Science, Irving K. Barber Faculty of (Okanagan)
Computer Science, Mathematics, Physics and Statistics, Department of (Okanagan)
DSpace
UBCO
Goetz, Geoffrey Owen
Master of Science - MSc
University of British Columbia
Modelling wildfire spread from data collected on various slopes and across different fuel types is an important way to understand how different parameters influence the speed at which these fires travel. Fitting this data to statistical models could be used to provide accurate fire spread predictions and can influence the way in which fire fighters may combat a wildfire. The objective of this thesis is to establish a model that describes the influence of slope on a fire's resulting rate of spread in three different fuel types. We present the analysis of data from both small scale experimental fires and test burns conducted in the field in a natural landscape. We use non-linear regression and generalized additive models to create this model, using current rate of spread models as our foundation. An analysis on the influence of slope on the rate of spread travelling downhill at the heel end of a fire is also provided.
A Statistical Investigation of howSlope affects a Wildfire’s Rate ofSpreadbyGeo↵rey Owen GoetzB.Sc., The University of British Columbia Okanagan, 2021A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE COLLEGE OF GRADUATE STUDIES(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Okanagan)October 2021© Geo↵rey Owen Goetz, 2021The undersigned certify that they have read, and recommend to theCollege of Graduate Studies for acceptance, a thesis entitled: A Statis-tical Investigation of how Slope affects a Wildfire’s Rate ofSpread submitted by Geoffrey Owen Goetz in partial fulfilment of therequirements of the degree of Master of ScienceDr. W. John Braun, Irving K. Barber Faculty of Science - CMPSSupervisor, Professor (please print name and faculty/school above the line)Dr. Rebecca Tyson, Irving K. Barber Faculty of Science - CMPSSupervisory Committee Member, Professor (please print name and faculty/school abovethe line)Dr. Warren Hare, Irving K. Barber Faculty of Science - CMPSSupervisory Committee Member, Professor (please print name and faculty/school abovethe line)Dr. David Scott, Irving K. Barber Faculty of Science - EEGSUniversity Examiner, Professor (please print name and faculty/school above the line)October 2021(Date Submitted to Grad Studies)Additional Committee Members include:Dr. Je↵ Andrews, Irving K. Barber Faculty of Science - CMPS(please print name and faculty/school above the line)Dr. Lael Parrot, Irving K. Barber Faculty of Science - Biology(please print name and faculty/school above the line)iiAbstractModelling wildfire spread from data collected on various slopes andacross di↵erent fuel types is an important way to understand how di↵er-ent parameters influence the speed at which these fires travel. Fitting thisdata to statistical models could be used to provide accurate fire spreadpredictions and can influence the way in which fire fighters may combat awildfire. The objective of this thesis is to establish a model that describesthe influence of slope on a fire’s resulting rate of spread in three di↵erent fueltypes. We present the analysis of data from both small scale experimentalfires and test burns conducted in the field in a natural landscape. We usenon-linear regression and generalized additive models to create this model,using current rate of spread models as our foundation. An analysis on theinfluence of slope on the rate of spread travelling downhill at the heel endof a fire is also provided.iiiLay SummaryThe Canadian Forest Fire Behaviour Prediction System [1] describedmany of models used for studying and predicting the size, growth, and inten-sity of wildfires which are still widely used in today in Canada. In particular,this system presents a model for a fire’s rate of spread, along with a modelfor how slope, which is known to increase a fire’s rate of spread, influencesthis rate. However, the model that is brought forward to describe the e↵ectof slope on a fire’s rate of spread is founded by averaging other models thatwere developed from informal observations or small scale experiments. Thissuggested slope-e↵ect provides one single model to describe the influence ofslope for every fuel type. Furthermore, all models from this system pertain-ing to grassland fuel types are made from Australian data. In this study, ourgoal was to use Canadian data for creating slope-e↵ect models for grasslandand forest stands consisting of Ponderosa Pine and Douglas Fir fuel typesalong with investigating slope-e↵ect on small scale lab fires where externalfactors can be mitigated. This study also looks into how slope may influencea fire’s rate of spread when travelling downhill. The models presented in thisstudy can be used to aid in creating more accurate fire behaviour predictionsoftware, which in turn may aid in suppression e↵orts of fire fighters.ivTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . vList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . xiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiiChapter 1: Introduction . . . . . . . . . . . . . . . . . . . . . . . 1Chapter 2: Literature Review . . . . . . . . . . . . . . . . . . . 42.1 Canadian Forest Fire Behavior Prediction System . . . . . . . 42.2 Fitting a Stochastic Fire Spread Model to Data . . . . . . . . 62.3 Applications of simulation-based burn probability modelling:a review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Rate of Spread Estimates . . . . . . . . . . . . . . . . . . . . 82.5 Locally Weighted Scatter plot Smoother (LOWESS) . . . . . 92.6 Generalized Additive Model (GAM) . . . . . . . . . . . . . . 11Chapter 3: Data Collection . . . . . . . . . . . . . . . . . . . . . 133.1 Field data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.1.1 British Columbia Wildfire Service (BCWS) . . . . . . 133.1.2 Canadian Forest Service (CFS) . . . . . . . . . . . . . 153.2 Lab Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2.1 Data Extraction and Segmentation . . . . . . . . . . . 17vTABLE OF CONTENTS3.3 Data Summary . . . . . . . . . . . . . . . . . . . . . . . . . . 18Chapter 4: Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 214.1 Field Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224.1.1 C7 Ponderosa Pine and Douglas Fir . . . . . . . . . . 234.1.2 O1-B Grassland . . . . . . . . . . . . . . . . . . . . . 294.2 Lab Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.2.1 Forward ROS . . . . . . . . . . . . . . . . . . . . . . . 364.2.2 Backwards Rate of Spread (BROS) . . . . . . . . . . . 444.3 Influence of Slope Presence on Resulting Downhill ROS . . . 46Chapter 5: Conclusion . . . . . . . . . . . . . . . . . . . . . . . . 49Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54Appendix A: Tables . . . . . . . . . . . . . . . . . . . . . . . . . . 55viList of TablesTable 3.1 Rates of spread measured in meters per minute (m/min)of both the forward and back burn flanks of the firesconducted in the O1-B grassland fuel type, with theiraverages, range and standard deviation (SD) from theBCWS test burn data and n denotes the number offires. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18Table 3.2 Rates of spread measured in meters per minute (m/min)of both the forward and back burn flanks of the firesconducted in the C7 Ponderosa pine and Douglas firfuel type, with their averages, range and standard de-viation (SD) from the BCWS test burn data . . . . . . 19Table 3.3 Rate of spread data measured in meters per minute(m/min) collected from the videos of the lab smoulderfires. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20Table 3.4 Back Burn Rate of Spread data measured in metersper minute (m/min) collected from the videos of thelab smoulder fires. . . . . . . . . . . . . . . . . . . . . 20Table 4.1 C7 and O1-B parameter estimates for a, b, c, d, and efor the complete ROS model (4.1) given in [12]. . . . . 21Table 4.2 Slope and fire type multipliers obtained by the GAM.The estimate for these parameters indicates the amountof influence they had on the log(ROS) for C7 data. . . 24Table 4.3 Statistics of resulting slope parameter of C7 BROSdata applied to a linear model. A p-value of 0.55 in-dicates that when trying to fit the data to a modelwith a slope influence, it could not do so with confi-dence. This tells us that there is nothing indicating anon-zero slope-e↵ect. . . . . . . . . . . . . . . . . . . . 29viiLIST OF TABLESTable 4.4 Estimates for the ROS and slope e↵ect parameters forthe model described in (4.7), along with their standarderror (SE), 95% confidence intervals and p-values. Pa-rameter a represents the part of the ROS explained byall influencing factors besides slope, and b representsthe influence of slope on the ROS. . . . . . . . . . . . 31Table 4.5 Estimate, standard error, 95% confidence interval andp-value for the slope e↵ect parameter resulting fromfitting the O1-B log(BROS) data to a linear model . . 35Table 4.6 Results for the intercept and slope factor from theGAM applied to the forward log(ROS) data for thelab fires. Both parameters prove to be significant asboth have a very small p-value. . . . . . . . . . . . . . 39Table 4.7 Estimate, confidence interval, and p-value for the in-tercept and slope parameter estimates resulting fromapplying a linear model to the ROS data for the smoul-dering lab fires . . . . . . . . . . . . . . . . . . . . . . 40Table 4.8 Estimates for slope e↵ect parameters d, and e of equa-tion 2.7 calculated by non-linear regression conductedon lab smoulder fire ROS data with their associatedstandard errors (S.E), confidence intervals, and p-values. 42Table 4.9 Results of 2-sample t-test conducted for all fuel typescomparing the means of flat terrain ROS and slopingterrain BROS. . . . . . . . . . . . . . . . . . . . . . . . 47Table A.1 Averages and standard deviation (SD) of the weatherdata variables which influence fire behaviour from theBCWS test burns conducted in the O1-B fuel type.The n column denotes the number of fires observed. . 55Table A.2 Averages and standard errors of the weather data vari-ables which influence fire behaviour from the BCWStest burns conducted in the O1-B fuel type. n denotesthe number of fires observed. . . . . . . . . . . . . . . 55viiiList of FiguresFigure 1.1 Di↵erent sections of a wildfire where the ⌦ representsthe ignition point. ROS is calculated by measuringthe distance (m) the head of the fire has travelledfrom the ignition point over a measured amount oftime in minutes. The BROS is similar but from theignition point to the heel edge of the fire. . . . . . . 3Figure 3.1 The flat tray fixed onto the metal breadboard withthe Olympus Stylus 600 camera positioned above theburn sheet on top of the wooden hinging platform in-side of a fume hood. In this image the burn platformis at a 40 angle . . . . . . . . . . . . . . . . . . . . . 16Figure 3.2 Images extracted using ↵mpeg from a video record-ing one of the lab smoulder fires. These images weretaken 5 seconds apart from one another as the burnprogressed. . . . . . . . . . . . . . . . . . . . . . . . 18Figure 4.1 Diagnostic plots from the GAM model applied to theC7 ROS data. At the top there is a linear QQ plot(left) and a residuals vs linear predictor (right). Onthe bottom there is a histogram of the frequency ofresiduals (left) and response vs fitted values plot scat-ter plot (right). . . . . . . . . . . . . . . . . . . . . . 25Figure 4.2 The fitted model (4.4) obtained by running a GAMon the C7 ROS master data set (green) surrounded byit’s associated 95% confidence bands (red) comparedwith the Van Wagner slope e↵ect model used in theCFFBPS (dashed black). . . . . . . . . . . . . . . . . 26ixLIST OF FIGURESFigure 4.3 Contour plot of log(ROS) as a function of wind speedand Fine Fuel Moisture, obtained from the general-ized additive model applied to the C7 data. The blackcurves represent the fitted curve whereas the red andgreen curves are the confidence bands surrounding thecurve. . . . . . . . . . . . . . . . . . . . . . . . . . . . 27Figure 4.4 LOWESS applied to the BROS data from the BCWSC7 data . . . . . . . . . . . . . . . . . . . . . . . . . . 28Figure 4.5 LOWESS function applied to ROS of BCWS datafrom test burns conducted in the O1-B grassland fueltype . