%% sample.tex VERSION 2 (February 9th, 2007 - revised February 12th, 2007) %% sampledocument.tex VERSION 3 (April 27th, 2007) %% A sample full paper template you modify and save as fullpaperdocument.tex %% to produce your camera-ready full paper %% +-------------+ %% | PAPER TITLE | %% +-------------+ %\title[ short-title ]{ full-title } \title% [The Random Spread Model]% short title (maximum of 50 characters) {The Random Spread Model}% full title %% +---------+ %% | AUTHORS | %% +---------+ \begin{Authors} \Author {Jean Serra} \Affil[ESIEE] {Laboratoire L2ASI, ESIEE, France \\ \email{serraj@esiee.fr}} \end{Authors} %% +-----------------------+ %% | ABSTRACT and KEYWORDS | %% +-----------------------+ %% Abstract and keywords are written inside two environments, %% the "Abstract" amd the "Keywords" environments, respectively. %% Note the uppercase "A" and "K". %% Keywords are separated with comma. The last keyword ends %% with a period. \begin{Abstract} The paper proposes the new stochastic model of a Random Spread for describing the spatial propagation of sequential events such as forest fires. A Random Spread is double Markov chain, whose each step is a (random) set operator $\beta $ combining a Cox process with a Boolean random closed set. Under iteration, operator $\beta $ provides the time evolution of the Random Spread, which turns out to be a birth-and-death process. Average sizes, and the probabilities of extinction are derived. The random spread model was applied to the analysis of the fires that occurred in the State of Selangor (Malaysia) from 2000 to 2004. It was able to predict all places where burnt scars actually occurred, which is a strongly significant verification. \end{Abstract} \begin{Keywords} \Index{Random Spread}, \Index{Boolean RACS}, \Index{Markov chain}, \Index{birth-and-death}, \Index{random closed sets}, \Index{forest fires}, \Index{simulations}, \Index{mathematical morphology}. \end{Keywords} %% +----------+ %% | SECTIONS | %% +----------+ \section{Introduction} In a large number of wild forests, such as typically in South East Asia, forest fires propagate less under the action of the wind, as in Mediterrenan countries, than under almost isotropic causes. For instance, the fires that occurred in the state of Selangor (Malaysia, see Figure~\ref{\fullpaperid:init}) from 2000 to 2004 seem to evolve randomly: the initial seats vanish sometimes, but may also give birth to new seats at some distance. The foresters use to describe the fire progress by means two key maps \cite{Serra2:HAI05}, namely the daily \textit{spread rate\ }of the fire,\ i.e. the radius $r$\ of the daily circular propagation of the fire, and the\textit{\ fuel consumption } $f_{w}$% , (a weighted version of the vegetation map). Both are depicted in Figure~\ref{\fullpaperid:init}. Clearly, a straightforward use of such key maps does lead to a pertinent description. By starting from \textit{any} \ point seat, one always arrives to burn the whole country in a finite time, under iteration, as both maps are positive. Indeed, one must use the rate information in some restrictive way, to be able to reach actual events. It is exactly the purpose of the present paper. %\FRAME{ftbpFU}{5.6982in}{2.2468in}{0pt}{\Qcb{a) State of Selangor in %Malaysia; b)\textit{\ }map of the spread rate, i.e.of the radius %$r$\ of the daily circular propagation of the fire; c) map $f_{w}$ %of the fuel %consumption.