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\DOI{10.5802/crmeca.299}
\datereceived{2025-04-02}
\daterevised{2025-04-30}
\dateaccepted{2025-05-04}
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\dateposted{2025-05-20}
\begin{document}

\begin{noXML}

\CDRsetmeta{articletype}{research-article}

\shortrunauthors

\title{Critical threshold for crossing a firebreak: mathematical model
and fire experiments}

\alttitle{Seuil critique de franchissement d'un coupe-feu : mod\`{e}le
math\'{e}matique et exp\'{e}riences de feu} 

\author{\firstname{Akahoua David Vincent} \lastname{Brou}\CDRorcid{0000-0003-3597-0931}\IsCorresp}
\address{Universit\'{e} Jean Lorougnon Gu\'{e}d\'{e} de Daloa, UFR Environnement, Laboratoire des Sciences et Technologies de l'Environnement, BP 150 Daloa, C\^{o}te d'Ivoire}
\email[A. D. V. Brou]{brou.akahoua@ujlg.edu.ci}

\author{\firstname{Tionhonk\'{e}l\'{e} Drissa} \lastname{Soro}\CDRorcid{0000-0003-2195-6194}}
\address{Universit\'{e} F\'{e}lix Houphou\"{e}t Boigny d'Abidjan, UFR Biosciences, Laboratoire des Milieux Naturels et Conservation de la Biodiversit\'{e}, 22 BP 582 Abidjan 22, C\^{o}te d'Ivoire}
\email[T. D. Soro]{soro.tionhonkele@ufhb.edu.ci}

\author{\firstname{Kouassi\nobreakauthor Kouassi\nobreakauthor Serge}\nobreakauthor\lastname{Yanga}}
\address{Universit\'{e} F\'{e}lix Houphou\"{e}t Boigny d'Abidjan, UFR Math\'{e}matiques et Informatique, Laboratoire de M\'{e}canique et d'Informatique, 22 BP 582 Abidjan 22, C\^{o}te d'Ivoire}
\email[K. K. S. Yanga]{yanga.serge@ufhb.edu.ci}

\keywords{\kwd{Critical probability threshold}
\kwd{Firebreak crossing risk}
\kwd{Bushfire experiments}
\kwd{Firebreak effectiveness}
\kwd{Logistic regression}
\kwd{Risk matrix}}

\altkeywords{\kwd{Seuil de probabilit\'{e} critique}
\kwd{Probabilit\'{e} de franchissement}
\kwd{Exp\'{e}riences de feu de brousse}
\kwd{Coupe-feu}
\kwd{R\'{e}gression logistique}
\kwd{Matrice de risque}}

\begin{abstract}
Firebreaks are commonly used to limit the spread of wildfires towards
sensitive areas. However, the risk of extreme fires increases the
likelihood that suppression capacities will be exceeded.
This study proposes an effective design framework for
firebreaks, based on the estimation of the probability of crossing as a
function of fire behaviour and weather conditions. A risk matrix is
established to support decisions. The probability of crossing is
calculated using logistic regression, with data from several bushfire
experiments. The results show that a probability greater than 35\%
indicates a high risk of crossing, while a probability less than 25\%
indicates a moderate risk. Above 45\%, the risk becomes severe. By
estimating the risks, we can improve the design and maintenance of
firebreaks to make them safer and more effective.
\end{abstract}

\begin{altabstract}
Les coupe-feux sont couramment utilis\'{e}s pour limiter la progression
des feux de v\'{e}g\'{e}tation vers des zones sensibles. Cependant, le
risque de feux extr\^{e}mes augmente les chances de d\'{e}passement des
capacit\'{e}s de suppression. Cette \'{e}tude propose un cadre de
conception efficace pour les coupe-feux, bas\'{e} sur l'estimation de
la probabilit\'{e} de franchissement en fonction du comportement du feu
et des conditions m\'{e}t\'{e}orologiques. Une matrice de risque est
\'{e}tablie pour soutenir les d\'{e}cisions. La probabilit\'{e} de
franchissement est calcul\'{e}e via une r\'{e}gression logistique, avec
des donn\'{e}es provenant de plusieurs exp\'{e}riences de feu brousse.
Les r\'{e}sultats montrent qu'une probabilit\'{e} sup\'{e}rieure \`{a}
35 \% indique un risque \'{e}lev\'{e} de franchissement, tandis qu'une
probabilit\'{e} inf\'{e}rieure \`{a} 25 \% indique un risque
mod\'{e}r\'{e}. Au-del\`{a} de 45 \%, le risque devient tr\`{e}s
\'{e}lev\'{e}. L'estimation des risques permet d'am\'{e}liorer la
conception et l'entretien des coupe-feux pour les rendre plus efficaces
et s\^{u}rs.
\end{altabstract}

\maketitle

\end{noXML}

\section{Introduction}

{\vspace*{-2pt}}

{\advance\baselineskip by -.03pt

The rapid development of our societies is leading to an increase in the
number of forest-habitat interfaces \cite{1}, raising concerns about an
increase in the risk of vegetation fires, as the use of fire is a
common practice among the population. Recent studies have observed an
increase in the frequency and intensity of fires, which has been
attributed to climate change \cite{2,3,4,5,6}. 

