1 Spatial Modelling of the Terrestrial Environment - Georeferencial

Characterizing Land Use in Urban Systems via Built-Form Connectivity Models 215
65% and 52.3% of the total OPEN SPACE and BUILT nodes, respectively. This suggests that
the land-cover containment hierarchy for the Orpington scene is, in fact, characterized by
two distinct patterns, rather than just the one postulated earlier.
It is also important to note that the overwhelming majority (97.6%) of OPEN SPACE nodes
occur at Levels 1 and 2 (165 and 83, respectively) in the containment hierarchy, although
the actual number of these that contain further nodes is quite small – 28 and 24 at Levels
1 and 2, respectively. In other words, 79.0% of the total number of OPEN SPACE nodes at
these two levels do not contain any other nodes. Closer inspection of Figures 10.5 and 10.6
reveals that many of this latter set of OPEN SPACE nodes constitute either: (i) small, grassy
roundabouts within the road network; (ii) small, isolated parcels of grass in wooded areas;
or (iii) small, isolated parcels of grass between buildings and the road network.
On the basis of the analysis outlined above, it is possible to define formally the containment
relations needed to identify distinct built-form constellations. Two hierarchical
containment patterns are of particular interest: the first is ROAD→OPEN SPACE→BUILT
(i.e., buildings wholly contained within areas of open space that are themselves wholly
contained within sections of the road network); the second is OPEN SPACE→BUILT (i.e.,
buildings wholly contained within areas of open space, where the latter are not wholly contained
within the road network). Critically, identification of constellations based on these
relations must avoid the inclusion of OPEN SPACE nodes that do not contain any BUILT
nodes (i.e., OPEN SPACE↛BUILT).
The relations that encapsulate these hierarchical containment patterns are defined more
formally as follows. First, the relation for all OPEN SPACE nodes that contain BUILT nodes
is given by:
r Open Space→Built = r(x, y) ∈ r containment
∧l(x) ↦→ Open Space
∧l(y) ↦→ Built
∧((l(x) ∧ l(y) ∈ g land cover ), ∀x, y ∈ N
(7)
where r(x, y) is an edge in the containment relation and l(x) and l(y) represent the labels
assigned to nodes x and y from the LAND-COVER group. Second, the relation for all ROAD
nodes that contain OPEN SPACE nodes is given by:
r Road→Open Space = r(x, y) ∈ r containment
∧l(x) ↦→ Road
∧l(y) ↦→ Open Space
∧((l(x) ∧ l(y) ∈ g land cover ), ∀x, y ∈ N.
(8)
It should be clear that, while r Open Space→Built and r Road→Open Space form new relations,
they are also sub-graphs of r containment defined in terms of the label set, L, in XRAG (i.e.,
r(x, y) ∈ r Open Space→Built ∧ r(x, y) ∈ r containment and r(x, y) ∈ r Road→Open Space ∧ r(x, y) ∈
r containment ).
Having derived r Open Space→Built and r Road→Open Space , it is possible to define a graph relation
for built-form constellations characterized in terms of ROAD→OPEN SPACE→BUILT
containment relationships:
G(N, r Road→Open Space→Built ) = (y, z) :r Road→Open Space→Built |
r(x, y) ∈ r Road→Open Space
∧r(y, z) ∈ r Open Space→Built , ∀x, y, z ∈ N
(9)