burn time (bottom right), and this correspondsto an even more heterogeneous fuel type. The 40 datasets have several variables, including the density ofthe trees and the moisture of the vegetation. Most of the datasets whose distribution resembles the top leftplot of Figure 3 come from the drier vegetation, while the all of the datasets that resemble top right plotare from plots with a wetter vegetation.One conclusion that our analysis reveals is that a single parameter for the burn time probability—manyCA models make this assumption—is not applicable since the burn time distributions are wide. However,these prior burn time distributions (Figure 3) can be improved by using the experimental data to create aposterior distribution. That is, by using a Bayesian approach, the burn times, which are related to the fueltype, can be incorporated.The probability of spread is described in terms of the number of burning neighbors. That is, given anewly ignited pixel, we determine how many of its neighbors are burning at the previous time step. Themost common occurring instance is only one burning neighbor, while eight burning neighbors happens lessfrequently. To analyze these probabilities for larger grids, we aggregated pixels and computed the sameprobabilities. At all the resolutions we observed that most pixels are ignited by only one burning neighbor,while the fewest are ignited by eight neighbors. Moreover, the probability of ignition (right plots of Figures 4and 6) verify, unsurprisingly, that the probability of ignition increases with the number of burning neighbors,and this probability distribution is essential for a CA model.More interestingly, a simple model for fire spread captures features of these transition probabilities. First,for the unaggregated pixels (Figure 5), the model does a good job of capturing the experimental data whenthere are three or more burning neighbors. However, the model severely underestimates the probability ofspread if only one or two neighbors are burning. In this case, we expect that much of the fire spread isnonlocal from convective heating and small-scale spotting, and our model does not account for these nonlocaleffects. This hypothesis is further supported by analyzing the aggregated cells. In these cases, the effectivespotting becomes less prevalent, and we expect that the leading cause of new ignition is neighboring pixels.In Figure 7 and Table 2, we see that the fit between the model and the experimental data improvesas larger pixels are considered. In particular, an RMS of less than 5% is achieved when considering pixelsthat are 10cm . This suggests that the resolution of the camera was high enough to observe the fine scaleprocesses of combustion and that some averaging over these processes is needed to produce stable statisticsof fire spread, a result that was confirmed by Loudermilk et al. [12]. The degree of averaging necessary wasfound to be 10cm in these experiments. 10he spatial structure of burn times (e.g. Figure 2) is a small-scale example of the fuel bed complexitythat can result from even the simplest of rules. For instance [20] describes a simulated tree population modelwith instantaneous fire spread and zero burn time. Real wildland systems have finite spread rate but theirmodel shows how the fuel itself (they simulated trees but it could be any fuel) can spontaneously developcomplex spatial structure.Differentiating between the three prevalent types of fire (i.e. head fire, backing fire, and flanking fire)is important because the different types of fire have different probabilities for spreading. 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Figure 6:
Left:
The number of instances where an unburnt aggregated pixel’s nearest neighbors did (orange) and did not(blue) cause ignition at different levels of aggregation.
Right:
The probability of an aggregated pixel igniting conditioned onthe number of its aggregated neighbors being on fire. p RMS1 0.5779 0.17342 0.3321 0.14593 0.2339 0.13554 0.1810 0.13335 0.1436 0.13016 0.1247 0.12967 0.1061 0.12848 0.0948 0.1287 (a) 3 × M p
RMS1 0.5843 0.13892 0.3256 0.11113 0.2306 0.10524 0.1765 0.10055 0.1378 0.10016 0.1188 0.09567 0.1028 0.09598 0.0896 0.0938 (b) 5 × M p
RMS1 0.6137 0.06432 0.3463 0.04433 0.2403 0.04264 0.1825 0.04325 0.1498 0.04066 0.1261 0.04367 0.1090 0.04368 0.0956 0.0450 (c) 10 ×
10 aggregatedgroups.
Table 2: The best probability of spread p for each choiceof the number M of neighbors that can ignite an unburntaggregated pixel. Also reported is the root mean square error. . . . . . . P r o b a b ili t y o f I g n i t i o n M = 1 M = 2 M = 7Experimental Data (a) 3 × . . . . . . P r o b a b ili t y o f I g n i t i o n M = 1 M = 2 M = 7Experimental Data (b) 5 × . . . . . . P r o b a b ili t y o f I g n i t i o n M = 1 M = 2 M = 7Experimental Data (c) 10 ×
10 aggregated groups.
Figure 7: The probabilities of ignition for a different numberof neighbors that can ignite an unburnt pixel (solid lines). Thecircled line is the experimental data from Figure 6.
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