Geoscience Reference
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observations. The user would have difficulty in relating the range of predictions with any degree of
confidence that they would match any particular observation.
7.10 Recognising Equifinality: The GLUE Method
If we accept that there is no single correct or optimal model, then another approach to estimating prediction
limits is to estimate the degree of belief we can associate with different models and parameter sets: this is
the basic idea that follows from recognising the equifinality of models and parameter sets (Beven, 1993,
2006a, 2009). Certainly we will be able to give different degrees of belief to different models or parameter
sets, and many we may be able to reject because they clearly do not give the right sort of response for
an application. The “optimum”, given some data for calibration, will have the highest degree of belief
associated with it but, as we discuss in this section, there may be many other models that are almost as
good. This can be seen in the dotty plots of Figure 7.8a which represent an application of TOPMODEL
to the Maimai catchment in New Zealand.
The dotty plots are scatter diagrams of parameter value against some single performance measure
value. Each dot represents one run of the model from a Monte Carlo experiment using many simulations
with different randomly chosen parameter values. They essentially represent a projection of a sample of
points from the goodness-of-fit response surface onto individual parameter dimensions. In Figure 7.8a,
the good models are those that plot near the top. For each parameter, there are good simulations across a
wide range of values. We commonly find with this type of Monte Carlo experiment that good simulations
go all the way up to the edge of the range of parameters sampled. There are generally also poor simulations
across the whole range of each parameter sampled. In another early study of this type, Duan et al. (1992)
show similar behaviour for a completely different model. Whether a model gives good or poor results
is not a function therefore of individual parameters but of the whole set of parameter values and the
interactions between parameters. As a projection of the response surface, the dotty plots cannot show the
full structure of the complex parameter interactions that shape the surface. In one sense, however, that
does not matter too much since we are really primarily interested in where the good parameter sets are
as a set .
All of these good parameter sets give different predictions, but if we associate a measure of belief with
each set of predictions (highest for the optimum, zero for those models that have been rejected) then we
can estimate the resulting uncertainty in the predictions in a conceptually very simple way by weighting
the predictions of all the acceptable models by their associated degree of belief. Such an approach allows
the nonlinearity of the response of acceptable models using different parameter sets to be taken into
account in prediction and uncertainty estimation.
This appears to lead quite naturally to a form of Bayesian analysis, but in the form proposed in the
original paper of Bayes (1763) more than in more recent usage. Bayes himself talked about the way
in which evidence (of some sort) might modify the odds on a particular hypothesis, given some prior
estimates of the odds (see, for example, Howson and Urbach, 1993). Thus, if any model and its associated
parameter set are considered to be a hypothesis about how the catchment system functions, then any
measure of belief in representing the evidence of how well the model fits the available observations
fits rather naturally into this form of Bayes (although the way in which different measures of belief are
combined need not be restricted to Bayes multiplication). This is the essence of the generalised likelihood
uncertainty estimation (GLUE) methodology proposed by Beven and Binley (1992), which has now been
used in a variety of hydrological modelling contexts with a variety of likelihood measures. Updating of
the model likelihood distributions as new calibration data become available is handled easily within this
Bayesian framework.
In the GLUE methodology, a prior distribution of parameter values is used to generate random pa-
rameter sets for use in each model using Monte Carlo simulation. An input sequence is used to drive
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