CELLULAR AUTOMATON MODEL – THE INFLUENCES OF A [631052]

CELLULAR AUTOMATON MODEL – THE INFLUENCES OF A
FOREST FIRE ON ITS NEIGHBORHOODS

Ioan Valentin Marcel POSEA
Spiru Haret University, Turnului Street, no. 7, 500 152, Brașov, Romania.

Abstract. The aim of the present paper is to produce a model for the propagation of a forest
fire analyzing the influences that the fire zone ha s on its neighborhoods. The model is a
Moore cellular automaton type. It depends on six pa rameters: the medium slope of the
elementary cell, the layer type, and the burning ti me of the fuel, the fuel type, the wind
direction and speed. In order to study the influenc es of various parameter configurations on
the system of vicinities of a fire cell we construc t some special directional correlation
functions. An application is elaborated based on re al data.

Keywords: mathematical modeling, cellular automata model, fo rest fire.

INTRODUCTION
The problem of simulating the propagation of fire i n a forest is one of great
importance and is extensively studied. The main mod els are classified as stochastic, based
mainly on laboratory experimental data, or determin istic, based mainly on physical laws of
conservation of energy in the system formed by the burning and the surrounding area. The
deterministic models are grouped in vector models a nd cellular automata.
Cellular automata type models are based on ideas of J. von Neumann and S. Wolfram.
From a theoretical view point, a cellular automaton is defined by the universe of the automata,
the system of vicinities of a cell, the state of a cell, and the transition rule.
The type of the system of vicinities determines the type of the cellular automaton.
Usually there are four types presented in Fig. 1.

Fig.1 a) Linear Automaton, where V1 and V2 are neig hbors of the active cell; b) J. von
Neumann cellular automaton (four neighbors of the a ctive cell); c) Moore automaton with
eight neighbors; d) Hexagonal automaton (6 neighbor s of the active cell).

Being a time dependent model, the associated lattic e of the automaton universe
generates at a given time the configuration of the machine. This configuration and the
transition rules determine the evolution of the sys tem.

EXPERIMENT
The type of the transition rules, deterministic or stochastic gives the type of the
cellular automaton. We shall consider a covering 20 x 26 grid of square subdomains (Fig. 2)
of the domain D in study.

Fig. 2 The universe of the cellular automata

We take as a cellular automaton universe all the sh aded cells except the remote one
which has the address (10:22). The parameters that will be used to determine the transition
rule proposed by the model will be: the bedding typ e; the wind speed and direction and the
average slope of the cell.
Each parameter will produce an index of contributio n in the transition rule.
Based on the data obtained in the domain we used th e following quantification of the
bedding type (Fig. 3) of each cell:
0 – bedding less
1 – irregular and interrupted bedding
2 – continuous and thin
3 – continuous and thick.

Fig. 3 Distribution of bedding type
In order to determine the contribution of this para meter we construct a special mark
correlation function. The general theoretical conte xt is [10]:
Let us consider a point process with n points on a sample surface. Assign to each point
a triplet formed by its Cartesian coordinates and t he value of the specific attribute considered.
We denote the set of points:
{ }n ii i i imyxp P,…, 2 , 1 ),, (= = (1)
The second order characteristics function is:
j i j i mm mmf ⋅=),( (2)
and the mark correlation function is given by:
[ ]
2),(
) (µh dmmf E
hkij j i
f=
= (3)
where: μ represent the mean value of the considered attribu te; E[X] – the mean of the random
variable X, and dij – distance between points i and j.
For our purpose we shall consider a correlation fun ction of bedding types dependent
on the eight directions attached to a cell in the u niverse of the cellular automata (Table 1).

