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interacting processes in a spatially heterogeneous
and temporally non-stationary environment. In
addition, it is very difficult to be secure about the
effective parameter values that appear in those
equations and that must be specified in any appli-
cation of such models, since they cannot gener-
ally be measured directly. It is also often the case
that we cannot be secure about the input data
that are used to drive the model, particularly
when spatial patterns of the inputs might be
important. Thus, we should expect that the
definition and application of such process-based
models will necessarily be uncertain (Beven,
2002a,b, 2006a). Since such models are often used
in practice to support decisions, it follows that
an appropriate assessment of uncertainty might
be important to the decision-making process
(Pappenberger & Beven, 2006; Beven, 2009).
The very few studies of the uncertainty of soil
erosion models are reviewed below. There have
also been relatively few studies of uncertainty in
the predictions of fully distributed process-based
hydrological models. This is, at least in part,
because it seems to be too great a problem to
address when such models can still require signifi-
cant run times and involve such a large number of
sources of uncertainty, including alternative for-
mulations of the process equations. Thus, there
are uncertainties in the descriptive equations of
the processes; there are uncertainties in the param-
eter values required for every spatial element;
there are uncertainties in the measurements of
the required inputs and the need to interpolate the
measurements in space and time; and there is
uncertainty in the observations used to evaluate
model predictions. Since each of these sources of
uncertainty can be difficult to characterize
a pri-
ori
, it follows that there is significant scope for
uncertainty about uncertainty estimation.
uncertainty in real-time forecasting (see Beven,
2009), since this is not common in applications of
erosion models). In the first type of simulation
application, there are no data with which to eval-
uate the model predictions so that only a forward
uncertainty analysis is possible. The results will
then depend totally on the assumptions made
about the different sources of uncertainty in the
modelling process. The second type of applica-
tion is where there are some data with which to
evaluate the model predictions. These data can
therefore be used to constrain the range of feasi-
ble models and therefore the resulting prediction
uncertainty. The results will then depend strongly
on how the data are incorporated into the evalua-
tion process. Different assumptions can be made.
Here we will differentiate between forward uncer-
tainty estimation, a formal Bayesian statistical
approach and the Generalised Likelihood Uncer-
tainty Estimation (GLUE) methodology proposed
by Beven and Binley (1992).
4.2.1
Forward uncertainty estimation
In forward uncertainty estimation, each source of
uncertainty must be defined in some way. The
prediction uncertainties then follow deductively
from the assumed definitions. The most common
form of forward uncertainty estimation is to
assume that a model structure is known, that the
input data are known, and to define statistical dis-
tributions for the model parameters on the basis
of prior knowledge. In simple cases, where a
model can be transformed to be linear in its
parameters (such as the USLE), the uncertainty
can be propagated analytically; but in the case of
nonlinear hydrological or erosion models it is nec-
essary to use approximate numerical methods in a
forward uncertainty analysis, such as Monte Carlo
simulation. This is often combined with Latin
Hypercube sampling to ensure that the parameter
space is sampled efficiently (see Beven, 2009).
Clearly, the resulting prediction uncertainties
depend only on the assumptions made about the
sources of uncertainty. Thus it is important that
the assumptions should be realistic in a particu-
lar application. This is the main problem with
4.2
Uncertainty About Uncertainty
Estimation
In trying to assess the uncertainty in model pre-
dictions we should distinguish between two types
of simulation applications (we will not deal here
with techniques for data assimilation to reduce