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30Figure 4.6 Curve of equation (4.8) with parameters estimatedthrough nonlinear regression fit to the O1-B ROS data 31Figure 4.7 The LOWESS curve of Figure 4.5 (red) along withthe curve obtained as a result of non-linear regressionapplied to the O1B ROS data (dashed blue). . . . . . 32Figure 4.8 The slope factor model obtained by non-linear regres-sion on the O1-B grassland fuel type (green) withit’s corresponding 95% confidence interval (red) com-pared to the Van Wagner model presented in theCFFBPS (dashed black). Here we are looking at thesuggested slope factor multiplier as the response vari-able and slope percentage as the explanatory variable. 33Figure 4.9 The Van Wagner model (dashed black) crosses intothe lower point-wise 95% confidence band (red) ofour regression model at a slope percentage of 34% atwhich it stays within the bounds of the interval. . . . 34Figure 4.10 LOWESS function applied to the back burn rate ofspread for the O1-B grassland data on the log scale . 35Figure 4.11 A generalized additive model applied to the lab ROSdata on the log scale. The dotted lines indicate the95% confidence interval bounds of the GAM curve(solid). The red line at s(pctSlope,1)= 0 is used toindicate whether or not the variance in the data issignificant. . . . . . . . . . . . . . . . . . . . . . . . . 36xLIST OF FIGURESFigure 4.12 Diagnostic plots of residuals for the GAM fitted modelof the ROS lab data. The top left plot is a QQ plotwhich shows that there are no strong deviations fromnormality as all the points stay close to the red line.The top right plot of residuals vs linear predictorshows that the residuals stay relatively constant andcentered around 0 even as the linear predictor (slope)increases. The histogram also o↵ers another indica-tion of the residuals following a fairly normal distri-butions and finally the response vs fitted plot showsthat the responses are changing as the predictions goup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38Figure 4.13 Rate of Spread vs the percent slope of the lab burnson the log scale with the LOWESS curve (red) included. 39Figure 4.14 Calculated linear model (green) from the lab ROSdata with its corresponding 95% confidence intervalbands (red) and the Van Wagner model suggested bythe CFFBPS (black). . . . . . . . . . . . . . . . . . . 41Figure 4.15 Observed forward rate of spread of the smoulderinglab fires vs percent slope. On the left the regressionmodel (red) is plotted through the data points, andon the right is the linear model (green). . . . . . . . 43Figure 4.16 Nonlinear regression model (green) and correspond-ing 95% confidence bands (red) plotted with the VanWagner slope e↵ect model (black). The Van Wagnermodel of the CFFBPS suggests a much stronger slopee↵ect for this fuel type than what was observed. . . . 44Figure 4.17 The results of fitting a GAM (black) to the BROS labdata on the log scale. The black dotted lines are the95% confidence intervals for this model, and the redhorizontal line at y = 0 indicates that there was noe↵ect of slope on the log(BROS). . . . . . . . . . . . 45Figure 4.18 Diagnostic plots of the GAM which was fit to thelog(BROS) data from the smouldering lab fires. Thedeviance residuals vs theoretical quantiles plot (QQplot) tells us that there were no significant deviationsfrom normality. The residual vs linear predictor tellsus that the residuals remained fairly constant andnear zero, and the histogram tells us that these resid-uals maintained normality. . . . . . . . . . . . . . . . 46xiAcknowledgementsI would first like to give thanks to my supervisors Dr. John Braun andDr. Rebecca Tyson for their continuous support and patience throughoutmy M.Sc program. I’m very grateful to have had the opportunity to workunder their supervision as their student and to have them pass on someof their knowledge over the last few years. I am particularly thankful forthe support Dr. Tyson provided me during my undergraduate studies andgiving me the opportunity to apply my knowledge to a research project, andfor introducing me to John when I expressed interest in studying wildfirefor my M.Sc.I would also like to thank the Tyson Lab (Tylab) research group, theComputational Ecology Reseach Group (CERG) and QSCU members fortheir support and insight throughout all of my years at UBCO. Also, Iwould like to thank Dr. Andrew Jirasek for allowing me to access his lab sothat I could conduct my smouldering experiments.I am also thankful to my friends at UBC Okanagan who I shared manylaughs and late night study sessions with. I will strangely miss all of thehair pulling assignments which lead up to grand AHA! moments. I enjoyedthe noon o’clock beers with Phil and Je↵ at the Well and will miss all of theconversations that came with those study breaks.To my roommates during all of my university studies: Adam, Gabriel,Bryce, Dawson, and Kelsey. The laughs and stories we shared and madewill never be forgotten and will always be held dear to me. And to Gradywho has always been a brother to me.To the Fire Devils Unit Crew, Merritt Initial Attack, and everyone elseat the Merritt Fire Zone. Thank you for all of your support over the yearsand for fearlessly standing next to me out on the fire line. My passion forwildfire would not exist without you. You will always be my fire family.Last but not least, I am forever grateful to my parents, brothers, andgirlfriend Megan for their constant love and support. Even when life wasflipped upside down, you were the anchor that kept me grounded and allowedme to keep pushing. Thank you.xiiDedicationTo my parents, brothers, and everyone who hasfought a fire alongside mexiiiChapter 1IntroductionWildfires burn on average 370,000 hectares of forest in British Columbiaeach year. These fires can spread at speeds up to 23 km/hour, burningvegetation, wildlife, or even buildings and communities that lie in its path.As was seen in the town of Rock Creek in 2015 where 30 structures werelost or in 2021 where an estimated 90% of the village of Lytton was lostin just a few hours. The risk of large devastating fires appears to be in-creasing in countries all over the world as evidenced by recent fires whichhad devastating impacts in Australia, the United States, and the Amazonrain forest. In British Columbia, a record breaking year was witnessed in2017 when 1.2 million hectares burned, 900,000 more hectares than any ofthe 10 preceding years. This record however, was broken the following yearwhen 1.35 million hectares burned. This increase in fire activity poses animmediate threat to timber harvesting, and threatens communities whichborder forested land. With the continuation of urban expansion into theserural areas where wildfires are most prevalent, this threat to communitiesand lives only increases and creates a risk for these fires to become evenmore devastating than they have already proven to be.Wildfire risk, and behaviour has been and continues to be studied bygovernment research teams, and independent researchers. This researchhas lead us to better understand how di↵erent vegetation reacts to vari-ous weather conditions such as temperature, wind, humidity, and precipi-tation levels. From this, di↵erent models have been established to aid inunderstanding the mechanics and behavioural characteristics of these firesso that we can better prepare communities and firefighters to prevent andsuppress these fires. These models are used within software programs suchas Prometheus [11], FARSITE [4], and Burn-P3 [8] so that researchers canexplore di↵erent scenarios and run multiple simulations to observe the re-sulting output of how a fire will react within a given landscape.When a wildfire burns up a slope it is known that its rate of spread (ROS)increases. This is due to the well known fact that heat rises, and that a flamethat is not being influenced by an air current will stand vertically regardlessof the orientation of the burning fuel. Thus, flames are closer to up-slope1Chapter 1. Introductionneighbouring fuels which allows this unburnt fuel to absorb more radiantheat, which pre-heats and dries the fuel. This radiant heat does not onlycome from the flames themselves, but it will also come from the resultingsmoke which also rises and heats up the unburnt fuels ahead of the fire frontas it travels uphill. This increases the flammability of these unburnt fuels byevaporating some of the moisture inside and brings their temperature closerto their ignition point, resulting in these fuels igniting more quickly once theflame front reaches them. Not only this, but with these fuels being closerto the flames themselves, the flames have a higher chance of coming intocontact with the fuel and directly transmitting the fire. This phenomenacan be exacerbated by wind as it could cause the flame to bend over theactual fire perimeter, extending the fire’s reach.Most research surrounding wildfire behaviour is centered on studyingrate of spread [5], however, most of these studies focus on the method usedfor measuring this rate, and less on the factors such as slope which influ-ence these rates. Furthermore, little research has been conducted looking atthe downhill rate of spread of the back burning fire perimeter. This studyaims to investigate the various influences a slope has on fire behaviour us-ing methods presented by the Canadian Forest Fire Behaviour PredictionSystem (CFFBPS) applied to three di↵erent fuel types. This research alsotakes into consideration data from Canadian grassland burn data, a fueltype within the CFFBPS whose analysis up to this point has been based onAustralian data.In this thesis, we study test burns conducted by both BC Wildfire Ser-vices and Canadian Forest Services as well as a fire smouldering experimentto determine a model that explains the influence of slope on a fire’s ROS.The data collected from these di↵erent experiments can be used to estab-lish a slope-e↵ect model that could be added to ROS models given in theCFFBPS [1] that are used in fire modelling software. This is the objectiveof this thesis, along with investigating the e↵ect of slope on the back burn-ing section of the fire travelling downhill. The back burning rate of spread(BROS) is the measurement of the fire’s growth (m/min) at the heel endof the fire where the fire’s intensity is the lowest. In the case of the testburns used throughout this thesis, the heel of the fire was the fire’s edgewhich travelled downhill, these di↵erent parts of a fire can be observed infigure 1.1 below. In figure ?? the ⌦ represents the ignition point. ROS iscalculated by measuring the distance (m) the head of the fire has travelledfrom the ignition point over a measured amount of time in minutes. TheBROS is similar but from the ignition point to the heel edge of the fire.Uncontrolled conditions such as weather and fuel moisture content for the2Chapter 1. Introductionfield studies vary across burns and an e↵ort is made to isolate these factorsfrom the influence of slope.Figure 1.1: Di↵erent sections of a wildfire where the⌦ represents the ignitionpoint. ROS is calculated by measuring the distance (m) the head of the firehas travelled from the ignition point over a measured amount of time inminutes. The BROS is similar but from the ignition point to the heel edgeof the fire.The first goal is to study the CFFBPS model for wildfire ROS, andthe suggested model put forward in that study for a slope-e↵ect. As thepresented slope e↵ect model within the CFFBPS is based o↵ of averagingother models, we wish to find out if their model is a good fit for modellingthis e↵ect. The second goal is to study the slope-e↵ect at the heel end of thefire as is travels downhill. The heel is the slowest moving edge of the fire andis often not the focus of studies. In conjunction with slope, a model is yetto be presented for a downhill back burning fire edge. Chapter 2 providesa literature review summarizing previous research that has been conductedstudying fire behaviour as well as summaries of di↵erent statistical methodsused in the analytical processing of the data acquired for this study. Thenin Chapter 3, an overview of the experimental methods used for collectingthe ROS data is highlighted for both the field and lab portions. This datais then analyzed in Chapter 4 with the use of the statistical software R [10]where the up-slope and down-slope data sets are analyzed independently ofone other. Finally, Chapter 5 discusses the results found during the analysisand how these findings can be used.3Chapter 2Literature