}}{\Qlb{init}}{inittriple.bmp}{\special{language %"Scientific Word";type "GRAPHIC";maintain-aspect-ratio TRUE;display %"USEDEF";valid_file "F";width 5.6982in;height 2.2468in;depth %0pt;original-width 13.2083in;original-height 5.1802in;cropleft %"0";croptop "1";cropright "1";cropbottom "0";filename %'inittriple.bmp';file-properties "XNPEU";}} \begin{figure}% [!tb] \begin{center} \includegraphics[width=\linewidth]{\fullpaperpdirectory/images/inittriple.pdf} \end{center} \caption{a) State of Selangor in Malaysia; b)\textit{\ }map of the spread rate, i.e. of the radius $r$\ of the daily circular propagation of the fire; c) map $f_{w}$ of the fuel consumption.} \label{\fullpaperid:init} \end{figure} \section{Stochastic models for growth} One can classify the proposed model as a discrete branching process which generalises the Galton-Watson process, since it involves a location of the new generation in space. Since C.\ J.\ Preston's pioneer work on\textit{\ spatial birth-and-death} \cite{Serra2:PRE77}, one finds in the literature several point stochastic processes for describing joint evolutions in space and time. Their characteristic functionals $Q(K)$ are generally inaccessible, but they yield significant simulations. A second class is the concern of "thick" structures, i.e. which do not reduce to isolated points. A particular case can be found in \cite{Serra2:Serra82:Book}, p.562, under the name of hierarchical RACS, with several variants such as the following: \textit{% the RACS at time }$t$\textit{\ is }$X_{t}$\textit{, and }$X_{t+dt}$\textit{\ is generated by adding to }$X_{t}$\textit{\ any boolean grain which occurs during }$[t,t+dt]$\textit{\ and whose center hits }$X_{t}$. In spite of its outward simplicity, this variant is not tractable, i.e. one cannot express the functional $Q_{t}$ of $X_{t}$\ by means of the initial conditions, as proved by D.Jeulin in \cite{Serra2:JEU91}. However, a number of phenomena, including forest fires, follows the same type of behavior. Each time that in the mineral, vegetal or animal worlds, seeds move and then develop a new colony,\ they involve some random sequential growth. But how to model it by tractable RACS? The trouble with the hierarchical RACS comes from that their evolution between steps $n$ and $% n+1$ refers to the \textit{whole past}, from $0$ to $n$. If we relax this condition, can we reach more tractable growth RACS? In addition, we must take into account that the space parameters which govern the evolution laws (e.g. the fuel amount for forest fires) usually vary from place to place, so that the new model should not be\textit{\ a priori} translation invariant, but accept some imposed heterogeneity. That are the questions we consider in this paper, by proposing the \textit{% Random Spread }RACS. \section{Reminders} The random spread model in presented in the framework of the Euclidean space $\mathbb{R}^{d}$ of dimension $d$. Denote by the $\mathcal{P}=\mathcal{P}(% \mathbb{R}^{d})$ (resp. $\mathcal{F}$, $\mathcal{K}$) the family of all sets (resp. closed sets, compact sets) of $\mathbb{R}^{d}$. Symbol $\mathcal{S}$ stands for the singletons of $\mathcal{P}(\mathbb{R}^{d})$ and the same symbol, e.g. $x$, is used for the points of $\mathbb{R}^{d}$\ and for the elements of $\mathcal{S}$. \paragraph{Set dilation} A \emph{structuring function }is an arbitrary mapping $x\longmapsto \delta (x)$ from $\mathcal{S}(\mathbb{R}^{d})$ into $\mathcal{P}(\mathbb{R}^{d})$. It extends into the \textit{dilation} $\delta $ from $\mathcal{P}(\mathbb{R}% ^{d})$ into itself \cite{Serra2:Serra88:Book}, \cite{Serra2:Heijmans90:Ronse}, \cite{Serra2:KIS01} by the relation \begin{equation} \delta (X)=\cup \left\{ \delta (x)\,\,,\,\,x\in \mathcal{S}\right\} \qquad \qquad X\in \mathcal{P}(\mathbb{R}^{d}). \label{\fullpaperid:Random dil 0} \end{equation}% In the following, all