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sort network (cont.)
operation junction: network x vertex — boolean
axiom junction (nl, v1) == inDegree (nl, v1) € 2 and
outDegree (n1, v1) 2 1
3.24 Split
Reverse to the junction, a single channel may split into two or
more separate channels. Such a situation occurs usually in a
river delta or when islands are formed. In terms of the di-
graph model, a split corresponds to a vertex with an in-degree
of 1 and an out-degree of at least 2.
sort network (cont.)
operation split: network x vertex — boolean
axiom split (n1, vl) == inDegree (nl, vl) = 1 and
outDegree (nl, v1) 2 2
Figure 6 depicts an example of a splits and its mappings onto
a di-graph.
A
ume ipn A
—»
Figure 6: Split pattern and its graph representation.
— — —»—
3.3 Lakes
À lake is a waterbody without a flow direction. River channels
carry water into and out of a lake; occasionally, lakes may
have no (observable) channels associated with them.
For lakes the same classification of vertices applies, i.e., a lake
may be a source, destination, junction, split, or simply an
auxiliary node of a river network. In addition to these
configurations, a lake may be the destination of several
channels, i.e., the lake vertex is of in-degree » 1 and out-
degree 0. Likewise, a lake may be the source of multiple river
branches, sometimes even of different river networks. Thus, a
lake may also have out-degree »1 and in-degree 0. Even the
combination of in-degree 0 and out-degree 0 is feasible for a
lake (without any visible channels); however, such a vertex
would be an isolated vertex and, therefore, would not be part
of any river network (Figure 7).
Figure 7: Configurations with a lake and their representa-
tions as graphs.
321
In terms of the graph representation for a river network, the
degree of a vertex does not provide any clue whether the
vertex is a lake or not, because a lake can have any
combination of in-degree n (0 X n) and out-degree m (0 X m).
3.4 Islands
An island separates a channel temporarily into two separate
channels that must join later on. In terms of the di-graph
model, an island is an ordered sequence of two vertices, the
first of in-degree 1 and out-degree 2, and the second of in-
degree 2 and out-degree 1. Figure 8 show examples of islands
and their graph representations.
Figure 8: Configurations with islands and their represent-
ations as graphs.
A formal analysis whether two vertices, V/ and V2, form an
island or not has to consider the following issues (for
simplicity, only nodes of degree 3 are considered, but the idea
generalizes to vertices of higher degrees):
. VI must be a split vertex in the network n;
. V2 must be a junction vertex in 1;
° the downstream paths from the final vertices of the
two downstream channels of the split VI must have a
junction in V2; and
° the upstream paths from the initial vertices of the
two upstream channels of the junction V2 must have
a split in VJ.
This operation can be easily expressed as an operation on a
partially ordered set (Birkhoff, 1967), made up by the set of
vertices and the orientation of the edges between the vertices
(such that downstream is x and upstream 2 ). In a partially
ordered set, an element u is an upper bound of a set A if a <u
for all a “ A. The least upper bound (lub) is then the smallest
element in the set of upper bounds of a given set. Reversely,
an element u is a lower bound of a set À if u <a for all a “ A
and the greater lower bound (glb) is the largest element in the
set of lower bounds of a given set. Applied to the
identification of an island in a channel graph, the glb and lub
define an island as follows:
sort network (cont.)
operation island: network x vertex x vertex — boolean
axiom island (nl, v1, v2) ==
split (n1, v2) and junction (n1, v2) and
(glb (finalVertex (downStreamChannell (n1, v1)),
finalVertex (downStreamChanneD (n1, v1))) = v2)
and
(lub (initialVertex (upStreamChannell (n1, v2)),
initial Vertex (upStreamChannel2 (nl, v2))) = v1)
