Spatial Modelling and Scaling Issues - Environmental Modelling: Finding Simplicity in Complexity - page 80

scales. Global-scale, soil-erosion modelling can provide a
quantitative and consistent approach to estimating ero-
sion. Such a methodology can be used to define regions
where erosion is potentially high and management is
needed (Drake et al ., 1999; Kirkby et al ., 2008).
Current erosion models are mainly developed from the
analysis of the results of plot-scale erosion experiments
(1m 2 to 30m 2 ). One of the physically based models is
developed by Thornes (1985) as:

pixel is homogenous with either vegetation or bare soil.
Erosion is then very high in bare areas and nonexistent
under the plants. When the spatial resolution of the image
is reduced to a grid exceeding the size of the field plant,
heterogeneity within a pixel occurs and predicted erosion
is reduced because some of the vegetation cover of the
field is assigned to the bare areas. Thus a scaling technique
must be employed to account for the original structure of
the vegetation cover.

kOF 2 s 1 . 67 e − 0 . 07 v E = (5.12)

5.6.2 Afractalmethodfor scalingtopographic
slope

where E is erosion (mmday -1 ), k is a soil-erodibility
coefficient, OF denotes overland flow (mmday -1 ), s is the
slope (m m -1 ), and v the vegetation cover (%).
In order to overcome the scaling problem, methods
have been developed to upscale this plot scale model by
downscaling topography and vegetation cover from the
global scale (10 arc minutes) to the plot scale (30m).
Global soil erosion has then been calculated by imple-
menting this upscaled erosion model.

When analyzing the regular decrease of slope values with
the increase of spatial resolution of DEMs, Zhang et al .
(1999) directly link the scaling of slope measurements
with the fractal dimension of topography. Focusing on
the difference in elevation between two points and the
distance between them, the variogram equation used to
calculate the fractal dimension of topography (Klinken-
berg and Goodchild, 1992) can be converted to the
following formula:

= γ d 1 − D

5.6.1 Sensitivityofbothtopographicslopeand
vegetationcover toerosion

Z p − Z q
d

(5.13)

Both the nonlinearity between model output and param-
eters and the spatial heterogeneity of parameters can
be identified by using sensitivity analysis. In order to
investigate the nonlinear effects of slope, we analyzed the
sensitivity of the soil-erosion model to the scale of the
slope measurement. To do so, a global 30 DEM was
degraded to lower resolutions using the pixel thinning
algorithm (see above) in order to create a set of slope
values at various scales. The changing slope as a function
of spatial resolution was used to calculate a set of erosion
values when averagemonthly values of both overland flow
and vegetation cover remained constant. When average
slope is reduced from 3.93% at 30 to 0.39% at 30 in the
area of Eurasia and northern Africa, estimated erosion is
reduced exponentially by two orders of magnitude (from
0.03 to 0.0007mmmonth -1 )(Zhang et al ., 1997).
When changing spatial distributions and scales of het-
erogeneous vegetation cover, the amount of soil loss
predicated by using the erosion model varies consider-
ably (Drake et al ., 1999). This variability occurs because
the negative exponential (nonlinear) relationshipbetween
vegetation cover and erosion combined with the hetero-
geneity of vegetation cover means that erosion is very
high in bare areas but very low once cover is greater than
40%. If we consider a high-resolution image of individual
plants with 100% cover surrounded by bare soil, each

where Z p and Z q are the elevations at points p and q , d
is the distance between p and q , γ is a coefficient and

D is fractal dimension. Because the equation

d
represents the surface slope it can be assumed that the
slope value s is associated with its corresponding scale
(grid size) d by the equation:
s = rd 1 − D

Z p − Z q

(5.14)

This result implies that if topography is unifractal in a
specified range of the measurement scale, slope will then
be a function of the measurement scale. If a research area
were taken as a whole to determine the parameters γ
and D, there would be only one value of scaled slope. It
is necessary in practice to keep the spatial heterogeneity
of slope values. After analyzing the spatial variation of γ

and D in different subareas, it was discovered that both
γ and D are mainly controlled by the standard deviation
of the elevations (Zhang et al ., 1999). Hence correlation
functions are established between γ and D and local
standard deviation. When the standard deviation in each
pixel is determined by moving a 3 × 3 pixel window, the

spatial distributions of both and D are then calculated.

Therefore, the slope values for each pixel with finer
measurement scales can be successfully estimated on the
basis of coarse resolution DEMs.

γ