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 oftenadvantageous to model sediment delivery processes at basin scale using a spatially distributed approach. In thispaper, a sediment delivery distributed (SEDD) model applicable at morphological unit scale, into which a basinis divided, is initially proposed. The model is based on the Universal Soil Loss Equation (USLE), in whichdifferent expressions of the erosivity and topographic factors are considered, coupled with a relationship forevaluating the sediment delivery ratio of each morphological unit. Then the SEDD model is calibrated bysediment yield and rainfall and runoff measurements carried out, at annual and event scales, in three smallCalabrian experimental basis. At event scale, the analysis showed that a good agreement between measured andcalculated basin sediment yields can be obtained using the simple rainfall erosivity factor; the agreement isindependent of the selected equations for estimating the topographic factors. The analysis developed at annualscale showed that the model reliability increases from the event scale to the annual scale. Finally, a Monte Carlotechnique was used for evaluating the effects of the uncertainty of the model parameters on calculated sedimentyield.

INTRODUCTION

According to Golubev (1982) the area of cultivated land inthe world is 14,300,000 km2 and in some regions of the worldcropland covers the major part of the total area. In cultivatedareas, bare soil is often exposed for most of the year withsparse crop vegetation existing for a few months. Computa-tions of Golubev (1982) suggest that soil erosion in the worldis 5.5 times more than during the preagricultural period.

According to Brown (1984) the world is currently losing 23billion t of soil from croplands in excess of new soil formationeach year (Walling and Quine 1992); therefore, accelerated soilerosion is a serious obstacle to the development of a sustain-able agriculture.

Long-term erosion monitoring is necessary to guide the de-velopment and testing of predictive models, to observe differ-ent scenarios due to changes in land use and climate. Moni-toring studies and modeling may be used in combination todevise soil conservation strategies.

Measured soil erosion rates and the processes responsiblefor the sediment transport depend on the spatial scale of ob-servation (Kirkby et al. 1996). Mean soil loss rates from fieldsize areas (100–10,000 m2) are usually much lower than thoseestimated by plot studies (Poesen et al. 1996).

At hillslope and basin scale (>10,000 m2), in addition to thesmaller scale erosion processes, mass movement, hillslopedeposition, and channel processes should be considered to es-tablish the total sediment budget. At hillslope and basin scale,representativeness problems limit the use of plot measure-ments. In fact, measurements from a bounded area providelittle information concerning the local variability of erosionrates and the redistribution processes of soil within a field(Walling and Quine 1991). Tracer techniques, such as 137-cesium measurements (Ritchie and McHenry 1990; Wallingand Quine 1991, 1992; Ferro et al. 1998) afford an alternativeto the use of plots and a means of overcoming the problemsof measurement representativeness and spatial variability.

1Prof., Dipartimento di Ingegneria e Tecnologie Agro-Forestali, Facoltadi Agraria, Universita di Palermo, Viale delle Scienze, 90128 Palermo,Italy. E-mail: vferro@unipa.it

2PhD, Instituto di Ecologia e Idrologia Forestale, Consiglio Nazionaledelle Ricerche, Via A. Volta, 106, 87030 Castiglione Scalo, Italy.

Note. Discussion open until March 1, 2001. To extend the closing dateone month, a written request must be filed with the ASCE Manager ofJournals. The manuscript for this paper was submitted for review andpossible publication on December 9, 1997. This paper is part of the Jour-nal of Hydraulic Engineering, Vol. 5, No. 4, October, 2000. qASCE,ISSN 1084-0699/00/0004-0411–0422/$8.00 1 $.50 per page. Paper No.17154.

At basin scale, predicting sediment yield (i.e., the quantityof sediment that is transferred in a given time interval fromeroding sources through the channel network to the basin out-let) can be carried out coupling a soil erosion model with amathematical function expressing the sediment transport effi-ciency of the hillslopes and the channel network (Renfro 1975;Kirkby and Morgan 1980; Walling 1983). Soil erosion modelsgenerally predict rill and interrill erosion; therefore, gully andchannel processes are to be separately modeled to obtain thetotal basin sediment budget.