structuring functions are supposed to be \textit{% compact}, i.e. \begin{definition} \label{\fullpaperid:regular dil}A structuring function $\delta :\mathcal{S}(\mathbb{R}% ^{d})\longmapsto \mathcal{P}(\mathbb{R}^{d})$ is said to be \emph{compact} when i) it is upper semi continuous from $\mathcal{S}$ into $\mathcal{K}$, ii) the union $\cup \{\delta _{-x}(x),x\in \mathbb{R}^{d}\}$ has a compact closure $\delta $% \begin{equation} \overline{\delta _{0}}=\overline{\cup \{\delta _{-x}(x),x\in \mathbb{R}^{d}\}% } \label{\fullpaperid:Random dil 37} \end{equation} By extension, the associated dilation $\delta :\mathcal{P}\longmapsto \mathcal{P}$ $\ $is also said to be compact. \end{definition} The \textit{reciprocal duality} between dilations plays an important role below. With each structuring function $\delta $, associate the \textit{% reciprocal structuring function} $\zeta $ by writing \begin{equation} y\in \zeta (x)\qquad \mathrm{if\,\ and}\text{ }\mathrm{only\,\ if}\qquad x\in \delta (y)\qquad \qquad x,y\in E. \label{\fullpaperid:Random dil 1} \end{equation} The algorithm that expresses $\zeta $ in function of $\delta $ is therefore% \begin{equation} \zeta (x)=\cup \{y:x\in \delta (y)\}. \label{\fullpaperid:Random dil 38} \end{equation} \paragraph{ Random Closed Sets (Euclidean case)} The results which follow, on RACS\ and Boolean RACS, are basically due to G.\ Matheron \cite{Serra2:Matheron75:RS}. Instructive introductions to RACS\ may be found in \cite{Serra2:MOD04},\ or \cite{Serra2:STO05}. Given an element $K\in \mathcal{K}$% , consider the class \begin{equation*} \mathcal{F}^{K}=\left\{ F:F\in \mathcal{F},\ F\cap K=\emptyset \right\} \end{equation*}% of all closed sets that miss $K$.\ As $K$ spans the family $\mathcal{K}$, the classes $\left\{ \mathcal{F}^{K},\,K\in \mathcal{K}\right\} $ are sufficient to generate the $\sigma $-algebra.\ Moreover, as set $\mathcal{F}$ is compact for the hit-or-miss topology, there exist probabilities, $\Pr $ say, on $\sigma _{f}$, and each triplet $(\mathcal{F},\sigma _{f},\Pr )$ defines a random closed set or RACS. \begin{theorem} \label{\fullpaperid:Th rand dil 2}The distribution of a RACS $X$ is \emph{uniquely} defined by the datum of the probabilities \begin{equation} Q(K)=\Pr \left\{ K\subseteq X^{c}\right\} \label{\fullpaperid:Random dil 5} \end{equation}% as $K$ spans the class $\mathcal{K}$ of the compact sets of $\mathbb{R}^{d}$% .\ Conversely, a family $\left\{ Q(K),\,K\in \mathcal{K}\right\} $ defines a (necessarily unique) RACS if and only if $1-Q(K)=T(K)$ is an alternating Choquet capacity of infinite order such that $0\leq T\leq 1$ and $% T(\emptyset )=0$.\ \ The mapping $Q:\mathcal{K\rightarrow }[0,1]$ is called the \emph{% characteristic functional }of the RACS. \end{theorem} The characteristic functional\emph{\ }$Q$ plays w.r.t. a random closed set$% \emph{\ X}$ the same role as the distribution function for a random variable $x$. If $\left\{ X_{i},i\in I\right\} $ stands for all possible realizations of the RACS $X$, and if $\psi :\mathcal{F}\longmapsto \mathcal{F}$ is semi-continuous, hence measurable, then the family $\left\{ \psi (X_{i}),\,i\in I\right\} $ characterizes in turn all realizations of a RACS, denoted by $\psi (X)$. This basic result allows us to play with RACS just as with deterministic sets, to intersect them, to dilate them, etc.. \paragraph{Boolean RACS\label{\fullpaperid:booleanmodel}} The Boolean RACS, is very popular and led to many variants (e.g., \cite{Serra2:Serra82:Book}, \cite{Serra2:AYA89}, \cite{Serra2:JEU00}, \cite{Serra2:STO05}). In the present study, we specify it as follows. Consider the two