Vegetation fire management policies are increasingly based on risk
assessment in order to~make appropriate decisions to reduce the risk of
fire outbreak and/or propagation \cite{7}. In general,~active
management of large fires is more costly than implementing risk
reduction techniques \cite{8,9,10}.


Fuel breaks are areas manipulated to reduce fuels and limit the spread
of vegetation fires \cite{11}. They are a common risk reduction method
that acts as a barrier to stop or slow the spread of fire \cite{12,13,14}.
The fuel breaks discussed in this study are those where fuel is
reduced to the point where the mineral soil is exposed. These breaks
are generally less than 20~m wide and are commonly referred to as
``firebreaks'' \cite{12,15}. It can be built before a fire to prevent
it or during a fire to contain it. Even if it does not stop the fire, a
firebreak can reduce the risk of a fire getting out of control because
the fuels provide the energy needed for the fire to spread \cite{16,17}.


The effectiveness of a firebreak depends on several parameters,
including its strategic location, adequate width and regular
maintenance. If the wind does not carry burning embers, or carries them
only for short distances, the firebreak can be effective in stopping
the fire. Otherwise, it will be ineffective \cite{18}. A quantitative
assessment of the risks associated with the effectiveness of a
firebreak can help decision-makers make rational choices about the
effective implementation of firebreaks. As risk is omnipresent,
decision-makers have indeed taken it into account in their decisions
\cite{19}.


Risk management allows the diffuse perception of vulnerability to be
replaced by rational knowledge, thus optimising the management of
risks. This knowledge is concretised by a global and coherent programme
based on tools and instruments for risk identification and management
\cite{20}.\looseness=-1

The aim of this study is to assess the risks associated with the
effectiveness of a firebreak. To do this, the probability of a
firebreak being breached is calculated using a logistic transport
equation. The probabilities allow uncertainty to be quantified and
expressed in terms of risk.

The safety assessment of vegetation fires recognises the importance of
the risk \cite{21,22,23,24,25}. However, the risk associated with
firebreaks is rarely assessed quantitatively. This study aims to fill
this gap.

Assessing the risk of crossing a firebreak is an essential element of
fire safety management. It allows the risks of fire spread to be
identified, the risks associated with the perceived danger to be
assessed and measures to eliminate or at least reduce these risks to be
implemented.


Evaluating the effectiveness of firebreaks based on the likelihood of
them being crossed has rarely been used to assess their effectiveness
\cite{26}. And where it is used, the critical probability threshold
is not specified.


Based on several experiments, Wilson~\cite{15} estimated
the probability of crossing a firebreak as a function of fireline
intensity, the presence of trees near the firebreak, and the width of
the firebreak. Gannon et~al.~\cite{27} analysed the
effectiveness factors of firebreaks in the national forests of southern
California using data on encounters between firebreaks and recent large
fires, using augmented regression. Ortega et~al.~\cite{26} 
quantitatively analysed the effectiveness of firebreaks during
wildfires in southern Spain. They developed models to predict the
potential effectiveness of firebreaks using machine learning
techniques, in particular logistic regression.

Although calculating the probability of a firebreak being crossed
allows a quantitative assessment of its effectiveness, existing models
do not specify the critical probability threshold above which the risk
of crossing is high.

}

The hypothesis of this study is that there is a probability threshold
above which the fire will cross the firebreak.

This work aims to answer the following question What is the probability
threshold above which the fire will cross the firebreak? Answering this
question requires three main steps. The first is to establish a model
that can be used to calculate this probability. This step is the
subject of the first section, which is devoted to the presentation of
this model. This model will then be applied to fire breaks from several
fire experiments carried out in the field. The third stage will consist
of analysing the results in order to establish a risk grid that will
allow decisions to be taken on the basis of the mathematical model.



\section{Methods}

\subsection{Study areas}

The tests on the effectiveness of the firebreaks were carried out
during experimental fires in the two main types of savanna in C\^{o}te
d'Ivoire: the Guinean savanna (in the Lamto Scientific Reserve) and the
Sudanese savanna (in the Como\'{e} National Park). In each of these
areas, the experiments were carried out between November and March
during the dry seasons of 
\mbox{2023--2024} and 
\mbox{2024--2025} in the shrub savanna,
where the herbaceous layer is dominant.


The Lamto Scientific Reserve is located in the transition zone between
forest and savanna in south-central C\^{o}te d'Ivoire, at the tip of ``V
Baoul\'{e}'', about 160~km north-west of Abidjan. It lies between
$6\text{\textdegree}10^{\prime}53^{\prime\prime}$ and 
$6\text{\textdegree}15^{\prime}20^{\prime\prime}$ north latitude and between
$4\text{\textdegree}58^{\prime}42^{\prime\prime}$ and 
$5\text{\textdegree}2^{\prime}53^{\prime\prime}$ west longitude in the
sub-prefecture of Taabo in the region of Agn\'{e}by-Tiassa. In Lamto,
the almost continuous herbaceous layer, 90\% of which is made up of
clumps of perennial grasses \cite{28}, facilitates the spread of
fire. These grasses are dominated by the Andropogoneae of the genera
\textit{Andropogon} and \textit{Hyparrhenia}~\cite{29}. In
November, the herbaceous layer reaches its maximum height, which can
exceed 3~m. Grasses and litter on the ground constitute the main
fuel during fires. The average grass load was 1.18~kg/m$^{2}$ with a
moisture content of 36\%. The average height of the herbaceous layer
was 0.6~m and the litter, which is dead vegetation, had a loading of
0.13~kg/m$^{2}$ with a moisture content of 4\%. 


