Geology Reference
In-Depth Information
sources of error. For example, input data (such as
rainfall) can be very well estimated in some
storms, but quite poorly estimated in others
because of poor siting of rain-gauges or anomalies
in radar rainfall data. These are generally non-
stationary errors rather than random errors, and
when fed through the model they are processed
non-linearly to give error series with complex
bias and correlation characteristics.
Thus, for this type of modelling there are good
reasons to believe that in real applications, with
poorly characterized input and model structural
errors, the statistical assumptions lead to over-
conditioning of the likelihood surface because
of an overestimation of the information content of
a series of residuals. Relaxing the assumptions to
make a more realistic assessment of residual
information, however, means either making fur-
ther statistical (and difficult to justify) assump-
tions about the nature of all sources of uncertainty
in the modelling process, or moving away from
the formal statistical definition of likelihood.
This was the original issue that led to the devel-
opment of the GLUE methodology first outlined
in Beven and Binley (1992) following the observa-
tion that use of the performance measures used
in the optimization of hydrological models often
suggested that there were many different models
that gave similar levels of performance. This led
to a rejection of the concept of the optimum
model in favour of the equifinality thesis in model
calibration (Beven, 1993, 1996a, 2001c; Beven &
Freer, 2001) that is the basis for the GLUE meth-
odology. The issue can be illustrated by the appli-
cation of soil erosion models to simulate real
datasets (Section 4.6).
used in GLUE (see Romanowicz
et al
., 1994,
1996). It is general in that respect. However, it is
the potential to use informal likelihood meas-
ures (including fuzzy and binary measures) and
different ways of combining likelihoods that
makes the GLUE methodology of interest. It is
much more flexible, but at the expense of adding
choice to the user and without the claims to
objectivity of formal Bayesian methods. However,
as noted above, that objectivity only holds if the
assumptions of the formal analysis are correct.
In its origins, GLUE is a development of the
Hornberger-Spear-Young General or Regionalised
Sensitivity Analysis (see Hornberger & Spear,
1981; Hornberger
et al
., 1985; Spear, 1997). This
form of sensitivity analysis was based on making
a large number of Monte Carlo realizations of a
model, then dividing the results into those that
gave acceptable simulations of the data available
(the
behavioural
models) and those that did not
(the
non-behavioural
models). A comparison of
the parameter distributions for the behavioural
and non-behavioural sets of models then could be
interpreted in terms of the sensitivity of the
results to individual parameters that had been
varied in the model realizations.
The GLUE methodology adds a likelihood
weighting and prediction step to this in which
each of the non-behavioural models is given a
likelihood of zero, while each of the models in
the behavioural set is given a likelihood based
on how well it has performed in the evaluation
process. Scaling the likelihoods such that the
cumulative sum is equal to 1 allows any likeli-
hood-weighted predictions over all models in the
behavioural set to be interpreted probabilisti-
cally. If an informal likelihood is used, this has a
quite different interpretation to the formal sta-
tistical probability, being the probability con-
tributed to the range of predictions by that
model, conditional on the assumed likelihood
measure (see discussion in Beven, 2009).
The choice of a likelihood in GLUE can be quite
subjective, and many people see this as reason
enough to reject it as an uncertainty estimation
methodology. A number of studies have shown
that for well-defined
hypothetical
examples where
4.2.3
Generalised Likelihood Uncertainty
Estimation (GLUE)
The GLUE methodology is based on the use of
informal likelihood measures in model evalua-
tion. Many such measures can still be given a
probabilistic interpretation (see Smith
et al
.,
2008) and there is no reason why formal statistical
likelihoods and Bayes combination of likelihoods
based on random residual assumptions cannot be