If the process is studied at mean annual temporal scale, thesediment transport efficiency of the hillslopes and the channelnetwork is usually represented by the spatially lumped conceptof 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 ac-tually leaving the basin to that eroded within it. According tothe upland theory of Boyce (1975) sediment delivery ratioSDRW generally decreases with increasing basin size becauseaverage slope decreases with increasing basin size, and largebasins also have more sediment storage sites located betweensediment source areas and the basin outlet. At mean annualtemporal scale, SDRW generally assumes values #1 (Renfro1975; Sedimentation 1975; Bagarello et al. 1991).

For a given event, SDRW can be >1 because sediments de-posited on the hillslopes or stored into the channel network,in some previous events, can be removed.

Sediment transport and deposition in the basin hillslopes arephysical processes distinct from transport and depositionwithin the channel network. Therefore hillslope sediment de-livery processes and channel processes should be consideredand modeled separately (Atkinson 1995).

At mean annual temporal scale, Playfair’s law of streammorphology (Boyce 1975) can be applied: over a long time astream must essentially transport all sediment delivered fromthe hillslopes to it. Therefore, at mean annual temporal scalethe sediment delivery ratio SDRW appears to be a measure ofthe efficiency with which materials eroded from hillslopes aredelivered to the stream system (Boyce 1975).

Leopold et al. (1964) suggested that materials eroded froma basin are only temporarily stored in the floodplains, andSchumm (1972) assumed that in a long time period a streamwill discharge essentially all of the sediment it receives. Inother words, for predicting the mean annual value of the basinsediment yield, the sediment delivery problem can be simpli-fied neglecting the channel component.

For a small basin, having an ephemeral channel networkand with no well-developed floodplains, the channel sediment

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FIG. 1. Scheme of Basin Discretized into Morphological Unit

delivery component also could be roughly neglected at eventscale. In other words, the channel network of a small basin isgenerally constituted of channels having narrow cross sectionsand short channel lengths, and therefore it can be assumed, atevent scale, that sediments eroded from hillslopes are notstored within the channel network.

Richards (1993) suggested that sediments are produced fromdifferent sources distributed throughout the basin and that sed-iment delivery processes may be modeled employing a spa-tially distributed approach considering individual fields (Kling1974).

To apply a spatially distributed strategy at the basin scalerequires the choice of a soil erosion model and a spatial dis-aggregation criterion for the sediment delivery process. Theexisting difficulties of physically based modeling (numerousinput parameters, differences between the scale of measure-ment of the input parameters and the scale of basin discreti-zation, uncertainties of the selected model equations, etc.) in-creased the attractiveness of a parametric soil erosion model,like Universal Soil Loss Equation USLE (Wischmeier andSmith 1965) or its revised version RUSLE (Renard et al.1994). For modeling the spatial disaggregation of the sedimentdelivery, 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 hillslopesediment delivery processes, Ferro and Minacapilli (1995) pro-posed a sequential approach. Basically, this approach followsthe sediment mass in a Lagrangian scheme and applies appro-priate delivery factors to each sequential modeling morpho-logical unit (Novotny and Chesters 1989). Neglecting thechannel 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 erodedparticles along the hydraulic path, from the considered area tothe nearest stream reach (Fig. 1). The travel time is assumedto increase as the length lp,i of the hydraulic path increases and

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as the square root of the slope sp,i of the hydraulic path de-creases. For evaluating the travel time tp,i of the particleseroded from a given morphological unit i, the travel times intoall morphological areas that are localized along the path be-tween the ith unit and the nearest stream reach have to besummed. This gives

Npl lp,i i, jt = = (1)p,i O

s sj=1Ï Ïp,i i, j

in which Np = number of morphological units localized alongthe hydraulic path j; li, j and si, j = length of slope of eachmorphological unit i localized along the hydraulic path j, re-spectively. The ratio is the travel time along the mor-l / si, j i, jÏphological unit i of the hydraulic path j; according to Chezy’suniform open channel flow equation, is directly pro-si, jÏportional to the flow velocity along the morphological unitlength li, j.