Como\'{e} National Park is located in the Sudanese and sub-Sudanese
savanna zone in the north-east of C\^{o}te d'Ivoire, about 570~km from
Abidjan, between $8\text{\textdegree}30^{\prime}$ and 
$9\text{\textdegree}37^{\prime}$ north latitude and between
$3\text{\textdegree}7^{\prime}$ and $4\text{\textdegree}25^{\prime}$
west longitude. It extends over five prefectures, including Bouna,
Nassian and T\'{e}hini (in the Bounkani region), Dabakala (in the
Hambol region) and Kong (in the Tchologo region). The herbaceous layer,
which constitutes the bulk of the fuel during the fires, is mainly
composed of perennial grasses (Poaceae), including \textit{Panicum
phramitoides Stapf}, associated with \textit{Andropogoneae} such as
\textit{Hyparrhenia subplumosa Stapf}, \textit{H.\ smithiana} (Hook.f.) Stapf,
\textit{Andropogon canaliculatus} Schumach., \textit{A.~chinensis}
(Nees) Merr., etc~\cite{30}. The average grass load was 0.54~kg/m$^{2}$
with a moisture content of 12\%. The average height of the herbaceous
layer was 0.5~m and the litter had a load of 0.22~kg/m$^{2}$ and a
moisture content of 1\%.


\subsection{Measurement of parameters during experiments} \label{sec2.2}

To assess the risk of a firebreak being crossed, we used several sets
of data from field-scale experimental bushfires. Fifteen experiments
were conducted in Lamto and eighteen in Como\'{e}. The experimental
plots, which varied in size, were surrounded by firebreaks. The biomass
on the firebreaks was completely removed. An example of a firebreak
around experimental plots established during the experiments is shown
in Figure~\ref{fig1}.

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig1}Aerial photograph of an experimental plot in the Lamto Reserve
(A) fire starting on one of the plots; (B) 10-metre-wide firebreak
separating two plots.}
\end{figure}

Several parameters were measured before, during and after the ignition
of each experimental plot. 

\subsubsection{Fuel characteristics} 

The characteristics of the vegetation were measured, in particular the
mass load, the height of the herbaceous layer, the length and diameter
of the grass stems and the water content. Vegetation samples taken from
each plot were used to measure certain parameters in the field and
others were analysed in the laboratory.

\subsubsection{Weather parameters} 

Weather parameters are measured at the moment of ignition by a weather
station located two metres above the ground. The main parameters
measured are wind speed, wind direction, ambient temperature and
relative humidity.

\subsubsection{Fire behaviour}

For each plot, the fire was lit with palm leaves on the windward side
over a distance of 20 to 50~m. 

Six thermocouples distributed throughout each plot were used to measure
the temperature of the flames and to estimate the residence time of the
fire. The thermocouple sensors were placed between 0 and 30~cm from
the ground. The height of the flames is estimated using paper tape
placed on certain shrubs and on iron stakes placed at regular intervals
in each plot. Images of the flames are also associated to determine
their length. 

The rate of fire spread is estimated from aerial photographs taken by a
drone and from terrestrial observations.

\subsubsection{Behaviour of firebreaks}

In this study, the firebreaks considered are those that have been
confronted with head fires. A ``head fire'' is a fire that spreads in the
direction of the wind. A firebreak is considered effective if the fire
does not cross it. In this case the notation is ``firebreak crossed:
NO''; In the opposite case, the notation is ``firebreak crossed: YES''.
In general, firebreaks were effective against flank fires (fires that
spread laterally). However, in order not to distort the statistical
calculations, these firebreaks were not taken into account in this
study. The effectiveness of the different firebreaks used in this paper
was recorded during the experiments. Figure~\ref{fig2} shows a firebreak
approached by a head fire during the experiments in the Lamto Reserve. 

 
\begin{figure}
\includegraphics{fig02}
\caption{\label{fig2}Image of an experimental fire at Lamto approaching
a firebreak.}
\end{figure}

The width of the firebreaks varied from 1.5 to 10~m. There are 33
firebreaks that meet the above criteria. Figure~\ref{fig3} shows a firebreak
crossed by a wind-driven fire in Como\'{e} National Park.

\begin{figure}
\includegraphics{fig03}
\caption{\label{fig3}Wind-driven fire beyond the firebreak in Como\'{e}
National Park.}
\end{figure}

The probability of the firebreak being crossed is calculated for the 33
firebreaks. Data collected in the field is used to estimate this
probability. This includes the length of the flame, the wind speed, the
width of the firebreak and the burning time of the flame. The burning
time is the time during which the temperature in the litter is above
500~K. This level of temperature has been found to be representative
of the onset of flaming combustion \cite{36,37,38,39}. The model used to
calculate the probability of the firebreak being crossed is presented
in the following section.


