SEDIMENT DELIVERY DISTRIBUTED (SEDD) MODEL
By Vito Ferro1 and Paolo Porto2
ABSTRACT: Because eroded sediments are produced from different sources throughout a basin, it is often
advantageous to model sediment delivery processes at basin scale using a spatially distributed approach. In this
paper, a sediment delivery distributed (SEDD) model applicable at morphological unit scale, into which a basin
is divided, is initially proposed. The model is based on the Universal Soil Loss Equation (USLE), in which
different expressions of the erosivity and topographic factors are considered, coupled with a relationship for
evaluating the sediment delivery ratio of each morphological unit. Then the SEDD model is calibrated by
sediment yield and rainfall and runoff measurements carried out, at annual and event scales, in three small
Calabrian experimental basis. At event scale, the analysis showed that a good agreement between measured and
calculated basin sediment yields can be obtained using the simple rainfall erosivity factor; the agreement is
independent of the selected equations for estimating the topographic factors. The analysis developed at annual
scale showed that the model reliability increases from the event scale to the annual scale. Finally, a Monte Carlo
technique was used for evaluating the effects of the uncertainty of the model parameters on calculated sediment
yield.
INTRODUCTION
According to Golubev (1982) the area of cultivated land in
the world is 14,300,000 km2 and in some regions of the world
cropland covers the major part of the total area. In cultivated
areas, bare soil is often exposed for most of the year with
sparse crop vegetation existing for a few months. Computations of Golubev (1982) suggest that soil erosion in the world
is 5.5 times more than during the preagricultural period.
According to Brown (1984) the world is currently losing 23
billion t of soil from croplands in excess of new soil formation
each year (Walling and Quine 1992); therefore, accelerated soil
erosion is a serious obstacle to the development of a sustainable agriculture.
Long-term erosion monitoring is necessary to guide the development and testing of predictive models, to observe different scenarios due to changes in land use and climate. Monitoring studies and modeling may be used in combination to
devise soil conservation strategies.
Measured soil erosion rates and the processes responsible
for the sediment transport depend on the spatial scale of observation (Kirkby et al. 1996). Mean soil loss rates from field
size areas (100–10,000 m2) are usually much lower than those
estimated by plot studies (Poesen et al. 1996).
At hillslope and basin scale (>10,000 m2), in addition to the
smaller scale erosion processes, mass movement, hillslope
deposition, and channel processes should be considered to establish the total sediment budget. At hillslope and basin scale,
representativeness problems limit the use of plot measurements. In fact, measurements from a bounded area provide
little information concerning the local variability of erosion
rates and the redistribution processes of soil within a field
(Walling and Quine 1991). Tracer techniques, such as 137cesium measurements (Ritchie and McHenry 1990; Walling
and Quine 1991, 1992; Ferro et al. 1998) afford an alternative
to the use of plots and a means of overcoming the problems
of measurement representativeness and spatial variability.
1
Prof., Dipartimento di Ingegneria e Tecnologie Agro-Forestali, Facoltà
di Agraria, Università di Palermo, Viale delle Scienze, 90128 Palermo,
Italy. E-mail: [email protected]
2
PhD, Instituto di Ecologia e Idrologia Forestale, Consiglio Nazionale
delle Ricerche, Via A. Volta, 106, 87030 Castiglione Scalo, Italy.
Note. Discussion open until March 1, 2001. To extend the closing date
one month, a written request must be filed with the ASCE Manager of
Journals. The manuscript for this paper was submitted for review and
possible publication on December 9, 1997. This paper is part of the Journal of Hydraulic Engineering, Vol. 5, No. 4, October, 2000. 䉷ASCE,
ISSN 1084-0699/00/0004-0411–0422/$8.00 ⫹ $.50 per page. Paper No.
17154.
At basin scale, predicting sediment yield (i.e., the quantity
of sediment that is transferred in a given time interval from
eroding sources through the channel network to the basin outlet) can be carried out coupling a soil erosion model with a
mathematical function expressing the sediment transport efficiency of the hillslopes and the channel network (Renfro 1975;
Kirkby and Morgan 1980; Walling 1983). Soil erosion models
generally predict rill and interrill erosion; therefore, gully and
channel processes are to be separately modeled to obtain the
total basin sediment budget.
If the process is studied at mean annual temporal scale, the
sediment transport efficiency of the hillslopes and the channel
network is usually represented by the spatially lumped concept
of basin sediment delivery ratio SDRW (Sedimentation 1975;
Walling 1983). In a given temporal interval (event, month,
year, etc.), SDRW is the ratio of the weight of sediments actually leaving the basin to that eroded within it. According to
the upland theory of Boyce (1975) sediment delivery ratio
SDRW generally decreases with increasing basin size because
average slope decreases with increasing basin size, and large
basins also have more sediment storage sites located between
sediment source areas and the basin outlet. At mean annual
temporal scale, SDRW generally assumes values ⱕ1 (Renfro
1975; Sedimentation 1975; Bagarello et al. 1991).
For a given event, SDRW can be >1 because sediments deposited on the hillslopes or stored into the channel network,
in some previous events, can be removed.
Sediment transport and deposition in the basin hillslopes are
physical processes distinct from transport and deposition
within the channel network. Therefore hillslope sediment delivery processes and channel processes should be considered
and modeled separately (Atkinson 1995).
At mean annual temporal scale, Playfair’s law of stream
morphology (Boyce 1975) can be applied: over a long time a
stream must essentially transport all sediment delivered from
the hillslopes to it. Therefore, at mean annual temporal scale
the sediment delivery ratio SDRW appears to be a measure of
the efficiency with which materials eroded from hillslopes are
delivered to the stream system (Boyce 1975).
Leopold et al. (1964) suggested that materials eroded from
a basin are only temporarily stored in the floodplains, and
Schumm (1972) assumed that in a long time period a stream
will discharge essentially all of the sediment it receives. In
other words, for predicting the mean annual value of the basin
sediment yield, the sediment delivery problem can be simplified neglecting the channel component.