The analysis carried out for 13 Sicilian and 2 Calabrianbasins (Ferro and Minacapilli 1995; Ferro 1997) to determinethe empirical cumulative distribution function of the traveltime showed that the SDRi ratio has the following expression:

Npl lp,i i,jSDR = exp(2bt ) = exp 2b = exp 2bi p,i S D F SO DG

s sj=1Ï Ïp,i i, j

(2)

in which b = coefficient. Taking into account Chezy’s scheme,the b coefficient lumps together the effects due to roughnessand runoff along the hydraulic path. Then b is affected by theroughness distribution along the flow path and is time depen-dent [i.e., for a given basin b is dependent on the temporalscale (event, annual, and mean annual)].

In this paper, a sediment delivery distributed (SEDD) modelbased on (2) and USLE, in which different expressions of theerosivity and topographic factors are considered, is proposed.The SEDD model is calibrated by using the sediment yieldmeasurements carried out in three small Calabrian experi-mental basins at the event and annual scales. Finally, a MonteCarlo technique will be used for testing the effect of modelparameters uncertainty on calculated sediment yield.

SEDD MODEL

The sediment production Yi[t], of each morphological uniti into which the basin is divided, is simply calculated by thefollowing equation:

Y = SDR A SU (3)i i i i

in which Ai = soil loss (t/ha) from the ith morphological unit,which has to be estimated by the selected erosion model; andSUi = area (ha) of the morphological unit.

Soil loss Ai is estimated by the following variation of theUSLE (Wischmeier and Smith 1965):

A = EF K L S C P (4)i t,i i i i i i

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 theprocedure of Wischmeier et al. (1971) (t h/kg m2); Ci = coverand management factor; Pi = support practice factor; and LiSi

= topographic factor. One uses two expressions for the topo-grahic factor (both from the RUSLE model). One is proposedby McCool et al. (1989)

mili

L S = (10.8 sin a 1 0.03) if tan a < 0.09 (5a)i i i iS D22.13mi

liL S = (16.8 sin a 2 0.5) if tan a $ 0.09 (5b)i i i iS D22.13

in which li = slope length of the ith morphological area; ai =slope angle; mi = fi/(1 1 fi ); and fi = sin ai/0.0896 (3 sin0.8ai

1 0.56) = ratio of rill to interrill erosion, and one is proposedby Moore and Burch (1986)

0.6 1.3l sin as,i i

L S = (6)i i S D S D22.1 0.0896

in which ls,i = ratio between the area of the ith morphologicalunit and the width measured along the contour line.

EXPERIMENTAL BASINS AND MEASUREMENTTECHNIQUE

The studied area is located near Crotone (35 m above sealevel (asl), 397099020 N, 177189100 E) in the ephemeral basinof Crepacuore stream, which drains to the Ionian sea (Fig. 2).Basin W1 is covered with a grass and shrub vegetation. BasinW2 was planted in 1968 with Eucalyptus occidentalis and wascoppiced twice in 1978 and 1990 (coppicing cycle equal to 12years). The forest cover is discontinuous, and 20% of the basinarea is bare. Basin W3 was covered with a Eucalyptus occi-dentalis high forest in the period 1968–1986. In 1986 the for-est cover was coppiced. The forest cover is uniform, and thepercentage of bare area is equal to 3% of the total basin area(Cantore et al. 1994). For each investigated basin, the basinarea AW (ha); the mean altitude Hm (m asl); the minimum el-evation Zmin (m asl); the maximum elevation Zmax (m asl); themean slope s (%); and the percentage of sand, silt, and clayof the soil are listed in Table 1. Water discharge, suspendedsediment concentration, and rainfall were measured at the out-let of each basin. Each basin is monitored by an H-flume weir(Brakensiek et al. 1979), and measurement of flow depth iscarried out at the end of a rectangular channel by a mechanicalrecording water level gauge. For each flood event, t, the basinhydrograph is available, and the runoff factor of Williams(1975) Rd,t (kg m2/ha h) can be calculated according to thefollowing relationship (Bagarello et al. 1990):