ReviewIn this chapter we will explore previous studies focusing on wildfire be-haviour models, along with a brief overview of some of the statistical meth-ods used to analyse the data for the study in this thesis. In the sectionsbelow we will see an overview of previous studies that focused on studyingwildfire behaviour such as the Canadian Forest Fire Behaviour PredictionSystem (CFFBPS) and provides the foundation for the models used in cur-rent fire modelling software. We will also look at a previous study thatused the same smouldering lab fire apparatus for data collection as done forpart of this thesis, along with a study focusing on applications of wildfiremodelling software and another study centered around determining precisemethods for measuring a fire’s rate of spread. Finally, theory behind twostatistical tools used for analysing data (LOWESS and GAM) is given.2.1 Development and Structure of the CanadianForest Fire Behavior Prediction SystemIn an unpublished discussion paper, a revised approach for predict-ing fire behaviour characteristics in specific fuel complexes was presented.The concept therein became known as the Canadian Forest Fire BehaviorPrediction System (CFFBPS). This system provides quantitative estimatesof wildfire characteristics such as rate of spread, intensity, fuel consumptionand fire description (surface or crown) from various types of input quan-tities, such as weather (relative humidity, wind speed, temperature, andprecipitation), topography (slope and aspect), and fuel type (vegetationclass). This system suggests equations for calculating these characteristicsand most importantly for this study, equations to describe a fire’s rate ofspread. Constructing these equations was done using data from fires burnedin a natural landscape, and the equation given in the CFFBPS used forcalculating a fire’s initial rate of spread (RSI) from a point source ignitionisRSI = a⇥ 1 exp(b⇥ISI) c (2.1)42.1. Canadian Forest Fire Behavior Prediction Systemwhere a,b and c are parameters that are fuel type specific. The InitialSpread Index (ISI) is a numeric rating of the expected rate of fire spread,based on wind speed and the fine fuel moisture code (FFMC). The FFMCis a numeric code ranging from 0-101 intended to rate the moisture contentof forest litter and other cured fine fuels. Like the rest of the Fire WeatherIndex (FWI) system components, ISI does not take fuel type into account.With W defined as the wind speed in km/h, the initial spread index isdefined asISI = 0.208f1(W )f2(F )where f1(W ) = exp0.05039⇥W and f2(F ) = 91.9e(0.1386)⇣1+ m5.314.93⇥107⌘wherem = 147.2⇥(101F)59.5+F with F indicating the FFMC.The FFMC is an indicator of how easily fine fuels will ignite. An FFMCvalue less than 60 indicates too much moisture for a fire to ignite, while avalue of at least 84 indicates sucient dryness for easy ignition.After the initial growth of a new fire, it becomes more established andburns deeper into the vegetation layer of the forest floor and is then influ-enced by other factors such as the amount of available fuel. This buildup offuels is referred to as the buildup e↵ect (BE)BE = exp⇣50⇥ ln(q)⇥⇣ 1BUI 1BUI0⌘⌘(2.2)where q is an assigned value between 0.7 and 1, relating the rate at whichincreasing dryness in depth might influence the ROS. The Buildup Index(BUI) is a numeric rating of the total amount of fuel available for combus-tion. It is based on the moisture content of both the loosely compactedorganic layers of moderate depth and of the deep compact organic layers. InEquation (2.2), BUI is the specific calculated value, and BUI0 is the averageBUI expected to be seen for that particular fuel type.The final adjustment that was made to equation (2.1) was the influenceof slope in the terrain. The slope factor (SF) that was added to this equationwas that of [12], however, modified to measure in terms of degrees ratherthan in ground slope percentage,SF = exp⇣d⇥ tan(Slope⇥ ⇡180)⌘e. (2.3)In the above slope factor equation, parameters d and e are numerical con-stants that are estimated in [12] to be 3.533 and 1.2 respectively. Thesevalues were obtained when plotting the results of five di↵erent slope fac-tor models and fitting an average line through them. Of these five models52.2. Fitting a Stochastic Fire Spread Model to Datahowever, two are suspected to be based on informal observations, and theother three were performed on small scale burns using 1.2m long beds of redpine needles. In their paper [12] states that the question remains of whetherthe results of these prior studies would remain accurate when applied tofull-scale forest fires. Because of this, these values were instead taken asparameters to be re-estimated in this thesis.With the RSI, BE, and SF equations established, we can create a fullrate of spread equation for a fully established fire on sloping terrain to be:ROS = RSI⇥ BE⇥ SF + ✏ (2.4)where ✏ represents error and assumed to be normal. Also in the CFFBPSis a suggested model for the back burn rate of spread (BROS). This BROSmodel however, is only described for the RSI portion of the above ROSequation. With the assumption that the buildup e↵ect and slope e↵ect maybe included in the model, the resulting BROS model is described asBROS = a⇥⇣1 exp(b⇥BISI)⌘c ⇥ BE⇥ SF (2.5)which is similar to equation (4.2) with the exception of the back burn ISI,(BISI) in place of the ISI in the RSI equation. The BISI is similar to theforward ISI from equation 2.3 except instead of f(W ) it is represented asBf(W ) and is simply the negation of the f(W ) exponentf1(W ) = exp0.05039⇥W.2.2 Fitting a Stochastic Fire Spread Model toDataIn the paper Fitting a Stochastic Fire Spread Model to Data [13], theauthors introduce the process that I used in my own research for conduct-ing smouldering fires within a fume hood. In addition, [13] developed theprocedure whereby the video footage of the smouldering burns is segmentedand processed for analysis.Burns were conducted in a fume hood where the burning fuel is a sheet ofwax paper coated in a 1:10 solution of potassium nitrate (KNO3) to water.The sheets soaked in the solution for an hour, ensuring that the potassiumnitrate attached to the paper, and were then dried before burning. Thepotassium nitrate coating promotes smouldering and inhibits the productionof a flame. The burn was initiated by a barbeque lighter burning a small62.3. Applications of simulation-based burn probability modelling: a reviewhole in the sheet through a hole drilled into the platform of which the sheetwas attached to. As the sheet burned, the fire was recorded by a cameradirectly above the sheet. The flame inhibition proved useful for eliminatinglight interference with the video footage, as flames would block the fireperimeter from view, thereby interfering with data collection, and makethe image processing more dicult. Another tactic used in this study toreduce the amount of interference in the video footage was to record all themicro fires in a dark setting as to avoid external light sources. In additionthe burns were conducted on a dark platform to ensure that there was noreflected light interfering with data collection. The multiple light controllingmeasures ensured that the only captured source of light in the videos wasthe burning fire perimeter.Once the micro fires were recorded, the videos were then put into theopen source software ↵mpeg, where still shots were extracted from the videoat every half second. This sequential set of images was then imported intothe program R where the images of the fires themselves were colour codedto represent the status of each individual cell. The cell states used wereburning, burnt out, and unburnt fuel that were coloured red, black, andgreen respectively. The spatial data was then extracted numerically, andeach colour (cell state) was given a numerical value (2 for burning, 1 forburnt out, and 0 for unburnt).From their work, [13] concluded that this method of data collection yieldsdata to which a stochastic fire spread model can be fit. They also claimedthat they were able to capture some of the stochastic nature of fire.2.3 Applications of simulation-based burnprobability modelling: a reviewBefore the creation of burn probability maps, fire behaviour analysts hadto manually estimate fire spread by drawing out their predictions on topo-graphical maps. This was a much less ecient way to model fire behaviourthan today’s use of software built on a foundation of mathematical mod-els, also known as burn probability models. There are many di↵erent burnprobability models. The most commonly used models in North Americaare Prometheus, Burn P3 (both used heavily in Canada), and FARSITE [4](used in the United States). In their review, Parisien et al. [7] explain thefunction and application of these pixel-based spatial models and how theyrun inputs such as weather, fuel, and topography through numerous di↵er-ent mathematical models. These simulated models each describe di↵erent72.4. Rate of Spread Estimatesaspects of wildfire, such as rate of spread, fire intensity (calculated as theenergy produced by the fire in kW/m), fire size, and the probability of firebeing transmitted from one pixel to the next. These models can be usedto analyse real, or theoretical fires where the researcher can investigate howaltering di↵erent parameters influences fire behaviour [7].The inputs that a user would enter into the software are: fuels, topog-raphy, ignition points, and weather. As the topography of an area is verydicult to alter, this input is not generally manipulated very often in re-gards to testing the e↵ects when specific parameters are changed. However,as this input relates directly to slope, establishing a more concrete rela-tionship between slope and ROS would benefit these models in being ableto more accurately predict fire behaviour characteristics. A model whichwould describe this relationship could easily be implemented into all of theburn probability models listed above.2.4 Influence of Measurement Approach onFlame-Front Rate of Spread EstimatesIn an e↵ort to understand the best approach researchers can take to mostaccurately measure a fire’s rate of spread, [5] examined various methods forcalculating rate of spread, coupled with di↵erent data collection methods.In their study they tested eight di↵erent methods for calculating the rate ofspread of a fire over three di↵erent methods for collecting the data namely:visual imagery camera, thermocouple temperature monitors, and infraredimagery. Some of the calculation methods were very similar in concept,di↵ering only in the method used to collect the data, which in turn wouldgive insight as to which data collection method o↵ered the most accurateresults.For the thermocoupling collection method, [5] used two di↵erent arrange-ments of the heat sensing thermocouples: a linear arrangement down thecenter of the burn platform and a grid formation over the entire platform.When collecting data with the visible and infrared cameras, the respectiverates of spread were calculated using a variety of image processing tech-niques. Rate of spread calculations for the visible imagery data were doneby recording the timestamps at the point where the fires reached specificthreshold locations across the burn platform, this was done from a groundlevel and overhead perspective of the fires. The thermal imagery videos wereused to conduct four di↵erent spread rate calculations: a geo-referenced ver-sion of the thermocouple grid array where rather than thermocouples, the82.5. Locally Weighted Scatter plot Smoother (LOWESS)brightness temperature of pixels were used, a leading perimeter methodintroduced by [9] as well as an altered version of this method with an in-crease threshold of brightness temperature, and finally a triangular methoddescribed by [6].In this study [5] found that the method of observation used for deter-mining the ROS has an impact on the resulting ROS estimates. However,all of the methods were in close agreement except two: the linear array ar-rangement of thermocouples, and the triangulation method of [6] that wasperformed on infrared data.2.5 Locally Weighted Scatter plot Smoother(LOWESS)Later in this thesis, when the burn data are analysed, a Locally WeightedScatterplot Smoother will be applied to determine a relationship betweenROS and slope. This is a non-parametric tool used for fitting a smoothcurve to data [3], i.e., its objective is to demonstrate a