As the climate changes, it is important to understand the effects on the environment. Changes in wildland fire risk are an important example. A stochastic lattice-based wildland fire spread model was proposed by Boychuk et al. (2007), followed by a more realistic variant (Braun and Woolford, 2013). Fitting such a model to data from remotely sensed images could be used to provide accurate fire spread risk maps, but an intermediate step on the path to that goal is to verify the model on data collected under experimentally controlled conditions. This paper presents the analysis of data from small-scale experimental fires that were digitally video-recorded. Data extraction and processing methods and issues are discussed, along with an estimation methodology that uses differential equations for the moments of certain statistics that can be derived from a sequential set of photographs from a fire. The interaction between model variability and raster resolution is discussed and an argument for partial validation of the model is provided. Visual diagnostics show that the model is doing well at capturing the distribution of key statistics recorded during observed fires.

The risk of large catastrophic wildland fires appears to be increasing in many countries as evidenced by recent high-profile wildfire events including the 2016 Fort McMurray fire in Canada and the “Black Saturday” bushfires of 2009 in Australia. Of heightened concern are climate change impacts on wildland fires. Weber and Stocks (1998) postulated that increasing temperatures could lead to an increased number of wildland fire ignitions, a longer fire season, and/or an increased number of days with severe fire weather. In regions of Canada, fire seasons are getting longer (Albert-Green et al., 2012) and fire risk has been shown to be increasing (Woolford et al., 2010, 2014). Annual area burned has increased and has been connected to human-induced climate change (Gillett et al., 2004). Studies that analyzed data output from climate model scenarios have suggested increased severity ratings (Flannigan and Van Wagner, 1991), area burned (Flannigan et al., 2005), ignitions (Wotton et al., 2010), and a longer fire season (Wotton and Flannigan, 1993).

Consequently, the development of accurate, spatially explicit fire spread models is of crucial importance for understanding aspects of fire behaviour and forecasting whether and how a wildland fire may spread. Many fire spread models are deterministic and although there have been some efforts to incorporate randomness into such models, there is also a strong need to develop stochastic fire spread models along with the statistical methodology for calibrating such models to data so that the uncertainty associated with where and when a fire might spread can be determined. A well-calibrated fire spread model can be used at the incident level for individual fire management or be coupled to fire occurrence and fire duration models in a simulation-based approach for longer-term strategic planning by wildland fire management agencies.

Taylor et al. (2013; Sect. 3) provided a detailed review of fire growth, discussing the physical process of fire growth, fire spread rate models, and spatially explicit fire growth models, including a discussion of deterministic fire spread models, such as Prometheus (Tymstra, 2005) and FARSITE (Finney, 2004), which play important roles in operational fire spread modelling by fire management agencies. Although these simulators are well established and used frequently in Canada, the United States, and in several other countries, their chief weakness is that they are not stochastic. Fire managers would benefit from probability maps to indicate where a currently burning fire may spread.

Burn-P3 (e.g. Parisien et al., 2005) used the deterministic Prometheus fire spread model in an ensemble-type simulation procedure which randomizes weather sequences in order to induce randomness to produce “burn probability maps”, typically a gridded map of the susceptibility of the landscape to be burned by wildfire over a large study area over the course of a year. We note that this kind of procedure may be more appropriate for studying fire risk on large temporal and spatial scales, such as producing an annual burn probability map for a region or district where wildland fires are managed. However, modelling at the incident level, namely quantifying whether or not a given fire may spread and where it may spread to along with estimating the uncertainty associated with the spread of a single fire, requires a different approach.

We note that deterministic models, such as Prometheus, can and are used at the incident level to model the spread of a single fire given local conditions. We also note that there has been some work to incorporate randomness into the Prometheus fire growth engine. For example, Garcia et al. (2008) attempted to introduce stochasticity to the Prometheus model via a block bootstrap procedure, and Han and Braun (2014) incorporated uncertainty through introducing an error component into the underlying model for rate of spread (ROS), as a parametric bootstrap. Nevertheless, much work remains to be done in order to make these procedures operational.

