3.5 Deltas 
A delta is a split that is not followed by a junction of the 
downstream channels or their subsequent channels (Figure 9). 
  
Figure 9: A river delta and its graph representation. 
The analysis is similar to the process of identifying islands; 
however, in lieu of searching for the least upper bound, it is 
the goal for a delta vertex that its downstream nodes do not 
have a common least upper bound. 
sort network (cont.) 
operation delta: network x vertex — boolean 
axiom delta (nl, vl) == 
split (n1, v1) and 
(glb (finalVertex (downStreamChannell (n1, v1)), 
finalVertex (downStreamChannel2 (n1, v1))) — (b 
3.6 Channel Pattern Analysis 
Table 1 shows the compilation of the features and their 
corresponding graph representations. 
inDe outDe 
source 1 
destination 0 
node 1 
1 
split >1 
  
Table 1: Summary of channel features and their vertex 
  
  
  
  
  
  
  
  
  
  
  
  
degrees. 
InDegree OutDegree Feature 
0 0 lake with no inlet or outlet 
1 0 destination 
lake with inlet 
0 1 source 
lake with outlet 
1 1 auxiliary node 
lake with inlet and outlet 
2 0 lake with 2 inlets 
0 2 lake with 2 outlets 
3 0 lake with 3 inlets 
2 1 junction of 2 rivers 
lake with 2 inlets and 1 outlet 
1 2 split 
lake with 1 inlet and 2 outlets : 
0 3 lake with 3 outlets 
Table 2: Classification of vertices according to their degrees. 
322 
Reasoning about these features will involve the reverse 
operation, deriving from a graph representation- the kind of 
feature that made up the graph. Table 2 shows an extended 
"inverted" table, classifying vertices by the number of links 
and their flow directions, and assigning the corresponding 
river features. Besides the features from Table 1, the 
corresponding lake-river patterns are included as well. 
4 Simplifications of River Graphs for Flow Inference 
The goal of the inference of the flow direction is to derive 
such a directed graph from an ordinary graph and additional 
metric information about the junction angles. In order to 
simplify this process, a few simplifications of the directed 
graph are possible by removing channels (and corresponding 
vertices) that are not necessary for the inference process. 
Removing a channel puts the network into a state as if the 
channel had never been inserted. 
sort network (cont.) 
operation remove: network x channel 
axioms remove (create, cl) == create 
remove (addChannel (nl, c1), c2) == 
if equal (c1, c2) then return n1 
— network 
else addChannel (remove (n1, c2), c1) 
isIn (remove (nl, c1), c1) == false 
4.] Elimination of Auxiliary Nodes 
Auxiliary nodes, connecting exactly two channels, can be 
eliminated, because they contain no significant information 
from which the flow direction can be inferred (Figure 10). 
  
Figure 10: Simplification by eliminating auxiliary nodes. 
After removing the upstream and downstream channel from 
an auxiliary node, the simplified channel, preserving the 
connectivity and the flow direction, must be inserted. 
sort network (cont.) 
operation merge: network x channel x channel — network 
axioms merge (cl, c2) == error if 
finalVertex (cl) <> initial Vertex (c2) 
merge (c1, c2) == error if 
(inDegree (finalVertex (c1) + outDegree (c1))) > 2 
merge (nl, cl, c2) == 
if not (isIn (n1, c1) and isIn (n1, c2)) 
then return n1 
else addChannel (make 
(initialVertex (c1), finalVertex (c2))), 
remove (nl, cl), remove (nl, c2). 
initial Vertex (merge (nl, cl, c2)) == 
initial Vertex (c1). 
finalVertex (merge (nl, cl, c2)) == final Vertex (c2). 
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