Agriculture, Ecosystems and Environment 81 2000 125–135
The development of land quality indicators for soil degradation by water erosion
M.J. Kirkby
a,∗
, Y. Le Bissonais
b
, T.J. Coulthard
c
, J. Daroussin
b
, M.D. McMahon
a
a
School of Geography, University of Leeds, Leeds LS2 9JT, UK
b
Institute Nationale pour Recherche Agronomique INRA Science du Sol, Avenue de la Pomme de Pin, B.P. 20 619, Ardon, 45 166 Olivet cedex, France
c
Institute of Geography and Earth Sciences, University of Wales, Aberystwyth, SY23 3DB, UK
Abstract
The paper describes a proposed methodology for estimating water erosion risk for large areas. The estimates are based on a one-dimensional hydrological balance and a physically based sediment transport model. Estimates of risk associated
with a given storm amount are integrated over the frequency distribution of daily rainfalls to provide a properly weighted estimate of the average annual risk. Soil factors are estimated from textural classes using qualitative pedo-transfer functions.
Climatic data are taken from interpolated gridded data. Topographic data are taken from global or local DEMs. The estimates are for sediment delivery to stream channels. The method has been applied to provide a preliminary estimate for France at a
resolution of 250 m, and could be applied globally at a resolution of 1 km. © 2000 Published by Elsevier Science B.V.
Keywords: Soil erosion; Land degradation; Regional modelling; Indicators
1. Introduction
National, European and International agencies need objective information at scales of 1:250,000
to 1:1,000,000 to compare levels of environmental risks, as an aid to economic planning and policy de-
velopment. Despite the increasingly effective use of remote sensing, many risk factors are not directly
accessible, particularly at this scale. At present, much more can be achieved by combining physically based
models with remotely sensed land cover, digital ele- vation models, gridded climate data and databases of
fixed soil and other characteristics. Early versions of one such proposed modelling approach are currently
∗
Corresponding author. Tel.: +44-113-233-3310; fax: +44-113-233-6758.
E-mail address: mikegeog.leeds.ac.uk M.J. Kirkby.
being implemented within EU Environment projects — MEDALUS and MoDeM — in close co-operation
with the Space Applications Institute of European Joint Research Centre JRC, Ispra, among others. A
preliminary erosion risk map of France is also be- ing prepared, in collaboration with JRC and Institute
Nationale pour Recherches Agronomiques INRA, Orléans, for comparison with CORINE methods
Briggs and Giordiano, 1992, and these methods are now being improved and taken forward in ongoing
work.
The objective of this work is to provide indicators of soil erosion risk at a regional scale, which are suit-
able for planning and policy for national to continental areas. Although risk is at present expressed on a quali-
tative scale, the values are closely related to estimated average rates of water erosion, calculated month by
month. Because of the uncertainties of the weather,
0167-880900 – see front matter © 2000 Published by Elsevier Science B.V. PII: S 0 1 6 7 - 8 8 0 9 0 0 0 0 1 8 6 - 9
126 M.J. Kirkby et al. Agriculture, Ecosystems and Environment 81 2000 125–135
these values cannot be regarded as forecasts, but are a weighted average based on the expected long-term
frequency of storms of all magnitudes.
2. Method
The proposed physical model is based on a one-dimensional soil-vegetation-atmosphere transfer
SVAT type scheme for surface hydrology, coupled where appropriate to a dynamic model for generic
vegetation growth, controlled by available water, po- tential evapotranspiration and temperature. It has been
developed from a basis in earlier work de Ploey et al., 1991; Kirkby, 1994; Kirkby and Cox, 1995 on soil
erosion. These variables are among the critical and dynamic controls for a many earth surface processes.
The methodology may be applied to a number of environmental processes. At present it is being ap-
plied to water erosion, salinisation, depth of the ac- tive unfrozen layer and peat mire accumulation. It is
also proposed to extend it to wind erosion in the near future.