For a small basin, having an ephemeral channel network
and with no well-developed floodplains, the channel sediment
JOURNAL OF HYDROLOGIC ENGINEERING / OCTOBER 2000 / 411
delivery component also could be roughly neglected at event
scale. In other words, the channel network of a small basin is
generally constituted of channels having narrow cross sections
and short channel lengths, and therefore it can be assumed, at
event scale, that sediments eroded from hillslopes are not
stored within the channel network.
Richards (1993) suggested that sediments are produced from
different sources distributed throughout the basin and that sediment delivery processes may be modeled employing a spatially distributed approach considering individual fields (Kling
1974).
To apply a spatially distributed strategy at the basin scale
requires the choice of a soil erosion model and a spatial disaggregation criterion for the sediment delivery process. The
existing difficulties of physically based modeling (numerous
input parameters, differences between the scale of measurement of the input parameters and the scale of basin discretization, uncertainties of the selected model equations, etc.) increased the attractiveness of a parametric soil erosion model,
like Universal Soil Loss Equation USLE (Wischmeier and
Smith 1965) or its revised version RUSLE (Renard et al.
1994). For modeling the spatial disaggregation of the sediment
delivery, the basin can be discretized into morphological units
(Bagarello et al. 1993) (i.e., areas of clearly defined aspect,
length, and steepness) (Fig. 1).
For modeling the within-basin variability of the hillslope
sediment delivery processes, Ferro and Minacapilli (1995) proposed a sequential approach. Basically, this approach follows
the sediment mass in a Lagrangian scheme and applies appropriate delivery factors to each sequential modeling morphological unit (Novotny and Chesters 1989). Neglecting the
channel sediment delivery component, Ferro and Minacapilli
(1995) proposed to calculate the sediment delivery ratio SDRi
of each morphological unit i into which the basin is divided.
In particular, SDRi depends on the travel time tp,i of the eroded
particles along the hydraulic path, from the considered area to
the nearest stream reach (Fig. 1). The travel time is assumed
to increase as the length lp,i of the hydraulic path increases and
as the square root of the slope sp,i of the hydraulic path decreases. For evaluating the travel time tp,i of the particles
eroded from a given morphological unit i, the travel times into
all morphological areas that are localized along the path between the ith unit and the nearest stream reach have to be
summed. This gives
tp,i =
lp,i
兹sp,i
冘
Np
=
j=1
␭i, j
(1)
兹si, j
in which Np = number of morphological units localized along
the hydraulic path j; ␭i, j and si, j = length of slope of each
morphological unit i localized along the hydraulic path j, respectively. The ratio ␭i, j /兹si, j is the travel time along the morphological unit i of the hydraulic path j; according to Chézy’s
uniform open channel flow equation, 兹si, j is directly proportional to the flow velocity along the morphological unit
length ␭i, j .
The analysis carried out for 13 Sicilian and 2 Calabrian
basins (Ferro and Minacapilli 1995; Ferro 1997) to determine
the empirical cumulative distribution function of the travel
time showed that the SDRi ratio has the following expression:
冉
SDRi = exp(⫺␤tp,i) = exp ⫺␤
lp,i
冊
兹sp,i
冋 冉冘 冊册
Np
= exp ⫺␤
j=1
␭i,j
兹si, j
(2)
in which ␤ = coefficient. Taking into account Chézy’s scheme,
the ␤ coefficient lumps together the effects due to roughness
and runoff along the hydraulic path. Then ␤ is affected by the
roughness distribution along the flow path and is time dependent [i.e., for a given basin ␤ is dependent on the temporal
scale (event, annual, and mean annual)].
In this paper, a sediment delivery distributed (SEDD) model
based on (2) and USLE, in which different expressions of the
erosivity and topographic factors are considered, is proposed.
The SEDD model is calibrated by using the sediment yield
measurements carried out in three small Calabrian experimental basins at the event and annual scales. Finally, a Monte
Carlo technique will be used for testing the effect of model
parameters uncertainty on calculated sediment yield.
SEDD MODEL
The sediment production Yi [t], of each morphological unit
i into which the basin is divided, is simply calculated by the
following equation:
Yi = SDRi Ai SUi
(3)
in which Ai = soil loss (t/ha) from the ith morphological unit,
which has to be estimated by the selected erosion model; and
SUi = area (ha) of the morphological unit.
Soil loss Ai is estimated by the following variation of the
USLE (Wischmeier and Smith 1965):
Ai = EFt,i Ki Li Si Ci Pi
(4)
in which EFt,i = erosivity factor for a given temporal scale t
(event, annual, and mean annual) of the ith morphological unit
(t/ha/unit of Ki); Ki = soil erodibility factor estimated by the
procedure of Wischmeier et al. (1971) (t h/kg m2); Ci = cover
and management factor; Pi = support practice factor; and Li Si
= topographic factor. One uses two expressions for the topograhic factor (both from the RUSLE model). One is proposed
by McCool et al. (1989)
Li Si =
FIG. 1.
Scheme of Basin Discretized into Morphological Unit
412 / JOURNAL OF HYDROLOGIC ENGINEERING / OCTOBER 2000
Li Si =
冉 冊
冉 冊
␭i
22.13
␭i
22.13
mi
(10.8 sin ␣i ⫹ 0.03)
if tan ␣i < 0.09
(5a)
(16.8 sin ␣i ⫺ 0.5)
if tan ␣i ⱖ 0.09
(5b)
mi
in which ␭i = slope length of the ith morphological area; ␣i =
slope angle; mi = fi /(1 ⫹ fi ); and fi = sin ␣i /0.0896 (3 sin0.8␣i
⫹ 0.56) = ratio of rill to interrill erosion, and one is proposed
by Moore and Burch (1986)
Li Si =
冉 冊冉 冊
␭s,i
22.1
0.6
sin ␣i
0.0896
1.3
(6)
in which ␭s,i = ratio between the area of the ith morphological
unit and the width measured along the contour line.
