Environmental Engineering Reference
In-Depth Information
analysis prior to any uncertainty estimation and
then consider only the most sensitive parameters
or boundary conditions in the analysis. There are
different ways of doing sensitivity analysis (see,
e.g., Saltelli et al. 2004), and what appears to be
sensitive is often dependent on the form of anal-
ysis and measure of sensitivity used (e.g. Pappen-
berger et al. 2008). This is, at least in part, because
local sensitivities and interactions between differ-
ent parametersmight differ in different parts of the
model space.
Another technique useful for distributed mod-
els is to set a pattern of distributed parameters or
boundary conditions a priori and then, rather than
varying individual values indifferent places, touse
a multiplier to change the values of the whole
pattern at once. Sensitivity or uncertainty analy-
ses are then carried out on the multiplier rather
than on local values. While thismay not always be
appropriate where spatial information is available
for use inmodel evaluation, it can clearly result in
a drastic reduction in the dimensionality consid-
ered (and consequently in the computer time
required to sample the model space). Even so, it
might still be difficult to obtain sufficient samples
to ensure an adequate sample of the likelihood
surface.
demonstrated for a range of distributed
models from SHE (McMichael and Hope 2007;
Vazquez et al. 2008) to Topmodel (e.g. Beven and
Freer 2001a; Blazkova et al. 2002; Freer et al. 2004),
groundwater models (Feyen et al. 2003) and
hydraulic models (e.g. Aronica et al. 1998;
Romanowicz and Beven 2003; Bates et al. 2004;
Pappenberger et al. 2005a, 2007a, 2007b) within
the Generalized Likelihood Uncertainty Estima-
tion (GLUE) methodology (Beven and Binley 1992;
Beven and Freer 2001a; Beven 2008). In GLUE, the
concept of an optimum model is rejected. Models
are treated as multiple working hypotheses about
catchment functioning that can be rejected by
different types of qualitative and quantitative
model evaluation. Those that survive the evalua-
tion (at least until new data become available) are
used in prediction.
This, however, presupposes that an adequate
number of acceptable models can be found by
systematic, guided or crudeMonte Carlo sampling
in the space of potential models as hypotheses
about how the system functions (while taking any
useful prior information about parameters and
boundary conditions into account). The potential
for doing so will depend on how many potential
sources of error need to be addressed, the computer
time needed for a single model run, the available
computer resource, and the shape of the likelihood
surface in themodel space. If there is a single, well-
defined peak in the likelihood surface then it will
be much easier to obtain an adequate sample of
high-likelihood models than if the surface is a
complex of multiple peaks and ridges of similar
likelihood scattered through the high-dimension-
al model space. Experience suggests that the latter
is often the case for even simple hydrological
models. Indeed, there are good physical reasons
for expecting this, especially in distributedmodels
when, for example, there might be a trade-off
between hydraulic conductivity or surface rough-
ness values in different parts of the flow domain in
producing similar output fluxes.
There are therefore good reasons to try and
reduce the dimensionality of the space in this form
of analysis to try to simplify the response surface.
One way of doing so is to carry out a sensitivity
Model likelihood issues
Clearly, the issues about sampling the likelihood
surface raised in the previous section depend very
much on the choice of how the likelihood of
a particular model is to be evaluated. This issue
has received a lot of attention in the recent hydro-
logical modelling literature. In fact, there is an
ongoing debate aboutwhether equifinality ofmod-
els is really an issue or whether it can be avoided by
the use of formal likelihoods within the theory of
Bayes statistics. Several authors have criticized
the GLUE methodology because it has generally
not used formal statistical likelihoods, even
though GLUE is generalized in the sense that the
user can choose to do so (as has been demonstrat-
ed, e.g., by Romanowicz et al. 1994, 1996). A
formal likelihood is then just one choice of many
forms of likelihood that could be used in model
