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is possible). Repeated application of Bayes equation would not lead to this end if the likelihood measure
was a linear function of the inverse error variance. The successive multiplications would result in the most
recent period of data having the greatest weight in the determination of the posterior likelihoods (which
may, of course, give the desired effect if the system is thought to be changing over time). However, the
use of a likelihood measure that is a linear function of the inverse exponential of the error variance would
result in an equivalence of final posterior likelihood. This type of choice has been the subject of some
discussion in the literature (Mantovan and Todini, 2006; Beven
et al.
, 2008; Smith
et al.
, 2008a). It is
intrinsically related to beliefs about the information content of a series of model residuals (see Box 7.2).
7.11 Case Study: An Application of the GLUE Methodology in
Modelling the Saeternbekken MINIFELT Catchment, Norway
The approach of the GLUE methodology is probably best understood by means of an example. In this
example, we consider only one model, TOPMODEL, in an extension of the application to the small
Saeternbekken MINIFELT catchment in Norway (used in the case study of Section 6.4). Although this
application is now a little old, it is still very interesting since it involves the use of spatial information
about catchment responses as well as discharge observations. There are still relatively few studies of
this type. The version of TOPMODEL used was based on the original exponential transmissivity profile
assumptions. It is worth pointing out that the choice of just a single model is equivalent to assigning a
positive prior likelihood of one to that model (my model!) and zero to all other models. This is, of course,
quite common practice, but there is no reason why more than one model structure should not be included
within the GLUE framework (apart from the computational expense of making even more Monte Carlo
simulations).
The use of both discharge and borehole measurements in conditioning the uncertainty in the predictions
of TOPMODEL for the Saeternbekken catchment has been considered by Lamb
et al.
(1998b). They
first studied the use of global (catchment scale) parameter values. Five parameters were varied in the
Monte Carlo simulations. The ranges chosen for each parameter reflect past modelling experience using
TOPMODEL and an initial analysis of recession curves in setting the range for the
m
parameter. An
example of the responses to the individual parameters has already been shown in the dotty plots of
Figure 7.7, which use the Nash-Sutcliffe efficiency as a likelihood measure.
Lamb
et al.
(1998b) chose to use a different likelihood measure, though also one based on the variance
of the residuals, as:
L
=
exp
−
Wσ
ε
(7.6)
where
W
is a weighting coefficient. This gives values close to zero for larger error variance but, as with
the Nash-Sutcliffe efficiency, has a limit of one if the error variance is very small. In the case of the
Saeternbekken catchment, a number of different measures of performance could be calculated using
discharge and different borehole observations. This form of performance measure can be combined
easily using Bayes equation, since multiplying the likelihood measures is equivalent to taking a weighted
average of the different residual variances within the exponential (see Equation (B7.2.3) in Box 7.2). In
this particular application, this allowed different weights to be used in the 1987 calibration period, when
one of the recording boreholes did not appear to produce hydrologically meaningful responses.
An impression of the sensitivity of the individual TOPMODEL parameters can be gained by plotting
the cumulative likelihood weighted distributions for each parameter after evaluating each parameter set
on data from the 1987 simulation period (see Figure 7.9). For each parameter, a rather wide initial range
of values was used. Because of its operation within the TOPMODEL structure, the transmissivity at
saturation was sampled on a log scale as
ln
(
T
o
). Lacking any prior information about the covariation of