EXPERIMENTAL BASINS AND MEASUREMENT
TECHNIQUE
The studied area is located near Crotone (35 m above sea
level (asl), 39⬚09⬘02⬙ N, 17⬚18⬘10⬙ E) in the ephemeral basin
of Crepacuore stream, which drains to the Ionian sea (Fig. 2).
Basin W1 is covered with a grass and shrub vegetation. Basin
W2 was planted in 1968 with Eucalyptus occidentalis and was
coppiced twice in 1978 and 1990 (coppicing cycle equal to 12
years). The forest cover is discontinuous, and 20% of the basin
area is bare. Basin W3 was covered with a Eucalyptus occidentalis high forest in the period 1968–1986. In 1986 the forest cover was coppiced. The forest cover is uniform, and the
percentage of bare area is equal to 3% of the total basin area
(Cantore et al. 1994). For each investigated basin, the basin
area AW (ha); the mean altitude Hm (m asl); the minimum elevation Z min (m asl); the maximum elevation Z max (m asl); the
mean slope s (%); and the percentage of sand, silt, and clay
of the soil are listed in Table 1. Water discharge, suspended
sediment concentration, and rainfall were measured at the outlet of each basin. Each basin is monitored by an H-flume weir
(Brakensiek et al. 1979), and measurement of flow depth is
carried out at the end of a rectangular channel by a mechanical
recording water level gauge. For each flood event, t, the basin
hydrograph is available, and the runoff factor of Williams
(1975) Rd,t (kg m2/ha h) can be calculated according to the
following relationship (Bagarello et al. 1990):
0.8776
Rd,t =
(qp,tVt)0.56
AW
(7)
where qp,t = peak flow rate of the flood event t (m3/s); Vt =
runoff volume (m3); and AW = basin area (ha). The sampling
device is a Coshocton Wheel (Parson 1954; Carter and Parson
1967) collecting a sample (⬃1/200) of the flow volume. Each
collected sample flows into appropriately-sized tanks. At the
end of each event, the collected suspension is well mixed.
One-liter suspension samples at different heights are collected,
and the suspended solid content concentration in the 1 L (g/
L) is determined by oven-drying at 105⬚C. The sediment yield
of each event is calculated by the product of the mean concentration Cm and the measured runoff volume. Rainfall is
measured by a recording rain gauge, and rainfall erosivity factor Rt (Wischmeier and Smith 1965) is calculated for each
rainfall event t. Forty-six selected events were measured at
basin W1, 52 events at basin W2, and 42 events at basin W3
in the recording period 1978–1994 (Avolio et al 1980; Callegari et al. 1994; Cantore et al. 1994; Cinnirella et al. 1998).
For each basin, the soil data (grain-size distribution, total
organic carbon, soil structure index, and permeability index)
used for estimating the soil erodibility factor were obtained
from 10 sampling sites. Then, for each basin, the corresponding 10 soil erodibility factor values were calculated by the
nomograph of Wischmeier et al. (1971). A characteristic basin
value Kb equal to the mean of the 10 values was assumed.
For basin W1 a single crop management factor Cb = 0.086
(Wischmeier and Smith 1978) and Kb = 0.5 were used. For all
morphological units of basins W2 and W3, Cb was set equal
to 0.164 (Cinnirella et al. 1998), whereas Kb = 0.55 for basin
FIG. 2.
TABLE 1.
Basin
(1)
AW
(ha)
(2)
W1
W2
W3
1.473
1.375
1.654
Experimental Calabrian Basins
Characteristic Data of Investigated Basins
Hm
Z min
Z max
(m asl) (m asl) (m asl)
(3)
(4)
(5)
155
128
114
90
85
85
122
103
98
s
(%)
(6)
Sand
(%)
(7)
Silt Clay
(%) (%)
(8)
(9)
53
35
24
14.0
14.6
20.7
44.5 41.5
49.2 36.2
45.5 33.8
W2 and Kb = 0.58 for basin W3 were used (Avolio et al. 1980).
Management works are not carried out in the experimental
basins, and therefore the support practice factor of each morphological unit Pi will be assumed equal to 1.
CALIBRATING SEDD MODEL
Modern understanding of rainfall erosion processes provides
for more process-based indices taking into account that soil
loss increases with rainfall amount, rainfall intensity, and runoff (Kinnell et al. 1994; Kinnell 1997).
The product of the storm rainfall kinetic energy E and the
maximum rainfall intensity measured over a 30-min time interval I30 has been used for the rainfall erosivity factor R in
USLE and RUSLE.
Raindrop impact is often involved in the detachment of soil
particles from soil surface, whereas splash by itself is not a
very efficient transport mechanism. However raindrop impact
induces turbulence in the overland flow and increases particle
transport (rain-induced flow transport) (Kinnell 1990, 1991).
When runoff develops, the erosion rate increases because of
the development of rills representing an efficient transport system (Kinnell et al. 1994). When rill erosion occurs, in modeling activities, the rill-interrill concept is often applied: the
sediments discharged with the flow result from raindrop and
overland flow detachment in the interrill areas together with
channel flow detachment in the rills.
According to these physical reasons and suggestions of Foster et al. (1977), the following erosivity factor EFt including
terms of rainfall and runoff erosivity was used:
EFt = aRt ⫹ bRd,t
(8)
JOURNAL OF HYDROLOGIC ENGINEERING / OCTOBER 2000 / 413
in which Rt = rainfall erosivity index of Wischmeier and Smith
(1965) of a given rainfall event t; Rd,t = Williams’s runoff
factor of a given runoff event corresponding to the rainfall
event t; and a and b = two numerical constants representing
the weight of each erosivity term (rainfall and runoff ) on the
total erosivity factor.
Different values of a and b (respecting the condition a ⫹ b
= 1) should correspond to a different ratio between interrill
and rill erosion (Kinnell 1997) even if Foster et al. (1977)
suggested that ‘‘research is needed to precisely define the relationship between rill erosion and runoff erosivity and that
between interrill erosion and rainfall erosivity.’’ Furthermore,
Rd,t also lumps the effects of sediment transport in rills and in
the interrill areas and therefore the runoff factor also represents
the effects of the sediment delivery processes (Williams 1975).