0.8776 0.56R = (q V ) (7)d,t p,t tAW

where qp,t = peak flow rate of the flood event t (m3/s); Vt =runoff volume (m3); and AW = basin area (ha). The samplingdevice is a Coshocton Wheel (Parson 1954; Carter and Parson1967) collecting a sample (;1/200) of the flow volume. Eachcollected sample flows into appropriately-sized tanks. At theend 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 1057C. The sediment yieldof each event is calculated by the product of the mean con-centration Cm and the measured runoff volume. Rainfall ismeasured by a recording rain gauge, and rainfall erosivity fac-tor Rt (Wischmeier and Smith 1965) is calculated for eachrainfall event t. Forty-six selected events were measured atbasin W1, 52 events at basin W2, and 42 events at basin W3in the recording period 1978–1994 (Avolio et al 1980; Cal-legari et al. 1994; Cantore et al. 1994; Cinnirella et al. 1998).

For each basin, the soil data (grain-size distribution, totalorganic carbon, soil structure index, and permeability index)used for estimating the soil erodibility factor were obtainedfrom 10 sampling sites. Then, for each basin, the correspond-ing 10 soil erodibility factor values were calculated by thenomograph of Wischmeier et al. (1971). A characteristic basinvalue 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 allmorphological units of basins W2 and W3, Cb was set equalto 0.164 (Cinnirella et al. 1998), whereas Kb = 0.55 for basin

FIG. 2. Experimental Calabrian Basins

TABLE 1. Characteristic Data of Investigated Basins

Basin(1)

AW

(ha)(2)

Hm

(m asl)(3)

Zmin

(m asl)(4)

Zmax

(m asl)(5)

s(%)(6)

Sand(%)(7)

Silt(%)(8)

Clay(%)(9)

W1 1.473 155 90 122 53 14.0 44.5 41.5W2 1.375 128 85 103 35 14.6 49.2 36.2W3 1.654 114 85 98 24 20.7 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 experimentalbasins, and therefore the support practice factor of each mor-phological unit Pi will be assumed equal to 1.

CALIBRATING SEDD MODEL

Modern understanding of rainfall erosion processes providesfor more process-based indices taking into account that soilloss increases with rainfall amount, rainfall intensity, and run-off (Kinnell et al. 1994; Kinnell 1997).

The product of the storm rainfall kinetic energy E and themaximum rainfall intensity measured over a 30-min time in-terval I30 has been used for the rainfall erosivity factor R inUSLE and RUSLE.

Raindrop impact is often involved in the detachment of soilparticles from soil surface, whereas splash by itself is not avery efficient transport mechanism. However raindrop impactinduces turbulence in the overland flow and increases particletransport (rain-induced flow transport) (Kinnell 1990, 1991).When runoff develops, the erosion rate increases because ofthe development of rills representing an efficient transport sys-tem (Kinnell et al. 1994). When rill erosion occurs, in mod-eling activities, the rill-interrill concept is often applied: thesediments discharged with the flow result from raindrop andoverland flow detachment in the interrill areas together withchannel flow detachment in the rills.

According to these physical reasons and suggestions of Fos-ter et al. (1977), the following erosivity factor EFt includingterms of rainfall and runoff erosivity was used:

EF = aR 1 bR (8)t t d,t

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 runofffactor of a given runoff event corresponding to the rainfallevent t; and a and b = two numerical constants representingthe weight of each erosivity term (rainfall and runoff) on thetotal erosivity factor.