relationship betweenthe explanatory and response variables. The non-parametric nature of thissmoother means that it will exhibit much more flexibility than smoothersthat are constructed from specific classes, such as polynomial models.The points (xi, yi) i = 1, . . . , n of the original data set are summarized bysmooth points (xi, yˆi), that portray the location of the estimated expectedvalue of Y given X = xi. This formation of new points is called smoothingthe scatterplot. Smoothing the scatter plot is done using locally weightedregression where for each xi, weights are defined using a non-negative weightfunction W for the explanatory variable x.An example of a weight function (e.g. [2]) is the bi-square function Band is defined as:B(x) =((1 x2)2 for |x| < 10 for |x|  1.The first step in the LOWESS process is to establish a new set of points(xi, y˙i). These points are established by locally weighted linear regressionand the weight function W . For each point of the original data set, thisregression is performed using the surrounding points within a set neighbour-hood around the focal point. The points closest to the focal point (currentpoint of interest that changes throughout the iterations of the LOWESSprocess) hold a larger weight, whereas the points furthest from the focalpoint within the window, hold less weight, and result in having a smaller92.5. Locally Weighted Scatter plot Smoother (LOWESS)influence on the fitted line that results from the regression. For each i, hiis the distance from xi to the rth neighbour, the point furthest away fromthe focal point at which any point further than the rth neighbour will haveno weight. The applied weight function wk is calculated using the followingequationwk(xi) = W (h1i (xk  xi)).This weight function is used to determine estimates for j(xi) wherej = 0, . . . , d. These coecients j(xi) are the values of j which minimizesnXk=1wk(xi)(yk  0  1xk  · · · dxdk)2. (2.6)From the original point (xi, yi) the resulting smoothed point (xi, y˙i) isobtained by the locally weighted regression described above, and y˙i is thefitted value of the regression.y˙i =Pdj=0 ˆj(xi)xjiIt is important to note that the above process and associated weights, dependonly on a point’s distance from the focal point along the x-axis. These pointsare preliminary as the next step in the process deals with the distancesbetween points along the y-axis.To compute yˆi, the same dth degree polynomial from above is used forweighted least squares, however, an additional weight kwk(xi) is now in-cluded wherek = B(ek/6s)and ek is the residuals from the original yi value and the current fitted valuey˙i and s is the median of the residuals |ei| of the surrounding neighbourhood.The above process may be repeated until the numerical change of thefitted values falls below a set threshold.LOWESS is often used to analyze data with a lot of noise as it focusesmore on the fitted curved rather than estimating specific parameters within apredetermined model. It is for this reason that it is an appropriate approachwhen exploring the fire spread data and relating rate of spread to slope.102.6. Generalized Additive Model (GAM)2.6 Generalized Additive Model (GAM)In a linear model, the response variable is modeled as a sum of linearterms of the covariates and takes the formyi = 0 +Xjjxji + ✏i, (2.7)whereas in a generalized additive model (GAM) the covariates are used thesum of smooth functions and has the model takes the formyi = 0 +Xjsj(xji) + ✏i (2.8)with errors ✏i assumed to be normal, i.e. ✏i ⇠ N(0,2). The GAM usespenalized splines, which are functions made from simpler functions calledbasis functions. Each basis function bk has an associated constant coecientk, and the resulting spline s(x) is created from the linear combination ofthese basis functions and coecientssj(xji) =KXk=1kbk(xi), (2.9)where K is the total number of basis functions. To avoid over fitting, thecomplexity of our fitted function is penalized. The complexity (W ) is de-termined by integrating the squared second derivative of the fitted functionover the range of our observed predictor variableW =ZR[s00(x)]2dx.The complexity is calculated in terms of the second derivative as it is thestandard form of determining the complexity for the GAM model, and wasthe method used for this study. The second derivative is used for determiningthis as it measures the rate of change in the slope of the line at any point.The complexity W is scaled using a smooth parameter  and is used to findthe spline coecient k that maximizes the penalized log-likelihoodLp() = L() WThe log-likelihood L() measures how well the model recreates the dataand is a function of the i parameter coecients in (2.9). Setting  = 0would result in Lp() = L(), an un-penalized model. This smoothing112.6. Generalized Additive Model (GAM)parameter can be selected in a number of possible ways although the defaultmethod used by the gam() function in R is the generalized cross validationscore (GCV). This score is built o↵ of the foundation of the ordinary crossvalidation (OCV) score where we wish to choose  such that our fittedfunction fˆ is as close as possible to the true function f [15]. The strategy forOCV is to calculate and average the squared di↵erences between fˆ [i]i , themodel fitted to all data points except yi, and the missing point yi itself.⌫0 =1nnXi=1(fˆ [i]i  yi)2 (2.10)The problem with OCV however, is that it is computationally expensivewhen dealing with large sets of data as the program would have to refit themodel for each of the n points. GCV was built to use the approach of OCVwhile solving this problem (2.10) can be rewritten as⌫0 =1nnXi=1(yi  fˆ(xi))2/(1Aii)2 (2.11)where fˆ is the estimate from fitting to all data, and A is the correspondinginfluence matrix. The matrix components Aii are often replaced by theirmean tr(A)/n, allowing us to rewrite (2.11) the following way⌫0 =nPni=1(yi  fˆ(xi))2(n tr(A))2 (2.12)which is the equation for the GCV score.12Chapter 3Data CollectionThere are three data sets that were used for this study: (1) field testburns conducted by British Columbia Wildfire Services (BCWS), (2) datafrom the Canadian Forest Services (CFS), and (3) data obtained from lab-oratory smoulder fires.3.1 Field dataTwo separate field data sets were used to study rate of spread and theslope e↵ect. These were gathered by the Canadian Forest Service (CFS)and by the British Columbia Wildfire Service (BCWS). Both of these datasets contain weather and fire growth measurements on real fires in outdoorenvironments where natural wildfires can occur. All of the data collectedby the BCWS was acquired on controlled test burns, whereas the CFS datacontained both controlled test burns, as well as well-documented wildfiredata.3.1.1 British Columbia Wildfire Service (BCWS)The BCWS data was gathered during the summer wildfire seasons from2014-2019, primarily during the early to mid summer months (May-July).These fires were conducted on crown land in areas within the Kamloops FireCenter with a total of 59 test burns. These fires were conducted in the C7fuel type that is made up of Ponderosa Pine (Pinus ponderosa) and DouglasFir (Pseudotsuga menziesi), as well as in the O1-B fuel type (grassland)both with various slopes. These fires were originally conducted to gain abetter understanding of possible fire behaviour that could be expected dur-ing that time, and therefore aid in Fire Center preparation plans. Accurateweather readings were also made during these test burns, and compared tothe weather received by weather stations, to see if the weather station datacan provide firefighters an accurate forecast of expected fire behaviour, asthese stations can be a large distance away from the burn site. It is impor-tant to note that these fires were mostly conducted between May and June133.1. Field dataas summer weather conditions increase the fire hazard too much for thesetests to be conducted with certainty of maintaining control. Influencingfactors such as time of day, or weather conditions (relative humidity, wind,and temperature), degree of fuel curing, and ISI, vary across the burns andare factors that could not be controlled. Many of these factors are not in-corporated into the model being investigated in this study, however, theseparameters were noted and summaries can be found in Appendix A.To conduct these test burns, fire crews would go out to a pre-determinedlocation with near continuous ground fuels (grass and forest litter). Thegrassland locations were categorized as an O1B fuel type, which consistedof matted and standing grass with near continuous cover. The burns donewhere the burning fuels consisted primarily of forest litter, were done inwhat is categorised as a C7 fuel type, which is mixture of Ponderosa pineand Douglas fir. Upon arrival, the local temperature and relative humid-ity were measured using a Bacharach Sling Psychrometer, a manual devicewith two thermometers (one wet one dry) specifically used to measure theseparameters. Then, a suppression line was established and tested to ensurewater is immediately available to maintain control and fully suppress theburn once needed.The ignitions were started by dropping a single match on the ground,the ignition point is clearly marked with a metal spike. If a fire fails to igniteafter 3 dropped matches, crews would then try a more direct approach suchas a lighter or a drip torch with a 3:1 diesel to gasoline mixture. At the timeof ignition the temperature and relative humidity were measured again withthe sling psychrometer while wind speed and direction are also measuredwith a Kestrel 1000 wind meter. All of these measurements were taken atthe 2, 5, 10, 20, and 30 minute intervals after the ignition has begun, oruntil the test is terminated. At these time intervals, the active fire is alsomeasured using a measuring tape where the head, back burn, and flankdistances were measured from the origin metal spike, to the perimeter ofthe fire.If the burn was conducted on a hill, GPS coordinates were taken fromthe lowest and highest points to give the elevation and distance that thefire travelled. This allows the angle in degrees to be calculated with thefollowing formula:Angle = 90 sin1⇣ab⌘(3.1)where a is the di↵erence in elevation, and b is the distance between the twopoints. Note that the coordinates o↵er the straight up elevation di↵erence,and the distance between the points are in a straight line with the GPS not143.2. Lab Datataking into account the slope. However, from this the slope was then roundedto the nearest factor of five, this would be a source of error within the dataand therefore most of the given slopes may not be exact representations ofthe slope that the test was conducted on. Also, it is unknown whether ornot these burns were conducted on slopes with varying degrees, in whichcase the resulting slope measurement would be an average across the entirefire.3.1.2 Canadian Forest Service (CFS)The other data set pertaining to field burn that was used is the CFSdata. This data set consists of 241 total burns in the Boreal forests ofNorth America between 1930 and 1986. These fires were done in all C1-7coniferous fuel types, as well as deciduous and mixed fuel stands all done onflat terrain. Of this data set, only the information gathered on the 13 burnsconducted in the C7 fuel type which were done between 1967 and 1985 wereconsidered. These were the only burns considered as it was the only fueltype that was common in both the CFS and BCWS data sets. Of these 13fires, five were done in BC and the other eight were in Ontario. Three of thefive BC fires grew to be crown fires, meaning that the fire travelled throughthe forest canopy rather than just through the surface fuels. The CFS datais the basis of the CFFBPS.3.2 Lab DataThe final data set that was used for this study consists of 44 smoulder-ing lab fires where a one sided sheet of wax paper was coated with a 1:10potassium nitrate (KNO3) and distilled water solution, following the sameprocedure as [13]. The wax paper was used so that the potassium nitrate isnot entirely absorbed by the paper and that it is still easy to ignite. Thesheets of wax paper were left to soak in the potassium nitrate solution forone hour to ensure that the solution could adhere to the sheet. After thepotassium nitrate was coated onto the sheets, the sheets were left to dry fora full day. Once the sheets were dried, it could be noticed that the potassiumnitrate