Earth Surface Processes and Landforms
Earth Surf. Process. Landforms 27, 1459–1473 (2002)
Published online in Wiley InterScience (www.interscience.wiley.com). DOI: 10.1002/esp.441
THE INFLUENCE OF LAND USE, SOILS AND TOPOGRAPHY ON THE
DELIVERY OF HILLSLOPE RUNOFF TO CHANNELS IN SE SPAIN
MIKE KIRKBY,1 * LOUISE BRACKEN2† AND SIM REANEY1
2
1 School of Geography, University of Leeds, Leeds, LS2 9JT, UK
Department of Geography, University of Durham, Durham, DH1 3LE, UK
Received 21 January 2002; Revised 5 April 2002; Accepted 5 April 2002
ABSTRACT
It is generally accepted that within particular physiographic and climatic regions catchments exhibit differences in their
hydrological response. These differences result from the interaction of spatial variability in catchment characteristics,
variability of rainfall inputs and surface and subsurface hydrological processes. These interactions are complex and
difficult to unravel. Hydrologically similar surfaces (HYSS) have been used to identify catchment areas that have a similar
response to rainfall and have been identified at a number of scales. HYSS have been identified at the subcatchment scale
for the Rambla de Nogalte in SE Spain. Areas with similar at-a-point hydrological storages were distinguished by using a
combination of geology, land use and topography. This mapping was compared with discharge estimates made throughout
the catchment following a seven-year return interval flood in September 1997. From this significant flood source areas
were identified from reaches showing rapidly increasing channel discharge, and associated with HYSS that combined
suitable internal characteristics with good connectivity to the main channel. This paper presents a simulation model that
has been developed to investigate the way in which the hydrological response of areas within a HYSS respond to changes
in source area, gradient, connectivity to the channel, storm size and intensity profile. This is one of the first studies
using a hillslope model to investigate spatial patterns of runoff-response in semi-arid areas and results have implications
for scaling up hydrological response, and on how the dynamics of runoff producing areas vary both under changing
storm conditions and over time. It is implicit in our results that the nature of stream–slope coupling differs substantively
between semi-arid and humid areas. Copyright  2002 John Wiley & Sons, Ltd.
KEY WORDS: connectivity; hill slope hydrology; runoff; semi-arid
INTRODUCTION
Flood discharge at a catchment outlet can generally be attributed to source areas that usually occupy only a
small fraction of the catchment area (Horton, 1933; Dunne et al, 1975). The most widely used conceptual
models are primarily concerned with the volume of runoff contributing to quick-flow in the flood hydrograph
(Beven and Kirkby, 1979; Beven, 1997), although there are also important issues of travel time between
rainfall and delivery to the outlet. There are key differences in the structure of this partial-area response
between humid and semi-arid areas, although some regions appear to show seasonal differences in response,
and/or responses that depend on storm amount or intensity (Hewlett and Hibbert, 1967; Dunne and Black,
1970; Berndtsson and Larson, 1987; Dunne et al. 1991). One of the most important contrasts between the two
types of response is that, in semi-arid climates, runoff generation is determined primarily by a one-dimensional
vertical water balance, whereas lateral subsurface flow provides a topographic coupling between areas under
humid climates.
In this paper we explore some of the factors which control the effective contributing area for semi-arid
climates, using variants of a simple ‘bucket’ model to separate some of the changes in runoff with soil type,
slope or catchment area and gradient. In all of these models, the runoff volume is estimated on a daily or
storm basis as a fixed proportion of the excess rainfall after a runoff threshold (or bucket depth) has been
* Correspondence to: M. Kirkby, School of Geography, University of Leeds, Leeds, LS2 9JT, UK.
† Née Bull.
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M. KIRKBY, L. BRACKEN AND S. REANEY
filled. Thus storm runoff is estimated as:
j D ˛r h
1
where r D storm rainfall (mm), h D runoff threshold (mm), ˛ D proportion of runoff over threshold and
j D storm runoff depth (mm). This expression represents an acceptable simplification of the rainfall–runoff
relationship that is adequate where only daily rainfall totals are available, as is commonly the case for many
areas. Clearly better representations of the infiltration process may be used where there is detail of intensity
variations within storms. Figure 1 shows laboratory data for cumulative runoff during steady sprinkling on
soils taken from the Guadalentı́n catchment, SE Spain, which indicates that this type of linear relationship
can be a reasonable approximation.