\subsection{Logistic regression model}

This section presents the mathematical model used to estimate the
probability of a firebreak being crossed. This model is based on a
logistic regression technique trained on data generated by a simple
physical model of vegetation fire spread. The latter is based on the
solution of a heat transfer equation in a network of two-dimensional
cells representing the vegetation. The main modes of heat transfer are
considered, namely: radiation from the flame and embers, convection of
hot smoke and heat loss by radiation. Flame radiation is based on the
solid flame model and consists of assuming that the flame is a
geometric body that emits radiation \cite{31,32}. This radiation
model has been successfully used by several authors to estimate the
acceptable safety distance for firefighters \cite{33,35}. In this
study, the flame is a radiating cylinder whose emissivity is expressed
as a function of its length. The mean flame temperature used in the
physical fire propagation model is 1083~K~\cite{36}. The
herbaceous layer, which is the vegetation used in this article,
receives radiation from the flame above it with an absorption
coefficient of 0.6 \cite{37}.

Apart from the emissivity of the flame, which is expressed as a
function of the length of the flame, the last two parameters have the
greatest influence on the radiation model and must be taken into
account when using the model to fight bushfires. The effect of wind on
fire spread is also considered in the model. The wind tilts the flame
towards the healthy fuel, promoting heat transfer between the flame and
the vegetation. It also transports hot gases over horizontal distances,
supplying oxygen to the combustion zone. This process contributes to
convective heat transfer to unburned vegetation in front of the fire
line. The simplified physical model of fire spread has been optimised
and evaluated on different types of fires. Details of the model have
already been published and can be found in the following articles:
\cite{38} and \cite{39}. 

Using the physical model described above, a database of several hundred
series has been created to build the mathematical model. The purpose of
the latter is to calculate the probability of a firebreak being
breached based on the behaviour of the fire and meteorological data.
The most relevant variables in the mathematical model are, in order,
the length of the flames, the width of the firebreak, the wind speed
and the burning time of the flames. The probability of the firebreak
being breached, which is related to these four variables, is given by
the following equation:
{\begin{eqnarray}\label{eqn1}
P(\mbox{firebreak crossed})=\frac{\exp (\beta ^{T}x)}{1+\exp (\beta ^{T}x)}
\end{eqnarray}}\unskip
where the vector of coefficients $\beta ^{T}$ and that of variables
$x$ are given by the following relationship:
{\begin{eqnarray}\label{eqn2}
\beta ^{T}x=-1.8925+0.055U_{w}+0.9418L_{\mathrm{fl}}
-0.4469W_{\mathrm{fb}}+0.0114t_{c}
\end{eqnarray}}\unskip
with $U_{w}$ the wind speed at 2~m above the ground, $L_{\mathrm{fl}}$
the length of the flame, $W_{\mathrm{fb}}$ the width of the firebreak
and $t_{c}$ the flame burning time.

This mathematical model has been successfully compared with reference
models from the literature \cite{40}. The present study focuses on
some possible applications of the model. For example, the determination
of the critical probability threshold above which a firebreak is
breached. For this reason, readers wishing to learn more about the
physical and technical aspects of the model are encouraged to consult
the reference \cite{40}.

To determine the critical probability threshold for breaching a
firebreak, the probability of breaching the firebreaks in the
experiments described in the previous section is calculated. This is
done using the variables collected during the experiments. Equations
(\ref{eqn1}) and (\ref{eqn2}) are those used for the probability
calculations.



\subsection{Statistical analyses}

Statistical analyses were performed using the Python programming
language (Python Software Foundation, 2025) from the Scipy statistical
analysis library (Scipy 1.15.1: Fundamental algorithms for scientific
computing). To determine whether the data in Section~\ref{sec2.2} were
normally distributed, we used the Shapiro--Wilks test (Stats v.3.6.2).
If the data were normally distributed, we used the Student's $t$-test to
analyse the difference between groups. However, if the data were not
normally distributed, we used the Mann--Whitney test to analyse the
difference between groups in terms of firebreak effectiveness. The
significance level used for these tests is 0.05. 