In the meantime, several other models have been considered by several other authors, including the stochastic lattice-spread model of Boychuk et al. (2007) that was studied by Braun and Woolford (2013), who also introduced an interesting variant of that lattice-spread model in their paper. This modified Boychuk model is used in our study herein where we address some statistical issues, studying the model from the point of view of data analysis not through operational implementation, which we illustrate through the analysis of some small experimental fires or “microfires”. We purposely restrict our analysis to such microfire data in order to study the model on data collected under controlled experimental conditions.

We study the modified version of the Boychuk et al. (2007) model as described
in Braun and Woolford (2013). In its simplest special case one assumes the
landscape to be flat with a fuel type and density that is homogeneous, the
weather conditions to be constant, and no wind. On this landscape we impose a
regular square

We note that the Boychuk model and its variant are much more general than described above. Nonhomogeneous conditions, due to changing weather, variations in fuel type, moisture content, and topography can be handled. These issues are described in detail in the Boychuk et al. (2007) paper.

The purpose of the current paper is to study the modified Boychuk model from the point of view of data analysis. In particular, we assume we have video data from a burning wildfire and investigate the following two questions. First, is it possible to fit the model to the data? And secondly, is it possible to carry out model assessment? We will argue in this paper that it is possible to show that the parameters for the simplest case of this model can be estimated from a sequence of pictures of a single fire. As in Zhang et al. (1992), we are studying the characteristics of the modified Boychuk model in the context of data collected on a tiny fire burned under very controlled conditions. Although Zhang et al. (1992) burned ordinary paper, we have chosen to use waxed paper in our experiment because it burns more cleanly. In an earlier paper (Braun and Woolford, 2013), we showed that the lattice grid cell size can be calibrated using given data, and we studied particular ways to assess the appropriateness of the model, but a general parameter estimation scheme was not proposed in that paper.

The rest of this paper proceeds as follows. In the next section, we describe our burning experiments and how we extract data from a video clip of a microfire. In Sect. 3, we describe a method to estimate the two parameters of the basic interacting particle model using data on numbers of burning grid sites and numbers of neighbouring unburned sites. We next summarize the results from the experiments and then provide specific tools to assess the fit of the model to the data. We conclude the paper with our observations and our ideas about future work on related problems.

We note that work has been carried out in incorporating fire rate of spread variability using variations in weather streams (e.g. Anderson et al., 2007); here, our focus is on what statisticians refer to as “unexplained” variation.

Data for testing the usefulness of the grid-based fire spread model were
obtained from a set of small-scale experimental fires. The experiments were
conducted under a fume hood in a laboratory, at a temperature of
20 ^{®} 600 camera,
which was suspended on a tripod approximately 48 cm above the paper, was
used to digitally record the experimental fire until most of the paper was
consumed.

A sequence of burn patterns on a sheet of wax paper observed at times of 1, 6, 11, 16, 21, 26, 31, 36, 41, and 46 s for the third microfire. Time increases from left to right, and then down.

The potassium nitrate treatment protocol was adopted for two reasons. First,
it prevented flaming and, in fact, induced smouldering combustion. This was
important from a lab safety standpoint. Second, flames tend to obscure the
pattern of combustion, and this leads to additional image processing issues
when extracting the data from the video footage. To further simplify the
image processing methodology, these experimental runs were conducted in
darkness. This resulted in video footage in which only burning sites were
visible; for example, see the screenshots from one of the experimental runs
that are displayed in Fig.

A total of 10 microfires were obtained, but only six were retained for further study, due to issues deemed to be unrelated to the study. These issues had to do with difficulties in processing the images from the video streams, due to the presence of fire-spotting, which produced small fires outside the perimeter of the original fire, and due to accidental changes in lighting, which interacted with the reflectivity of the waxed paper. In both cases, this rendered data that will be studied in the future, but which were not amenable to the very quick and simple data extraction and image segmentation procedures that will be described later. In future work, we plan to develop new image segmentation methodology to handle these situations, but as our real focus here is on applying a stochastic model to real data, we feel this is beyond the scope of the current paper.