primitives of i/ a Poisson process $\mathcal{J}(\theta )$, whose intensity measure $\theta $ is upperbounded,\ i.e. $\theta (dx)\leq \overline{\theta }.dx$ with $% \overline{\theta }<\infty $ ii/ a compact, and deterministic, structuring function $\delta :\mathcal{S}(% \mathbb{R}^{d})\rightarrow \mathcal{K}(\mathbb{R}^{d})$ called "primary grain". \bigskip The Boolean RACS $X$ is constructed in two steps. First, take a realization $% J$ of Poisson points, which provides the set of points $x_{j},$ $x_{j}\in J$% . Second, take the union $X$ of all primary grains whose centers belong to the Poisson realization, i.e. \begin{equation*} X=\cup \{\delta _{x_{j}}\,,x_{j}\in J\} \end{equation*} This union generates a realization of the Boolean RACS $X$. The characteristic functional $Q$ of RACS $X$ derives easily form the above definition \cite{Serra2:Matheron75:RS}, and equals \begin{equation} Q(K)=\exp -\int \theta (dz)1_{\delta (z)\cap K}=\exp -\int \theta (dz)1_{z\cap \zeta (K)}=\exp -\theta \lbrack \zeta (K)]. \label{\fullpaperid:Lantu 2} \end{equation}% where $\zeta $ is the dilation reciprocal of $\delta $. If we restrict the previous Poisson points to those which occur in a given compact set $X_{0}$, then this comes back to change the intensity $\theta (dx)$ into $\theta ^{\ast }(dx)=$ $\theta (dx).1_{X_{0}}(x)$, the Boolean structure being preserved, so that% \begin{equation} \Pr \{K\subseteq X_{1}^{c}\}=Q_{1}(K)=e^{-\theta ^{\ast }[\zeta (K)]}=e^{-\theta \lbrack \zeta (K)\cap X_{0}]}. \label{\fullpaperid:Lantu 8} \end{equation} \section{Random Spreads\label{\fullpaperid:Random spread}} \subsection{Definition} The Random Spread model generalizes Matheron's Boolean RACS\ by introducing a genetic dimension, namely the successive steps, according to which the $% (n+1)^{th}$ Boolean RACS depends on the realization of the $n^{th}$ one. Consider an initial random seat $I_{0}$ made of an a.s. locally finite number of initial point seats in $\mathbb{R}^{d}$. The fire evolution from $% I_{0}$ is the concern, on the one hand, of the fire the initial seats provoke, or\textit{\ fire spread} $X_{1}=\delta (I_{0})$, and on the other hand of the generation of subsequent \textit{seats spread} $I_{1}=\beta (I_{0})$. These secondary seats will develop new fires in turn. Both aspects refer to some compact dilation $\delta $. We propose to model the seats spread $\beta (I_{0})$ by picking out, randomly, a few points in each dilate $\delta (x_{i})$, for all points $x_{i}\in I_{0}$. The double\ spread process is then written for the spread of \begin{eqnarray} \text{the fire }X_{1}(I_{0}) &=&\delta (I_{0})={\LARGE \cup }% \{\delta (x_{i}),\text{ \ }x_{i}\in I_{0}\} \label{\fullpaperid:Lantu 13} \\ \text{the seats }I_{1}(I_{0}) &=&\beta (I_{0})={\LARGE \cup }% \{\left( \delta (x_{i})\cap J_{i}\right) ,\text{ \ \ }x_{i}\in I_{0},\text{ \ }J_{i}\in \mathcal{J}(\theta )\} \label{\fullpaperid:Lantu 11} \end{eqnarray}% where a different realization\ $J_{i}$ of the Poisson points process $% \mathcal{J}(\theta )$ is associated with each point $x_{i}$. Therefore, each point $x_{i}$ of the set $I_{0}$ induces a bunch of seats $\delta (x_{i})\cap J_{i}$ independent of the others. These two equations mean that though the fire from a seat $x$ does burn the zone $\delta (x)$ around $x$, only a few points of the scar $\delta (x)$ remain active seats for the next step. \bigskip Under iteration, Rel.