The runoff threshold (h in Equation 1) consists of several elements which are able to store rainfall, and so
delay the onset of runoff:
1.
2.
3.
4.
above-ground interception on plant leaves and stems
interception on litter
depression storage within surface roughness features
Storage within the soil.
The storage capacity of these elements depends on a number of factors, including plant cover/biomass and
soil type. However, it is worth noting here that there appears to be a general dependence on slope gradient,
with the runoff threshold generally decreasing with gradient, other things being equal. First, there is a direct
effect on depression storage, which is explored at greater length below. Second, steeper slopes generally have
thinner soils, so that the soil storage element decreases with gradient. Third, vegetation responds to thinner
60
50
Runoff (mm)
40
30
Marl #1
Marl #2
Brown schist
20
10
0
0
5
10
15
20
25
30
35
40
45
Rainfall (mm)
Figure 1. Sprinkler experiment relationship between cumulative simulated rainfall and cumulative runoff for soils from the Nogalte
catchment. Marl #1 and Brown Schist were runs on dry soil. Marl #2 was obtained for a second run on the marl after wetting
Copyright  2002 John Wiley & Sons, Ltd.
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RUNOFF DELIVERY IN SE SPAIN
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soils by producing a sparser cover, so that the plant and litter interception also tend to decrease with slope,
although this effect may vary significantly with land management, for example if trees are found only on steep
slopes. The final effect of gradient, reinforcing the direct storage effects, is that drainage density generally
increases with gradient (Dietrich and Dunne, 1993; Kirkby et al., in press), so that the distance to the nearest
channel tends to be less on steeper slopes.
For many humid regimes, the partial contributing area is associated with variable areas of saturation or
near-saturation, generally around channel heads and along channels, where shallow subsurface flow creates
a gradient of increasing soil moisture downslope (Hewlett and Hibbert, 1967). The lateral movement of
subsurface flow provides a clear pattern of saturated areas that is directly linked to the catchment topography.
For equilibrium with a steady net antecedent rainfall of iŁ the inflow and outflow for a point in the landscape
is given by the expression:
q D iŁ a D KfD
2
where q D overland discharge per unit contour width (l m1 h1 ), a D area drained per unit contour length
(m),  D local slope gradient (m m1 ), K D saturated surface lateral hydraulic conductivity (m h1 ) and
fD is the mean effective depth flow depth (in mm), expressed as a decreasing function of D, the soil
moisture deficit below saturation.
Inverting this relationship, the local deficit, D (mm) is given as:
a 3
D D f1 iŁ
K
in which the term (a/K) is termed the soil-topographic wetness index (in h) (Beven, 1986).
For the semi-distributed TOPMODEL (Beven and Kirkby, 1979), f1 is assumed to take a particularly
simple logarithmic form, which leads to:
a D D m lniŁ m ln
4
K
where m is the effective depth parameter (in mm) (Km is the transmissivity of the saturated soil).
In this case, the deficit contains an intensity-dependent term and differences in deficit across the area are
constant over time and related directly to the soil-topographic wetness index. The runoff threshold at any
point (h in Equation 1) is identified with the saturation deficit (D in Equations 3 and 4), and all storm rainfall
above this threshold is assumed to generate overland flow runoff (i.e. ˛ D 1 in Equation 1). Thus knowledge
of antecedent conditions and the frequency distribution of the wetness index forms a basis for estimating
runoff and contributing area from any storm.
For typical semi-arid areas, most rainfall is lost to overland flow or evapotranspiration, and little is available
for subsurface flow, so that the corresponding mechanisms of connection between points in the landscape
are absent. Generation of runoff is therefore determined independently for each potential source point, and
the role of the landscape is in determining the connectivity between these sources and any chosen outlet
point. Following this line of reasoning, there has been a considerable amount of research in dividing a
catchment into areas with different hydrological response. Hydrological response units (HRUs) have been
defined as distributed, heterogeneously structured entities having a common climate, land use and underlying
pedological–topographical–geological association controlling their hydrological dynamics (Flügel, 1995).