The open-source program ffmpeg (Libav, 2010–2013) was used to freeze-frame
each movie at approximately half-second intervals to obtain clear image
captures with timestamps. These captured images were then saved as JPEG
files, readable in the R system (R Core Team, 2018). For illustrative
purposes, the images for the third experimental run are shown in
Fig.

Thresholded patterns for the third microfire. Unburned areas are coded as the light grey (green, if in colour), burning regions are at an intermediate grey shade (or red), and burnt regions are black.

This image segmentation problem was greatly simplified because of the use of
darkness in the experimental setup. Each grid cell is unburned until it burns
(and is lit up in the video footage) and is burnt out for all remaining time.
Therefore, the time(s) at which each grid cell burns can be identified from
the time series of the red, green, and blue (rgb) measurements. The series
corresponding to red is most useful for this purpose. Prior to burning, the
values are at or near 0; thus all images corresponding to these times can be
set to “green”. After burning, the values are again at or near 0; these can
be set to “black”. The values at the time(s) of burning can be set to
“red”. The resulting patterns, corresponding to the images in Fig.

Finally, the colour-coded images were converted to a numeric matrix
corresponding to the green, red, and black pixels of the image array. We
assigned the colour green to the value 0, red to the value 1, and black to the
value 2. The numbers of 0, 1, and 2 were counted, which gave the numbers of
unburned fuel sites, the number of burning sites, and the number of burnt-out sites.
In addition, the number of fuel sites neighbouring burning sites at each time
point were also counted. To illustrate, the counts for each of these
statistics during the first 30 s are listed in Table

Counts of the six statistics for the third microfire at a sequence of times (measured in seconds): numbers of burning sites, burnt-out points, and the four nearest-neighbour counts.

We now develop the methodology required to fit the grid-based fire spread
model to the data extracted from a sequence of images of a growing fire.
Referring to data such as in Table

These equations hold for both the original Boychuk model as well as the variant introduced by Braun and Woolford (2013). This result seems surprising since the Braun and Woolford variant is non-Markovian, while the Boychuk model is. However, it is important to note that the differential equations are for population level quantities. They are also only a partial description of the process dynamics. However, they contain enough process information to allow for construction of moment estimators for the process parameters as we now demonstrate.

The notation in the ensuing discussion can be simplified by making the
substitution

We can estimate the functions

Substituting these local linear kernel-smoothing-based estimates

An estimator for

It is important to note that the pixel size induced by the camera specifications has nothing to do with the size of the grid cells imposed on the lattice underlying the stochastic spread model. Thus, a scale parameter must be selected, which is used to expand or contract the default pixel sizes so that the spread model process can be used as a realistic approximation for the actual data. Specifically, we adjust the scale factor until the range of the observed data matches the range of the simulated data by using the estimated parameters.

In this paper, slightly different amounts of scaling are used for six
replicates of the dark smouldering experiment in order to match the pixel
size for the camera and the grid cell size on the lattice. The scale factors
for all replicates are listed in Table

Estimates of stochastic spread model parameters,

A simulation run of the model fit to the observed data from Figs. 1 and 2. Colour coding is the same as described in Fig. 2.

A second simulation run of the model fit to the observed data from Figs. 1 and 2.

As noted earlier, the sequence of fire images is displayed in Fig.

The remaining rows of Table

By simulating from the fitted model and re-estimating the parameters, it is
possible to assess bias and variability of the parameter estimates. We have
chosen to display the results of this assessment for the third microfire in
Fig.

The box plots also show that the distributions of the parameter estimates are approximately symmetric, and the amount of variability is of the same order as the bias. That is, if these simulations were viewed as a parametric bootstrap, the standard errors for the two estimators are approximately 0.01. Thus, the bias dominates the MSE, and the root-MSE is approximately 0.022 for both estimators.

Comparisons of parameter estimates between simulations and observed
data for the third microfire. The box plots represent samples of estimates
based on 100 simulations of the fitted stochastic spread model. The solid
horizontal line represents the estimated value of

Box plots for the other five fires (not shown) are quite similar to what appears
in Fig.

To assess the adequacy of the grid-based fire spread model for this
particular data set, we again simulated realizations of fire spread at the
estimated parameter values, using the chosen grid resolution. The results are
the sequence of pictures in Fig.