(\ref{\fullpaperid:Lantu 13}) and (\ref{\fullpaperid:Lantu 11}) become% %\begin{eqnarray} \[ X_{2}(I_{0}) =\delta (I_{1})=\cup \{\delta (y_{k}),\text{\ }y_{k}\in I_{1}\}= \cup \{\delta (y_{k}),\text{ \ \ }y_{k}\in \delta (x_{i})\cap J_{i} \text{ };\text{ \ }x_{i}\in I_{0}\} \label{\fullpaperid:Lantu 25} \\ \] \[ I_{2}(I_{0}) = \beta (I_{1})=\beta \lbrack \beta (I_{0})]={\LARGE \cup }_{i} {\large \{}{\LARGE \cup }_{k}[\delta \left( \delta (x_{i})\cap J_{i}\right) ]\cap J_{k}{\large \}},\text{\ \ }x_{i}\in I_{0}\}. \notag \] %\end{eqnarray} \bigskip Figure~\ref{\fullpaperid:propagations4} depicts the first three steps of a random spread, for which: %\FRAME{ftbpFU}{3.3278in}{1.7893in}{0pt}{\Qcb{Three generations of %fires stemming from point $x_{0}=$ $I_{0}$. Note the genertation of %new %fires in already burnt areas.}}{\Qlb{propagations4}}{propagations4.bmp}{% %\special{language "Scientific Word";type "GRAPHIC";display %"USEDEF";valid_file "F";width 3.3278in;height 1.7893in;depth %0pt;original-width 7.9442in;original-height 3.6668in;cropleft %"0";croptop "1";cropright "1";cropbottom "0";filename %'propagations4.bmp';file-properties "XNPEU";}} \begin{figure}% [!tb] \begin{center} \includegraphics[width=3.8597in]{\fullpaperpdirectory/images/propagations4.png} \end{center} \caption{Three generations of fires stemming from point $x_{0}=$ $I_{0}$. Note the generation of new fires in already burnt areas.} \label{\fullpaperid:propagations4} \end{figure} \begin{itemize} \item the initial seat $I_{0}$ is the point $x_{0}$,\ and the first spread, or front, the dark grey disk $X_{1}(I_{0})=\delta (x_{0})$; \item then two Poisson points, namely\ $y_{1}$ and $y_{2}$, fall in $\delta (x_{0})$. They generate% \begin{eqnarray*} \text{ }X_{2}(x_{0}) &=&\delta (y_{1})\cup \delta (y_{2})=\delta (I_{1})% \text{, in medium grey,} \\ \text{and }I_{1}(x_{0}) &=&\{y_{1}\}\cup \{y_{2}\}; \end{eqnarray*} \item then again a new Poisson realization generates one point, $z_{1}$ in $% \delta (y_{1})$, and another Poisson realization the three points $% z_{2,1},z_{2,2},$ and $z_{2,3}$ in $\delta (y_{2})$, hence% \begin{eqnarray*} X_{3}(x_{0}) &=&\delta (z_{1}){\LARGE \cup }[\delta (z_{2,1})\cup \delta (z_{2,2})\cup \delta (z_{2,3})]=\delta (I_{2})\text{, in light grey.} \\ \text{and }I_{2}(x_{0}) &=&\{z_{1}\}{\LARGE \cup }[\{z_{2,1}\}\cup \{z_{2,2}\}\cup \{z_{2,3}\}]. \end{eqnarray*} \end{itemize} The doublet spread $(X_{n},I_{n})$ of order $n$, is the $n^{th}$ element of the chain depicted in Figure~\ref{\fullpaperid:chain}, which is of Markov type: as soon as $% I_{n}$ is known, the previous links do not serve in the creation of link $% (X_{n+1},I_{n+1})$. This Markov type assumption means that the fire of tomorrow can only be caused by points seats stemming from the zone which burns today. What burnt yesterday, before yesterday, etc., has no longer importance. In this space-time process, the successive sets $% X_{1},X_{2}...X_{n}$ are at least as descriptive as the seats $% I_{1},I_{2}...I_{n}$ themselves, because they show the extension of the fire front through the time steps. The $(n+1)$ step is obtained by the two induction relations% \begin{eqnarray} X_{n+1}(I_{0}) &=&\delta \lbrack I_{n}(I_{0})] \label{\fullpaperid:Random dil 25} \\ I_{n+1}(I_{0}) &=&\beta \lbrack I_{n}(I_{0})] \notag \\ I_{n+1}(I_{0}) &=&{\LARGE \cup }\{\left( \delta (x_{i,n})\cap J_{i,n}\right) ,\text{ \ \ }x_{i,n}\in I_{n},\text{ \ }% J_{i,n}\in \mathcal{J}(\theta )\} \label{\fullpaperid:Random dil 26} \end{eqnarray} We observe that, in Rel.(\ref{\fullpaperid:Lantu 11}), and the ulterior ones, each small zone $dz$ intervenes as many times as $dz$ belongs to different $\delta (x_{i})$.