Each experimental basin was divided into morphological
units, and for each one the topographic factor Li Si was calculated by (5) and (6).
According to (2)–(4) and (8), the sediment yield Yi,t of each
i morphological unit for a given event t has the following
expression:
Yi,t = (aRt ⫹ bRd,t)Kb Cb Li Si [exp(⫺␤t tp,i)]SUi
TABLE 2.
Values of ␤m Coefficient of Each Investigated Basin
Eq. (5)
Eq. (6)
a = 1;
a = 0.3; a = 1;
Basin b = 0 a = b = 0.5 b = 0.7 b = 0
(1)
(2)
(3)
(4)
(5)
W1
W2
W3
0.0201
0.0157
0.0165
0.0135
0.0073
0.0114
0.0082
0.0032
0.0085
0.0418
0.0310
0.0197
a = 0.3;
a = b = 0.5 b = 0.7
(6)
(7)
0.0314
0.0212
0.0143
0.0231
0.0163
0.0110
(9)
in which the ␤ coefficient, for a given basin, assumes a different value ␤t for each event. The basin sediment production
of a given event t, Yb,t , is equal to the sum of the sediment
yields produced by all morphological units into which the basin is divided (sediment balance equation)
冘
Nu
Yb,t = (aRt ⫹ bRd,t)Kb Cb
Li Si [exp(⫺␤t tp,i)]SUi
(10)
i=1
in which Nu = number of morphological units into which the
basin is divided.
According to (10), for applying the distributed model, the
a, b, and ␤ coefficients have to be determined. The coefficients
a and b were fixed according to the following hypotheses for
the total erosivity factor.
a = 1;
b=0
a = b = 0.5
a = 0.3;
b = 0.7
(11a)
(11b)
(11c)
The first hypothesis corresponds to the classic scheme of
USLE where the erosion process is assumed to be mainly due
to rainfall (interrill erosion). Notwithstanding, the USLE rainfall factor does not consider runoff directly, the measurements
carried out to determine USLE ensure that the rainfall factor
is the driving force of sheet and rill erosion (Renard and Ferreira 1993). The second hypothesis gives equal weight for the
two hydrological processes of rainfall (interrill erosion) and
and runoff (rill erosion), and the third hypothesis assumes that
the runoff processes (rill erosion) is more important than the
rainfall processes for hillslope soil erosion.
Taking into account that Rd,t was used originally by Williams
(1975) to modify USLE for calculating basin sediment yield
without employing SDRW , the hypothesis a = 0 and b = 1 or
other hypotheses in which the weight of the runoff erosivity
term is prevailing can be neglected.
For each basin, for the different hypotheses, and for each
rainfall-runoff event t, the corresponding ␤t value was calculated by (10) in which Yb,t is set equal to the measured basin
sediment yield Ym,t . For each basin, for each hypothesis, and
for each topographic factor equation [(5) and (6)], Table 2 lists
the median value ␤m of the calculated ␤t values. For each hypothesis and the expression of the topographic factor proposed
by McCool et al. (1989) [(5)], the measured basin sediment
yield Ym,t at the event scale and the Yc,t values calculated by
414 / JOURNAL OF HYDROLOGIC ENGINEERING / OCTOBER 2000
FIG. 3. Comparison for USLE Model [Eqs. (5) and (10)] between Measured and Calculated Sediment Yield Values at Event
Scale
TABLE 3.
ity Factor
Results of Analysis of Three Hypothesis for ErosivEq. (5)
Eq. (6)
a
(1)
b
(2)
r
(3)
MSE
(t 2)
(4)
r
(5)
MSE
(t 2)
(6)
1.0
0.5
0.3
0.0
0.5
0.7
0.710
0.748
0.771
7.515
5.528
4.299
0.710
0.748
0.771
7.510
5.532
4.299
(10), using as estimate of ␤t the median value ␤m, are plotted
in Fig. 3.
For each hypothesis and for each selected topographic factor
equation [(5) and (6)] the relationship between calculated and
measured sediment yield values were also compared using the
correlation coefficient r and the mean square error (MSE) (Table 3). Even if the third hypothesis [(11c)] is characterized by
the maximum r value and the minimum square error, measurements of peak flow rate and runoff volume at the basin
outlet are often unavailable, and therefore q p,t and Vt should
be estimated by a hydrological model.
The good agreement between Ym,t and Yc,t also checked for
the USLE hypothesis (a = 1 and b = 0) has encouraged the
choice of this hypothesis [(11a)] and to restrict the following
analysis to this choice. The USLE-based hypothesis is also
justified as the sediment delivery processes will be simulated
by the sediment delivery ratio of each morphological unit
[(2)].
EVENT CALIBRATION
At first, for each basin, the predictive capability of the
SEDD model for the USLE hypothesis (a = 1 and b = 0) was
tested at the event scale. Different expressions for L i Si were
used and ␤m was estimated using only the first 15 events of
each recording period.
Using these ␤m estimates listed in Table 4, the agreement
between measured and calculated sediment yield values was
controlled for the remaining set of Ne —15 events, in which
Ne = total number of events available for each investigated
basin. This agreement was characterized by r and MSE values
[r = 0.69, MSE = 8.01 (t2) for (5) and r = 0.69, MSE = 8.19
(t2) for (6)], which are comparable with those listed in the first
row of Table 3.
To determine whether the agreement between measured and
calculated sediment yield is independent of the 15 events selected for calibrating the model (i.e., for calculating ␤m), the
estimate stability of the ␤m coefficient was controlled. Therefore, for each basin the cumulative distribution function of the
␤t coefficient (Fig. 4) was used for drawing 15,000 random
sequences, each having a sample size equal to 15. For each
sequence, the median value ␤m was calculated, and the
15,000␤m values were used to estimate the mean ␮(␤m). The
␮(␤m) values listed in Table 5 are very close to the previous
␤m estimates (listed in Columns 2 and 5 of Table 2) obtained
using all available ␤t values.