Water erosion is directly controlled by 1. Climate — through the distribution of storm events
2. Vegetation — via crown cover, providing protec- tion from rainsplash impact and via root mat
strength 3. Soil properties — texture, organic matter and struc-
ture influence both water storage and resistance to sediment detachment and transport erodibility
4. Topography — through hillslope length repre- senting the collecting area for overland flow
and through gradient as the driver for sediment traction
A number of these factors act through agricultural land use, which is itself influenced by economic and social
factors Fig. 1.
2.1. Scientific rationale for estimating erosion risk The combined erosion index Fig. 2 is obtained as
an estimate of mean soil loss in tonne ha
− 1
, as a prod- uct of terms which are primarily dependent on soil, cli-
matevegetation and topography, integrating over the frequency distribution for each month to give
Y = p
2
k
R
2r R exp
−h r
H
2
exp −2r H
L .
1
Fig. 1. Factors influencing water erosion rates.
where Y is the sediment loss, R the total monthly rain- fall, r
the mean rain per rain-day, h the runoff thresh- old, p the proportion of runoff above threshold, Θ the
tractive stress threshold, k the soil erodibility, H the mean slope relief and L the mean slope length.
Daily rainfall intensity is represented above by a simple exponential distribution above. Better fits
may be obtained using the sum of two exponential curves, or the more general Gamma distribution, as
is indicated below. Such an expression makes use of widespread daily rainfall data. This cannot reflect
Fig. 2. Sources of data for estimating soil erosion potential.
M.J. Kirkby et al. Agriculture, Ecosystems and Environment 81 2000 125–135 127
detailed differences in storm intensity profiles, but does contain information about the frequency distri-
bution of daily rainfalls, represented by the mean rain per rain-day r
above, and by its standard deviation in Eqs. 9 and 10 below. Over the frequency dis-
tribution, Eq. 1 gives due weighting to the greater impact of large flows, and the dominant event size
for erosion can be derived de Ploey et al., 1991 as a daily rainfall of r
+ 2h, which has a recurrence
interval of 1N exp 1+2hr
years. 2.1.1. Runoff threshold
A one-dimensional dynamic model for hydrology is based on 50 km or less gridded climate data, as
average monthly values, historic sequences or fu- ture GCM-based scenarios. Monthly values can be
used to grow potential ‘natural’ vegetation and soil organic matter within the SVAT, to provide crop
states for each specified land use, or, in combination with a socio-economic model, to forecast land use in
response to climatic and economic factors.
Forecast values can be compared with, or replaced by monthly land cover maps derived from AVHRR
satellite mosaics, using corrected NDVI and surface temperature estimates. Differences between cover ob-
tained from these two approaches provide one impor- tant and direct index of change.
The runoff threshold and proportion of subsequent runoff are simplifications of cumulative infiltration
and runoff curves, illustrated in Fig. 3 for a USDA experimental catchment in Oklahoma for which data
is available. Runoff Threshold is generally estimated from the crown cover, soil organic matter and soil
texturestructure characteristics, since data like that of Fig. 3 is not widely available. It may be seen that this
relationship shows a large amount of scatter, and this is typical of such data. The large amount of scatter
around the threshold point is readily understood in terms of the generally unknown variations in rainfall
intensity combined within the simplified relationship, as can be seen when comparable theoretical curves
are generated from a semi-empirical infiltration ex- pression such as the Green-Ampt or Philip equation.
The threshold represents the effects of surface storage in random roughness and plough furrows, the dy-
namic evolution of soil crusting and moisture storage within the upper soil layers. Surface storage changes
rapidly over the year on cultivated land, as raindrop impact crusts newly ploughed land, and reduces fur-
row roughness, and this is one source of the scatter shown in a single grouped plot like that of Fig. 3a.
The runoff model, for a daily rainfall of r, is then used in the simple form:
j = pr − h 2
q = xj where j is daily runoff and q the overland flow dis-
charge per unit flow strip width This relationship is recognised as a simplification
which neglects the variations in rainfall intensity in the course of a storm, and the duration of bursts of intense
rainfall during which runoff accumulates downslope. Work is in progress on improving these approxima-
tions in Eq. 2. Preliminary results indicate that dis- charge increases less than linearly downslope, and that
an equilibrium discharge is reached after a distance which increases with mean intensity.
