Geoscience Reference
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(a)
(b)
(c)
FIGURE 2.4
Variations in configuration for regular neighbourhoods. (a) Develop if there is only one cell
developed in the Moore neighbourhood. (b) Develop if there are one or two cells developed in the Moore
neighbourhood. (c) Develop if there is only cell developed in the von Neumann neighbourhood.
development were ρ = 0.2, then a random number between 0 and 9999, say, is drawn. If the num-
ber is 2000 or greater, the cell is not developed; if less, it is developed. It is usually argued that
probabilities reflect uncertainty about the decision in question or variety in preferences which is
captured through noise in the transition rule.
We show these kinds of automata generated by the probabilistic rule in Figure 2.5. In
Figure 2.5a, we show the pattern using the rules to generate Figure 2.4a where the probability
threshold is fixed at 0.2 with the boundary condition at
r
= 40 which is reached after 66 time
iterations. If we were to tighten this threshold, then all that would occur is that the same kind
of pattern would be generated faster as the probabilities are independent on each cycle of the
automata; that is, if a cell has a 20% chance of being developed and it is not, it still has a 20%
chance of being developed the next time, and so on. Although the patterns are different in terms
of the
time
each cell is developed even when the probability thresholds are different, the space
gradually fills up, and in the limit, the entire space is filled with
D
= 2. If sparser structures with
1 <
D
< 2 are to be generated, then it is necessary to make the sequence of probabilities depen-
dent; for example, if the threshold is ρ = 0.2 the first time and the cell is not selected, then the
