Geology Reference
In-Depth Information
extent), and data support (the extent to which a
sample is averaged in space or time). The discus-
sion above suggests that all of these uses of scale are
intricately linked, and thus require an integrated
conceptual framework in modelling studies. Scaling
is the movement between different scales of infor-
mation. We differentiate between upscaling and
downscaling. The former refers to the extrapolation
of smaller to larger data extents, spacings or sup-
ports in terms of measurement scales, and to
extrapolation from smaller to larger geographical
areas in terms of model scale. Changes in process
scale mean that upscaling is not necessarily a
straightforward statistical procedure, further emp-
hasizing the need for an integrated conceptual
framework. Downscaling is the interpolation of
larger to smaller data extents, spacings or supports
in terms of measurement scale, or the interpola-
tion of subgrid- or element-scale patterns or sub-
timestep results in terms of modelling scale. While
upscaling and downscaling are generally considered
to be issues of model parameterization, the con-
founding effect of process scaling again suggests
that in reality the case is not so simple. Model eval-
uations as well as the presentation of results to dif-
ferent audiences will depend on the appropriate
up- and downscaling of model results.
This lengthy introduction to the problems
and questions of scale and scaling in erosion mod-
els has demonstrated a number of common prob-
lems with the ways in which erosion rates are
measured and interpreted. Consideration of proc-
ess, measurement and modelling scales requires
an integrated approach that allows the consistent
movement between these different domains.
Scaling erosion models requires integrated process-
based
and
empirical (statistical) approaches. In the
rest of this chapter, we evaluate the scaling of ero-
sion processes, measurements and models within
such an integrated framework.
issue has implications for the scaling of erosion
properties and parameters. Straightforward alge-
braic manipulation demonstrates why erosion at
larger scales is not simply a matter of summing
the predicted erosion at smaller scales (and vice
versa for downscaling). One approach to scaling is
linearization of the underlying equations as a way
of estimating the different scaling effects, but this
approach introduces errors that propagate through
the model domain, and affect the results both
quantitatively and qualitatively (e.g. Mokrech
et al
., 2003). In this section we take an alternative
approach, to use process characteristics of an ero-
sional system to evaluate the patterns of erosion
relationships with changing scale in a very sim-
ple set of conditions.
Parsons
et al
. (2004) investigated the patterns
that should emerge on homogeneous slopes in
single process domains. They used the measure-
ment of particle travel distance as the underlying
concept for characterizing scale-related differ-
ences (see also Kirkby, 1991, 1992), and avoided
process-related scaling issues as noted above by
considering sediment flux as the basic charac-
terization of erosion rate. For particles of a given
diameter
d
[L], the flux
j
d
(x)
[M T
−1
] will vary
with distance downslope
x
as a direct function of
the rate of entrainment (
E
d
(x)
[M T
−1
]) and the
rate of deposition (
D
d
(x)
[M T
−1
]) at each point on
the slope:
d
j
d
()
xExDx
=
()
-
()
(6.1)
d
d
dx
Making the assumption that all particles of a
given size travel the same distance,
L
d
[L], then
the deposition at a specific point is the same as
the entrainment rate that distance upslope, so
Equation (6.1) can be rewritten as:
d
j
d
()
xExExL
=
()
-
(
-
)
(6.2)
d
d
d
dx
6.2
Process-Based Scaling in
Simple Conditions
It will be seen below that this assumption of
uniform distance of movement, which is unreal-
istic in all but the most exceptionally uniform
conditions, is not critical for the results of this
A significant issue with all erosion models is that
they are non-linear, often highly so. Beyond mak-
ing them difficult to work with numerically, this
