Geoscience Reference
In-Depth Information
To “calibrate” the Markov chain, namely to calculate the probabilities of
transition between two states that allow the observed changes to be reproduced,
the author uses the transition matrices observed at the aggregate level, that is to say,
the matrices crossing the land cover at the dates t and t+1: at the intersection of the
modality k at t and the modality k′ at t+1 is the number of pixels
9
whose land
cover has shifted from the modality k to the modality k′: this quantity is denoted by
∆N
kk′
(t).
The ratio of this amount of area having recorded a transition from the state k to
the state k′ (∆N
kk′
(t)) to the number of pixels whose land cover is k at t (noted N
kt
),
expresses the share of the area occupied by the modality k that has experienced a
change of land cover in favor to the modality k′. This ratio can be assimilated to the
probability of transition from a state k toward a state k′, noted p
k,k′
:
p
k,k′
=∆N
kk′
(t)/ Nkt with
Σ
k,k′
p
k,k′
=1
In the example, the transition matrices associated with the two periods are quite
different because these periods correspond to different phases of the urban sprawl
and have very contextual values, such as the expansion of the collective built area
from the 1960s, or the construction of highways. The choice was then made to
smooth these values by calculating the transition probabilities from a weighted
average of the probabilities calculated from the two matrices. This is the same as
taking into account the two states previously observed. Figure 3.13 shows the results
of the shares of land cover simulated for 2015 from several weighting systems
associated with the two periods. Such an approach consists of varying the weighting
system and adopting the situation corresponding to the best calibration.
As it is used here, the implementation of the Markov chain is deterministic since
its objective is to extend a tendency, hypothetically in the event where change would
obey the same mechanisms as between 1955 and 1995. With regards to models of
the land cover change, the Markov chains are used to predict the amount of change
at the global level but also to simulate the localization of these changes. In
the example, the Markov chain is coupled with a model of potential to locate the
changes taking into account the states of the neighboring pixels. Thus, once the
number n of pixels that will change status is known, the potential model allows us to
calculate for each pixel a score of “change potential” in function of its state and the
states of the neighboring pixels: the potential state of a pixel is influenced by the
neighboring states inversely to their distance. A weighting system formalizes this
influence according to the rules, specific to each state: for example, a built-up pixel
will have a strong influence on a non-built-up pixel in its neighborhood, whereas a
forest pixel will have a low influence on the change of a built-up pixel. The space is
9 This can also be measured by the area of polygons.