2.1.2. Erodibility and traction thresholds These depend on soil and vegetation characteris-
tics. As vegetation cover changes in a given rainfall regime, sediment loss is directly related to runoff, as
illustrated in Fig. 4 for loess soils in Mississippi. High traction thresholds are usually associated with strong
grass or tree root mats andor very stony soils. This approach, linked to studies of fluvial sediment trans-
port, appears to be generally applicable to overland flow, as indicated by small scale flume experiments,
but relies on locally relevant parameter values for soil and land cover combinations.
2.1.3. Topographic factors Topographic factors in the sediment loss equation
are chosen to be consistent Fig. 5 with data from the long-term erosional forms of hillslope profiles, with
the location of stream heads which evolve over pe- riods of 10s to 100s of years Dietrich and Dunne,
1993; Poesen et al., 1997, and with erosion plot ex- periments over 1–20 years, including data sets such as
that shown in Fig. 4. This approach provides a robust formulation for a wide range of current and possible
changed conditions.
Hillslope profiles indicate the integrated impact of the distribution of rain storms, although not necessar-
ily for today’s climate or land use conditions. Back
128 M.J. Kirkby et al. Agriculture, Ecosystems and Environment 81 2000 125–135
Fig. 3. Runoff and Sediment Yield for USDA Catchment C5, Oklahoma from Kirkby, 1998. a Overland flow runoff vs. storm rainfall. b Sediment yield vs. overland flow runoff in each storm.
analysis of the forms requires assumptions, not gen- erally justified in detail, that the landform is either
declining exponentially towards a base level Kirkby, 1971 or that it is being lowered at a uniform rate
Hack, 1960, perhaps in balance with tectonic up- lift. The latter assumption is simpler to implement.
If the slope profile is expressed in a functional form, then possible sediment transport expressions are con-
strained as follows:
For a slope profile form z=fx and a sediment transport law assumed to be of form S=φxg
n
, the expression φx is constrained to the form:
φ x = Tx[f
′
x]
− n
3 where x is distance downslope, z the elevation, g the
local slope gradient, T the constant rate of denudation and the
′
indicates differentiation. A simple example illustrates the way in which these
three types of evidence may be evaluated and com- bined. This is not identical to the form finally adopted
below, but one which is more readily manipulated mathematically, for purposes of illustration.
M.J. Kirkby et al. Agriculture, Ecosystems and Environment 81 2000 125–135 129
Fig. 4. The relationship between sediment yield, runoff and vegetation type, holly springs, Mississippi data from Meginnis, 1935 in Kirkby, 1998.
For a long-term sediment transport law of the form: S = κ
h g +
x u
xg u
− θ
i 4
for constants κ, u and θ is associated with the convexo–concave slope form:
Fig. 5. Sources of data for sediment transport dependence on topography.
z = u
2 Tu
k +
θ ln
1 + x
2
u
2
− ln
1 + x
2
u
2
where x is the slope length.
In some areas with a relatively unchanging climate, these methods are in accord with cosmogenic dating,
although this accordance is not expected in areas of actively accelerated soil erosion.
The position of stream heads in permanent channels and ephemeral gullies give the best data for Traction
thresholds. Valleys grow where there is positive feed- back which concentrates erosion, giving a downslope
zone of unstable growth, where:
x ∂S
dx S in general
5 and for the sediment transport law in Eq. 4,
x u Erosion plot data provides some systematic informa-
tion on the effect of individual storms, which may be represented by a disaggregated form of Eq. 4,
roughly replacing distance by discharge as:
S
∗
= κ
′
g + q
q
∗
qg q
∗
− θ
′
6 for appropriately modified constants κ
′
, q
∗
, and θ
′
, where q is overland flow discharge per unit contour
width.