\section{Results}
\subsection{Data characteristics}

The field measurements used in this study are summarised in
Table~\ref{tabI}. The width of the firebreaks varied from 1.5 to 10~m,
with an average of 5.11~m. The length of the flames varied from 1.1 to
6~m, with an average of 3~m. The wind speed varied from 0.12 to
5.4~m/s, with an average of 1.5~m/s. The estimated burning time was on
average 15~s, although it could exceed 25~s on some plots with a high
litter load.


\begin{table}
\caption{\label{tabI}Mean values of the variables measured during the
experiments}
\begin{tabular}{cccccc}
\thead
\parbox[t]{1.5cm}{\centering Fire number} &amp;
\parbox[t]{2.1cm}{\centering Wind speed (m/s)} &amp;
\parbox[t]{2.1cm}{\centering Flame length (m)} &amp;
\parbox[t]{2.1cm}{\centering Combustion time (s)} &amp; 
\parbox[t]{2cm}{\centering Firebreak width (m)} &amp;
\parbox[t]{2cm}{\centering Firebreak breached} \vspace*{2pt}\\
\endthead
{p1}  &amp;  1.86 &amp;  1.9 &amp;  15 &amp;  1.5 &amp;  NO \\
{p2}  &amp;  0.59 &amp;  1.1 &amp;  15 &amp;  1.5 &amp;  NO \\
{p3}  &amp;  0.12 &amp;  1.6 &amp;  15 &amp;  1.5 &amp;  NO \\
{p4}  &amp;  1.56 &amp;  1.2 &amp;  15 &amp;  1.5 &amp;  NO \\
{p5}  &amp;  0.79 &amp;  2.2 &amp;  15 &amp;  3   &amp;  NO \\
{p6}  &amp;  1.01 &amp;  1.9 &amp;  15 &amp;  3   &amp;  NO \\
{p7}  &amp;  2.05 &amp;  2   &amp;  15 &amp;  5   &amp;  NO \\
{p8}  &amp;  1.29 &amp;  3.2 &amp;  15 &amp;  5   &amp;  NO \\
{p9}  &amp;  0.41 &amp;  3.1 &amp;  15 &amp;  5   &amp;  NO \\
{p10} &amp;  0.55 &amp;  2.8 &amp;  15 &amp;  5   &amp;  NO \\
{p11} &amp;  0.21 &amp;  2.1 &amp;  15 &amp;  10  &amp;  NO \\ 
{p12} &amp;  0.9\0 &amp;  2.9 &amp;  15 &amp;  10  &amp;  NO \\ 
{p13} &amp;  2.35 &amp;  3.2 &amp;  15 &amp;  10  &amp;  NO \\ 
{p14} &amp;  1.5\0 &amp;  2.3 &amp;  15 &amp;  10  &amp;  NO \\ 
{p15} &amp;  1.52 &amp;  2.7 &amp;  15 &amp;  10  &amp;  NO \\ 
{p16} &amp;  1.27 &amp;  2.9 &amp;  15 &amp;  10  &amp;  NO \\ 
{p17} &amp;  1    &amp;  3.5 &amp;  15 &amp;  10  &amp;  NO \\ 
{p18} &amp;  1.27 &amp;  1.7 &amp;  15 &amp;  1.5 &amp;  YES \\ 
{p19} &amp;  2.1\0  &amp;  2.9 &amp;  15 &amp;  1.5 &amp;  YES \\ 
{p20} &amp;  1.87 &amp;  2.6 &amp;  15 &amp;  2   &amp;  YES \\ 
{P21} &amp;  1.6\0 &amp;  3   &amp;  15 &amp;  2   &amp;  YES \\ 
{p22} &amp;  0.43 &amp;  3.1 &amp;  15 &amp;  3   &amp;  YES \\ 
{p23} &amp;  1.23 &amp;  3.2 &amp;  15 &amp;  3   &amp;  YES \\ 
{p24} &amp;  1.17 &amp;  3.2 &amp;  15 &amp;  3   &amp;  YES \\ 
{p25} &amp;  2.29 &amp;  3.5 &amp;  15 &amp;  2.5 &amp;  YES \\ 
{p26} &amp;  1.55 &amp;  3.3 &amp;  15 &amp;  3   &amp;  YES \\ 
{p27} &amp;  0.72 &amp;  3.1 &amp;  15 &amp;  5   &amp;  YES \\ 
{p28} &amp;  1.13 &amp;  3.2 &amp;  15 &amp;  5   &amp;  YES \\ 
{p29} &amp;  1.33 &amp;  3.2 &amp;  15 &amp;  5   &amp;  YES \\ 
{p30} &amp;  5.4\0  &amp;  5.7 &amp;  15 &amp;  10  &amp;  YES \\ 
{P31} &amp;  2.4\0  &amp;  4.4 &amp;  15 &amp;  10  &amp;  YES \\ 
{p32} &amp;  3.57 &amp;  6   &amp;  15 &amp;  10  &amp;  YES \\ 
{p33} &amp;  2.15 &amp;  4.6 &amp;  15 &amp;  10  &amp;  YES  
\botline
\end{tabular}
\end{table}

