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In-Depth Information
of surface flows in Chapter 5), then the width function can be used directly as a transfer function for
routing runoff inputs into the channel.
This type of routing algorithm has been used, for example, in the TOPMODEL software of Chapter 6.
It has the advantage that it requires only the network width function, which can be derived directly from
maps or digital terrain data, and a single parameter, the wave speed in the channels. It has the disadvantage
that it does not deal explicitly with routing on hillslopes and that there is no dispersion of the form of
the width function in routing the flow to the outlet since, for a unit input everywhere along the channel,
the shape of the resulting hydrograph will directly reflect the width function. This would be obscured,
of course, if different patterns of runoff were being produced in different parts of the catchment. Both of
these limitations can be relaxed. It is not difficult in an analysis of digital terrain data to derive a distance
to the nearest channel for every point on the hillslopes of a catchment, so that this type of approach can be
extended to routing surface runoff (at least) to the channel, perhaps using a different velocity. Secondly,
Mesa and Mifflin (1986) and Naden (1992) have shown that a diffusive routing algorithm in each reach
can be implemented relatively easily at the expense of introducing an additional parameter.
Diffusive network width function algorithms have also gained some recent popularity within the
macroscale hydrology field to allow the routing of runoff at the continental scales in a computationally
efficient way (see, for example, Gong
et al.
, 2009). The model of Naden (1992) was applied to the Thames
and Severn catchments by Naden (1993) and to the Amazon and Arkansas-Red River catchments using
GCM-generated rainfall inputs by Naden
et al.
(1999). It is also included as a component of the UP
macroscale model discussed in Chapter 8. At this scale, it is generally necessary to estimate the routing
parameters required. For a uniform channel, the effective wave velocity,
c
, and the dispersion parameter,
D
, used in this model may be approximately related to the characteristics of the flow at a site as:
3
2
v
o
c
=
(4.10)
1
−
F
o
2
4
q
o
2
S
o
D
=
(4.11)
where
v
o
is the mean velocity at a reference discharge
q
o
in a channel of bed slope
S
o
and Froude number
F
o
. In a large basin, the discharge characteristics and channel dimensions will change downstream.
Work by Snell and Sivapalan (1995) and Robinson
et al.
(1995) showed how effective values for these
parameters over the whole network could be related to the at-a-site and downstream hydraulic geometry
of the channels. The parameter estimates have to be considered very approximate and further work needs
to be done to both evaluate the routing procedures and improve them by taking full account of the variation
in the parameters within these large networks. A recent application of this approach in the context of
macroscale rainfall-runoff modelling is provided by Gong
et al.
(2009).
4.5.2 The Geomorphological Unit Hydrograph (GUH)
Related to the use of the network width function as a transfer function for routing runoff is the geomor-
phological unit hydrograph concept. This idea was initiated by Ignacio Rodriguez-Iturbe in a series of
papers (summarised in Rodriguez-Iturbe, 1993) that explored the cause and effect linkages between hill-
slope form, runoff production, channel growth and network development. The network is a reflection of
the runoff-producing mechanisms of the hillslopes operating over a long period of time, but development
of the network has a feedback effect on the form of hillslopes and consequent runoff production. These
geomorphological linkages result in structural regularities in the form of catchments and it should be
possible to take advantage of these regularities in making hydrological predictions. The regularities have
been studied by geomorphologists for a long time and are summarised in Horton's laws that express the
expected relationships between channel numbers, upstream areas, lengths and slopes for different orders