\ Therefore, the measure% \begin{equation} \tau _{n+1}(dz)=\theta (dz)\sum \{1_{\delta (x_{i,n})}(z),\text{ \ }% x_{i,n}\in I_{n}\} \label{\fullpaperid:Random dil 24} \end{equation}% turns out to be a realization of the Cox process of intensity $\tau _{n+1}$, if $I_{n+1}$ is finite. Now, are we sure that all these handlings and derivations really lead to a RACS, in Matheron-Kendall's sense, i.e. to something we can characterize by a functional $Q(K)$?\ Are we sure, for example, that the intensity $\tau _{n+1}$ is always a.s.finite? The answer is positive, as soon as the structuring function $\delta $ is \textit{compact% }, in the sense of Definition \ref{\fullpaperid:regular dil}. Then one can state the following \cite{Serra2:SER07} %\FRAME{ftbpFU}{4.4529in}{0.6538in}{0pt}{\Qcb{A chain of random spreads.}}{% %\Qlb{chain}}{chain.bmp}{\special{language "Scientific Word";type %"GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file %"F";width 4.4529in;height 0.6538in;depth 0pt;original-width %9.903in;original-height 1.4304in;cropleft "0";croptop "1";cropright %"1";cropbottom "0";filename 'chain.bmp';file-properties "XNPEU";}} \begin{figure}% [!tb] \begin{center} \includegraphics[width=4.4529in]{\fullpaperpdirectory/images/chain.pdf} \end{center} \caption{The Markov chain of the spreads.} \label{\fullpaperid:chain} \end{figure} \begin{proposition} Let $I_{0}$ be a a.s. locally finite set of points, let $\theta $ be a Poisson intensity, and $\delta :\mathbb{R}^{d}\rightarrow \mathcal{K}(% \mathbb{R}^{d})$ be a structuring function. If function $\delta $ is compact, then both families $\{X_{n}(I_{0}),n>0\}$ and $\{I_{n}(I_{0}),n>0\}$ of fire spreads and seats spreads are RACS. \end{proposition} \bigskip Since the two spreads $I_{n}$ and $X_{n}$ are RACS, we shall characterize them by their functionals $Q_{n}(K)$. In the "Boolean-Cox RACS" $X_{n}$, the primary grain \textit{only }is independent of step $n$, since at each time $% n $, the intensity $\tau _{n}$ is a new one. This circumstance simplifies the theoretical study of the time evolution. Also, it suggests to find induction relations between $Q_{n}(K)$ and $Q_{n+1}(K)$ that reflect the two definitions by induction of Rel.(\ref{\fullpaperid:Random dil 25}) and (\ref{\fullpaperid:Random dil 26}). \subsection{Characteristic functional} The additivity property of the random spread allows us to take for $I_{0}$ a point initial seat, $x_{0}$ say, of dilate $X_{1}=\delta (x_{0})$, and whose intersection of the dilate with Poisson points $J$ provides the first random set $I_{1}=$ $\delta (x_{0})\cap J$. The functional $Q_{n}(K\mid x_{0})$\ of the random fire spread $X_{n}(x_{0})$, i.e. the probability that set $K$ misses the $n^{th}$ spread $X_{n}(x_{0})$ of initial seat $x_{0}$, satisfies an induction relation between steps $n$ and $n+1$. The compact set $K$ lies in the pores of the $(n+1)^{th}$ spread if and only if none of the points $% y\in \delta (x_{0})$ can develop a $n^{th}$ spread that hits $K$. For a given $y$ $\in $ $\delta (x_{0})$, this elementary probability is \begin{equation} dQ_{n+1}(K\mid x_{0}\mid y)=1-\theta (dy)+\theta (dy)Q_{n}(K\mid y),\text{ \ \ \ \ \ }dy\in \delta (x_{0}). \label{\fullpaperid:Random dil 12} \end{equation}% As the events occurring in disjoint $dy$ are independent, we obtain $% Q_{n+1}(K\mid x_{0})$ by taking the infinite product inside $\delta (x_{0})$% , i.e.% \begin{equation} Q_{n+1}(K\mid x_{0})=\exp -\int_{\delta (x_{0})}\theta (dy)[1-Q_{n}(K\mid y)]. \label{\fullpaperid:Random dil 3} \end{equation}% Each step involves an exponentiation more than the previous one.