Similar definitions have been used for hydrologically similar units (HSUs: Karnoven et al., 1999), grouped
response units (GRU: Kouwen et al., 1993) and representative elementary areas (REAs: Wood et al, 1988,
1990). HRUs have been identified over a range of scales from plots (Bergkamp et al., 1996) and hillslopes
(Cerda, 1995) to catchments (Imeson et al., 1992; Yair, 1992). Fitzjohn et al. (1998) considered that HRUs
were scale independent, within a nested hierarchy, which increases in complexity with scale.
However, the issues of at-a-point runoff generation and effective connection to the outlet have not always
been clearly distinguished, and it is proposed to separate them more explicitly, using the concept of hydrologically similar surfaces (HYSS: Bull et al., in press). HYSS identify at-a-point properties, and deal with
Copyright  2002 John Wiley & Sons, Ltd.
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M. KIRKBY, L. BRACKEN AND S. REANEY
connectivity and other scaling issues separately and explicitly. A HYSS is associated with an area within
which the at-a-point runoff generation is similar, or belongs to the same frequency distribution. In this respect
it is similar to other distributed bucket models, such as VIC (Wood et al., 1992). From observation, the
areas which produce most runoff are found on steep slopes, either abandoned after agriculture or with sparse
vegetation, and with strongly crusting soils (Bull et al., 2000). However, these areas are not necessarily well
connected to the channel network, and may not therefore contribute to floods from larger areas. Furthermore
channels also suffer from infiltration losses, which may be enhanced by valley floor terracing and/or irrigation,
so that floods generated within a small storm area may not travel very far downstream.
Here we explore the interaction of topography with at-a-point HYSS response, and in particular the role
of slope length within an area. The two main factors investigated are the influence of variability in runoff
thresholds and the influence of storm duration, both of which influence the relationship between runoff and
slope length, for a given at-a-point response. These effects will be used to show how the apparent runoff
threshold changes with slope length, and how this interaction changes across the frequency distribution of
storm sizes. Finally we will return to the practical issue of how HYSS, identified from local measurements,
can be combined with topographic factors to create practical hydrological response units which can be mapped
in the field, or from GIS and remotely sensed data.
THE UNIFORM BUCKET MODEL
The simplest version of a bucket model is for containers of constant size, during an idealized storm of longcontinued uniform intensity i, and with overland flow at constant routing velocity c, and on a hillslope with
parallel contours (i.e. no flow convergence). In this case Equation 1 gives the runoff from every bucket at
any stage of the storm, and runoff (discharge per unit area) is given by:
j D ˛r h
if x < xŁ
xŁ
j D ˛ r h if x ½ xŁ
x
5
where j D runoff per unit contour width (mm h1 , x D distance downslope (m) and xŁ D cr/i is the distance
(m) to equilibration which increases during the storm. Once equilibrium is established, the percentage runoff
is constant downslope.
This runoff model may be integrated over the frequency distribution of rainstorms, which is taken here as
exponential. This is a reasonable first-order model if the distribution is calculated for each month separately
(de Ploey et al., 1991), giving:
Nr D r 0 D
N0
expr 0 /r0 r0
6
where Nr D r 0 is the probability density of days with rainfall >r 0 , N0 is the total number of rain-days, r0
is the mean rain per rain-day (D R ł N0 ) (mm) and R is total mean rainfall (mm).
Total accumulated runoff (discharge per unit area, in mm) is then:
N0
JD˛
r0
1
r h expr/r0 dr D ˛R exph/r0 7
rDh
which is also independent of distance downslope.
The runoff coefficient (ε D runoff ł rainfall) is clearly strongly dependent on storm rainfall and runoff
threshold. For an individual storm of r, Equation 1 shows that it is:
ε D j/r D ˛1 h/ror zero, whichever is less
Copyright  2002 John Wiley & Sons, Ltd.
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While over the distribution of storms (say for a single month or season), we have:
ε D J/R D ˛ exph/r0 9
THE VARIABLE BUCKET MODEL
A more interesting distribution arises where runoff thresholds are distributed, and we will initially assume
that they follow an exponential distribution, with a mean size of h0 and a probability density:
1
h0
exp ph D h D
h0
h0
0
10
Then the runoff from a storm of rainfall r is:
jD
0
1
0
˛r h ph D h D ˛
h0 D0
h0
r
r h0
0
exp dh D ˛ r h0 C h0 exp h0
h0
h0
11
This expression should be compared with Equation 1. It shows some at-a-point runoff, even at small rainfalls,
from the shallowest buckets, while for large rainfalls it gives a similar depth of runoff, tending towards
j D ˛(r h0 , as in Equation 1.