does not coat the sheet uniformly, instead, there were small patchesthroughout where clusters of the chemical bonded to the paper. These smallpockets of fuel added an element of stochastic behaviour to the smoulders,which could represent larger patches of forest catching and spreading morerapidly. The potassium nitrate is used as it promotes a constant smoulderyet inhibits the production of any actual flame. This ensures that all of the153.2. Lab Datasmouldering fire perimeter is not obstructed from view by a flame, an issuewhich is brought up in the study of [5] on the use of visible imagery cameras.This was also required as these smoulders were conducted in a fume hood,an environment that would have promoted a larger combustion as it cyclesair.The dried sheets of wax paper were fixed to a 15.25x10.25x0.75 flat trayusing tape, leaving room between the sheet and the tray to allow enoughoxygen for burning. The tray was painted black with non-flammable spraypaint to minimize the amount of reflected light that is captured by the cam-era. The black paint, along with ensuring that all lab burns were conductedin the dark were precautions that were taken to minimize the amount of lightinterference. With all of these precautions in place to reduce the amountof noise in the data, it can be assumed that the only light captured by thecamera was of the progressing fire perimeter. The tray itself was placed on ametal breadboard that sat on a wooden device that hinges so that the angleof the burn can be adjusted between 5 and 50 degrees at 5 degree intervals.An Olympus Stylus 600 camera was attached to the breadboard by metalpegs 48cm above the burn sheet with a directly overhead view. This entireapparatus was placed inside of a fume hood (Figure 3.1).Figure 3.1: The flat tray fixed onto the metal breadboard with the OlympusStylus 600 camera positioned above the burn sheet on top of the woodenhinging platform inside of a fume hood. In this image the burn platform isat a 40 angle163.2. Lab DataIt is important to note that as the camera was attached to the breadboardsitting on the wooden hinging platform, which allowed the camera to havea direct overhead view of the burn regardless of the angle. The ignitionstarted by using a barbeque lighter, igniting the sheet from below througha hole which was drilled into the tray for this purpose. Igniting the sheetsin this way was done to reduce the amount of interfering light which wouldhave come from the flame of the lighter itself.Four separate burns were conducted for each of the 11 di↵erent angles(0,5,10,...50). The experiments were conducted to investigate the impact ofslope on a fires rate of spread. For these trials, the slope of the burn wasthe only parameter that changed, and as these tests were conducted in acontrolled environment we can be confident that any significant variationin the ROS data across all the burns would be due to slope, and not someother external influencing factor.3.2.1 Data Extraction and SegmentationTo analyse the data from the videos, we first used the open source pro-gram ↵mpeg to capture the clearest frame of the video at every half-second.These freeze-frames were saved as JPEG files so that they would be read-able in the R software. As previously stated, measures were taken to reducereflective light such as black paint and conducting the trials in the dark. Bytaking these precautions, problems with segmenting the images are avoidedas the only visible light is that of the fire, meaning there are only blackand non-black coloured cells. Figure 3.2 shows the still images extratedfrom the burn videos using the ↵mpeg program, the figure shows a grid ofthe smouldering burn perimeter growth where each image is taken 5 sec-onds apart. Each pixel is assigned a numerical value based on its colour,with black being 2 (and near black being close to 2) and a clearly nonzerovalue when the pixel burns. The pixel will return to a 2 value when it isburnt-out. By tracking pixels across the sequence of images we can followalong with what state (unburnt, burning, burnt-out) they are in and at whattimes. Segmentation follows easily as each pixel is classified into the land-scape as ”unburnt fuel” (green), ”burning” (red), and finally ”burnt-out”(black). Therefore, the time(s) at which each pixel transitions from unburntto burning can be identified from the time series of the red, green and black(rgb) measurements.173.3. Data SummaryFigure 3.2: Images extracted using ↵mpeg from a video recording one ofthe lab smoulder fires. These images were taken 5 seconds apart from oneanother as the burn progressed.3.3 Data SummaryBelow are a series of tables highlighting the averages, ranges and stan-dard errors of the ROS and BROS samples for the three data sets. Forthe BCWS and lab smoulder data sets, this information is categorised interms of slope. Tables highlighting di↵erent weather parameters which alsoinfluence fire behaviour can be found in Appendix A.Rate of Spread Data (m/min) From Field Burns O1-BSlope AverageROSRangeROSSDROSAverageBROSRangeBROSSDBROSn0 3.18 0.8-5.2 1.72 1.11 0.5-2.3 0.65 105 4.38 1.5-8.9 2.94 1.95 0.45 - 6 1.75 810 11 11 0 2.1 2.1 0 1Table 3.1: Rates of spread measured in meters per minute (m/min) of boththe forward and back burn flanks of the fires conducted in the O1-B grasslandfuel type, with their averages, range and standard deviation (SD) from theBCWS test burn data and n denotes the number of fires.183.3. Data SummaryRate of Spread Data (m/min) From Field Burns C7Slope Average Range SD Average Range SD nROS ROS ROS BROS BROS BROS0 1.2 0.3-2.9 0.74 0.8 0.3-1.7 0.4 115 1.63 1.25-2 0.26 1.01 0.5-1.6 0.33 710 1.6 1.3-1.9 0.42 0.6 0.5-0.7 0.14 215 3.4 3.4 0 2.5 2.5 0 120 2.34 1.5-3 1.52 0.61 0.3-1 0.25 730 0.8 0.8 0 0.5 0.5 0 1Table 3.2: Rates of spread measured in meters per minute (m/min) of boththe forward and back burn flanks of the fires conducted in the C7 Ponderosapine and Douglas fir fuel type, with their averages, range and standarddeviation (SD) from the BCWS test burn dataIt is important to note that these field burns were conducted acrossmultiple years, at di↵erent times of the year, di↵erent locations, and underdi↵erent weather conditions. All of these factors play a role in fire behaviouras it influences the burn characteristics of the fuels. As these factors influ-ence vegetation types di↵erently, the data must be analysed in terms ofvegetation. Varying weather and fuel characteristics are incorporated in themodel for rate of spread suggested in the CFFBPS used in this study.Certain fire weather parameters such as FFMC, ISI, and BUI for theBCWS test burn data are obtained from the nearest weather station towhere the test burn was conducted, and not on site of the burn itself likethe wind, temperature, and fire growth parameters. These parameters aredisplayed in the tables of Appendix A.193.3. Data SummaryROS Data (m/min) From Lab BurnsSlope Average ROS Range ROS SD0 0.027 0.023 - 0.032 0.0045 0.021 0.0122-0.0267 0.006410 0.032 0.022-0.055 0.01515 0.039 0.031-0.05 0.009520 0.029 0.0261-0.032 0.002425 0.029 0.013-0.039 0.01230 0.0315 0.027-0.04 0.005835 0.032 0.021-0.043 0.01240 0.045 0.04-0.057 0.008245 0.0458 0.035-0.055 0.009150 0.046 0.038-0.05 0.0059Table 3.3: Rate of spread data measured in meters per minute (m/min)collected from the videos of the lab smoulder fires.BROS Data (m/min) From Lab BurnsSlope Average BROS Range BROS SD0 0.026 0.011 - 0.041 0.0125 0.018 0.014-0.024 0.003910 0.031 0.027-0.034 0.003915 0.026 0.02-0.035 0.006720 0.032 0.018-0.057 0.01725 0.02 0.017 - 0.026 0.00430 0.022 0.019 - 0.029 0.004735 0.021 0.015 - 0.027 0.004940 0.03 0.019-0.055 0.01645 0.025 0.02-0.03 0.004650 0.022 0.015-0.031 0.0069Table 3.4: Back Burn Rate of Spread data measured in meters per minute(m/min) collected from the videos of the lab smoulder fires.20Chapter 4AnalysisIn this chapter we will go over the the analysis that was conducted on thedata sets described in Chapter t3. Here we will see how the LOWESS andGAM tools described in Chapter 2 were implemented and interpret theirresults. Analysis for the ROS and BROS data are presented individually foreach of the three fuel types (C7, O1-B, and potassium nitrate wax sheets).Finally, we will briefly investigate whether or not the presence of a slopeinfluences a fire’s back burning rate of spread (BROS) downhill.We have seen in section 2.1 the statistical model presented in the Cana-dian Forest Fire Behaviour Prediction System (CFFBPS) to model a fire’srate of spread:ROS = a⇥ 1 exp(b⇥ISI) c⇥BE⇥ exp⇣d⇥ tan(Slope⇥ ⇡180)⌘e+ ✏ (4.1)where BE is the buildup e↵ect of equation (2.2). This equation can berewritten in terms of the initial rate of spread (RSI), buildup e↵ect (BE)and slope factor (SF) as we saw in Equation (2.4) whereROS = RSI⇥ BE⇥ SF. (4.2)Estimates for the a,b,c,d, and e parameters for this model are given in theCFFBPS and are given in Table 4.1 below, for the C7 and 01-B fuel types.Fuel type a b c d eC7 45 0.0305 2 3.533 1.2O1-B 250 0.035 1.7 3.533 1.2Table 4.1: C7 and O1-B parameter estimates for a, b, c, d, and e for thecomplete ROS model (4.1) given in [12].Of these five parameters, only d and e pertain to the pre-existing slopee↵ect model. As previously mentioned, [12] first introduces the slope e↵ectmodel used in the CFFBPS, these parameters were obtained by fitting an214.1. Field Dataexponential curve through the average of five models. These five modelswere brought forward between the years 1958 and 1972, of which two arespeculated to be based on informal observations while the other three werebased on small scale fuel beds.The goal of the current study is to develop a model for the slope-e↵ect for-mulated from the data sets described in Chapter 3, and investigate whetherthe suggested slope-e↵ect parameter estimates (d and e of Table 4.1) in[1] match our findings. There are three main components which determineROS in a homogeneous fuel type environment: wind, fuel moisture, andslope. To analyse our ROS data, a generalized additive model was usedusing the gam() function in R [14] which follows the process described inSection 2.6. A GAM provides a flexible way of modelling ROS in terms ofthese three influential factors. However, a generalized additive model couldonly be fit to a master C7 data set consisting of the BCWS and CFS dataand to the smouldering lab data. The individual C7 and O1-B data setsfrom the BCWS test burns contained insucient data for the function to beapplied.4.1 Field DataAs mentioned in Section 3.1.1, weather and fire measurements were taken2, 5, 10, 20, and 30 minutes after ignition, or at these intervals until thefire was extinguished for the BCWS data. Almost all of these burns wereextinguished much earlier than the full 30 minutes with the earliest timebeing 2 minutes and 40 seconds, leaving only the 2 minute rate of spreadmeasurement available for all burns. Because of this, the rate of spread dataafter 2 minutes was used for the analysis as it would provide the most data.To ensure that only using the 2 minute ROS data would be a good repre-sentation of the data, a two sample t-test was used to compare the 2 minuterate of spread data to that of the 5 minute rate of spread data. This testresulted in a t-value of 0.55 and a p-value of 0.58. Having a p-value thislarge indicates that the null hypothesis of the two data sets (2 minute and 5minutes ROS data) being the same cannot be rejected, suggesting no di↵er-ence in the mean ROS using the 2 minute or 5 minute measurements. Thisindicates that the 2 minute data may accurately represent the behaviour ofthe observed burns.224.1. Field Data4.1.1 C7 Ponderosa Pine and Douglas FirForward Rate of SpreadTo study the ROS behaviour for the C7 fuel type, a master data set wascreated by combining the BCWS and CFS burns allowing us to analyse 44individual fires together. Modelling the ROS for this data was done on thelog-scale with the use of the gam() function where a bivariate smoothingfunction was applied to the wind and FFMC variables, with slope and firetype entering the model as a sum of linear terms. The gam() function fitthe data to the following modellog(ROS) = 0⇥Slope+1⇥Fire Type+2Xs(Wind+FFMC)+✏ (4.3)where s is a linear combination of spline basis functions whose coecientsare to be estimated from the data. The bi-variate smoothing was done sothat we could isolate slope from the other two driving factors of ROS, namelywind and FFMC.The fire type refers