that has physically-meaningful parameters, it is possible that measurements could be used to define empirical prior dis-
tributions. Although statistically appealing, this approach could risk over-interpretation of the physical significance
of the model parameters, which are often defined in hydro- logical models at different scales than those at which
suitable physical properties are measured.
3,15
In the case of TOPMODEL, the exponential saturated zone store is
areally integrated, as are flows at the outlet, allowing estimation of the value of the store parameter m directly
from flow recession curve data.
20
However, in the following analysis, m was treated as if its value could not be deter-
mined a priori. It will become apparent that there may still be uncertainty about the m parameter, especially when
simulations of
variables other
than streamflow
are considered.
4.2 Definition of the likelihood measure
In practice, the choice of parameter sampling distribution may not be critical since the prior distributions can be mod-
ified by comparison of simulated data with observations, and this comparison is applied to complete sets of model
parameters. A likelihood measure, L, is used to make the comparison, where, as noted above, the term ‘likelihood’ is
interpreted fairly broadly to indicate goodness-of-fit. The measure chosen for this study was
L ¼ exp ¹
W j
2 e
j
2 o
5 where W is a weighting factor, to be discussed below, j
2 e
is the variance of the simulation errors and j
2 o
the variance of the observed data. The function L would
equal unity if the observed and simulated data were the same
and reduces
towards zero
as the
similarity decreases.
4.3 Rejection of non-behavioral simulations
There may be many parameter sets that lead to acceptable simulations of a given set of observations. Definitions of
acceptability may vary according to context. For example, if a hydrological model is evaluated in terms of some
simple goodness-of-fit function, less good values of this function may be considered acceptable for simulations of
a time period known to pose particular difficulties for the model.
Allowance can be made within GLUE for the explicit definition of acceptability by the rejection of simulations
for which the likelihood measure falls below a chosen threshold. The use of a rejection threshold is similar in
approach to the Generalised Sensitivity Analysis GSA procedure of Spear and Hornberger
28
which divides multi- ple model simulations into a set that are judged to reproduce
the behavior of the system and a set that are rejected as ‘non-behavioral’.
Previous application of TOPMODEL to the Minifelt
19,21
suggested that very different values would be expected for the chosen goodness-of-fit measure eqn 5 when
evaluated for simulations of discharge and water level data, leading to difficulties in the specification of a
consistent rejection threshold. As a response to this, the approach of Binley and Beven
7
has been followed, where model
simulations were
ranked according
to the
likelihood measure and only the best 10 retained as ‘behavioral’.
It is recognised that the choice of a ‘best 10’ rule to reject relatively poor simulations is somewhat arbitrary, and
that the definition of the rejection rule may affect the uncer- tainty bounds computed using the GLUE method. In fact,
tests demonstrated that relaxation of the rejection threshold to define a larger proportion of the total number of simula-
tions as ‘behavioural’ would cause only slight modifications of uncertainty bounds. The reason for this lack of great
sensitivity to the rejection rule is that although the best parameter sets achieved values of L conditioned on flow
data of 0.8, the majority of simulations achieved only extremely small values, particularly when conditioned on
a combination of observed variables. Therefore, even if a larger number of parameter sets were retained as ‘beha-
vioural’, predictions associated with many of the retained parameter sets would fall within the tails of the cumulative
distributions of L and have little effect on the location of uncertainty bounds, computed according to the method
described below.
4.4 Construction of uncertainty bounds