However, the calculation of at-a-point runoff is partially misleading, since not all runoff reaches the base
of a slope without infiltrating into a deeper bucket along its flow path. There is therefore a term which
is dependent on slope length, and is scaled to the correlation distance for independence in the distribution
(Equation 10). The path length (m) of connected runoff may be considered as a form of random walk, the
length of which may be computed numerically, and can be approximated by the expression:
D 0Ð32xc
2
1 2
0Ð863
12
where xc is the correlation distance (in m) and is the probability of at-a-point runoff, equal to 1 expr/h0 from Equation 10.
Where > 0Ð5, a similar distribution applies to areas that are not delivering runoff, for which the mean
disconnected pathway is:
1 2 0Ð863
13
0 D 0Ð32xc
2
Thus the total expected runoff is approximately given by:
1
r
1 exp for r h0 ln2
˛ r h0 C h0 exp 2
h0
x
0 r
1
1 C exp for r ½ h0 ln2
j D ˛ r h0 C h0 exp 2
h0
x
jD
14
This relationship is illustrated in Figure 2 for individual storms of uniform intensity. It may be seen that
the runoff coefficient decreases downslope for small storms [r h0 ln(2)], and increases slightly for larger
storms. Distances are scaled by the correlation distance for variations in storage capacity, which are not well
documented. Observation suggests that appropriate values are of the order of 1 m for cultivated land, and
5–10 m for rangeland/mattoral, related to the regularities or patch sizes in the land surface.
Copyright  2002 John Wiley & Sons, Ltd.
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M. KIRKBY, L. BRACKEN AND S. REANEY
Figure 2. Relationship between mean travel distance of connected flow (number of cells) and probability, p of flow generation within
any cell. The scale of the x-axis is in 2p/1 p, so that mean travel distance becomes very large as p approaches 0Ð5. The relationship
has been computed from Equation 14
Although variations in runoff threshold sharply reduce the runoff coefficient for small storms, its effect
weakens, and is eventually reversed for large storms. At-a-point runoff can be integrated over the frequency
distribution (using Equation 6) to give:
JD
0
1
r
r
˛R
˛
r h0 C h0 exp exp dr D
r0
h0
r0
1 C h0 /r0
15
and the runoff coefficient:
εD
˛
1 C h0 /r0
16
which may be compared with Equations 7 and 9 respectively.
The effect of limited connectivity can be integrated over the frequency distribution (using Equation 6), and
this has been done numerically. The results are illustrated in Figure 3. It may be seen that, except where
rainfall intensity (r0 ) is much less than the runoff threshold (h0 ), the frequency of large storms soon outweighs
the reduction in runoff downslope for small storms, and the integrated response shows little or no decrease
in runoff downslope. It is concluded that the effect of runoff patchiness alone is insufficient to account for
the decrease in runoff downslope, which is observed over a wide range of storm sizes and areal scales.
The size of the surface depression storage is one important component in the runoff threshold (Moore and
Larson, 1979; Zobeck and Onstad, 1987). Examining the response of randomly generated surfaces therefore
gives some insight into the variability of thresholds within a HYSS. For a given surface, the amount of depression storage is strongly affected by the local slope (Onstad, 1984). This relationship has been investigated
using a two-dimensional grid-based hydrological model for mattoral and ploughed surfaces.
The mattoral surface was simulated using independent random heights drawn from an exponential distribution, superimposed upon a uniform gradient. The relationship between the mean of the exponential distribution
and the random roughness coefficient (RR) (Allmaras et al., 1966) is given by:
RR D 0Ð6578˛
17
where RR is the random roughness (mm) and ˛ is the mean of the exp. distribution (mm).
Copyright  2002 John Wiley & Sons, Ltd.
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RUNOFF DELIVERY IN SE SPAIN
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Figure 3. Runoff for different slope lengths summed over the assumed exponential distribution of storm sizes and frequencies, for a
range of mean runoff thresholds, h, and mean rain per rain-day, r0
Figure 4. Depression storage verses gradient. Both axes are scaled by alpha, the mean departure from the average surface elevation (in
mm). It may be seen that a single relationship approximately describes the five data sets shown, which are summarized values from a
large number of simulated random surfaces
Copyright  2002 John Wiley & Sons, Ltd.