Different values of a and b (respecting the condition a 1 b= 1) should correspond to a different ratio between interrilland rill erosion (Kinnell 1997) even if Foster et al. (1977)suggested that ‘‘research is needed to precisely define the re-lationship between rill erosion and runoff erosivity and thatbetween interrill erosion and rainfall erosivity.’’ Furthermore,Rd,t also lumps the effects of sediment transport in rills and inthe interrill areas and therefore the runoff factor also representsthe effects of the sediment delivery processes (Williams 1975).

Each experimental basin was divided into morphologicalunits, and for each one the topographic factor LiSi was cal-culated by (5) and (6).

According to (2)–(4) and (8), the sediment yield Yi,t of eachi morphological unit for a given event t has the followingexpression:

Y = (aR 1 bR )K C L S [exp(2b t )]SU (9)i,t t d,t b b i i t p,i i

in which the b coefficient, for a given basin, assumes a dif-ferent value bt for each event. The basin sediment productionof a given event t, Yb,t, is equal to the sum of the sedimentyields produced by all morphological units into which the ba-sin is divided (sediment balance equation)

Nu

Y = (aR 1 bR )K C L S [exp(2b t )]SU (10)b,t t d,t b b i i t p,i iOi=1

in which Nu = number of morphological units into which thebasin is divided.

According to (10), for applying the distributed model, thea, b, and b coefficients have to be determined. The coefficientsa and b were fixed according to the following hypotheses forthe total erosivity factor.

a = 1; b = 0 (11a)

a = b = 0.5 (11b)

a = 0.3; b = 0.7 (11c)

The first hypothesis corresponds to the classic scheme ofUSLE where the erosion process is assumed to be mainly dueto rainfall (interrill erosion). Notwithstanding, the USLE rain-fall factor does not consider runoff directly, the measurementscarried out to determine USLE ensure that the rainfall factoris the driving force of sheet and rill erosion (Renard and Fer-reira 1993). The second hypothesis gives equal weight for thetwo hydrological processes of rainfall (interrill erosion) andand runoff (rill erosion), and the third hypothesis assumes thatthe runoff processes (rill erosion) is more important than therainfall processes for hillslope soil erosion.

Taking into account that Rd,t was used originally by Williams(1975) to modify USLE for calculating basin sediment yieldwithout employing SDRW, the hypothesis a = 0 and b = 1 orother hypotheses in which the weight of the runoff erosivityterm is prevailing can be neglected.

For each basin, for the different hypotheses, and for eachrainfall-runoff event t, the corresponding bt value was calcu-lated by (10) in which Yb,t is set equal to the measured basinsediment yield Ym,t. For each basin, for each hypothesis, andfor each topographic factor equation [(5) and (6)], Table 2 liststhe median value bm of the calculated bt values. For each hy-pothesis and the expression of the topographic factor proposedby McCool et al. (1989) [(5)], the measured basin sedimentyield Ym,t at the event scale and the Yc,t values calculated by

414 / JOURNAL OF HYDROLOGIC ENGINEERING / OCTOBER 2000

TABLE 2. Values of bm Coefficient of Each Investigated Basin

Basin(1)

Eq. (5)

a = 1;b = 0

(2)a = b = 0.5

(3)

a = 0.3;b = 0.7

(4)

Eq. (6)

a = 1;b = 0

(5)a = b = 0.5

(6)

a = 0.3;b = 0.7

(7)

W1 0.0201 0.0135 0.0082 0.0418 0.0314 0.0231W2 0.0157 0.0073 0.0032 0.0310 0.0212 0.0163W3 0.0165 0.0114 0.0085 0.0197 0.0143 0.0110

FIG. 3. Comparison for USLE Model [Eqs. (5) and (10)] be-tween Measured and Calculated Sediment Yield Values at EventScale