For each estimate criterion of L i Si , the relationship between
TABLE 4. Values of ␤m Coefficient Estimated by First 15
Events of Recording Period Assuming USLE Hypothesis (a ⴝ 1
and b ⴝ 0)
␤m
Basin
(1)
Eq. (5)
(2)
Eq. (6)
(3)
W1
W2
W3
0.0150
0.0141
0.0194
0.0338
0.0291
0.0228
FIG. 4. Probability Distribution of ␤t Coefficient for Investigated Experimental Basins
TABLE 5. Statistical Parameters ␮(␤m) and ␴(␤m) of Sequences Generated by Monte Carlo Technique
Basin
Model
(1)
Parameter
(2)
W1
(3)
W2
(4)
W3
(5)
Eq. (5)
␮(␤m)
␴(␤m)
␮(␤m)
␴(␤m)
0.0206
0.0110
0.0421
0.0174
0.0166
0.0060
0.0318
0.0071
0.0192
0.0043
0.0225
0.0046
Eq. (6)
JOURNAL OF HYDROLOGIC ENGINEERING / OCTOBER 2000 / 415
␤m listed in Table 2 (Columns 2 and 5) and the mean value of
the clay percentage (CL) of the soils falling in each basin is
shown in Figs. 5(a and b). This correlation can be justified by
taking into account that a relationship between particle grainsize distribution and sediment delivery exists. According to
Williams (1975) when the median particle size increases from
0.001 to 0.1 mm, the delivery ratio decreases from 0.37 to
0.06, and he proposed the following expression for SDRi :
SDRi = exp(⫺␾D1/2
50 tp,i)
(12)
in which D50 = median particle size; and ␾ = routing coefficient. According to (12) the grain-size distribution affects the
␤m coefficient but in a manner that cannot be readily quantified
because travel time and particle size are also correlated.
For each selected expression of the topographic factor and
for each basin, ␤t was correlated with the event runoff coefficient, RCt equal to the ratio between the runoff Dt (mm) and
the corresponding rainfall At (mm), in order to provide an estimate criterion of ␤t . The relationship between ␤t and RCt ,
shown in Fig. 6 as an example for basin W3, may be approximated by the following mathematical form:
␤t = c ⫹ d ln RCt
(13)
in which c and d = coefficients.
The correlation in (13) can be explained by taking into account that for increasing values of RCt the hillslope sediment
transport and SDRi values generally increase. In other words,
a more efficient hillslope sediment transport, which is characterized by high SDRi values, is obtainable, according to (2)
(Fig. 6) by small ␤t values. The agreement between measured
and calculated sediment yield values is shown, as an example
for the topographic factors calculated by (5), in Fig. 7 and is
characterized by r = 0.765 and MSE = 11.679 (t2). The com-
FIG. 6. Relationship between ␤t and Runoff Coefficient RCt
for W3 Basin and Topographic Factors Calculated by Models of:
(a) McCool et al. (1989); (b) Moore and Burch (1986)
FIG. 7. Comparison, for Eq. (5), between Measured and Calculated Sediment Yield, at Basin Scale, with ␤t Estimated by
Eq. (13)
parison between Fig. 7 and Fig. 3(a), and the corresponding
statistical parameters r (Table 3, first row), show that (13) allows a reliable ␤t estimate and a good agreement between Ym,t
and Yc,t .
ANNUAL CALIBRATION
FIG. 5. Relationship between ␤m and Soil Clay Percentage for
Expressions of Topographic Factors of: (a) McCool et al. (1989);
(b) Moore and Burch (1986)
416 / JOURNAL OF HYDROLOGIC ENGINEERING / OCTOBER 2000
For testing the applicability of the SEDD model at annual
scale, for each year (10) was applied using the annual value
of the rainfall erosivity factor [(8) with a = 1 and b = 0)]
calculated adding all Rt values of the rainfall events happening
during the year. The model was calibrated (i.e., ␤m coefficient
was calculated) using the available data set of a basin (e.g.,
basin W1) and the other two data sets (e.g., basins W2 and
W3) were used to validate the model comparing the calculated
sediment yield values Yc,a with the measured ones Ym,a.
For L i Si calculated by (5) and (6) for each basin, Table 6
lists the ␤m coefficient, r, and MSE values corresponding to
the comparison between measured Ym,a and calculated Yc,a annual values of sediment yield.
The comparison between Fig. 8 and Fig. 3(a), and the corresponding statistical parameter r, shows that the reliability of
the sediment delivery distributed modeling increases with the
temporal scale (from the event scale to the annual scale). This
result can be explained by taking into account that the accuracy of the soil erosion model is dependent on which temporal
scale is used (Risse et al. 1993). Further research to improve
the agreement at the event scale between measured and calculated sediment yield values could hypothesize an expression
of the sediment delivery ratio of each morphological unit also
lumping the channel network delivery processes.
TABLE 6. Values of ␤m Coefficient and Statistical Parameters
of Model Validation at Annual Scale
Basin
used for
calibration
(1)
Eq. (5)
␤m
(2)
r
(3)
MSE
(4)
␤m
(5)
r
(6)
MSE
(7)
W1
W2
W3
0.0132
0.0072
0.0142
0.70
0.55
0.75
42.3
79.6
26.5
0.3090
0.0210
0.0173
0.77
0.55
0.75
22.8
30.4
91.3
Eq. (6)
EFFECT OF PARAMETER UNCERTAINTY ON
CALCULATED SEDIMENT YIELD
The recognized random variability of hydrologic variables
(Haan 1977; Bogardi et al. 1985) in addition to the circumstance that input model parameter values are only estimates
(as the actual values are not known with certainty) has suggested the inclusion of uncertainty analysis in modeling activities (Hession et al. 1996).