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M. KIRKBY, L. BRACKEN AND S. REANEY
The amount of depression storage in the model was determined numerically using a method similar to that
used by Onstad (1984) in which rainfall at constant intensity is applied to a sealed surface until runoff reaches
equilibrium with rainfall. This procedure gives the total average depth of depressions, which are then used
to route overland flow dynamically during simulated storm events, using a multiple flow algorithm and with
the left and right edges of the grid joined to form a repeating unit.
The empirical relationship between slope, roughness and depression storage in the model has been investigated over the range of slopes from 1° to 40° and for ˛ = 4, 6, 8, 12 and 16 mm. The results are shown in
Figure 4.
The trend in all of the data series is for a decrease in the depression storage with an increase in slope. As
the roughness increases the amount of depression storage also increases. With greater roughness, the mean
and maximum size of depression increases leading to the increase in storage. All of the data series show a
steady decline in depression with gradient.
A curve has been fitted to the data of the form:
hD /˛ D 0Ð11 exp0Ð020/˛
18
where hD is the depression storage (in mm) and  is the slope gradient. This relationship allows prediction
of the depression storage of a mattoral surface of a defined slope and roughness.
The furrows in the ploughed field sites were simulated using a sine wave parameterized from field measurements and a roughness component. The added random roughness component was generated in the same
way as for mattoral surfaces with the mean held constant at 5 mm. The angle of the furrow in relation to
the contour was varied from 0° to 90° in increments of 15° . The overall uniform surface gradient was then
added to this surface. The furrow troughs were aligned at the edges of the simulated plot to ensure that water
was free to run along the length of the furrows without obstruction. Example results from the simulation are
shown in Figure 5.
Figure 5. Depression storage associated with random roughness, imposed on regular furrows, inclined at different angles to the contour.
The unploughed case is also included for comparison
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RUNOFF DELIVERY IN SE SPAIN
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Furrows exactly parallel to the contours have far greater depression storage than a purely random roughness
pattern. This is because the water must increase in depth to the point where it can breach the furrow top. At
any other furrow orientation, water is able to flow along the furrows to the outflow, and depression storage
volume is always somewhat less than for the random surface, mainly due to the concentration of water and
flow along the furrow bottoms. This flow concentration enables the flow to overcome the micro-roughness
and connect with the plot outflow. The greater organization in the surface leads to greater connectivity of
overland flow.
STORM DURATION
A second important effect on downslope changes in discharge is the limited duration of intense bursts of rainfall in storms. Figure 6 shows the duration curve for rainfall intensity at one site in the Nogalte catchment,
measured with a 0Ð2 mm tipping bucket rain gauge (Shannon, 2000, personal communication). As an illustration of the effect of storm duration, the average duration of showers with an intensity exceeding 20 mm h1
is 3 minutes. For an assumed overland flow routing velocity (compatible with data from Emmett, 1978) of
0Ð1 m s1 , the flow will travel 36 m downslope before re-infiltration. This is equivalent to the equilibration
distance, xŁ in Equation 5. Large storms, with a 5–10 year recurrence interval, deliver 50–200 mm over a
two-day period, but this is generally broken up into showers of 10–20 mm over periods of 15–30 minutes,
each containing peak intensities of over 100 mm h1 (Figure 7).
To analyse the interaction of storm duration and amount, it is helpful to simplify the problem, by associating
storms with a dominant intensity, iŁ , which may be allowed to vary in a number of ways, but will be taken
as constant to begin with. During a storm, both the total rain volume and its duration are increasing, and
both of these affect the interaction between runoff and slope length. Combining variable storage and storm
duration effects shows that there are two zones on the hillslope. The upslope zone is defined by:
x < cT h0 /iŁ 19
where c is the routing velocity (m h1 and T is the storm duration (h).
In this zone, the decrease in runoff is primarily associated with the variability of runoff thresholds, which
is with the patchiness of the surface. Beyond this distance, the effects of storm duration predominate, and
discharge tends towards an upper limit set by rainfall and the runoff threshold. It is possible to compute the
Figure 6. Duration curve for rainfall intensities for rain gauge S1 in the Nogalte basin, measured at 1 minute time resolution
Copyright  2002 John Wiley & Sons, Ltd.
