81
Figure 5: Scar function s — f w x r 2 — u/nk whose thresholds
estimate the burnt scar zones.
spontaneously, almost surely, in a finite time. When u > 1, then
the two curves of Figure 4b intersect at point (p, p), where p > 0
is solution of the equation p = 1 — exp up. There is a non zero
probability, namely p, of an infinite spread.
Figure 6: a) Two thresholds of function s for 1/irk = 190, in
dark grey, and l/nk = 200, in black (the simulations suggest
value 193); b) map of the actual burnt areas. Note the similarity
of the sets, and of their locations.
* * • #
Suppose now that both functions 9 and 5 vary over the space,
and let Z be the set of all points where u(z) > 1. If xo G
Z, then there is every chance that the fire invades the connected
component of 6(Z) that contains point xo.
Figure 7: Series of structuring functions 5 for the five levels of
spread rate.
3 SCAR PREDICTION
4 FIRE SPREAD SIMULATIONS
We now confront the observed scars of the fires with the stochas
tic model. The forestry services call scar of the fire the cumula
tive process
Y(xo) = U {A’p(xo), 1 < p < oo} . (19)
We will now match the data of the actual scars with the random
spread model. Section 2.4 has shown the crucial role played by
the weight u (x). In Selangor’s case, the numerical expression of
u is determined from the two maps of Figure 2 is as follows
u (x) = 6 (dz) l c(x) (z) = k / f w (z) l c(a: ) (z) dz,
4.1 Calibrations
For simulating the Selangor forest fires by random spreads, we
must calibrate the two input parameters <5 and 9 (Hairi Suliman et
al. 2005).
The digital scale for r, in Figure 2b, is of 0.433 km/pixel. Now,
according to the Canadian forestry commission, the median value
for a fire spread equals 0.6 metres/minute (Forestry Canada2003)
(this number is of course the concern of Canadian forests, but it
gives a likely order of magnitude for Malaysia). In the simula
tions below, the day is taken as the unit step of time, i.e. if set X
is the fire at day n, then /3(X) represents the fire at day (n + 1).
Moreover, the following equivalences hold
U (X) ~ 7Tkf w (x) r 2 (x)
inter-pixel distance/day =
0.443 x 10 3
1440
m/mn = 0,307 m/mn.
We will take for the seasonal factor k the value 1.64 x 10 -3 ,
which is obtained by counting the hot spots on satellite images
(see next section), so that 1/ nk = 193. The scar function s —
f w r 2 is depicted on the map of Figure 5, and the condition u >
1 gives s = u/nk > 193. The two sets above thresholds 190
and 200 are reported in Figure 6a, side by side with the burnt
areas (Figure 6b). In Figure 6a, the fire locations A to E predicted
by the model point out regions of actual burnt scares. Such a
remarkable result could not be obtained from the maps f w and r
taken separately: the scare function s = f w r 2 means something
more, which corroborates the random spread assumption. Region
F is the only one which seems to invalidate the model. As a matter
of fact this zone is occupied by peat swamp forest, or rather, was
occupied. It is today the subject of a fast urbanization, linking the
international airport of Kuala Lumpur to the administrative city
of Putra Jay a. Finally, on the whole, the random spread model
turns out to be realistic.
Therefore the average rate of spread over the digital map r of
Figure2b must be ~ 2 inter-pixel/day, which amounts to assign
values 1 to 5 to the five grey levels of the map. The five possible
values of r(x) generate the five discs S(x) of radii r(x) that are
depicted in Figure7.
The fuel amount f w of Figure 2a corresponds to the permanent,
or static, vegetal substratum for the fuel consumption, whereas
all dynamic causes are regrouped in the factor k, constant over
the space and which takes the meaning of a seasonal coefficient.
9(x, t) = k(t)fuj{x). (20)
Indeed, the Malaysian climate, where rainy and dry seasons al
ternate, makes the total number of high spots vary considerably,
as shown in Figure8(for the sake of display, the ’’points” appear
as small rhombs, but they are actual points in all computations).