TABLE 3. Results of Analysis of Three Hypothesis for Erosiv-ity Factor

a(1)

b(2)

Eq. (5)

r(3)

MSE(t 2)(4)

Eq. (6)

r(5)

MSE(t 2)(6)

1.0 0.0 0.710 7.515 0.710 7.5100.5 0.5 0.748 5.528 0.748 5.5320.3 0.7 0.771 4.299 0.771 4.299

TABLE 4. Values of bm Coefficient Estimated by First 15Events of Recording Period Assuming USLE Hypothesis (a 5 1and b 5 0)

Basin(1)

bm

Eq. (5)(2)

Eq. (6)(3)

W1 0.0150 0.0338W2 0.0141 0.0291W3 0.0194 0.0228

(10), using as estimate of bt the median value bm, are plottedin Fig. 3.

For each hypothesis and for each selected topographic factorequation [(5) and (6)] the relationship between calculated andmeasured sediment yield values were also compared using thecorrelation coefficient r and the mean square error (MSE) (Ta-ble 3). Even if the third hypothesis [(11c)] is characterized bythe maximum r value and the minimum square error, mea-surements of peak flow rate and runoff volume at the basinoutlet are often unavailable, and therefore qp,t and Vt shouldbe estimated by a hydrological model.

The good agreement between Ym,t and Yc,t also checked forthe USLE hypothesis (a = 1 and b = 0) has encouraged thechoice of this hypothesis [(11a)] and to restrict the followinganalysis to this choice. The USLE-based hypothesis is alsojustified as the sediment delivery processes will be simulatedby the sediment delivery ratio of each morphological unit[(2)].

EVENT CALIBRATION

At first, for each basin, the predictive capability of theSEDD model for the USLE hypothesis (a = 1 and b = 0) wastested at the event scale. Different expressions for LiSi wereused and bm was estimated using only the first 15 events ofeach recording period.

Using these bm estimates listed in Table 4, the agreementbetween measured and calculated sediment yield values wascontrolled for the remaining set of Ne—15 events, in whichNe = total number of events available for each investigatedbasin. 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 firstrow of Table 3.

To determine whether the agreement between measured andcalculated sediment yield is independent of the 15 events se-lected for calibrating the model (i.e., for calculating bm), theestimate stability of the bm coefficient was controlled. There-fore, for each basin the cumulative distribution function of thebt coefficient (Fig. 4) was used for drawing 15,000 randomsequences, each having a sample size equal to 15. For eachsequence, the median value bm was calculated, and the15,000bm values were used to estimate the mean m(bm). Them(bm) values listed in Table 5 are very close to the previousbm estimates (listed in Columns 2 and 5 of Table 2) obtainedusing all available bt values.

For each estimate criterion of LiSi, the relationship between

FIG. 4. Probability Distribution of bt Coefficient for Investi-gated Experimental Basins

TABLE 5. Statistical Parameters m(bm) and s(bm) of Se-quences Generated by Monte Carlo Technique

Model(1)

Parameter(2)

Basin

W1(3)

W2(4)

W3(5)

Eq. (5) m(bm) 0.0206 0.0166 0.0192s(bm) 0.0110 0.0060 0.0043

Eq. (6) m(bm) 0.0421 0.0318 0.0225s(bm) 0.0174 0.0071 0.0046

JOURNAL OF HYDROLOGIC ENGINEERING / OCTOBER 2000 / 415

FIG. 5. Relationship between bm and Soil Clay Percentage forExpressions of Topographic Factors of: (a) McCool et al. (1989);(b) Moore and Burch (1986)

bm listed in Table 2 (Columns 2 and 5) and the mean value ofthe clay percentage (CL) of the soils falling in each basin isshown in Figs. 5(a and b). This correlation can be justified bytaking into account that a relationship between particle grain-size distribution and sediment delivery exists. According toWilliams (1975) when the median particle size increases from0.001 to 0.1 mm, the delivery ratio decreases from 0.37 to0.06, and he proposed the following expression for SDRi:

1/2SDR = exp(2fD t ) (12)i 50 p,i

in which D50 = median particle size; and f = routing coeffi-cient. According to (12) the grain-size distribution affects thebm coefficient but in a manner that cannot be readily quantifiedbecause travel time and particle size are also correlated.

