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(
LS
) and the rainfall-runoff erosivity (
R
) factor in
RUSLE. Each of these studies described the spa-
tial variability in one of the six model parameters
by using experimental semi-variograms to define
the variability, which is then input to the model.
Thus, an effort is made to quantify one type of
error, for one parameter at a time, albeit over the
entire space being modelled. Results demonstrate
that it is important to understand the spatial error
associated with estimates of input parameters to
models such as the RUSLE, and that if such
'uncertainty' is not considered then it is impos-
sible to 'construct error budgets for the predic-
tions of soil loss' (Wang
et al
., 2002). Such work
definitely represents progress in the modelling of
error associated with erosion model output.
However, both of these numerical experiments
were carried out in isolation of each other - the
error surrounding the
R
factor and its influence
on model output was not assessed alongside that
of the
LS
factor to produce a combined error asso-
ciated with the interaction of both parameters.
Moreover, the error associated with the remain-
ing four parameters in the model was not assessed
at all. The next step might be to extend the
approach to include a full error analysis, con-
ducted to assess the total error associated with
parameterisation of the model, in order that the
user can understand the dominant source of error
from input to the model.
The work of Mokrech (2001) described an in-
depth approach to forward modelling uncertainty
of the Thornes erosion model (Thornes, 1990) by
understanding the propagation of parameter error
within the model structure. The Thornes model
is parsimonious, containing only four real para-
meters, and as such is a good choice of model to
apply at large spatial scales, but also to under-
stand how the inevitable error that surrounds
often poorly constrained parameters may influ-
ence model output using a forward uncertainty
analysis.
Two approaches are suggested: (1) classical
error propagation, whereby error associated with
each parameter is moved through the model in an
analytical sense; and (2) Monte Carlo simulation,
whereby a stochastic description of the form of
model error is described
a priori
for each
parameter, in this example assuming a Gaussian
distribution, to provide a range of parameter
values for each parameter within the model.
The former approach does not allow for a robust
assessment of the error associated with non-linear
models when compared to the latter, which is
employed using many realisations of the model
to provide a distribution of model outputs con-
sistent with the assumptions made about sources
of uncertainty in the forward analysis. This error
propagation approach is employed in the Mokrech
(2001) study without consideration of spatial auto-
correlation or covariation in the varied para-
meters. As Heuvelink (1998) recognized, this
consequently provides a rather narrow descrip-
tion of model error. To overcome this, Mokrech
tried to represent the distribution function of the
model error in a spatially explicit way; however,
such an approach still does not take into account
autocorrelation or interaction of parameters, and
so may not map the true uncertainty associated
with model predictions very well.
The two approaches of analytical error propa-
gation and Monte Carlo simulation are compared,
and it is concluded that the analytical approach
suggests the greater 'uncertainty' in the model
output, although this term is used somewhat
loosely here as this uncertainty is not assessed
against any observed data (such as hydrographs or
sediment yield data). Finally, it is concluded that
the spatial scale of the input data (a 30-m grid is
used) may also affect the output uncertainty sur-
rounding the model predictions. Such a conclu-
sion is not surprising as no attempt is made to
understand the quality of the spatial data at rep-
resenting the dominant processes that operate at
this spatial scale, and whether or not the proc-
esses that can be represented by 30-m data are
actually the processes responsible for the erosion
being modelled (see Brazier
et al
., 2005, and
Chapter 5 for a discussion of this).
When compared with the first-order sensitiv-
ity analyses described in Section 4.5.1, the for-
ward error analyses detailed above represent a
significant advance in understanding soil erosion
models. Not only is consideration given to error