\subsection{Behaviour of firebreaks}

The firebreaks studied were those reached by the head fire, i.e.\ the
fire spreading in the direction of the wind. A total of 33 firebreaks
were analysed, as shown in Table~\ref{tabI}. Of these 33 firebreaks, 16 were
crossed by the fire, i.e.\ 48\% of the total. The width of these
firebreaks varied from 1.5 to 10~m, with an average of 4.8~m.
Firebreaks that stopped the fire were between 1.5 and 10~m wide,
with an average width of 6~m and a median width of 5~m. Those
that were not crossed had an average width of 4.8~m and a median
width of 3~m. There were 17 firebreaks that were not crossed,
representing 52\% of the total. Wind speeds for crossed firebreaks
ranged from 0.43~m/s to 5.4~m/s, with a mean of 1.89~m/s and a median
of 1.57~m/s. For uncrossed firebreaks, it ranged from 0.12~m/s to 2.35
m/s, with a mean of 1.12~m/s and a median of 1.02~m/s. The length of
the flames for the breached firebreaks varied from 1.7 to 6~m,
with an average of 3.54~m and a median of 3.2~m. The length
of the flames for the unbroken firebreaks varied from 1.1~m to 3.5~m,
with a mean of 2.38~m and a median of 2.3~m. The Shapiro--Wilk test
showed that the data for each group were not normally distributed. The
Mann--Whitney test showed that there was no significant difference
between the data for the width of crossed and uncrossed firebreaks
(test statistic ${=}$ 158.0; $p = 0.425$). For wind speed, this test showed a
significant difference between the two groups (test statistic ${=}$ 75.5; $p
= 0.031$). The same is true for flame length, where the test showed a
significant difference between the two data groups (test statistic ${=}$
47.5; $p = 0.001$).

\begin{figure}
\includegraphics{fig04}
\caption{\label{fig3A}Graph  illustrating the probability of crossing
based on observed firebreak effectiveness and burning time.}
\end{figure}

\subsection{Probability of crossing a firebreak}

Experimental data on flame length, firebreak width, wind speed and
burning time were used to calculate the probability of firebreak
crossing using Equations~(\ref{eqn1}) and (\ref{eqn2}). As mentioned in
the previous section, the average burning time is 15~s, but it
can exceed 25~s under certain conditions. For this reason, the
probability of crossing has been presented for two burning times: 
15~s and 30~s (double the first). This probability is
shown in Figure~\ref{fig3A} as a function of the observed effectiveness
of the firebreaks. The behaviour of the firebreaks is indicated by
``yes'' if the fire crossed the firebreak and ``no'' if it did not. The
calculated probabilities vary between 0.01 and~0.7. For a burning time
of 15~s, 64\% of the 33 experimental firebreaks have a
probability of less than 0.3, 12\% have a probability between 0.3 and
0.45 and 24\% have a probability greater than 0.45. With a burning time
of 30~s, 48\% of the firebreaks have a probability of less than
0.3, 24\% have a probability between 0.3 and 0.45 and 27\% have a
probability greater than 0.45.

The Shapiro--Wilk test performed on both probabilities showed that the
data were normally distributed (test statistic ${=}$ 0.948 and 0.948; $p =
0.113$ and 0.118). Therefore, a Student's $t$-test was performed to test
for the existence of a difference between the two probability groups.
The results showed that the two probability groups were not
significantly different (test statistic ${=}$~${-}0.638$; 
\mbox{$p$-value} ${=}$ 0.525).

42\% of the firebreaks have a probability of crossing of less than 0.25
for a burning time of 15~s and 36\% for a burning time of 
30~s. Firebreaks with a probability of crossing between 0.25 and
0.35 represent 24\% of the total for a burning time of 15~s and
27\% for a burning time of 30~s. Firebreaks with a probability of
crossing greater than 0.35 represent 33\% for a 15~s burning time
and 36\% for a 30~s burning time. Of the firebreaks with a
probability of crossing greater than 0.35, 91\% were actually crossed
during the fire tests. More than 80\% of firebreaks with a probability
of crossing less than 0.25 were effective in the tests. Between 0.25
and 0.35, there are almost as many crossed firebreaks as not crossed
firebreaks. 



\section{Discussion}
\subsection{Experimental data}

The width of firebreaks that were breached by the fire was compared to
the width of firebreaks that were not breached. The same comparison was
made for flame length and wind speed. While statistical tests showed no
significant differences in firebreak width, they did show significant
differences in wind speed and flame length. Means and medians were
calculated for wind speed and flame length. The trend is the same for
mean and median. The average length of the flames in the uncrossed
firebreaks is 2.4~m, compared to 3.5~m in the crossed firebreaks. The
average length of the flames that did not cross the firebreaks is less
than that of the flames that did. This trend is also reflected in the
wind speed: the average wind speed for fires that crossed the
firebreaks is 2~m/s, compared to 1.1~m/s for fires that did not cross
the firebreaks. These results are consistent with the mathematical
model defined by Equations~(\ref{eqn1}) and (\ref{eqn2}). In fact, the
coefficients of the different explanatory variables indicate that the
probability of crossing the firebreak increases with the length of the
flame and/or the wind speed.


The behaviour of firebreaks described above is consistent with
observations made by several authors in the literature \cite{41}. For
example, Wilson~\cite{15} found that the length of the
flames had a significant effect on the effectiveness of the firebreak.
Other researchers have estimated that more intense fire behaviour
(longer flames) significantly reduces the effectiveness of the
firebreak \cite{11,42,27,26}. The effect of flame length on
firebreak effectiveness has led some researchers to size the effective
width of firebreaks according to flame length \cite{33,34,35,43,44}.


