Geoscience Reference
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runoff routing that will control the relationship between the volume of rainfall falling on the catchment
in a storm, effectively
AR
, and the discharge at the hydrograph peak. In addition, the coefficient
C
is
required to take account of the nonlinear relationship between antecedent conditions and the profile of
storm rainfall and the resulting runoff generation. Thus
C
is not a constant parameter, but varies from
storm to storm on the same catchment and from catchment to catchment for similar storms. The easiest
way to get a value for
C
is to back-calculate it from observations of rainfall and peak discharge (the very
simplest form of model calibration). Predicting the correct value for a different set of conditions, perhaps
more extreme than those that have occurred before, or for a catchment that has no observations is a much
more difficult task.
Similar difficulties persist to the present day, even in the most sophisticated computer models. It is
still difficult to take proper account of the nonlinearities of the runoff production process, particularly
in situations where data are very limited. It is still easiest to obtain effective parameter values by back-
calculation or calibration where observations are available; it remains much more difficult to predict the
effective values for a more extreme storm or ungauged catchment. There are still problems of separating
out the effects of runoff generation and routing in model parameterisations (and in fact this should be
expected because of the real physical interactions in the catchment).
However, it is not impossible to make predictions, even with such simple models. Even in the pre-
computer era, the Rational Method evolved into the Graphical Estimation technique (see the work of
Linsley, Kohler and Paulhus (1949) or Chow (1964) for full details). This was an attempt to summarise
a wide range of analyses carried out for catchments in the USA into a set of graphs or
nomograms
that
could be used to predict peak discharges under different rainfall and antecedent conditions (Figure 2.1).
This approach has been used as a design tool for many years and has been put into mathematical form
by, for example, Plate
et al.
(1988).
2.2 Practical Prediction: Runoff Coefficients and Time
Transformations
In Chapter 1 and Section 2.1, the problem of separating the effects of runoff generation and runoff routing
have been raised. This differentiation of two sets of processes was the essence of the first attempts to
model hydrographs, starting back in the 1890s. It must be remembered that all the calculations at that time
had to be done by hand, without benefit even of electronic calculators. At this time, the word “computer”
meant a human being who did calculations. The calculating aids available were limited to log tables.
Thus the calculations had to be simple.
In a paper recently re-discovered by Charles Obled, a French engineer, Edouard Imbeaux (1892),
working on floods in the catchment of the Durance in south-east France, was perhaps the first to attempt
to use a distributed hydrological model. His idea was to split the catchment up into zones on the basis
of travel time to the catchment outlet. Zone 1 would be the area for which runoff could reach the outlet
within one time step (e.g. one hour). Zone 2 would be the area with a travel time of two time steps, and
so on (see Figure 2.2). Imbeaux argued that if the production of runoff could be calculated for each area
then it was a relatively simple matter to route that runoff to the catchment outlet to obtain a prediction
of the hydrograph. Snowmelt is also an important issue and Imbeaux came up with an early version of
the degree-day method of predicting snowmelt, taking account of the effect of elevation on temperatures
in zones at different distances from the outlet. Different antecedent conditions, different rainfall rates,
and different snowmelt conditions would result in different amounts of runoff production and, after the
routing, different hydrographs. The resulting time-area diagram represents the delays for runoff from
each portion of the catchment. A similar concept was used in the USA by Ross (1921, also reproduced in
Loague, 2010), Zoch (1934), Turner and Burdoin (1941), and Clark (1945), and in the UK by Richards