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M. KIRKBY, L. BRACKEN AND S. REANEY
Figure 7. An example storm for site S1 in the Nogalte catchment, recorded by a 0Ð2 mm tipping bucket rain gauge
runoff, and this can be approximated by the following relationship for storm runoff volume from a rainstorm
of r:
c
r
iŁ x Ł
q D ˛ r r h0 C h0 exp Ð 1 exp 20
iŁ
h0
c r
where iŁ is storm shower dominant intensity (mm h1 ) and xŁ is the distance downslope (m), or the connected
travel distance (from Equation 12), whichever is less.
In Figure 8, this expression is drawn for the following reasonable values:
ž
ž
ž
ž
ž
correlation distance D 5 m
routing velocity D 0Ð1 m s1
dominant intensity D 20 mm h1
mean runoff threshold D 50 mm
mean rain per rain-day D 10 mm.
As storm rainfall is increased, the transition between the response to local patchiness and the response to
storm duration migrates downslope, corresponding to the greater duration in Equation 19. When the effects are
integrated over the frequency distribution of storms, using Equation 10 as before, the weighted average curve
(broken line) shows a more continuous decline in runoff with slope length, as the patchiness and duration
effects combine in various possible ways, according to the parameters used (Figure 8). However, it remains
true that local patchiness in runoff thresholds is the dominant effect on short slopes, whereas storm duration
effects become increasingly important on longer slopes.
In the simulations reported above, it has been assumed that runoff thresholds are exponentially distributed,
so that their coefficient of variation is 100 per cent. Some experiments have also been made with normally
distributed values, allowing the coefficient of variation to be set arbitrarily. As the coefficient of variation is
changed, the integrated response downslope always shows a strong response to storm duration downslope,
and the effect of changed coefficient of variation is on the slope of the response at small distances. Observed
responses suggest that runoff coefficient decreases approximately as distance0Ð3 for experimental plots, and
this is compatible with the original choice of coefficient of variation D 100 per cent, suggesting that the
exponential model provides a satisfactory working hypothesis.
Copyright  2002 John Wiley & Sons, Ltd.
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RUNOFF DELIVERY IN SE SPAIN
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Figure 8. Runoff as a function of slope length, taking account of the variation in runoff threshold and storm duration effects incorporated
in Equation 20. It may be seen that the runoff coefficient decreases with slope length across the full range of values
INTERPRETATION: HYSS AND HRU
Formally the HYSS response denotes the at-a-point runoff generation. Thus Equation 20 may be split into the
HYSS component and the topographic component, which converts it to the HRU response of the complete
equation. Thus the HYSS response is given by Equation 11 for a single storm, or by Equation 15 for the
integrated response, weighted over the frequency distribution.
r
jS D ˛ r h0 C h0 exp h0
˛R
JS D
1 C h0 /r0
21
22
where the suffix S indicates the at-a-point response of the surface.
The topographic term KU , which converts local response to the response of the HRU through the expression
j D KU jS , is then taken from Equation 20 as:
KU D
1
[1 expˇy]
y
23
In this expression, y D iŁ x/cr D x/cT is a dimensionless ratio of downslope distance to overland-flow
equilibration distance, which determines the effect of storm duration on the distance term. ˇ is the ratio of
connected flow path to downslope distance, which is always 1 and shows the effect of patchiness at low
storm rainfalls (Figure 9). For small distances, Equation 23 behaves like KU D ˇ, while at large distances it
behaves like KU D 1/y, showing the two asymptotic behaviours.
Although this analysis largely separates the influence of at-a-point HYSS runoff generation from the effects
of topography, it can be seen that this separation is not complete, because the term ˇ still retains some
dependence on the local mean storage capacity, h0 . Nevertheless the approach presented provides an explicit
theoretical basis for relating topographic factors to the changes in runoff coefficients observed. Thus it shows
how the runoff coefficient decreases with slope length and increases with storm duration T.