For evaluating the uncertainty of model parameters on calculated sediment yield, Monte Carlo techniques can be applied. Repeated simulations are performed with the model using randomly selected input parameter values. For each
simulation the input parameter values are chosen using their
predetermined probability distribution. The simulation process
is repeated for a number of iterations sufficient to estimate a
probability distribution of the output variable.
According to MacIntosh et al. (1994) the major types of
uncertainty are knowledge uncertainty and stochastic variability. Knowledge uncertainty is due to incomplete understanding
of studied phenomena, inadequate measurement of system
properties, and input data availability. Stochastic variability is
due to random variability of the studied natural environment
and can be divided into temporal and spatial variability. In the
uncertainty analysis, the knowledge uncertainty and stochastic
variability has to be distinguished in order to detect their effects on output uncertainty. Stochastic variability, being a natural property of the system, can be quantified but cannot be
reduced. The knowledge uncertainty can be reduced by decreasing the possible range of parameter estimates through appropriate physical measurements. In other words, knowledge
uncertainty can be used as indicator of the benefits due to
additional measurements.
When modeling a basin discretized into morphological
units, the actual correlation structure of the system sediment
yield model-basin is not known. In other words, some factors
of the USLE model are physically related (R and C factors),
and others are dependent on the discretization level of the
basin. In the following, taking into account that the correlation
structures are dependent on many factors (studied site, scale,
considered property, etc.) and that at present additional research is needed to determine the appropriate level of correlation at basin scale (Hession et al. 1996), the stochastic variability of the basin sediment yield Yb,t calculated by SEDD,
at the event and annual scales, will be studied for each experimental basin, neglecting the correlation between the different RUSLE factors.
FIG. 8. Comparison, for Eqs. (6) and (10), between Measured
and Calculated Sediment Yield Values at Annual Scale
JOURNAL OF HYDROLOGIC ENGINEERING / OCTOBER 2000 / 417
To perform the Monte Carlo simulations, a probability distribution must be assigned for the following uncertain parameters: rainfall erosivity, soil erodibility, crop factor, ␤t coefficient, or runoff coefficient. For each basin, the measured event
rainfall erosivity values Rt were found to be lognormally distributed (Fig. 9) with mean ␮ and standard deviation ␴ listed
in Table 7.
The comparison between the empirical cumulative frequency distribution of crop factor of each event Ct , calculated
by Cinnirella et al. (1998), and a lognormal distribution (LN2)
having mean value and standard deviation equal to the sample
values (Table 7) is shown in Fig. 10.
For determining the frequency distribution of ␤t coefficiency, for each basin having known values of Kb and Cb, the
sediment yield Yt and the rainfall erosivity Rt measured for
each event t were first used to calculate by (10) the corresponding ␤t coefficient.
For basin W1 ␤t is distributed according to a beta distribution,
whereas for basins W2 and W3 the Gauss law agrees with the
empirical frequency distribution of ␤t variable (Fig. 4).
The soil erodibility factor was treated as having only knowledge uncertainty representing the range of possible values
available from the literature. The knowledge uncertainty, due
to the difficulty of establishing an appropriate value for use in
this study’s model for the examined soil type, was considered
by a uniform distribution that describes K values ranging from
FIG. 9. Probability Distribution of Event Rainfall Factor Rt for
Basin W1
TABLE 7. Parameters of Cumulative Distributions Functions
(CDF) of Characteristic Variables at Event Scale
Basin Variable
(1)
(2)
W1
Rt
Kb
Cb
RCt
␤t
␤t
W2
W3
Ri
Kb
Cb
RCt
␤t
␤t
Rt
Kb
Cb
RCt
␤t
␤t
CDF
(3)
Parameters
(4)
␮ = 1.5929; ␴ = 0.9962
Kmin = 0.4; Kmax = 0.6
␮ = ⫺2.6538; ␴ = 1.3392
␣ = 1.3592; ␥ = 2.4071
A = 0.012; B = 0.703
Beta
␣ = 0.9689; ␥ = 4.5144
A = ⫺0.0121; B = 0.218
Beta
␣ = 0.9613; ␥ = 5.5818
A = ⫺0.0079; B = 0.4249
LN2
␮ = 1.9154; ␴ = 1.1355
Uniform Kmin = 0.4; Kmax = 0.6
LN2
␮ = ⫺1.9745; ␴ = 1.3853
Beta
␣ = 0.962; ␥ = 1.7096
A = 0.032; B = 0.525
Gauss
␮ = 0.0166; ␴ = 0.0187
Gauss
␮ = 0.0318; ␴ = 0.0220
LN2
␮ = 2.0811; ␴ = 1.1116
Uniform Kmin = 0.4; Kmax = 0.6
LN2
␮ = ⫺1.9751; ␴ = 0.9661
Beta
␣ = 1.2526; ␥ = 2.9635
A = 0.017; B = 0.552
Gauss
␮ = 0.0192; ␴ = 0.0135
Gauss
␮ = 0.0225; ␴ = 0.0143
LN2
Uniform
LN2
Beta
LS model
(5)
Eqs.
Eqs.
Eqs.
Eqs.
(5)
(5)
(5)
(5)
and
and
and
and
(6)
(6)
(6)
(6)
(5)
(5)
(5)
(5)
and
and
and
and
(6)
(6)
(6)
(6)
Eq. (5)
Eq. (6)
Eqs. (5)
Eqs. (5)
Eqs. (5)
Eqs. (5)
and
and
and
and
(6)
(6)
(6)
(6)
Eq. (5)
Eq. (6)
Eqs.
Eqs.
Eqs.
Eqs.
Eq. (5)
Eq. (6)
418 / JOURNAL OF HYDROLOGIC ENGINEERING / OCTOBER 2000
FIG. 10.
Probability Distribution of Crop Factor for Basin W2
a minimum Kmin = 0.4 to a maximum Kmax = 0.6. The topographic factor was treated as a constant deterministic value of
each morphological unit, under the assumption that the lengths
and the slope of the units are controlled.