The data collected also shows that the width of the firebreak is an
essential parameter for its effectiveness. The wider the firebreak, the
more effective it is. However, the costs associated with constructing
and maintaining firebreaks, as well as the constraints associated with
forest protection, require an optimal design of effective firebreak
widths \cite{11}. The aim is to create firebreaks that are both
economical and effective. Decision support tools, such as the one
presented in this paper, can help to achieve this goal.

\subsection{Critical probability threshold}

The effectiveness of firebreaks observed in the experiments is
highlighted by the theoretical probability of their effectiveness (see
Figure~\ref{fig3A}). By examining Figure~\ref{fig3A}, we can see the existence of a
demarcation line between the probabilities of firebreaks being crossed
and those not being crossed by the fire. These results therefore reveal
the existence of a critical threshold of crossing probability. Indeed,
when the probability of crossing is greater than 0.35, the risk of the
firebreak being crossed is high. Conversely, when the probability of
crossing is less than 0.25, the risk of crossing the firebreak is
moderate. Consequently, the probability threshold that marks the
transition from effectiveness to ineffectiveness is around 0.3.


The critical probability threshold was determined by plotting the
probability of the firebreak being crossed against its observed
effectiveness. A chi-square test was performed to confirm this result.
The aim is to test the validity of the assertion that there is a
critical probability threshold above which the firebreak is crossed,
and that this threshold can be set at 0.3. To do this, we formulated
the null hypothesis H0 as follows H0 ``The probability threshold that
marks whether the firebreak is crossed or not is 0.3''. The alternative
hypothesis is given by H1 ``there is no probability threshold marking
the crossing or not of the firebreak''. The chi-square ($\chi ^{2}$)
statistic is given by Equation~(\ref{eqn3}),
{\begin{eqnarray}\label{eqn3}
\chi ^{2}=\sum _{i=1}^{33}\frac{(P_{\mathrm{obi}}-P_{\mathrm{thi}})^{2}}{P_{\mathrm{thi}}}
\end{eqnarray}}\unskip
where $P_{\mathrm{obi}}$ is the observed probability of crossing for
experiment $i$, calculated using Equations~(\ref{eqn1}) and
(\ref{eqn2}), and $P_{\mathrm{thi}}$ is the theoretical probability of
crossing for experiment $i$. This probability is expected to be less
than 0.3 when the firebreak has not been crossed and greater than 0.3
when the firebreak has been crossed. We set the theoretical probability
to $0.3\pm 0.1$. We will therefore carry out the following test:
$P_{\mathrm{thi}}=0.2$ when the firebreak is not crossed and
$P_{\mathrm{thi}}=0.4$ when the firebreak is crossed.

The calculated chi-squared statistic and the critical value of the
chi-squared at 1 degree of freedom, at the 0.05 confidence level, are
given in Table~\ref{tabII}. 
At this threshold, the H0 hypothesis is
accepted if the calculated chi-squared statistic is less than the
critical value of the chi-squared. Note that the critical value of the
chi-squared is given by the chi-squared distribution table.

\begin{table}
\caption{Chi-square test 
results\label{tabII}}
\begin{tabular}{lccc}
\thead
Chi-square test  &amp; Calculated &amp; Critical value &amp;  $p$-value  \\ 
\endthead
The chi-square statistic &amp; 2.256 &amp; 3.841 &amp; 0.05 
\botline 
\end{tabular}
\end{table}

The calculated chi-square statistic is 2.256, while the critical value
is 3.841. The calculated chi-square statistic is therefore well below
its critical value. Therefore, the H0 hypothesis is not rejected. This
means that the probability threshold of 0.3 corresponds to the limit
beyond which the firebreak is no longer effective in stopping the fire.
The statistical test therefore confirms the observation made after
examining Figure~\ref{fig3A}.

The fuel break examined in this article consists of removing all
vegetation down to the bare ground (or firebreak). In bushfire
management, this technique aims to prevent the spread of fire even in
the absence of suppression measures. The mathematical model developed
is valid for wind speeds below 10~m/s, which do not carry burning
debris over medium or long distances \cite{39}. The wind speeds used
in this study were moderate (maximum of 5.4~m/s). These are wind speeds
measured at 2~m above the ground. Taking these conditions into
account, the results obtained above allow us to establish a risk scale.
This is the risk of the firebreak, which is designed to stop the fire
in the absence of firefighting agents, being crossed. If the
probability of a firebreak being breached is less than 10\%, the risk
of it being breached is low. If the probability of a firebreak being
breached is less than 25\%, the risk of it being breached is moderate.
If the probability is between 35\% and 45\%, the risk becomes high, and
above 45\% it becomes severe. This risk hierarchy is illustrated in
Figure~\ref{fig4}, where the probability of a firebreak being breached
is shown as a function of the ratio of the width of the firebreak to
the length of the flames. The higher the ratio of the width of the
firebreak to the length of the flames, the more effective the
firebreak.

 
\begin{figure}
{\vspace*{-3pt}}
\includegraphics{fig05}
{\vspace*{-4pt}}
\caption{Risk mapping.\label{fig4}}
{\vspace*{-3pt}}
\end{figure}

