Geoscience Reference
In-Depth Information
suggested using simple linear and logarithmic functions to represent different parts of the discharge
range (Lambert, 1969, 1972). Then, at each time step during an event, average rainfall (less an estimate
of evapotranspiration if necessary) is added to the relative catchment storage (the absolute storage does
not need to be estimated) and, using the representation of the storage-discharge function, an incremental
change in discharge is easily predicted. The discharge is subtracted from the storage, and the model is
ready for the next time step. This is about as simple a rainfall-runoff model as is possible. Calibration
of the model is simply a matter of deriving the master storage-discharge curve for as wide a range of
discharges as possible. There are effectively no other parameters (see the similar more recent use of this
concept in Section 4.2).
In continuous simulation, such a model would not be very accurate since we know that there is not a
simple relationship between storage and discharge. This is why we use more complex models that attempt
to reflect the complexities of the rainfall-runoff processes more realistically. In real-time forecasting,
however, the ISO model can be used in a way that reduces the impact of the errors inherent in using
such a simple model. Its first advantage is in initialising the model at the start of an event. In general,
the best index of the antecedent wetness of a catchment area is the discharge at the start of an event. The
ISO model is easily initialised if the discharge is known at the first time step, since this can be used to
infer the initial relative storage. The second advantage is that, as soon as errors are detected between the
observed and predicted discharges, the model can be reinitialised using the current measured discharge.
This procedure can be implemented at every time step as soon as the measured discharges are received
and used to update the predictions into the future.
Thus this is also the very simplest possible adaptive modelling scheme. No complex mathematics is
required for the adaptation, it is easily understood, easily calibrated and easily implemented. It can be a
very effective real-time forecasting scheme but clearly has some limitations. In particular, extrapolation
outside the range of measured recession curves is uncertain. In the model, the relationships used for
the upper and lower sections of the master recession curves are simply assumed to continue for more
extreme conditions. In addition, adaptive use of the model to change the current relative storage invalidates
any overall water balance for the model, but this is not important in real-time forecasting if it leads to
improved predictions.
In small catchments, this is effectively the model to beat for real-time forecasting!! More complex
models that are less easy to make adaptive may not necessarily produce better real-time flood forecasts.
8.4.2 Adaptive Transfer Function Models for Real-Time Forecasting
It is worth noting that the simplest linear ISO model element is effectively a first-order transfer function
model, equivalent to Equation (B4.3.1). The difference is that the inputs used with the ISO model are
the rainfalls, whereas the transfer function models of Section 4.3.2 use the rainfalls filtered in some way
to produce an effective rainfall. For a filter that is directly and only dependent on the current discharge,
such as that used by Young and Beven (1994), simple ISO model type updating can still be used directly,
but this is not possible where effective rainfall components introduce additional storage elements. There
is, however, a simple way of making such models adaptive, as noted above, by using an adaptive gain or
multiplier. The best initial estimate of the gain parameter is normally 1.0 but it is then allowed to vary as
the event progresses to correct for any differences detected between the forecasts and the observations
supplied to the flood warning system. If an underprediction is detected, the gain can be increased for the
next time step; if overprediction, the gain can be reduced. The changes in the gain are filtered so that the
changes from time step to time step stay relatively smooth. The adaptive gain approach is a simple way of
compensating for any errors in the data or transfer function model structure that might affect the accuracy
of the forecasts. It will generally lead to greatly improved forecasts. One example of a simple but successful
adaptive algorithm that can, in fact, be applied to any deterministic model output is given in Box 8.1.