to whether the ROS was measured on surface fires,burning only the ground litter, or whether they were crown fires, where thefire travelled through the forest canopy as well as the forest floor. This wasincluded as crown fires burn with much higher intensity, and the CFS datacontained three measurements on crown fires. These three crown fire burnshad significantly larger associated ROS measurements than the controlledtest burns, and had these fires not been classified by fire type, their ROSmeasurements would have been seen as significant outliers, especially con-sidering that they occurred on flat terrain. Isolating these factors attributestheir own contribution to the resulting ROS, and allows us to view theirinfluence separate from slope.The results of the GAM suggests the following model for the slope e↵ectin a C7 fuel type.Slope E↵ect = e(0.019+✏)⇥Slope (4.4)where ✏ is approximately normal with mean 0 and variance 2 = 0.00915.A 95% confidence interval for the slope e↵ect can be calculated from(e0.00186⇥Slope, e0.03773⇥Slope) (4.5)The above confidence interval implies that the true slope e↵ect multiplierfor a fire burning within a C7 fuel type likely exists within these bounds.The value of the above model 4.4 at a given slope would be included as theslope-e↵ect multiplier in the overall ROS Equation (4.2).234.1. Field DataTable 4.2 highlights the standard errors and p-values for the slope andfire type resulting from the GAM test.Parameter Estimate Standard Error p-valueSlope 0.0198 0.00915 0.0384Fire Type -0.936 0.43837 0.0409Table 4.2: Slope and fire type multipliers obtained by the GAM. The es-timate for these parameters indicates the amount of influence they had onthe log(ROS) for C7 data.These values show that the exponential factor for slope is marginallysignificant with a p-value of 0.0384 as well as confirms the fact that surfacefires spread more slowly than crown fires. To ensure that the GAM modelthat was applied is indeed a good representation of the data, we can look atdiagnostic plots of the residuals Figure 4.1. Of the four plots of Figure 4.1the top left Figure is a linear QQ plot which tells us that once the factors inthe model are taken into account, there are no strong deviations or severedepartures from normality. The residuals vs linear predictor also shows thatthere are no serious outliers as they form a nice grouping around y = 0.The response vs fitted values plot indicates that as predictions go up, theresponses are matching, and finally the histogram shows that the residualsare fairly normal, with a slight left skew.244.1. Field DataFigure 4.1: Diagnostic plots from the GAM model applied to the C7 ROSdata. At the top there is a linear QQ plot (left) and a residuals vs linearpredictor (right). On the bottom there is a histogram of the frequency ofresiduals (left) and response vs fitted values plot scatter plot (right).The diagnostic plots in Figure 4.1 indicate that the GAM was able tosuccessfully model the data for the C7 fires. We see from the plots thatnormality was maintained throughout once the factors (wind, FFMC, slope,and fire type) were taken into account. Another good indication of theGAM’s ability to model the data is that the residuals are randomly scatteredaround 0 in the top right plot. This suggests us that the model was not overor under fitted and that it is close to the observed points.In Figure 4.2 both the Van Wagner and resulting GAM slope e↵ect mod-els are plotted, with the inclusion of the 95% confidence interval around ourcalculated model. We see that the Van Wagner slope e↵ect model (dashedblack) is greater than the GAM suggested model (green) and its surrounding95% confidence interval (red).254.1. Field DataFigure 4.2: The fitted model (4.4) obtained by running a GAM on the C7ROS master data set (green) surrounded by it’s associated 95% confidencebands (red) compared with the Van Wagner slope e↵ect model used in theCFFBPS (dashed black).When comparing the model obtained by the GAM to the model sug-gested by Van Wagner [12] used by the CFFBPS, we can clearly see thatthe pre-existing Van Wagner model suggests a much stronger slope influencethan what was observed from the data used for this thesis. The results of theGAM test also give insight into how wind and FFMC influence the resultingROS as we can see in the contour plot of Figure 4.3.264.1. Field DataFigure 4.3: Contour plot of log(ROS) as a function of wind speed and FineFuel Moisture, obtained from the generalized additive model applied to theC7 data. The black curves represent the fitted curve whereas the red andgreen curves are the confidence bands surrounding the curve.Looking at Figure 4.3, if we follow the plot across the x-axis we see thatas wind increases, the resulting log(ROS) increases significantly from -1 atthe top left to a magnitude 0.5 at a windspeed of around 20km/h. Thismagnitude increases up to 2 at wind speeds of 40km/h. Looking along thevertical axis, we note that the FFMC does not have the same impact level aswind as the curves are quite vertical with the exception of the region arounda wind speed of 10km/h. The smooth terms of the GAM model have anassociated p-value of 0.007, indicating that wind has a strong influence ona fire’s ROS.274.1. Field DataBackburn Rate of Spread (BROS)The CFS data set did not contain information about the back burningprogression of the fire perimeter, meaning that all data used to investigatethis aspect of the fire was taken from the BCWS data alone. Because of thisthere was not enough data to be able to run a GAM. Instead, a LOWESScurve was used to gain insight as to which model would best represent thedata. This smoother was obtained using the lowess() command in R, whichfollows the procedure highlighted in Section 2.5 and the resulting fit can beseen in Figure 4.4.Figure 4.4: LOWESS applied to the BROS data from the BCWS C7 dataLooking at Figure 4.4, there is no obvious simple functional form toexplain what sort of relationship may be present between the rate of spreadand the respective slope. A linear model produces the following results whenapplied:BROS = 0.006⇥ Slope + 0.88 (4.6)where the estimates for the slope parameter are displayed in Table 4.3.284.1. Field DataEstimate Standard Error 95% Confidence Interval p-value-0.006 0.01 0.006± 8.68⇥ 103 0.55Table 4.3: Statistics of resulting slope parameter of C7 BROS data appliedto a linear model. A p-value of 0.55 indicates that when trying to fit thedata to a model with a slope influence, it could not do so with confidence.This tells us that there is nothing indicating a non-zero slope-e↵ect.From these results we see that when trying to fit the data to the mostlikely model, there is no clear indication of the data being fit well. Fromthis we are left to suggest that for the C7 fuel type, based o↵ of the dataobtained for this study, that slope did not have an influence on the measuredback burning down-slope rate of spread.4.1.2 O1-B GrasslandIn the behaviour prediction system, there is no body of Canadian firebehaviour data for grassland fuel types [1]. Instead, all spread rate estimatesare based on regression functions derived from Australian grassland fire data.Because of this, the CFS data contained no information about grasslandfires, thus not allowing us to create a master data set as we did for theC7 fuel type. The analysis below was conducted on data obtained throughlocal Canadian grassland test burns in the areas surrounding Kamloops B.Cconducted by the British Columbia Wildfire Service (BCWS).Forward Rate of Spread (ROS)Similarly to the C7 BROS data, there was not enough data from grass-land burns to run a GAM model. Therefore analysis for these data was doneby first performing a lowess() to the data to gain insight into a potentialmodel. The lowess() function in R resulted in the curve seen in Figure 4.5.294.1. Field DataFigure 4.5: LOWESS function applied to ROS of BCWS data from testburns conducted in the O1-B grassland fuel typeFigure 4.5 displays a plot of the lowess estimate of the regression functionrelating expected ROS to slope (in degrees) with the observed data overlaid.We see that there appears to be a positive relationship between the rateof spread and slope within the data. This relationship appears to followeither linear or exponential growth in ROS. When fitting the data to bothof these models, the exponential model proved to be a better fit, whichwas consistent with what was observed in the C7 fuel type, along with theCFFBPS proposed model. The data were fitted through the use of non-linear regression with the nls() function in R. The data was used to fit thefollowing nonlinear regression modellog(ROS) = a⇥ exp(b⇥%Slope) + ✏ (4.7)where the a parameter encompasses all ROS influencing factors other thanslope such as wind, RH, temperature, etc. b is the slope e↵ect parameter, and✏ is the noise term. Table 4.4 highlights the resulting parameter estimatesalong with the associated 95% confidence interval and p-value for the testof the hypotheses that the true parameters are nonzero.304.1. Field DataParameter Estimate SE 95% Confidence Interval p-valuea 2.546 0.58 2.546 ± 1.186 3.65⇥ 104b 0.134 0.0362 0.134 ± 0.084 1.62⇥ 103Table 4.4: Estimates for the ROS and slope e↵ect parameters for the modeldescribed in (4.7), along with their standard error (SE), 95% confidenceintervals and p-values. Parameter a represents the part of the ROS explainedby all influencing factors besides slope, and b represents the influence of slopeon the ROS.From Table 4.4 we see that there is a nonzero slope e↵ect that is signifi-cant with a p-value of 1.62⇥ 103. This suggests the following slope e↵ectfor this fuel type:Slope E↵ect = exp((0.134 + ✏)⇥%Slope) (4.8)where ✏ is approximately normal with mean 0 and standard deviation ˆ =0.00162 Figure 4.6 shows the fitted curve suggested from the nls() outputwhich resulted in the parameter estimates seen in Table 4.4 overlaying theobserved data on log(ROS) and slope (in degrees).Figure 4.6: Curve of equation (4.8) with parameters estimated through non-linear regression fit to the O1-B ROS data314.1. Field DataWhen we combine both Figure 4.6 and Figure 4.5 in Figure 4.7 we cansee that the nonlinear least squares model (dashed blue) does in fact followthe general form of the LOWESS curve (red).Figure 4.7: The LOWESS curve of Figure 4.5 (red) along with the curveobtained as a result of non-linear regression applied to the O1B ROS data(dashed blue).Now that we have a model that fits the data available for this study,we wish to see if the pre-existing model in the CFFBPS is an accuraterepresentation of the slope e↵ect observed in our data. Figure 4.8 shows theVan Wagner [12] model from the CFFBPS (dashed black) in comparison tothe model obtained through non-linear regression (green) with its associatedpointwise 95% confidence bands (red).324.1. Field DataFigure 4.8: The slope factor model obtained by non-linear regression on theO1-B grassland fuel type (green) with it’s corresponding 95% confidenceinterval (red) compared to the Van Wagner model presented in the CFFBPS(dashed black). Here we are looking at the suggested slope factor multiplieras the response variable and slope percentage as the explanatory variable.In Figure 4.8 we are looking at the slope e↵ect multiplier to be added tothe ROS equation as we increase slope. We also see that the Van Wagner[12] model (dashed black) appears to be an under-estimate of the slope ef-fect in comparison to the model we obtained through non-linear regression.However, this is only true up until the point where we reach a slope percent-age of 34% (roughly 18.75 degrees) at which point the Van Wagner modelfalls within the 95% confidence interval of our model (Figure 4.9) and staysbetween the bounds for all other slope values.334.1. Field DataFigure 4.9: The Van Wagner model (dashed black) crosses into the lowerpoint-wise 95% confidence band (red) of our regression model at a slopepercentage of 34% at which it stays within the bounds of the interval.Back burn ROS (BROS)Analysis for the back burn rate of spread for the grassland fuel typefollows the same process that was followed for the C7 BROS data. Thatis, we start by observing the results of applying a LOWESS function to getan idea of what relationship may exist between slope and BROS, and thenfollow up with fitting the data to a model that fits the LOWESS results theclosest. We can see the result of this function application