Each Monte Carlo simulation was carried out drawing at
random a value from each of the distributions of rainfall erosivity, soil erodibility, crop factor, and ␤t coefficient. These
values were then used with the constant topographic factor of
each morphological unit as input to the model [(10)], whose
output Yc,t represented one iteration of the simulated scenario.
The resampling of rainfall erosivity, soil erodibility, crop factor, and ␤t was repeated 15,000 times, resulting in 15,000 estimates of the output basin sediment yield Yc,t . For each basin,
the 15,000 values of Yc,t were used to define the theoretical
probability distribution P(Yc,t) to compare with the empirical
frequency distribution F(Ym,t) of the measured values Ym,t of
the basin sediment yield. For each basin and for the case of
topographic factors calculated by (6), the agreement between
the distributions F(Ym,t) and P(Yc,t) is shown in Fig. 11. The
mean value ␮ and the standard deviation ␴ estimate of the
sediment yield (shown as an example for basin W1 in Fig. 12)
were essentially constant for iterations >3,000. The agreement
between the frequency distribution of the measured sediment
yield values and the generated ones remained acceptable when
the resampling was carried out using a constant ␤t value equal
to ␤m.
The Monte Carlo technique was also applied to resampling
of runoff coefficient RCt values, which are distributed according to a beta distribution (Table 7) and calculating ␤t by (13).
The agreement between the generated frequency distribution
P(Yc,t), with the topographic factors calculated by the expression of Moore and Burch (1986) and ␤t estimated by (13), and
the measured F(Ym,t) distribution (Fig. 13), shows the ability
of the model to reproduce the empirical distribution of basin
sediment yield.
The parameter uncertainty analysis was also developed at
annual scale, even though the basins have slightly different
recording periods (for basins W1 and W3 the recording period
is 13 years, and for basin W2 the recording period is equal to
14 years). At the annual scale, for each basin, the rainfall erosivity factor is lognormally distributed, and the normal distribution agrees with the empirical frequency distribution of the
␤t coefficient. Table 8, which lists the statistical parameters,
shows a good agreement between the mean value of the two
distributions, whereas basin W1 underestimates the median
value m and basin W3 underestimates the standard deviation
␴. Finally, for basin W1 the theoretical distribution is more
skewed than the empirical one, and for basins W2 and W3 the
empirical values of G are greater than the theoretical ones.
FIG. 12. Mean Value and Standard Deviation of Generated
Sediment Yield Sequences
FIG. 11. Comparison, for Each Basin, between Measured Sediment Yield Frequency Distribution and That Generated by
Monte Carlo Technique Using Distribution of ␤t Coefficient
In conclusion, the parameter uncertainty analysis showed
that the SEDD model, even if the ␤t coefficient is estimated
by (13), is able to generate a theoretical distribution function
of basin sediment yield, which has a satisfactory agreement
with the cumulative distribution function of the measured
value.
CONCLUSIONS
For evaluating the quantity of sediment that is transferred,
in a given time interval, from eroding sources through the
hillslopes and the channel network to the basin outlet, a plot
soil erosion model can be coupled with a distributed mathematical function expressing the sediment transport efficiency.
At first, for a basin discretized into morphological units, a
distributed model based on RUSLE, in which different expressions of the erosivity and topographic factors are consid-
FIG. 13. Comparison, for Each Basin, between Measured Sediment Yield Frequency Distribution and That Generated by
Monte Carlo Technique, Using ␤m Estimated by Eq. (13)
JOURNAL OF HYDROLOGIC ENGINEERING / OCTOBER 2000 / 419
TABLE 8. Statistical Parameters of Measured and Theoretical (Generated by Monte Carlo Technique) Distribution of Basin Sediment
Yield at Annual Scale
EQ. (5)
␤ (Median Value)
Parameter
(1)
EQ. (6)
␤ (from Theoretical Law)
␤ (Median Value)
␤ (from Theoretical Law)
W1
(2)
W2
(3)
W3
(4)
W1
(5)
W2
(6)
W3
(7)
W1
(8)
W2
(9)
W3
(10)
W1
(11)
W2
(12)
W3
(13)
Empirical
Theoretical
13
12,886
14
13,006
13
12,886
13
12,886
14
13,006
13
12,886
13
12,886
14
13,006
13
12,886
13
12,886
14
13,006
13
12,886
Empirical
Theoretical
10.473
8.170
27.808
23.916
8.249
5.698
10.473
11.782
27.808
25.084
8.249
6.129
10.473
8.398
27.808
24.234
8.249
5.641
10.473
11.403
27.808
24.392
8.249
6.393
Empirical
Theoretical
10.606
4.348
13.032
12.793
4.956
3.323
10.606
5.693
13.032
12.346
4.956
3.076
10.606
4.469
13.032
12.794
4.956
3.290
10.606
5.684
13.032
11.764
4.956
3.056
Empirical
Theoretical
COV
Empirical
Theoretical
G
Empirical
Theoretical
9.007
9.459
39.646
27.581
10.412
6.042
9.007
14.809
39.646
31.095
10.412
7.447
9.007
9.696
39.646
28.120
10.412
5.943
9.007
13.841
39.646
30.386
10.412
7.900
0.860
1.158
1.426
1.153
1.262
1.060
0.860
1.257
1.426
1.240
1.262
1.215
0.860
1.155
1.426
1.160
1.262
1.053
0.860
1.214
1.426
1.246
1.262
1.236
1.224
1.866
2.597
1.860
2.941
1.665
1.224
2.063
2.597
2.004
2.941
1.957
1.224
1.838
2.597
1.874
2.941
1.657
1.224
1.942
2.597
1.997
2.941
1.955
N
␮
m
␴
Note: COV = coefficient of variation.
ered, coupled with the sediment delivery ratio of each morphological unit was proposed. The model neglects the channel
sediment delivery component and can be applied at mean annual temporal scale or at event scale for a small basin having
an ephemeral channel network and with no well-developed
floodplains. The analysis, carried out by the sediment yield
measurement at the event scale in three Calabrian experimental
basin, initially used an erosivity factor that, as suggested in
the literature, linearly weights the rainfall factor of Wischmeier
and Smith and the runoff erosivity factor of Williams (1975).