Copyright  2002 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 27, 1459–1473 (2002)
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M. KIRKBY, L. BRACKEN AND S. REANEY
Figure 9. The combined effect of HYSS and topographic response to create the response from a hydrological response area (HRU). ˇ
is the ratio of connected flow path length to downslope length, which is influenced by both slope length and gradient
DISCUSSION AND FIELD VALIDATION
Conceptually the model is based on variations in runoff production. Hence for small storms there are separate
patches of runoff, and only patches connected to the channels contribute to channel flow, but as the storm
volume increases, so does the path length of runoff from the source areas. Hence in larger storms more runoff
is likely to enter channels. For large storms, some dry patches still exist, although these are relatively few,
and there is no contribution from dry patches adjacent to channels. Spatial variations in patchiness of runoff
are also caused by deviations in rainfall intensity.
Results from the variable bucket model demonstrate that:
ž
ž
ž
ž
ž
ž
the runoff coefficient decreases as slope length increases;
except at low runoff thresholds, runoff decreases more rapidly with slope length;
runoff falls off with distance downslope more slowly as storage increases;
the runoff coefficient increases with storm duration;
the effect of large storms becomes increasingly dominant on long slopes;
the effect of runoff patchiness alone is insufficient to account for the decrease in runoff downslope, which
is observed over a wide range of storm sizes and areal scales.
In the field mapping of areas promoting high runoff for the Rambla de Nogalte the most important features
identified were land use, rock/soil type and gradient (Bull et al., 2000). These features were then used to
identify HRUs within a GIS database (Bull et al., in press), and were validated against variations in discharge
throughout the catchment. The justification of why land use, geology and gradient were used to explain
areas promoting high runoff shall be briefly revisited, followed by a discussion of implications from the
model results.
Copyright  2002 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 27, 1459–1473 (2002)
RUNOFF DELIVERY IN SE SPAIN
1471
Land use defines the surface depression storage, and the proportion of the soil surface that is bare of
vegetation, and therefore exposed to raindrop impact and crusting. This factor influences interception, surface
and subsurface storage capacities directly. The land use type with the lowest runoff threshold appears to
be recently abandoned land, where the surface was bare, tillage features had been largely destroyed by rain
splash, and crusting was widespread with sparse vegetation cover. In contrast mattoral rangeland had patches
of vegetation with high infiltration capacity, and therefore a high runoff threshold. Rangeland also has a patch
structure with a larger correlation distance than that associated with cultivated fields.
Rock and soil types also directly influence runoff thresholds, primarily through differences in susceptibility
to crust formation and in crust strength. It has, however, also been noted that drainage density responds to
differences in soil type in semi-arid areas (e.g. Melton, 1957). It seems likely that soil and land use factors
interact, so that low runoff thresholds are strongly reinforced for, say, abandoned land on strongly crusting
soils, whereas the absence of either of these factors greatly increases the threshold.
Runoff generally increases with gradient for two main reasons. First, steep slopes generally have lower
runoff thresholds, and second they are associated with a high density of channels. The first of these factors
operates through reduced depression storage, generally thinner soils and consequent sparse vegetation, so that
some aspects of gradient directly influence the at-a-point (HYSS) response. Channel density, however, can
be seen as a direct response to topography, because of the widely observed positive relationship between
gradient and drainage density (e.g. Dietrich and Dunne, 1993; Kirkby et al., in press). In considering the
effective slope length within any area, a distinction should be drawn between channelled and unchannelled
flow. Channelways may, according to circumstances, either provide a store of highly permeable debris, or an
eroded path with very little storage capacity.
It is interesting to compare the above categories with the theoretical analysis presented in this paper,
particularly characteristics such as slope length, and roughness, that were not explicitly identified from the
field survey. It appears that factors influencing runoff production can be divided into those that essentially
describe HYSS classes, and are directly related to the runoff threshold at a point (such as land use and geology),
and those which are topographic factors that are connected to the HRU coefficient (such as gradient).
Model results suggest that the most relevant influence of topography is in terms of slope length, with runoff
coefficient decreasing with slope length. It is argued that slope length or area does not appear as a major field
control on runoff because the significant distances are not total slope length, but rather distance to the nearest
channel. The positive relationship between gradient and drainage density suggests that, other things (such as
rock and soil type) being equal, the distance to the nearest channel decreases with increasing gradient, so
that gradient may appear as a stronger topographic control than distance per se. This argument is supported
by the fact that in semi-arid catchments drainage densities are relatively high and distances to the divide
therefore relatively short, and hence most slope lengths may also be short. Therefore runoff coefficients may
not decline strongly with distance until channel gradients are low enough to promote in-channel deposition.