Such plots are limited in their usefulness. In this case, we can see that there are differences between the simulated pictures and the actual data, but it is difficult to tell if these differences are due to the variability we are trying to model or if these are failures of the model itself.

Comparisons between data from 100 simulation runs and observed data
from six microfires.

As another check on the appropriateness of the model, we can compare the
burning cell counts, burnt-out cell counts, and nearest-neighbour statistics
for the original data with simulated data from the fitted model (using data
from the third microfire only). Figure

What is evident from this set of plots is that the range and distribution of
simulated counts of burning sites (based on the observed data from one fire)
match the observed range and distribution of burning sites for other fires
very well. Except for a location shift in the distribution of simulated burnt-out sites and neighbourhood statistics, we also see similarities in the range
and distribution in these cases. The location shift is likely due to the
estimation bias discussed earlier. Note that by increasing both

These observations provide strong evidence that the model is doing well at capturing distributional behaviour in these dimensions. Further assessment of model goodness of fit is provided in the Supplement submitted along with this paper.

This work is part of an ongoing investigation into the suitability of a simple grid-based interacting particle system for stochastically modelling forest fire spread. Ultimately, we wish to fit such a model to sequences of satellite-based photographs of wildfires. Then, simulations of the model could be used to produce the maps of fire spread risk that are in demand by forest fire managers. Before this can happen, experiments under other conditions on slope and different kinds of fuel and wind conditions must be carried out. This paper represents the analysis of one such set of experiments, and the results appear promising.

What can be firmly concluded is that the parameters for the simplest case of the model can be estimated from a sequential set of photographs from a fire, using differential equations for the moments of certain statistics derivable from a video clip of a fire. A critical element of this estimation is that of scale. We have shown that the “natural” grid cell size can be determined, at least crudely, from a characteristic of the fire: the ratio of burnt-out area to the square of the burning area. Information about the scale is also likely related to the variability of the fire spread; this is an issue that can be addressed by studying an ensemble of experimental fires conducted under the same conditions. It should be noted that we have thresholded individual images; other methods taking account of the time sequence at each pixel (and at neighbouring pixels) may lead to more accurate counts of neighbourhood statistics and burning and burnt-out sites, given that flames and/or smoke cause distortions. The paper by Fang et al. (2007) indicates another possible approach that could be adopted.

We have developed some goodness-of-fit methods. A simple visual assessment based on comparing burn patterns simulated from the fitted model with the observed pattern is a useful, if limited, first step. This method of assessment gives some assurance that the model appears to reasonably fit the data. However, such a comparison is highly subjective and will not necessarily generalize to cases where, for example, the assumption of isotropy is invalid. The very nature of a stochastic model leads to different possible patterns under the same conditions. Hence, the following question arises: how different can the patterns be from the observed pattern before one might conclude that the model has failed?

What is needed, in general, is a metric for scoring spatial burn pattern maps that evolve over time in terms of their shape and boundary characteristics. In this paper, we have proposed the four nearest-neighbourhood statistics as belonging to such a set of measures. On the basis of an informal bootstrap procedure applied to these statistics, we have a fair degree of confidence that the model is capturing some of the stochastic behaviour of the actual fire.

For the purposes of reproducible research, we have uploaded the data (frozen video frames) and R code to the Open Science Framework repository. See Braun et al. (2018). The frozen video frames can be downloaded and unzipped, and then the file called MainDriver.R can be run in R to process the data as carried out in this paper.

The supplement related to this article is available online at:

DGW and WJB developed the modelling methodology and under their direction, XJW and JRJT designed and carried out the experiments. XJW and WJB designed and carried out the simulation experiments that were used to assess the model.

The authors declare that they have no conflict of interest.

The authors gratefully acknowledge support from the Natural Sciences and Engineering Research Council of Canada through individual Discovery Grants to W. John Braun and Douglas G. Woolford and the Canadian Statistical Sciences Institute through its Collaborative Research Team funding program. The authors are also grateful to Andrew Jirasek for facilitating access to his laboratory where the experiments were conducted.

This paper was edited by Christopher Paciorek and reviewed by two anonymous referees.