The probability that the firebreak will be crossed depends on its
width, the length of the flames, the speed of the wind, and the burn
time. Of these variables, the width of the firebreak is the one over
which the firefighters have the most control. It is obvious that a very
wide firebreak has an almost zero risk of being crossed \cite{11,17}.
In fact, according to Equations~(\ref{eqn1}) and (\ref{eqn2}), as the
width of the firebreak tends towards infinity, the probability of it
being crossed tends towards 0. However, the cost of constructing and
maintaining firebreaks forces decision makers to optimise the design of
these devices in terms of width. The challenge for decision makers is
to create firebreaks with widths that are both economically viable and
effective in stopping fires~\cite{11}. In order to achieve this
objective, the present model can prove to be an indispensable tool for
the manager, providing him with quantitative analysis elements. 

Due to the low wind speeds recorded during the experiments, the
results presented in this article are valid for wind speeds below 6~m/s
at a height of 2~m above the ground. However, the mathematical model
and methodology can be applied to fire tests with higher wind speeds,
which will make it possible to determine critical thresholds for the
probability of crossing for higher winds.

Policies to combat the use of firebreaks before or during a fire will
only be effective if there is a real will to act in the face of risk.
In other words, for every risk of crossing a firebreak that is studied
and assessed, there must be a political will to decide and act
\cite{45}. The measures taken to manage the risks of crossing a
firebreak often have a significant cost. However, these costs must be
compared with the costs of the consequences of not addressing this
risk. The model presented here can be used as a decision-making tool to
select the optimal width of the firebreak. Our study revealed the
existence of a probability threshold above which the firebreak is
breached. The risk matrix established here is valid for moderate winds
(wind speed less than 6~m/s at 2~m above the ground). One of the
perspectives of this study is to conduct experiments with high wind
speeds and use the mathematical model given by Equations~(\ref{eqn1})
and (\ref{eqn2}) to establish a risk matrix for wind speeds around
10~m/s at 2~m above ground level and/or above 10~m/s. For high wind
speeds, the probability threshold established in this article may
change.


Estimating the risks associated with firebreaks can help to design and
maintain more effective and safer firebreaks. Risk management needs to
be an integral part of the work of decision-makers and firefighters.
This approach means that less time is spent fighting fires. According
to Barthelemy~\cite{20}, risk management makes it possible to
replace a diffuse perception of vulnerability with rational knowledge.
A company that manages its risks is a company that knows how to protect
itself from hazards over which it has little control. The debate on the
effectiveness of firebreaks is partly due to a lack of awareness of the
dangers associated with crossing them.

The management of vegetation fires has become more complex due to
climate change. Fire regimes have changed in size, intensity, severity
and frequency, far outstripping firefighting capabilities
\cite{46,47,48}. In this scenario of global change, risk management
plays a key role in protecting forests and habitats \cite{1,26}.


According to Syphard et~al.~\cite{1}, the predominant approach
to mitigating fire risk is the construction of firebreaks. The model
presented in this paper for the quantitative estimation of firebreak
effectiveness and the critical threshold of effectiveness can serve as
a risk management decision support tool in vegetation fire management
policies. These tools are becoming essential for the experienced
manager due to the extreme phenomena that are increasingly observed
during vegetation fires. In extreme fires, a firebreak designed without
evaluating the variables that influence its effectiveness is usually
crossed by the fire \cite{27,26}.

In the present study, the variable with the greatest influence on the
effectiveness of fuel cutting is the length of the flames. In the case
of intense fires, where the length of the flames exceeds the
predictions, it is necessary to combine extinguishing with fuel
breaking. In fact, the combination of extinguishing agents (flame
length reduction) and firebreaks effectively reduces the risk of
firebreaks being crossed. Some studies have shown that in many contexts
firebreaks alone do not stop fires \cite{18,27,26}. 


\section{Conclusion}

This study demonstrated the existence of a critical probability
threshold above which a firebreak is breached. The estimate of this
threshold is based on data from several field bushfire experiments.
Variables measured during the experiments, such as flame length, wind
speed, flame burning time and firebreak width, were used to calculate
this probability. By comparing the observed effectiveness of firebreaks
with the breaching probabilities calculated by our model, we were able
to establish a hierarchy of breaching risks. The results of this study
fill a gap in our knowledge of critical thresholds for firebreak breach
probability.

In addition, the study proposes a rule for the objective and effective
design and maintenance of firebreaks. These results provide
decision-makers with valuable tools for optimising the design of
firebreaks, taking into account economic constraints, and thus
maximising their effectiveness in preventing the spread of forest
fires.

The wind speeds used in this study are relatively low. The risk
hierarchy presented in this article is therefore valid for wind speeds
below 6~m/s at 2~m above the ground. The next step in this study is to
apply the same method to fire experiments with wind speeds close to or
above 10~m/s. One of the main limitations is the unpredictability of
the wind.


\section*{Declaration of interest}
The authors do not work for, advise, own shares in, or receive funds
from any organization that could benefit from this article, and have
declared no affiliations other than their research organizations.

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