For each selected expression of the topographic factor andfor each basin, bt was correlated with the event runoff coef-ficient, RCt equal to the ratio between the runoff Dt (mm) andthe corresponding rainfall At (mm), in order to provide an es-timate criterion of bt. The relationship between bt and RCt,shown in Fig. 6 as an example for basin W3, may be approx-imated by the following mathematical form:

b = c 1 d ln RC (13)t t

in which c and d = coefficients.The correlation in (13) can be explained by taking into ac-

count that for increasing values of RCt the hillslope sedimenttransport and SDRi values generally increase. In other words,a more efficient hillslope sediment transport, which is char-acterized by high SDRi values, is obtainable, according to (2)(Fig. 6) by small bt values. The agreement between measuredand calculated sediment yield values is shown, as an examplefor the topographic factors calculated by (5), in Fig. 7 and ischaracterized by r = 0.765 and MSE = 11.679 (t2). The com-

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FIG. 6. Relationship between bt 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 Cal-culated Sediment Yield, at Basin Scale, with bt Estimated byEq. (13)

parison between Fig. 7 and Fig. 3(a), and the correspondingstatistical parameters r (Table 3, first row), show that (13) al-lows a reliable bt estimate and a good agreement between Ym,t

and Yc,t.

ANNUAL CALIBRATION

For testing the applicability of the SEDD model at annualscale, for each year (10) was applied using the annual valueof the rainfall erosivity factor [(8) with a = 1 and b = 0)]

calculated adding all Rt values of the rainfall events happeningduring the year. The model was calibrated (i.e., bm coefficientwas calculated) using the available data set of a basin (e.g.,basin W1) and the other two data sets (e.g., basins W2 andW3) were used to validate the model comparing the calculatedsediment yield values Yc,a with the measured ones Ym,a.

For LiSi calculated by (5) and (6) for each basin, Table 6lists the bm coefficient, r, and MSE values corresponding tothe comparison between measured Ym,a and calculated Yc,a an-nual values of sediment yield.

The comparison between Fig. 8 and Fig. 3(a), and the cor-responding statistical parameter r, shows that the reliability ofthe sediment delivery distributed modeling increases with thetemporal scale (from the event scale to the annual scale). Thisresult can be explained by taking into account that the accu-racy of the soil erosion model is dependent on which temporalscale is used (Risse et al. 1993). Further research to improvethe agreement at the event scale between measured and cal-culated sediment yield values could hypothesize an expressionof the sediment delivery ratio of each morphological unit alsolumping the channel network delivery processes.

EFFECT OF PARAMETER UNCERTAINTY ONCALCULATED SEDIMENT YIELD

The recognized random variability of hydrologic variables(Haan 1977; Bogardi et al. 1985) in addition to the circum-stance that input model parameter values are only estimates(as the actual values are not known with certainty) has sug-gested the inclusion of uncertainty analysis in modeling activ-ities (Hession et al. 1996).

For evaluating the uncertainty of model parameters on cal-culated sediment yield, Monte Carlo techniques can be ap-plied. Repeated simulations are performed with the model us-ing randomly selected input parameter values. For eachsimulation the input parameter values are chosen using theirpredetermined probability distribution. The simulation processis repeated for a number of iterations sufficient to estimate aprobability distribution of the output variable.

