Geoscience Reference
In-Depth Information
cell is already in one state or the other and that the emphasis is on changing states, not upon
further growth. Of course, this is often a matter of interpretation for the idea of developed and
non-developed states might be seen in a similar way where the focus is on change rather than
growth. We will explore all three of these starting points in the examples below in Sections 2.7
and 2.8, but before we do any of this, we need to step back, digress somewhat and consider how
CA models might be special cases of some more generic modelling structure. To progress this,
we will introduce the formal framework based on reaction-diffusion. After this, we need to say
something about the morphologies that we are generating and give some brief outline of fractal
geometry. At that point, we will return to the theme of CA modelling and pick up on various
examples that demonstrate the logic of this approach to GeoComputation.
2.5 THE GENERIC MODEL: REACTION AND DIFFUSION
The reaction-diffusion structure can be written generically as
∑
xt
(
+= +
1
)
α
xt
()
β
xtPz t
()
+
λ
()
(2.1)
k
k
j
jk
k
j
where
x
k
(
t
+ 1),
x
j
(
t
) is the state of the system in location or cell
k
,
j
at times
t
+ 1,
t
, respectively
P
jk
is a transition value, in this case a probability, between cell
k
and
j
whose cells sum to 1 across
their rows
z
k
(
t
) is an exogenous input which influences the state of cell
k
The first term α
x
k
(
t
) is the reaction, the second term the diffusion and the third term the external
driver. Each of these forces is weighted by α, β, λ, respectively, with these weights summing
to 1. It is easy to show that the state of the cells is in fact probabilities of the occurrence of some
attribute when α + β + λ = 1, 0 ≤ α, β, λ ≤ 1, and the transition probabilities are in the classic form
where
1
∑
∑
Pt
jk
() .
=
1 Then, if we sum Equation 2.1 over
k
, noting that the vectors
zt
k
()=
k
k
∑
and
xt
j
() ,
=
1 then
j
∑∑∑∑ ∑
xt
(
+=
1
)
α
xt
()
+
β
xt P
()
+
λ
z t
()
k
k
j
jk
k
.
k
k
j
k
k
(2.2)
=++=
αβλ
1
The probability structure is thus conserved through the reaction, the diffusion and the external driver.
We can develop several different model representations using the structure in Equations 2.1
and 2.2. Whether or not the process converges to a steady state depends on the extent to which its
influences
- reactions, diffusions and external events - dominate the dynamics. In the absence of
external drivers, there is a strong chance that the process will converge if the actions and reactions
are well-defined and stable transformations. Let us assume that the state variable
x
k
(
t
) is a probability.
Then in the structure which consists of simply reaction and diffusion with closure to the outside
world, that is, λ
z
k
(
t
) = 0, the final probability can be computed from solving
+= +
∑
xt
(
1
)
α
xt
()
β
xtP
() .
(2.3)
k
k
j
jk
j
