Geoscience Reference
In-Depth Information
perpetuates the generally incorrect conceptual link with the idea that runoff is generated by an infiltration
excess mechanism. This nomenclature should be discouraged as should the idea that the effective rainfall
is the same water that forms the discharge hydrograph. Tracer information generally suggests that it is
not (as discussed in Section 1.5 and Chapter 11).
Despite all these limitations, the unit hydrograph model works for predicting discharges. As noted
above, it has the basic functional components needed. There are modern variants of the technique that
work extremely well both in short-term flood forecasting and in prediction over longer periods (see
Chapter 4). There are also now variants linked into Geographical Information Systems (GIS) that have
returned to concepts similar to the original Ross time-area diagram formulation. The approach could
be considered as the “model to beat” for discharge prediction and we will return to it a number of times.
2.3 Variations on the Unit Hydrograph
As experience in the application of the unit hydrograph approach was gained, a number of difficulties
with the approach came to be appreciated, both in calibration to a particular catchment and in prediction.
Two major problems arise in calibration. One has already been mentioned, that of hydrograph separation.
However, once a technique has been chosen, the volume of storm runoff arising from the calibration
storms can be calculated. Because of the linearity assumption, comparison of this runoff volume with
the storm rainfall volume means that the runoff coefficient for each calibration storm can be calculated
exactly. Thus, in calibration at least, the choice of a method for separating the effective rainfall from the
total rainfall is not such a critical issue, because it can be ensured that the volumes match. This is one
reason why the very simple
index method continues to be used today.
Once effective rainfall and storm runoff time series are available, a second problem in calibration
arises from numerical difficulties in calculating the unit hydrograph. If the unit hydrograph is treated as a
histogram (Figure 2.6a), then each ordinate of the histogram is an unknown to be determined, effectively
a parameter of the unit hydrograph. However, the histogram ordinates are strongly correlated, especially
on the recession limb; with the errors inherent in the time series of effective rainfalls and storm runoff, this
makes for a mathematically ill-posed problem. Attempts at a direct solution tend to lead to oscillations,
sometimes extreme oscillations, in the unit hydrograph ordinates that are physically unacceptable as a
representation of the catchment routing.
A number of ways of avoiding such oscillations have been tried, including imposing various constraints
on the shape of the hydrograph (e.g. Natale and Todini, 1977), and superimposing data frommany storms
and determining an average unit hydrograph by a least squares procedure (e.g. O'Donnell, 1966). This
latter approach is still used in the DPFT-ERUHDIT model of Duband
et al.
(1993).
Another approach is to reduce the number of parameters to be determined. This can be achieved by
specifying a particular mathematical form for the unit hydrograph. The simplest possible shape, with
only two parameters, is the triangle (if the base time and time to peak are specified, then the mass balance
constraint of having a hydrograph equal to the unit volume means that the peak height of the triangle
can also be calculated). The triangle was chosen as a simple model in procedures for predicting the
response of ungauged catchments in the UK
Flood Studies Report
(NERC, 1975). It has been retained in
the revised procedures in the UK
Flood Estimation Handbook
(IH, 1999), see also Shaw
et al.
(2010).
This is not the only two parameter model that can be used, however. One of most well-known and
widely used models is the so-called Nash cascade, which can be visualised as a sequence of N
linear
stores
in series, each with a mean residence time of K time units (Nash, 1959). The resulting mathematical
form for the unit hydrograph
h
(
t
) is equivalent to the Gamma distribution:
t
K
N
−
1
exp(
−
t/K
)
K
(
N
)
h
t
=
(
)
(2.2)
