Geology Reference
In-Depth Information
2.7
Temporal and Spatial Scale Interactions
model can become very sensitive to the length
of the individual spatial units. The slope length of
the individual land units may then need to be
restricted within a certain range, as is the case
with the Revised MMF model (Morgan, 2001;
Morgan & Duzant, 2008) which routes annual
calculations of sediment transport over the land-
scape (see Chapter 13).
When it comes to assessing erosion over long
time periods or large areas, the inefficiency of
continuous simulation models raises the question
of whether there is an alternative approach.
Firstly, there is the issue that as temporal and spa-
tial scales change, the relative importance of the
factors that influence erosion also changes. Long-
period and large-area models therefore need to
emphasize the processes and factors that are most
relevant to those scales. They may also require
different equations to express those relationships
compared with those used at hillslope or plot
scales. At large spatial scales the important con-
trols are climate, lithology and vegetation (Kirkby,
1998). If a large area is divided into land units, the
model needs to simulate how erosion varies in
relation to these three factors, whereas other fac-
tors, like slope, surface roughness and soil proper-
ties, can be represented by average values for each
land unit. Averaging can take various forms
depending on the nature of the relationship
between each factor and erosion. Arithmetic aver-
ages may be appropriate if the relationship is lin-
ear, but where non-linear relationships apply, the
root mean square value, the logarithmic mean,
the geometric mean, or some value which is
exceeded by a given percentage of the statistical
population of values, may be more meaningful
alternatives. Examples of the latter are often used
in soil descriptions, which can be expressed by a
grain size value at which a certain percentage of
the soil (e.g. 5, 10 or 35%) is finer. At present most
of the models at large spatial scales operate on a
large grid cell basis to give an output of sediment
yield for each cell but with no indication of its
fate. Sometimes the output is simplified into
classes of different levels of erosion rate, with
each class being described by terms such as slight,
moderate or severe. In other cases, recourse is
So far it has been shown that erosion models exist
for a range of spatial scales from the small plot to
the large catchment, and a range of temporal
scales from the single storm to thousands of
years. Although the model user should be able to
select an appropriate model at the scale required
to address a specific problem, in reality model
choice is more complex. The challenge for
model developers and users is how to scale up
from a storm to several years, and from a plot to
a large catchment. The approach commonly
adopted for temporal scaling is to take a storm or
daily simulation model and run it consecutively
for many storms or many days. Thus, CREAMS
and WEPP can be operated by running daily simu-
lations for more than 7000 consecutive days in
order to give output for a 20-year period or more,
from which average annual values of soil loss can
then be calculated. Some modellers have run con-
tinuous simulation models for periods as long as
100 years (Lee, 1998). For spatial scaling, the
approach is to use distributed models to simulate
the movement of water and sediment over very
large numbers of land units in order to accommo-
date large catchments. There are two problems
with these approaches. Firstly, they are not very
efficient. Secondly, they can become very unsta-
ble mathematically when the time step between
successive calculations by the model becomes
greater than the value of the ratio of the slope
length between successive calculations to the
speed of the flow wave. The exact value of this
ratio depends on the mathematical method used
to solve the equations for routing runoff and sedi-
ment over the landscape (e.g. finite difference or
finite element methods). Generally, the shorter
the time steps and the shorter the space steps, the
more stable the model will be. Such a situation is
usually achieved by undertaking calculations at
shorter time steps and space steps than is given
by the model output. Thus, even in a daily simu-
lation model, calculations may be made every ten
minutes and for transport distances of every 5 to
10 m within each land unit. Where time and space
scales prevent such detailed calculations, the