in Figure 4.10.344.1. Field DataFigure 4.10: LOWESS function applied to the back burn rate of spread forthe O1-B grassland data on the log scaleFrom Figure 4.10, we see that if there was an influence of slope on thelog(BROS) for our data of O1-B fuel type, then it would be best describedas a linear model. Fitting this data to a linear model results in the followingoutput:log(BROS) = 0.091⇥ Slope 0.095 (4.9)where the error, confidence interval and p-values for the resulting slope e↵ectparameter are given in Table 4.5Estimate Standard Error 95% Confidence Interval p-value0.091 0.05 0.091 ± 0.105 0.08Table 4.5: Estimate, standard error, 95% confidence interval and p-value forthe slope e↵ect parameter resulting from fitting the O1-B log(BROS) datato a linear modelWe see from Table 4.5 that the p-value resulting from this test is greaterthan 0.05, suggesting that we cannot conclusively say that there is a lineare↵ect of slope on the resulting BROS.354.2. Lab Data4.2 Lab DataThe lab data gathered in this study was done in a controlled settingwhere the only measurable di↵erence among trials was the changing angleof the burn platform. The uneven coating of KNO3 provided a level ofstochasticity similarly to how pockets of fuel in a forest environment wouldinfluence a fire’s behaviour.4.2.1 Forward ROSFirst when analysing the lab data for determining a slope e↵ect modelfor the forward ROS, the following GAM was applied to the lab ROS dataon the log scalelog(ROS) =Xs(%Slope) + ✏ (4.10)where the smoothing function s() described in Section 2.6 was applied tothe percent slope. We can observe the resulting plot of the resulting fittedmodel in Figure 4.11.Figure 4.11: A generalized additive model applied to the lab ROS data onthe log scale. The dotted lines indicate the 95% confidence interval boundsof the GAM curve (solid). The red line at s(pctSlope,1)= 0 is used toindicate whether or not the variance in the data is significant.364.2. Lab DataFigure 4.11 plots the GAM model resulting from the ROS data of the labfires. The solid black line is the plot of the smooth functions of the percentslope, on the scale of the linear predictor. The dotted black lines indicatesthe pointwise 95% confidence bands surrounding these smooth functions,and the red horizontal line at s(pctSlope,1)= 0 is used to indicate whetheror not the influence of the slope parameter was nonzero. As the red linepasses through both the smooth functions and the 95% confidence bands,we can say that the e↵ect of slope on the resulting log(ROS) was nonzeroand that slope had a significant e↵ect on the resulting ROS. Had there notbeen any e↵ect from the slope, this line would pass within the upper andlower confidence bands.To ensure that the GAM model applied to the data is a good indicatorfor this relationship, we check the diagnostic plots of the residuals producedon this model in Figure 4.12. The top left plot is a QQ plot which shows thatthere are no strong deviations from normality as all the points stay close tothe red line. The top right plot of residuals vs linear predictor shows that theresiduals stay relatively constant and centered around 0 even as the linearpredictor (slope) increases. The histogram also o↵ers another indication ofthe residuals following a fairly normal distributions and finally the responsevs fitted plot shows that the responses are changing as the predictions goup.374.2. Lab DataFigure 4.12: Diagnostic plots of residuals for the GAM fitted model of theROS lab data. The top left plot is a QQ plot which shows that there areno strong deviations from normality as all the points stay close to the redline. The top right plot of residuals vs linear predictor shows that theresiduals stay relatively constant and centered around 0 even as the linearpredictor (slope) increases. The histogram also o↵ers another indication ofthe residuals following a fairly normal distributions and finally the responsevs fitted plot shows that the responses are changing as the predictions goup.The diagnostic plots above confirm that the GAM function was able tosuccessfully model the data for the lab fires. The above plots show us thatnormality was maintained throughout and that the residuals showed no clearindication of deviation or significant outliers. The results of the model arehighlighted in Table 4.6.384.2. Lab DataParameter Estimate Standard Error p-valueIntercept -3.708 0.077 2⇥ 1016Slope 5.3⇥ 103 0.001 7.84⇥ 105Table 4.6: Results for the intercept and slope factor from the GAM appliedto the forward log(ROS) data for the lab fires. Both parameters prove to besignificant as both have a very small p-value.We have the following model for the log(ROS) of the lab fires.log(ROS) = 3.708 + 5.36x103x%Slope (4.11)The above model obtained by the GAM suggests a linear model, whichdi↵ers from the previous ROS models we have seen when analysing the fieldburn data. To ensure that this is an accurate representation of the lab fires,a LOWESS function was also applied to the log(ROS) data which resultedin Figure 4.13.Figure 4.13: Rate of Spread vs the percent slope of the lab burns on the logscale with the LOWESS curve (red) included.In Figure 4.13, we see that the LOWESS curve suggests a positive linearrelationship between the rate of spread and the slope. This allows us to beconfident that the rate of spread from these fires can in fact be modelled394.2. Lab Datalinearly. When the data is fit to a linear model using the lm() function inR, we obtain the following model for the log(ROS) and slope relationship:log(ROS) = 3.708 + 5.36x103x% Slope (4.12)which is a direct match to the GAMmodel above. The linear model producesa residual standard error of 0.29 on 42 degrees of freedom, the resulting 95%confidence intervals and p-values can be found in Table 4.7 below.Parameter Estimate 95% Confidence Interval p-valueIntercept 3.708 (0.0202, 0.0299) 2x1016Slope 5.36x103 (1.02⇥ 104, 2.55⇥ 104) 7.84x105Table 4.7: Estimate, confidence interval, and p-value for the intercept andslope parameter estimates resulting from applying a linear model to the ROSdata for the smouldering lab firesIn Figure 4.14 we compare this linear model with the model in the CFF-BPS, we see that similarly to what was observed in the C7 fuel type, theCFFBPS model is a significant over estimate of the slope e↵ect suggestingit is not a good fit for explaining the increase of ROS for this data. Thissignificant di↵erence may however be due how much slower the observedspread rates were for the lab fires than what was observed in the field data.404.2. Lab DataFigure 4.14: Calculated linear model (green) from the lab ROS data withits corresponding 95% confidence interval bands (red) and the Van Wagnermodel suggested by the CFFBPS (black).It is important to note that the intercept value obtained by the linearmodel was scaled up to a value of 1 so that the models could be comparedpurely for the slope e↵ect. As the Van Wagner model is exponential, a zerovalued slope results in a scalar value of 1.When fitting a model to the log(ROS) data, the GAM and LOWESSboth suggested a linear model for the slope e↵ect within the lab data. Thiswould again lead to the slope e↵ect having an exponential e↵ect, and whenthe data was fit to the following simplified ROS modelROS = ↵⇥ exp⇣d⇥⇣%Slope100⌘e⌘(4.13)where ↵ would incorporate the RSI section of the ROS model suggested bythe CFFBPS. As the only measurable factor that di↵ers between burns isthe angle of the burn platform, it can be assumed that the only cause fordi↵erences in ROS would be a slope e↵ect. Because of this assumption, theexpected ROS for the no slope trials can be used in place of the RSI inequation of equation (4.1). This isolates the slope e↵ect when the ROS datawas fit to equation (4.3). We can see from Table 3.5 that ↵ will then havea value of 0.0267. The parameters d and e are still the same slope e↵ect414.2. Lab Dataparameters from before in equation (4.2).When fitting the lab ROS data to Equation (4.13) using nonlinear regres-sion, we obtain the estimates for the slope factor parameters from equation2.7; and these are displayed in Table 4.8Parameter Estimate S.E 95% Confidence Interval p-valued 0.479 0.05 (0.364 , 0.581) 2.7⇥ 1011e 1.151 0.38 (0.531 , 2.183) 0.004Table 4.8: Estimates for slope e↵ect parameters d, and e of equation 2.7calculated by non-linear regression conducted on lab smoulder fire ROSdata with their associated standard errors (S.E), confidence intervals, andp-values.In Table 4.8 we see that fitting the data with non-linear regression ap-pears to have made a good fit with the data, and this is confirmed whenplotting this suggested ROS model to the lab data itself as seen in Fig-ure 4.15a. In Figure 4.15b we include the linear model plotted through thedata for a comparison between the two models.424.2. Lab DataFigure 4.15: Observed forward rate of spread of the smouldering lab fires vspercent slope. On the left the regression model (red) is plotted through thedata points, and on the right is the linear model (green).We see that both the regression and linear models fit the data quitewell, and that the curve of the regression model is rather small causingthe appearance of the curve to be near-linear. When we compare either ofthese models to the suggested slope e↵ect model suggested by the CFFBPS(Figure 4.16), we see that the Van Wagner model from the CFFBPS suggestsa significantly higher slope e↵ect than both the regression and linear models.434.2. Lab DataFigure 4.16: Nonlinear regression model (green) and corresponding 95%confidence bands (red) plotted with the Van Wagner slope e↵ect model(black). The Van Wagner model of the CFFBPS suggests a much strongerslope e↵ect for this fuel type than what was observed.4.2.2 Backwards Rate of Spread (BROS)Similarly to how the analysis for the forward ROS of the lab fires wasconducted, we will analyse the BROS data first by using a GAM. Unlikeall other fuel types, this data set contained enough information for thisfunction to be applied. Again, we will run this function on the log scaleof the BROS data to reduce the amount of noise within the data. Whenapplying this function we obtain the following fitted model and confidenceintervals (Figure 4.17).444.2. Lab DataFigure 4.17: The results of fitting a GAM (black) to the BROS lab data onthe log scale. The black dotted lines are the 95% confidence intervals forthis model, and the red horizontal line at y = 0 indicates that there was noe↵ect of slope on the log(BROS).In Figure 4.17 we note that the horizontal red line passing throughs(pctSlope,1) = 0 stays within the the bounds of the 95% confidence in-tervals. This is an indication that there is no e↵ect of slope on the BROSwithin our data set as it tells us that the case of no slope e↵ect lies withinthe confidence intervals and cannot be rejected. Figure 4.18 highlights thediagnostic information about the fitting procedure and results produced byusing the gam.check() function.454.3. Influence of Slope Presence on Resulting Downhill ROSFigure 4.18: Diagnostic plots of the GAM which was fit to the log(BROS)data from the smouldering lab fires. The deviance residuals vs theoreticalquantiles plot (QQ plot) tells us that there were no significant deviationsfrom normality. The residual vs linear predictor tells us that the residualsremained fairly constant and near zero, and the histogram tells us that theseresiduals maintained normality.The results of the GAM suggest that the model was in fact an adequatefit, and accurately represents the data. From this we can again concludethat there did not appear to be an e↵ect of slope on the resulting BROS.4.3 Influence of Slope Presence on ResultingDownhill ROSEven though we saw in all previous sections that there was no indicationof slope-e↵ect on the resulting downhill back burning rate of spread (BROS),464.3. Influence of Slope Presence on Resulting Downhill ROSthat is not to say that the presence of a slope did not have an influence on thisrate. Although these two categories of influence may sound synonymous toone another, they are fundamentally di↵erent. No indication of slope e↵ectmeant that as the slope increased, there was no noticeable change to therespective BROS, whereas the influence of the presence of a slope refers toan overall di↵erence in a fire’s downhill spread rate.The lack of slope e↵ect on a fire’s BROS noticed across all data setssuggests that the downhill rate of spread is constant within its