Geoscience Reference
In-Depth Information
Fig. 12.12 Width function
w
=
w
(
s
+
)
for the channel network
shown in Figure 11.1. In this
case, the variable
s
+
is the
topological distance from the
outlet of the catchment,
scaled with the longest
(topological) stream length
from the outlet.
0.3
w(s
+
)
0.2
0.1
0
0
0.2
0.4
0.6
0.8
1
s
+
means of what is now usually called (see Kirkby, 1976) the
width function
(
s
);
this function can be defined as the density function of the channel flow distances in
the catchment from the outlet, and it describes the (normalized) number of links, or
channel segments, as a function of distance
s
from the outlet. This distance variable
s
has variously been taken as the actual distance along the channels, as geometric distance,
that is the piecewise straight line joining junctions, or as the number of links, that is the
topological distance. (On average, these distances are related; apparently (Shreve, 1974),
in large networks the longest topological distances differ only slightly from the longest
geometrical distances.) The number of links is strongly related with basin area; moreover,
average stream velocities are known (Wolman, 1955; Pilgrim, 1977; Rodriguez-Iturbe
et al
., 1992) to remain relatively constant in the downstream direction, in spite of the
decreasing slopes. Therefore, as the distances from the outlet can be taken to be roughly
proportional to the travel times, the width function concept is essentially equivalent with
the time-area function. However, because the width function is based on well-defined
morphological characteristics of the river network, it can be determined more objectively
and is therefore better suited for analysis. The correspondence between this concept and
the time-area function as a unit response function was probably first pointed out by
Surkan (1969), who used it to study the effect of stream channel pattern on the flow at
the outlet of the basin. Subsequently, the width function has proved to be a useful tool for
studying the stochastic properties of stream networks (see Kirkby, 1976; Veneziano
et al
.,
2000) and implicitly some of their translatory response characteristics (see Gupta and
Waymire, 1983; Troutman and Karlinger, 1985; Rodriguez-Iturbe and Rinaldo, 1997).
w
=
w
Example 12.5. Construction of a width function
Consider again the hypothetical catchment shown in Figure 11.1. The number of channel
links at topological distances 1, 2, 3, etc., from the outlet can readily be counted; they
are respectively 1, 2, 2, 4, 6, 8, 4, 4, 2, 2. The density at each distance can be calculated
by dividing the number of links by 35, that is the total number of channel links in this
catchment. The results are shown in Figure 12.12.