The analysis showed that the agreement between measured Ym,t
and calculated Yc,t sediment yield values is independent of the
selected equations for estimating the topographic factors and
can be simple obtained by using the rainfall erosivity factor.
At the event scale the model was calibrated using the first
15 events of the recording period to estimate the ␤m coefficient. A good agreement between measured and calculated sediment yield values was obtained validating the model by the
remaining set of events for each basin. The analysis also
showed that this agreement is independent of the 15 events
used for estimating the ␤m coefficient.
The median value ␤m of the ␤t coefficient estimated by all
events of the recording period was then correlated with the
event runoff coefficient [(13)] by taking into account that the
efficiency of the sediment transport is linked to hillslope runoff. The analysis showed that (13) allows a reliable estimate
of the ␤t coefficient because by using (13) a good agreement
between measured and calculated sediment yield values was
obtained.
The applicability of the SEDD model was also tested at the
annual scale, and the analysis showed that the reliability of
the sediment delivery distributed approach, measured by the
correlation between measured and calculated annual sediment
yield values, improves with the temporal scale (from the event
scale to the annual scale).
To represent the model limitations and uncertainties, the effect of parameter uncertainty on calculated sediment yield was
also studied. Monte Carlo simulations were performed using
predetermined probability distributions of the input parameters.
The resampling of the model factors produced 15,000 estimates of the output basin sediment yield Yc,t , which defined
the theoretical probability distribution P(Yc,t), to compare with
420 / JOURNAL OF HYDROLOGIC ENGINEERING / OCTOBER 2000
the empirical frequency distribution F(Ym,t) of the measured
values Ym,t .
For each studied basin, the uncertainty analysis developed
at event scale, showed that the SEDD model, even if the ␤t
coefficient is estimated by the runoff coefficient, is able to
generate a theoretical distribution of sediment yield P(Yc,t) having a satisfactory agreement with the cumulative distribution
function of the measured values.
ACKNOWLEDGMENTS
Research is supported by a grant from the Ministero Università e
Ricerca Scientifica e Tecnologica, Governo Italiano, ‘‘Produzione di
sedimenti a scala di bacino e tecniche di ricostruzione naturalistica del
sistema alveo-versante’’ and quota 60%. Both authors set up the research,
analyzed the results, and participated in writing the paper.
APPENDIX I.
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APPENDIX II.
NOTATION
The following symbols are used in this paper:
A
Ai
At
AW
a
B
b
Cb
Ci
=
=
=
=
=
=
=
=
=
Cm
Ct
Dt
D50
E
EFt
EFt,i
=
=
=
=
=
=
=
fi
G
Hm
I30
=
=
=
=
Kb
Ki
L i Si
lp,i
mi
Ne
Np
=
=
=
=
=
=
=
Nu =
parameter of beta distribution (Table 7);
soil loss from each morphological unit;
rainfall depth of each event t;
basin area;
numerical constant;
parameter of beta distribution (Table 7);
numerical constant;
cover and management factor of each basin;
cover and management factor of each morphological
unit;
mean suspension concentration;
cover and management factor of each event;
runoff depth of each event t;
median particle size;
storm rainfall kinetic energy;
erosivity factor for given rainfall event t;
erosivity factor for given temporal scale of each morphological unit;
ratio of rill to interrill erosion for ith morphological unit;
skewness;
mean altitude;
maximum rainfall intensity measured over 30-min time
interval;
soil erodibility factor of each basin;
soil erodibility factor of each morphological unit;
topographic factor of each morphological unit;
length of hydraulic path;
slope length exponent for ith morphological unit;
number of events of each investigated basin;
number of morphological units localized along hydraulic path;
number of morphological units into which basin is divided;
JOURNAL OF HYDROLOGIC ENGINEERING / OCTOBER 2000 / 421
Pi
q p,t
R
RCt
Rd,t
Rt
r
SDRi
SDRW
SUi
s
si, j
=
=
=
=
=
=
=
=
=
=
=
=
sp,i
tp,i
Vt
Yb,t
Yc,t
Yi
Yi,t
=
=
=
=
=
=
=
support practice factor of each morphological unit;
peak flow rate of flood event t;
rainfall erosivity factor in the USLE;
runoff coefficient of each event t;
runoff factor of Williams of each event t;
rainfall erosivity factor of each event t;
coefficient of correlation;
sediment delivery ratio of each morphological unit;
basin sediment delivery ratio;
area of each morphological unit;
mean slope;
slope of each morphological unit i localized along hydraulic path j;
slope of hydraulic path;
travel time of each morphological unit;
runoff volume of flood event t;
basin sediment yield for given event t;
basin sediment yield calculated for given event t;
sediment production of each morphological unit;
sediment yield of each morphological unit for given
event t;
422 / JOURNAL OF HYDROLOGIC ENGINEERING / OCTOBER 2000
Ym,t
Z max
Z min
␣
␣i
␤
␤m
␤m,15
␤t
␥
␭i
␭i, j
=
=
=
=
=
=
=
=
=
=
=
=
␭s,i =
␮(␤m)=
␴(␤m) =
␾ =
basin sediment yield measured for given event t;
maximum elevation;
minimum elevation;
parameter of beta distribution (Table 7);
slope angle of each morphological unit;
coefficient;
median yield of calculated ␤t values;
␤m value corresponding to first 15 historical events;
value of ␤ coefficient of given event t;
parameter of beta distribution (Table 7);
slope length of ith morphological unit;
length of each morphological unit i localized along hydraulic path j;
ratio between area of the morphological unit and width
measured along contour line;
mean value of 15,000␤m values generated by Monte
Carlo techniques;
standard deviation of 15,000␤m values generated by
Monte Carlo techniques; and
routing coefficient.