Gradient is not only an indirect measure of slope length but also directly influences depression storage
due to spatial variations in roughness, and may increase overland flow velocity. Changes in roughness and
depression storage are related to land use at the small scale, but also relate to catchment morphology at
the larger scale in terms of the valley shape and network. Small degrees of roughness at the field scale
are introduced by ploughing and vegetation, and can have a significant effect on runoff production. Plough
direction (with or perpendicular to contours) influences both the size of depression storage and the connectivity
of runoff-producing areas. However, as slope gradient increases, plough furrows of similar dimensions will
result in a decrease in depression storage and greater runoff. This interpretation is supported by model results.
Large-scale catchment roughness introduced by valley morphology and in-channel disturbances, such as
bedrock outcrops and check dams, has less importance for runoff generation, but is still vital in understanding
how semi-arid catchments fire and connect to produce flood waves. Often flood waves are short lived and
disconnected and do not flow for the whole length of a river because of large-scale roughness but also
transmission losses. Hence in attempting to define the conditions necessary to promote a continuous flood
wave down the whole of a main channel, large-scale roughness needs to be considered.
Roughness and depression storage are linked to the concept of connectivity. Connectivity can be defined
as how variable source areas and channels are connected to each other, and can be considered at a variety of
Copyright  2002 John Wiley & Sons, Ltd.
Earth Surf. Process. Landforms 27, 1459–1473 (2002)
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M. KIRKBY, L. BRACKEN AND S. REANEY
scales. At the finest scale connectivity may refer to the connection of patches on a hillslope and routes such
as rills that encourage removal of the water. Connectivity also refers to the flow of water from one hillside
into the main channel, or at coarser scales may refer to the flow of water from a small headwater producing
runoff to the main trunk channel. Connectivity is a function of the potential runoff produced by an area,
valley floor erosion or deposition, and the existence of structures that either prevent runoff reaching channels
such as dams and terraces or those that encourage connection such as roads and tracks. Connectivity is more
important for semi-arid areas where runoff production is more patchy than for humid areas with continued
rainfall onto expanding saturated areas. Connectivity within a basin, or lack of it, explains why some key
runoff-producing areas rarely generate floods at the catchment scale (Bull et al., 2000).
If the principles presented here are applied at catchment scales, two sets of conditions combine to influence
catchment flood runoff. The first is the distribution of effective storage capacities across the HRUs of the
catchment, and the second is the variation in storage capacity along the main channelways. HRU response is
produced, as described above, as the intersection of HYSS and hillslope-scale topography, including internal
connectivity. This response can be summarized as a runoff threshold for each unit, although travel time
factors also come into play. Where a catchment shows a wide range of runoff thresholds, then there is a
more than linear increase in discharge with storm rainfall until all HRUs are generating runoff. The response
of channelways is also important, with variable capacity to absorb small floods within valley-bottom gravels
along depositional reaches. On a regional scale there are also strong interactions with tectonic and erosional
conditions. Reaches undergoing active incision tend to have rocky beds and steep valley walls, both promoting
greater runoff. Reaches with natural or anthropogenically increased aggradation show the opposite effect, with
gentler side slopes which have low runoff potential and a high storage capacity in the valley floor.
CONCLUSIONS
This study has shown that HYSS are important for trying to decipher the important factors promoting hillslope runoff, i.e. slope, versus land use, versus geology. This paper has made progress by dividing factors
encouraging runoff into those affecting runoff thresholds and those affecting connection of runoff to trunk
channels. It has also demonstrated the nature of the relationship of runoff coefficients to slope length and
storm duration. However, the interrelationships between factors promoting runoff are still only poorly understood and remain difficult to disentangle. This study highlights the need for a better understanding of how
different areas within a catchment fit together and how these mosaics and the factors causing runoff change
temporally. This needs to include an understanding of how factors causing runoff vary over different seasons,
with land use change and with slope (both gradient and length). Another challenge for the variable bucket
model is to use the concepts and results presented here to scale up this research from the hillslope to the
small catchment. We feel it is important to encourage more research into these types of questions to enable
an accurate understanding of the production and movement of water within semi-environments. Once the
hydrological response is better understood, models can be accurately developed to include sediment transport
as well.
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