\ We find for example for the first steps that \begin{eqnarray} Q_{2}(K) &=&\exp -\theta \lbrack \zeta (K)\cap \delta (x_{0})], \label{\fullpaperid:Lantu 19} \\ Q_{3}(K) &=&\exp -\int_{\delta (x_{0})}\theta (dy){\large [}1{\large -}% e^{-\theta \lbrack \zeta (K)\cap \delta (y)]}{\large ]}. \notag \\ Q_{4}(K) &=&\exp -\int_{\delta (x_{0})}\theta (dy){\large [}1{\large -}\exp \{-\int_{\delta (y)}\theta (dz)[1-e^{-\int_{\delta (z)}\theta (dw)1_{\zeta (K)}(w)}]\}{\large ].} \notag \end{eqnarray}% where $Q_{2}$, but neither $Q_{3}$ nor $Q_{4}$, is equivalent to the Boolean RACS functional of Eq.(\ref{\fullpaperid:Lantu 8}). The seat spread $I_{n+1}$ satisfies the same induction relation (\ref{\fullpaperid:Random dil 3}) as the fire spread $X_{n+1}$% . The only change holds on the first term, for which it suffices to replace $% \zeta (K)$ by $K$ in Rel.(\ref{\fullpaperid:Lantu 19}).We can summarize the main results on the characteristic functional by stating: \begin{theorem} Let - $\beta $ be the random spread of parameters $(\theta ,\delta )$, - $I_{1}=\beta (x_{0})$ be the random seat spread stemming from point $x_{0}$ of dilate $X_{1}=\delta (x_{0}),$ - $I_{2}=\beta (I_{1})$ and $X_{2}=\delta (I_{1})$ be the iterated seat spread and its fire spread, - $I_{n+1}=\beta (I_{n})$ be $n^{th}$ iteration of $\beta $, and $% X_{n+1}=\delta (I_{n})$ the associated fire spread, then the characteristic functionals of both RACS $I_{n+1}$ and $X_{n+1}$ satisfy the induction relation% \begin{equation*} Q_{n+1}(K\mid x_{0})=\exp -\int_{\delta (x_{0})}\theta (dy)[1-Q_{n}(K\mid y)], \end{equation*} with initial terms \begin{eqnarray*} Q_{1}(K) &=&\exp -\theta \lbrack K\cap \delta (x_{0})]\text{ \ for the seat spread }I_{1}\text{ and } \\ Q_{2}(K) &=&\exp -\theta \lbrack \zeta (K)\cap \delta (x_{0})]\text{ \ for the fire spread }X_{2}. \end{eqnarray*} \end{theorem} \subsection{Discussion} A Markov type assumption of order one underlies the random spread model, since it suffices to know the $n^{th}$seats for constructing the $(n+1)^{th}$ ones. This assumption just allowed us to achieve the formal calculus of the functionals $Q_{n}$ and $R_{n}$. But it does not prevent us from the comeback of new seats on already burnt areas, after a few steps. The successive spreads may turn around $\delta (x_{0})$, as depicted in Figure~\ref{\fullpaperid:propagations4}. \bigskip The trouble can partly be overcome by a Markov assumption of order two, i.e. by making depend the $(n+2)^{th}$ seats of both the $(n+1)^{th}$ and $n^{th}$ steps. We can impose for example to keep the second seats if they fall in the $\delta (x_{i}),x_{i}\in I_{1}$ but \emph{not }in $\delta (x_{0}) $, and so on. The new version of the induction relation (\ref{\fullpaperid:Random dil 3}) must now involve two successive terms. Given the two points $y$ and $z$ the equation (\ref{\fullpaperid:Random dil 12}) of the elementary functional becomes% \begin{equation*} dQ_{n+1}(K\mid x_{0}\mid y\mid z)=1-\theta (dz)1_{\delta (y)\backslash \delta (x_{0})}(z)+\theta (dz)1_{\delta (y)\backslash \delta (x_{0})}(z)Q_{n}(K\mid z), \end{equation*} which gives, by integration in $z$% \begin{equation*} Q_{n+1}(K\mid x_{0}\mid y)=\exp \left\{ -\int_{\delta (y)\backslash \delta (x_{0})}\theta (dz)[1-Q_{n}(K\mid z)]\right\} \end{equation*} and finally% \begin{eqnarray*} Q_{n+2}(K &\mid &x_{0})=\exp \left\{ -\int_{\delta (x_{0})}\theta (dy)[1-Q_{n+1}(K\mid x_{0}\mid y)]\right\} \\ &=&\exp \left\{ -\int_{\delta (x_{0})}\theta (dy){\LARGE [}1-\exp \{-\int_{\delta (y)\backslash \delta (x_{0})}\theta (dz)[1-Q_{n}(K\mid z)]\}% {\LARGE ]}\right\} {\LARGE .