respectivefuel type. As the slope increased, the BROS did not increase or decrease.To investigate whether or not there was an influence due to presence of aslope, we must see if there is a significant di↵erence between the BROS ofthe slope fires and the ROS from the burns conducted on flat terrain. Inthe Table 4.9, we see the results when conducting a 2-sample t-test for eachfuel type comparing the means of the downhill BROS and the ROS of theburns conducted on a flat terrain. This t-test compared the forward ROS ofthe burns conducted on flat terrain and the down-slope BROS of all slopes.Fuel Type t-value p-value 95% Confidence IntervalC7 1.886 0.071 (-0.22,4.81)O1-B 2.518 0.023 (0.237 , 2.785)Lab burn 0.134 0.9 (-0.017,0.019)Table 4.9: Results of 2-sample t-test conducted for all fuel types comparingthe means of flat terrain ROS and sloping terrain BROS.We see in Table 4.9 there is a very strong similarity between the ROSand BROS within the smouldering lab fires, a strong di↵erence between theROS and BROS for the O1-B fires, and a moderate similarity between thetwo rates for the C7 fires. This may suggest that the presence of a slopehas an influence on the fire’s behaviour when burning down slope. Thesuspected way in which the slope may impact the fire behaviour is that thefire grows primarily by a conductive transfer of heat between the burningfire perimeter and the unburnt neighbouring fuel. In the field burns, theflames at the head of the fire pre-heat un-burnt neighbouring fuels throughradiant heat and convection, bringing unburnt fuels closer to their ignitionpoint as well as evaporating the moisture allowing them to ignite more easily.This pre-heating e↵ect increases as slope increases, and is also present whenburning on flat terrain. In the lab smoulder fires, heat transfers primarilythough conduction, meaning that the neighbouring fuels are heated oncedirectly exposed to the burning edge. This is due to the lack of flame which474.3. Influence of Slope Presence on Resulting Downhill ROSwould otherwise emit radiant heat onto the surrounding fuels.From Table 4.9 we can see that for the O1-B fuel type, there is a cleardi↵erence between the downhill BROS and the ROS of flat terrain burns withthe back burning spread rates down-slope being significantly slower. Thissuggests that the fire behaviour between these two were di↵erent. However,we cannot conclude that this e↵ect is present in the C7 fuel as the associatedp-value is not below the 0.05 threshold although it is close and more researchis required to determine whether or not slope influences the fire behaviourfor this fuel type. For the smouldering lab fires, it is clear that the flatand down-slope back burning spread rates are the same. This could be anindication that the method in which the burning perimeters travelled at thedownhill heel end of the fire was due to conductive heat transfer.48Chapter 5ConclusionThis thesis provides a deeper investigation into the relationship betweentopographical slope and a wildfire’s rate of spread using local Canadiandata. The use of the local BCWS data on the O1-B grassland fuel type forinvestigating ROS is in itself a new concept which has yet to be done upuntil this point. The idea of investigating whether or not slope is influentialon the back burn rate of spread is also a topic which has yet to be studiedcarefully. This study confirms the long standing notion that there is a pos-itive relationship between slope and a wildfire’s ROS. However, it suggeststhat the influence of slope varies across di↵erent fuel types, a result that hasnot been presented up until this point.From our data, it was not possible to confirm or deny the RSI parame-ters a, b, and c of Table 4.1. This was due to the high level of noise presentin the data which may have been a result in measuring error within thefield burn data collection. It could also be due to the lack on inclusion ofother influencing parameters that are not included in the CFFBPS model ofequations (2.1) and (2.3). Another source of error in the data (particularlythe BCWS data), was that parameters such as FFMC and ISI are obtainedfrom the nearest weather station and not at the specific location of the testburn. Not only that, these parameters given by the weather stations arecalculated using data gathered at noon, but projected for 1700hr, the peakburning hour of the day. This allows for inaccurate measurements specificfor the times of these burns that are primarily conducted early afternoon.Further research on a more carefully obtained data source would be neces-sary to suggest appropriate values for these parameters.The current slope e↵ect model suggested by Van Wagner [12] that isused within the CFFBPS [1] consistently failed to match the results wefound for our study. In the C7 fuel type this suggested slope e↵ect was anover estimate, and may lead to fire growth models such as the ones describedin the paper [7] highlighted in Section 2.3 to project a larger, faster travellingfire. The consequence of this may be that fire suppression crews choose towork the fire using di↵erent tactics that may take longer, and allow morearea to burn. However, when the Van Wagner model was compared to the49Chapter 5. Conclusionmodel obtained using the Canadian BCWS grassland data, this slope e↵ectwas an under-estimate for angles less than 18.75 degrees. This underestimatemay lead to fire crews being taken o↵ guard by the spread rate of a fire andcould put fire personnel in danger. For angles greater than 18.75, this modeldid prove to match the data used for this study. The overall take away fromthese findings is that the additional slope-e↵ect multiplier that is includedto calculate the overall rate of spread should have parameters which are fueltype dependent, similarly to what was seen for the RSI model of Equation(2.1).From the analysis, we can also conclude that the fire type also playsan important role in a fire’s resulting ROS. We saw this directly in theC7 data with the help of the GAM fitting with the inclusion of fire typeand noticed a moderate significance in the ROS between surface and crownfires. We also see this between the lab smoulder fires and the field testburns. The di↵erence between these is that the smoulder fires spread dueto a conduction heat transfer from the fire perimeter to the unburnt fuelwhereas the field data also has the influence of convection and radiant pre-heating of neighbouring fuels.Consistent across all of the three fuel types studied in this thesis (C7,O1-B, and the potassium nitrate coated wax sheets) was that a slope-e↵ecton the downhill BROS was not observed. That is to say, as the degree ofslope increased, a change in BROS was not noticed. This is a new result andcould present itself useful for potential future studies that may look at howa wildfire grows down slope. Although there was no evidence that changingthe degree of slope influenced the rate at which the downhill fire progressed,that is not to say that the presence of any slope had no resulting influencewhen compared to the trials conducted on a flat terrain. Looking at thedata presented in section 4.3 we do see that the downhill rates of spread areslower than burns conducted on flat terrain. This idea is supported throughtheory, as the idea behind why fire travels more rapidly uphill is because ofa stronger influence of radiant and convectional heat due to the property ofheat rising. This influence would be even less than the burns travelling ona flat terrain. This idea is also supported within our data as for the fieldburns, the average BROS was slower than the forward ROS on flat terrain,however, this was not seen in the lab data where the average forward ROSof flat terrain was similar to the average BROS of all burns across all elevenangles. Further research would be required to determine the cause of thisobservation, and the data would need to be fit to a flexible model that wouldincorporate factors such as wind and FFMC. The presence of this within ourdata may suggest however that a fire perimeter travelling downhill does so50Chapter 5. Conclusionthrough the direct contact of burning and unburnt fuel (conduction), as thatis the primary di↵erence between these burn styles in this study. This couldpossibly suggest that radiation and convection do not influence this rate theway it does on flat or uphill burns, however this subject would require moreresearch to confirm.51Bibliography[1] Canada. Forestry Canada Fire Danger Group. Development and Struc-ture of the Canadian Forest Fire Behavior Prediction System. TechnicalReport ST-X-3, Forestry Canada, Science and Sustainable DevelopmentDirectorate, Ottawa, 1992. ! pages iv, 2, 22, 29, 49[2] W.S. Cleveland. Robust locally weighted regression and smoothing scat-terplots. Journal of the American Statistical Association, 74(368):829–836, 1979. ! pages 9[3] W.S. Cleveland. LOWESS: A Program for Smoothing Scatterplotsby Robust Locally Weighted Regression. The American Statistician,35(1):54–54, 1981. ! pages 9[4] M.A. Finney. Farsite, fire area simulator–model development and eval-uation / mark a. finney. Technical report, U.S. Dept. of Agriculture,Forest Service, Rocky Mountain Research Station, 1998. ! pages 1, 7[5] J Johnston, M Wheatley, M Wooster, R Paugam, G. Davies, and K De-Boer. Flame-front rate of spread estimates for moderate scale exper-imental fires are strongly influenced by measurement approach. Fire(Basel, Switzerland), 1(1):16, 2018. ! pages 2, 8, 9, 16[6] D. J. McRae, J. Z. Jin, S. G. Conard, A. I. Sukhinin, G. A. Ivanova,and T. W. Blake. Infrared characterization of fine-scale variability inbehavior of boreal forest fires. Canadian journal of forest research,35(9):2194–2206, 2005. ! pages 9[7] M.A Parisien, D.A. Dawe, C Miller, C A. Stockdale, and O. B. Ar-mitage. Applications of simulation-based burn probability modelling:a review. International journal of wildland fire, 28(12):913–926, 2019.! pages 7, 8, 49[8] M.-A. Parisien and Northern Forestry Centre (Canada). Mapping wild-fire susceptibility with the Burn-P3 simulation model. Natural Resources52BibliographyCanada, Canadian Forest Service, Northern Forestry Centre, 2005. !pages 1[9] R. Paugam, M. J. Wooster, and G. Roberts. Use of handheld thermalimager data for airborne mapping of fire radiative power and energyand flame front rate of spread. IEEE transactions on geoscience andremote sensing, 51(6):3385–3399, 2013. ! pages 9[10] R Core Team. R: A Language and Environment for Statistical Com-puting. R Foundation for Statistical Computing, Vienna, Austria. !pages 3[11] R.W. Wotton B.M. Taylor S.W. Armitage O.B. Tymstra, C. Bryce.Development and Structure of Prometheus: the Canadian WildlandFire Growth Simulation Model. Technical report, 2010. ! pages 1[12] C.E. Van Wagner. E↵ect of slope on fire spread rate. Synthesis ofKnowledge of Extreme Fire Behavior, 1, 1977. ! pages vii, 5, 6, 21,26, 32, 33, 49[13] X.J. Wang, J.R.J. Thompson, W.J. Braun, and D.G. Woolford. Fittinga stochastic fire spread model to data. Advances in Statistical Clima-tology, Meteorology and Oceanography, 5(1):57–66, 2019. ! pages 6, 7,15[14] S.N Wood. Generalized Additive Models: An Introduction with R.Chapman and Hall/CRC, 2 edition, 2017. ! pages 22[15] S.N. Wood. Generalized Additive Models: An Introduction with R,Second Edition. CRC Press LLC, Philadelphia, PA, 2017. ! pages 1253Appendix54Appendix ATablesWeather Data From Field Burns O1-BSlope AverageRHSE RH AverageFFMCSDFFMCAveraceISISE ISI n0 27.59 13.18 92.2 2.7 11.21 3.9 105 27.51 7.5 88.92 5.79 8.54 5.06 810 36.1 0 91 0 6 0 1Table A.1: Averages and standard deviation (SD) of the weather data vari-ables which influence fire behaviour from the BCWS test burns conductedin the O1-B fuel type. The n column denotes the number of fires observed.Weather Data From Field Burns C7Slope AverageRHSE RH AverageFFMCSDFFMCAveraceISISE ISI n0 32.98 13.18 88.85 9.26 7.95 6.86 115 32.02 7.5 90.84 3.15 8.37 4.84 710 32.7 3.82 91.5 2.12 9 4.24 215 58 0 89.2 0 5.7 0 120 40 12.7 88.86 2.96 6.29 2.93 730 25 0 93 0 11 0 1Table A.2: Averages and standard errors of the weather data variables whichinfluence fire behaviour from the BCWS test burns conducted in the O1-Bfuel type. n denotes the number of fires observed.55
Thesis/Dissertation
eng
Mathematics
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Braun, John, 1963-
Tyson, Rebecca
A statistical investigation of how slope affects a wildfire's rate of spread
Text