79 
IEANS OF 
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>nes. This model 
ead” /3, describes 
ige sizes, and the 
•ed in the State of 
hich is a strongly 
lap of the spread 
ipagation of the 
laysian type for- 
; behaviour of a 
' observed (Car- 
itres of forestry, 
dyzed the situa- 
of pertinent pa- 
(fuel maps, soil 
r observations : 
ry Canada2003). 
ig this informa- 
rate of spread, 
front of the fire, 
ing et al. 2003). 
1 synthesized in 
lot only for the 
luse their maps 
lot spots which 
cription lies in 
lomena are in- 
2), gas proper- 
(Bak et al. 1990), 
annihilation of opposite fires (Bak et al.2001), irreversible ther 
modynamics (Sero-Guillaume et al.2002), etc. These physical 
analyses aim to go to the core of fire modelling. But, symptomat 
ically, the papers which develop this physics do not pursue their 
approach up to actual predictions of burnt areas. Indeed, several 
factors interfere when an actual fire propagates, making risky the 
practical use of one specific physical PDE. 
This practical impossibility suggests us to replace the determin 
istic physical PDE’s by the probabilities of fire propagation, via 
a model of random closed set (in brief, RACS). Such probabili 
ties can be expressed as a continuous time process. However, the 
day-night alternation as well as the periodic passing of the satel 
lites make more realistic a discrete approach where the time is 
digitized into successive steps. 
Finally, a few software programs, such as FARSITE, FlamMap, 
BEHAVE, etc., can also be used to predict future fire growth by 
simulations and to compute possible parameters of fires. Here 
we make the approach more explicitly dependent on the two key 
maps of the foresters. 
1.2 Forest fires in Southeast Asia 
The major vegetation fires in Southeast Asia during the El Nino 
event of 1997 triggered a worldwide interest due to its massive 
environment disaster when a blanket of thick, brown haze en 
veloped much of Southeast Asia. The cause of the haze was due 
to the forest and bush fires that lasted several months, deliberately 
lit by farmers to clear land for oil palm plantations in Sumatra 
and the development of the ‘mega rice’ project in Kalimantan, 
magnified by the drought brought by the El Nino phenomenon 
(ADB1999). 
Peninsular Malaysia still retains approximately 5.97 million hectares 
of natural forest that mainly consists of 89% of dipterocarp forests 
(Gantz2002). Forests constitute 45.4% of the land areas (FA02002). 
Most of the forest fires in Malaysia take place during the dry 
spells from January to March, and from June to August (Gantz2002). 
Forest fires in the natural forests of Peninsular Malaysia are gen 
erally low but occur more frequently in secondary forests, peat 
swamp forests and forest plantations (Gantz2002). However, in 
cidences of uncontrolled forest fires have been increasing since 
1991 as a result of land clearing activities that involve open burn 
ing due to human negligence or uncontrolled fires that encroach 
into the neighbouring forestland and for cultivation in peat forests 
(Abdullah et al.2003). A total of 35 cases of forest fires were re 
ported from 1991 to 2002 that covered 4,143 hectares (Abdullah 
etal.2003). As a precaution during the dry months, a ban on open 
burning was enforced by the local authorities to reduce the impact 
of local haze following the transboundary haze episodes from the 
vegetation burning advected from Sumatra and Kalimantan. 
Amongst the factors that constitute to burning of forests are those 
that are due to natural causes and due to man whq still execute 
the slash and bum practice for land clearing. The prolonged dry 
condition caused by the natural phenomenon of El Nino exacer 
bated the conditions where the smoke produced from the continu 
ous controlled and uncontrolled burning activities in Sumatra and 
Kalimantan were not doused by heavy rainfall (Mahmud 1999). 
2 RANDOM SPREADS 
2-1 Definition 
Interpret the two basic maps of Figure 2 as follows: 
Figure 3: Three generations of fires stemming from point xo = 
Io- Note the generation of new fires in already burnt areas. 
i/ the fuel consumption map f w is proportional to the intensity 9 
of the Poisson process J(9), with sup# < oo; 
ii/ the daily spread at point x is the disc 8(x) = ttr(x) 2 , where 
r(x) is the value of the spread rate map at point x. When X is 
a set then the union U{¿(x) , x € X} is denoted by <5(X) and 
called dilate of X. 
Consider an initial random seat Io made of an a.s. locally finite 
number of initial point seats in R 2 . The fire evolution from Io is 
the concern, on the one hand, of the fire the initial seats provoke, 
or fire spread X\ = 8(Io), and on the other hand of the gener 
ation of subsequent seats spread I\ = 0(h)- These secondary 
seats will develop new fires in turn. Both aspects refer to some 
compact dilation 8. We propose to model the seats spread 0(h) 
by picking out, randomly, a few points in each dilate 8(xi), for 
all points Xi € Io. The double spread process is then written 
-for the fire spread: 
Xi(I 0 ) = 8(I 0 ) = U{5(xj), Xi £ 7 0 } (1) 
-for the seats spread: 
7i(7 0 ) = 0(h) = U{(5(xi) n Ji) ,::xi€ 7 0 ,:: Ji £ J(6)} 
(2) 
where a different Poisson points realization Ji is associated with 
each point Xi. Therefore, each point Xi of the set 7o induces a 
bunch of seats 8{xf) n Ji independent of the others. These two 
equations mean that though the fire from a seat x does bum the 
zone <5(x) around x, only a few points of the scar 8(x) remain 
active seats for the next step. 
Under iteration, Relation (1) and (2) become 
* 2 (7o) = 8(h) = U{(%fc),:: yk € h} 
= U{<%A:),:: y k £ 8(xi) fl Ji;:: Xi £ 7 0 } 
(4) 
Hio) = p(h) = 
= Ui{Ufc[(5 (¿(Xi) n Ji)] n Jk},:: X* £ 7 0 }. 
Figure 3 depicts the first three steps of a random spread, for 
which: 
• the initial seat Io is the point xo, and the first spread, or 
front, the dark grey disk Xi(7o) = <5(xo);