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lower exceedance probability events rather uncertain. Fitting any statistical distribution also does not
take any account of how the runoff generation processes might change under more extreme conditions
(Wood
et al.
, 1990; Struthers and Sivapalan, 2007). Rainfall records, however, tend to be longer and more
widespread. Thus, the possibility of using rainfall-runoff models to estimate frequencies of flow extremes
is attractive.
In a seminal paper on the dynamics of flood frequency, Pete Eagleson (1972) outlined a method of
“derived distributions” for the calculation of flood frequency. Eagleson's idea was to look at the flood
frequency problem as a transformation of the probability distribution of rainstorms into a probability
distribution of flood peaks using a rainfall-runoff model. The advantage of this is that rainfall data is
available for many more sites and generally much longer periods than data for stream discharges. The
disadvantages are that rainstorms vary in intensity, duration and storm profile so that, if it is necessary
to model a sequence of rainstorms, a more complicated stochastic model is required; the results will be
very dependent on the type of rainfall-runoff model and the set of parameter values used. As we have
already seen, this is an important source of uncertainty in model predictions. Observed rainfalls had been
used previously with conceptual rainfall-runoff models to simulate sequences of flood peaks (Fleming
and Franz, 1971) and were shown to be competitive with statistical methods. Eagleson's initial proposal,
however, was not so much concerned with practical prediction for individual sites as to provide a structure
for thinking about the problem.
In the method of derived distributions, the probability density function of flood magnitudes is derived
from the density functions for climate and catchment variables through a model of catchment response to
a rainstorm. Eagleson managed to come up with a set of simplifying assumptions that allowed (in a paper
of only 137 equations) the analytical integration of the responses over all possible rainstorms to derive
the distribution of flood peak magnitudes. The method has been extended by later workers to include
different stochastic characterisations of the rainstorms and different runoff-producing mechanisms at the
expense of losing the mathematical tractability that allows analytical solutions. This is not perhaps so
important given current computer resources when a simple rainfall-runoff model can easily be driven
by stochastic inputs for periods of hundreds or thousands of years to derive a distribution of predicted
flood magnitudes numerically. However, Eagleson's (1972) attempt is of interest in that it provided the
framework that later models have followed of starting with a stochastic rainstorm model which is used to
drive, in either a probabilistic or numerical way, a runoff generation model, which might then be linked
to a routing model to predict the flood peak at a desired location.
8.6.1 Generating Stochastic Rainstorms
One possibility for the inputs to a rainfall-runoff model for estimating flood frequency is simply to
use a long record of observed rainfalls (Calver and Lamb, 1996; Lamb, 1999). However, many studies,
including the Eagleson (1972) paper, have chosen to use a stochastic rainfall generator. Eagleson's
rainstorm model was based on independent distributions for storm duration and mean storm intensity at
a point. Both probability distributions were assumed to be of exponential form. In his study, Eagleson
assumes that the storms are independent so that in calculating flood frequencies he only needs to know the
average number of storms per year. Other later studies have taken account of the variation in antecedent
conditions prior to different events and this requires the introduction of a third distribution of inter-
arrival times between storms. To allow the analytical calculations, Eagleson's stochastic rainstorms are
of constant intensity over a certain duration but, in using rainfall statistics from individual stations, he
did take account of the difference between point rainfall estimates and the reduction in mean intensity to
be expected as catchment area increases using an empirical function from the US Weather Bureau.
Later studies that have used direct numerical simulation, either on a storm by storm basis or by
continuous simulation, have not been so constrained in their rainstorm model component. Beven (1987)