130 M.J. Kirkby et al. Agriculture, Ecosystems and Environment 81 2000 125–135
A form of this kind can be formally re-aggregated to a long term relationship through integration over
the frequency distribution of storm flows. These three types of data on transport rates are all
consistent with the proposed form, which is there- fore known to respond robustly across a range of time
scales. The form presently adopted is: At the storm level:
S
∗
= κgr
2
+ µqg − θ
2
where r is the storm rainfall. If integrated over an exponential rainfall frequency
distribution, the long-term equivalent is: S = 2p
2
Rr κg + µ exp
− h
r xg
2
exp −2Φ 7
where Φ=θ 2r gx, R is the total rainfall and r
the mean rain per rain-day.
In the erosion risk model, these effects of topogra- phy are integrated through local hillslope relief, which
is obtained from 50 to 800 m DEM grids as the stan- dard deviation of elevation within a 1 km radius of
each point. This measure is both relevant to the model and stable over a range of DEM resolutions. Relief,
H, obtained in this way is a good estimate of the term xg which appears in Eq. 7. Similarly slope length,
L, may be estimated from the frequency of gradi- ent reversals on high quality DEMs, but is generally
considered to be a more conservative parameter than relief. Ignoring the small first term in Eq. 7, the total
estimated sediment loss delivered to the slope base is:
Y = S
L =
2µRr exp
− h
r H
2
exp −θ2r H
L 8
The final terms, in H and L, are the topographic com- ponents of the erosion indicator. It should, however,
be noted that relief is inversely related to soil erodibil- ity through lithology and texture, so that a relief factor
must be combined with good soil type data.
2.1.4. Potential natural vegetation model In most applications it is preferable to derive land
cover directly from existing maps such as CORINE land cover, or up-dateable remote sensing methods,
such as those based on AVHRR for 1 km resolution. It is, however, sometimes useful to compare these
‘actual’ cover distributions with the ‘potential’ vege- tation, based on climatic and other environmental con-
straints alone. The proposed vegetation growth model for this purpose is a dynamic carbon balance model,
responding to monthly actual evapo-transpiration us- ing a water use efficiency WUE approach. Vegeta-
tion is generic, but a survival model for cover gives some indication of morphology. The budget provides
a simultaneous estimate of SOM, which is an impor- tant variable in forecasting the runoff threshold.
Differences between computed potential unculti- vated vegetation and remotely sensed or surveyed
actual land cover give a direct measure of total human disturbance. Agricultural landuse may, in this scheme,
also be simulated directly by removing cropped ma- terial from the budget at harvest time, in a way which
reflects farming practice interacting with inter-annual variability. Over a restricted climatic range, this may
be related to a fixed cultivation timetable, but a more generic approach is to relate harvest-time to the pe-
riod of maximum biomass. Material harvested not only limits further growth, but also intervenes in the
transfer of leaf-fall from plant to soil organic matter.
2.1.5. Frequency distribution of erosive storms Summation over storms is better achieved by fitting
the distribution of daily rain amounts to a sum of two exponential distributions rather than a single exponen-
tial as used for illustrative purposes above, using the monthly values for number of rain days, mean rain-day
and its standard deviation. Thus, the frequency den- sity, Nr, for a daily rainfall in excess of r is
N r = N µ
r
1
exp −
r r
1
+ 1 − µ
r
2
exp −
r r
2
9 where µ is a fraction, normally between 0 and 1, N
the total frequency of rain-days=N0, and r
1
,r
2
are rainfall intensity parameters.
The mean and variance of this distribution are: ¯
r = µr
1
+ 1 − µr
2
σ
2
= r
2 1
µ2 − µ−2r
1
r
2
µ1 − µ + r
2 2
1 − µ
2
10 It has sometimes been found convenient in practice
to solve these equations iteratively, assuming a fixed ratio r
2
:r
1
= 5.0.
M.J. Kirkby et al. Agriculture, Ecosystems and Environment 81 2000 125–135 131
Integration of the storm runoff and sediment trans- port equations can be carried out analytically over this
distribution to give the form of the erosion risk ex- pression. The overall effectiveness of the erosion risk
estimates may be compared with observed empirical relationships between sediment loss and climate, with
a minimum in temperate zones. A model compari- son Kirkby, 1995 with a transect across the southern
United States shows fair qualitative agreement with empirical summaries Langbein and Schumm, 1958.
3. Results and discussion
