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useful analytical results. For discussions of alter-
native distribution functions, see Yang and Sayre
(1971), Grigg (1970), Kirkby (1991), Wainwright &
Thornes (1991), and Wainwright
et al
. (1995). In
portions of the slope where the entrainment rate
changes slowly with distance (as it does some
distance from the divide), Equation (6.5) with
an exponential distribution function implies that
erosion scales as a negative exponential function
of distance.
As Wainwright
et al
. (2001) pointed out, this
result implies that the
apparent
sediment deliv-
ery ratio should approximate an exponential
form, a result that has been obtained empirically
by a number of authors (e.g. Williams, 1977;
Onstad & Bowie, 1977; Ferro & Minacapilli,
1995). It is also consistent with observations in
loess catchments that sediment delivery approxi-
mates unity (Walling, 1983), in that the travel dis-
tance for very fine particles is high. However,
there are difficulties in assuming that this result
justifies the use of the SDR approach to scaling.
Parsons
et al
. (2004) noted that the SDR is prob-
lematic in that it implies that landscapes are far
younger than they are known to be geologically,
and that it cannot account for differences in land-
scape appearance where land-use change has
occurred on different timescales. Subsequently,
they note that this means that Playfair's Law
(whereby a valley is proportional in size to its
stream, and therefore the sediment load carried
past a point on the stream is proportional to the
channel length upstream of that point) is violated,
and that storage on slopes and floodplains must
occur indefinitely, although such stores are not
observed (Parsons
et al
., 2006b). They suggest
that this problem has arisen precisely because of
the ways in which erosion estimates are made,
and in particular, the assumption that fluxes can
be converted to specific yields simply by dividing
through by the catchment area. The problem
with this assumption has already been covered in
the introduction. Thus, we believe that the use of
SDRs for scaling erosion rates should be avoided,
because it produces results that
must be incor-
rect
in one or more parts of the landscape, even if
results are 'correct' at the catchment outlet.
Scaling properties covered in this section have
made a series of very simplistic assumptions
about the way the landscape works, and have only
considered characteristics in single, simple rain-
fall events. For single events such as these, we
suggest that the sorts of relationships presented
above
could
be used for downscaling erosion
rates. However, in most landscapes the surface
conditions and rainstorms generating erosion will
be more complicated, and other techniques will
also need to be employed. Some of these are
addressed in the next two sections. Upscaling ero-
sion estimates requires a recognition that due to
the simplifications used, as spatial patterns (and
process domains) change, the analytical (or semi-
analytical) approaches presented will break down.
Erosion rates between events are also not
independent, so upscaling in time requires recog-
nition that the starting conditions for the next
event are a function of the conditions at the end
of the previous event, as modified by other proc-
esses (e.g. bioturbation, or human land manage-
ment). Spatial and temporal scaling thus require a
unified framework, and some of the themes
required to develop this framework are covered in
the next three sections.
6.3
Statistical Scaling in Simple Conditions
Spatial representation of data was discussed above
in terms of scale triplets: data extent, spacing or
resolution, and support. Sampling strategies for
representing erosion variability as a means of
scaling rates must account for the characteristics
of all of these elements. For example, the
sediment-budget approach (e.g. Walling & Collins,
2008) may cover a large extent, but has relatively
poor support in that the values obtained are aver-
aged across the whole catchment. As discussed
above, such spatial averaging is not straightfor-
ward to downscale for process-based reasons.
Plot-based studies will tend to have a poor resolu-
tion, even if they have a large extent in some
cases, because they are time-consuming to set up,
maintain and monitor. Studies using erosion pins
tend to have an even poorer resolution, and poor