Table 1. Characteristics of cellular automata
Direction N N-E E S-E S S-V V N-V

θ 1 2 3 4 5 6 7 8

Using the same test function and the values of the attribute given by Fig. 3 we
construct the mark correlation function as:
[ ]
2),() (
µθθ=→=j i direction mmf Ekj i
f (4)
Using the algorithm from underneath, we obtained th e results presented in Table 2 and
Fig. 4:
For each direction from 1 to 8
For each cell in the universe
Calculate and store the correspo nding value of the test function
End for
Averaged the values of the test funct ion
Determine the value of the correlatio n function
End for

Table 2. Experimental results
θ 1 2 3 4 5 6 7 8
) (θfk 1,1113 1,0949 1,0897 1,1036 1,1113 1,0949 1,0897 1, 1036

Fig. 4 Polar graph of the correlation function

This function presents a natural symmetry with resp ect to the origin. The contribution
of the parameter bedding type to the model is given by the normalized values of the function
) (θfk and will be denoted by ).(θc This value is independent of the active cell.
The parameter wind speed and direction are consider ed constant during the simulation
and are obtained for v = 0 km/h, v = 8 km/h, v = 36 km/h, v = 58 km/h.
If we consider that the intensity of the wind is gr ouped in 5 categories and the
direction the wind is blowing from is a then 8 mod) 4 (+=a b and the contribution of the
parameter wind direction and speed is given by:
( )



±= ⋅±=⋅=⋅
=
otherwiseb ib ib i
v
02 05 , 01 1 , 02 , 0
θθθ
θ (5)
The value of this function is also independent of t he active cell.
A discrete quantification of the mean slope of the cells is given by Fig. 5

Fig. 5 Mean slope

The contribution of the parameter slope is given wi th respect to the active cell.
For each active cell a we consider p(a,θ) the difference between the mean slope of the
neighbor on direction θ and the mean slope of the active cell. The value o f the contribution of
this parameter is denoted by s(a,θ) and represents the normalized value of p(a,θ) .
Number of intervals of time for complete combustion of fuel in each cell
(corresponding to the number of steps in the algori thm in which a cell can be considered to
burn) denoted by t(i,j) is quantized in this model using the bedding type and fuel model and is
given in Fig 6.

Fig. 6 Time intervals for complete burning

In order to define the transition rules we consider seven different types of states for a
cell and for each we attach a color:
o ST0 – no fuel in the cell; attached parameter is 0 and the color is green;
o ST1 – cell unaffected by fire; the value of the atta ched parameter is smaller than 1 and
the color is white;
o ST2 – weak influences on the characteristics of the cell; the parameter take values
between 1 and 2, attached color being yellow;
o ST3 – moderate influences on characteristics of the cell; parameter take values
between 2 and 3.5 and the color is orange;
o ST4 – state of major influence on cell characterist ics, igniting combustible material
imminent. The attached parameter values are between 3.5 and 5, with brown color;
o ST5 – burning cell, considered active cell; value o f the parameter is 5 and red color
attached (Fig. 7);
o ST6 – cell total burned; without parameter attached and black color.

Fig 7. Parameters of the system of vicinities of th e active cell
The algorithm used is (Fig. 8):
For all ca
If ca(i,j) is in ST5 then
For 1=θ to 8
p ) (θ= p ) (θ+ ) , () ( ) ( θθθ cas v c ++
if 5) (≥θp then 5) (=θp

Next
t(ca)=t(ca)-1
if 0)(<cat then ca(i,j) is ST6
Next

Fig. 8. The algorithm
Considering two starting active cells the evolution of the model in 1, 3, 5 and 10 time
intervals is presented in Fig. 9.

Fig. 9. Evolution of the model

CONCLUSIONS
In this paper we propose a model that can estimate the performance of a fire
depending on wind speed and terrain slope. In this respect, it is determined a model of cellular
automaton type for simulating of litter fire for U. B. V Noua.
Mathematical modeling offers a number of advantages . After simulation we can see
how the fire spreading from one cell to another.
In terms of fire risk management, especially global warming interested reflected by
increasing the annual and monthly average temperatu res. For this purpose, in mathematical
modeling has considered these aspects, as well as o ther characteristic issues.

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