} \end{eqnarray*} More generally, if the random spread of order $n$ satisfies a Markov assumption of order $n$, one then obtains a new representation of the hierarchical RACS (\cite{Serra2:Serra82:Book}, p.562, \cite{Serra2:JEU91}, Ch. 6), probably more tractable. \subsection{Spontaneous extinction\label{\fullpaperid:extinct}} The fire which stems from the point seat $x_{0}$ may go out, spontaneously, after one, two, or more steps. The description of this phenomenon does not involve any particular compact set $K$. Denote by $g(n\mid x_{0})$ the probability that the fire extinguishes after step $n$. This event occurs after the first step when no Poisson point falls inside set $\delta (x_{0})$% , i.e. with the probability% \begin{equation*} \ g(1\mid x_{0})=1-\exp -\theta \lbrack \delta (x_{0})]. \end{equation*} The proof by induction that allowed us to link $Q_{n+1}$ with $Q_{n}$ in Rel.(\ref{\fullpaperid:Random dil 3}) applies again, and gives, for a spontaneous extinction after step $n+1$, the\ probability \begin{equation} g(n+1\mid x_{0})=1-\exp {\Large \{}-\int_{\delta (x_{0})}\theta (dy)g(n\mid y){\Large \}}. \label{\fullpaperid:Lantu 7} \end{equation} For example, the probability of an extinction after the third step is given by the expression: \begin{equation*} g(3\mid x_{0})=1-\exp -\int_{\delta (x_{0})}\theta (dy){\LARGE [}1{\large -}% \exp {\LARGE \{}-\int_{\delta (y)}\theta (dz)[1-\exp \{-\int_{\delta (z)}\theta (dw)\}{\large ]}{\LARGE \}].} \end{equation*} Suppose for the moment that $\theta $ and $\delta $ are translation invariant. The extinction probabilities no longer depend on\ $x_{0},y$, etc., and reduce to $g(n+1\mid x_{0})=g(n+1)$, $g(n\mid y)=g(n)$, etc.. Similarly, the integral $\bar{u}=$ $\int_{\delta (z)}\theta (dx)$ is independent of $z\in $ $\mathbb{R}^{d}$, so that% \begin{equation} g(n+1)=1-\exp -\bar{u}g(n). \label{\fullpaperid:Random dil 40} \end{equation} %\FRAME{ftbpFU}{4.7772in}{1.695in}{0pt}{\Qcb{a) the weight $\bar{u}<1$, then $% %g(n)\rightarrow 0$, as $n\rightarrow \infty $. b) the weight $\bar{u}\geq 1$% %, then $g(n)\rightarrow p$, as $n\rightarrow \infty $. }}{\Qlb{extinction}}{% %extinction.bmp}{\special{language "Scientific Word";type %"GRAPHIC";maintain-aspect-ratio TRUE;display "USEDEF";valid_file %"F";width 4.7772in;height 1.695in;depth 0pt;original-width %10.1114in;original-height 3.5691in;cropleft "0";croptop %"1";cropright "1";cropbottom "0";filename %'extinction.bmp';file-properties "XNPEU";}} \begin{figure}% [!tb] \begin{center} \includegraphics[width=4.4529in]{\fullpaperpdirectory/images/extinction.png} \end{center} \caption{a) the weight $\bar{u}<1$, then $% g(n)\rightarrow 0$, as $n\rightarrow \infty $. b) the weight $\bar{u}\geq 1$% , then $g(n)\rightarrow p$, as $n\rightarrow \infty$.} \label{\fullpaperid:extinction} \end{figure} As $n\rightarrow \infty $, the behavior of $g$ depends on $\bar{u}$. If $% \bar{u}<1$, which corresponds to Figure~\ref{\fullpaperid:extinction}a, then $% g(n)\rightarrow 0$, i.e. the fire extinguishes spontaneously, almost surely, in a finite time. When $\bar{u}\geq 1$, then the two curves of Figure~\ref{\fullpaperid:extinction}b intersect at point $(p,p)$, where $p>0$ is solution of the equation $p=1-\exp \bar{u}p$. There is a non zero probability, namely $p$, of an infinite spread. \bigskip Suppose now that both functions $\theta $ and $\delta $ vary over the space, and let $Z$ be the set of all points where $u(z)\geq 1$. If $x_{0}\in $ $Z$, then there is every chance that the fire invades the connected component of $% \delta (Z)$ that contains point $x_{0}$. Such typical behaviour will be used now for predicting the scars of forest fires in the State of Selangor. \section{Scar prediction\label{\fullpaperid:scars}} The forestry services call \emph{scar} of the fire the cumulative process% \begin{equation} Y(x_{0})=\cup \left\{ X_{p}(x_{0}),1