According to MacIntosh et al. (1994) the major types ofuncertainty are knowledge uncertainty and stochastic variabil-ity. Knowledge uncertainty is due to incomplete understandingof studied phenomena, inadequate measurement of systemproperties, and input data availability. Stochastic variability isdue to random variability of the studied natural environmentand can be divided into temporal and spatial variability. In theuncertainty analysis, the knowledge uncertainty and stochasticvariability has to be distinguished in order to detect their ef-fects on output uncertainty. Stochastic variability, being a nat-ural property of the system, can be quantified but cannot bereduced. The knowledge uncertainty can be reduced by de-creasing the possible range of parameter estimates through ap-propriate physical measurements. In other words, knowledgeuncertainty can be used as indicator of the benefits due toadditional measurements.

When modeling a basin discretized into morphologicalunits, the actual correlation structure of the system sedimentyield model-basin is not known. In other words, some factorsof the USLE model are physically related (R and C factors),and others are dependent on the discretization level of thebasin. In the following, taking into account that the correlationstructures are dependent on many factors (studied site, scale,considered property, etc.) and that at present additional re-search is needed to determine the appropriate level of corre-lation at basin scale (Hession et al. 1996), the stochastic var-iability of the basin sediment yield Yb,t calculated by SEDD,at the event and annual scales, will be studied for each ex-perimental basin, neglecting the correlation between the dif-ferent RUSLE factors.

TABLE 6. Values of bm Coefficient and Statistical Parametersof Model Validation at Annual Scale

Basinused for

calibration(1)

Eq. (5)

bm

(2)r

(3)MSE(4)

Eq. (6)

bm

(5)r

(6)MSE(7)

W1 0.0132 0.70 42.3 0.3090 0.77 22.8W2 0.0072 0.55 79.6 0.0210 0.55 30.4W3 0.0142 0.75 26.5 0.0173 0.75 91.3

FIG. 8. Comparison, for Eqs. (6) and (10), between Measuredand Calculated Sediment Yield Values at Annual Scale

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TABLE 7. Parameters of Cumulative Distributions Functions(CDF) of Characteristic Variables at Event Scale

Basin(1)

Variable(2)

CDF(3)

Parameters(4)

LS model(5)

W1 Rt LN2 m = 1.5929; s = 0.9962 Eqs. (5) and (6)Kb Uniform Kmin = 0.4; Kmax = 0.6 Eqs. (5) and (6)Cb LN2 m = 22.6538; s = 1.3392 Eqs. (5) and (6)RCt Beta a = 1.3592; g = 2.4071

A = 0.012; B = 0.703Eqs. (5) and (6)

bt Beta a = 0.9689; g = 4.5144A = 20.0121; B = 0.218

Eq. (5)

bt Beta a = 0.9613; g = 5.5818A = 20.0079; B = 0.4249

Eq. (6)

W2 Ri LN2 m = 1.9154; s = 1.1355 Eqs. (5) and (6)Kb Uniform Kmin = 0.4; Kmax = 0.6 Eqs. (5) and (6)Cb LN2 m = 21.9745; s = 1.3853 Eqs. (5) and (6)RCt Beta a = 0.962; g = 1.7096

A = 0.032; B = 0.525Eqs. (5) and (6)

bt Gauss m = 0.0166; s = 0.0187 Eq. (5)bt Gauss m = 0.0318; s = 0.0220 Eq. (6)

W3 Rt LN2 m = 2.0811; s = 1.1116 Eqs. (5) and (6)Kb Uniform Kmin = 0.4; Kmax = 0.6 Eqs. (5) and (6)Cb LN2 m = 21.9751; s = 0.9661 Eqs. (5) and (6)RCt Beta a = 1.2526; g = 2.9635

A = 0.017; B = 0.552Eqs. (5) and (6)

bt Gauss m = 0.0192; s = 0.0135 Eq. (5)bt Gauss m = 0.0225; s = 0.0143 Eq. (6)

FIG. 9. Probability Distribution of Event Rainfall Factor Rt forBasin W1

To perform the Monte Carlo simulations, a probability dis-tribution must be assigned for the following uncertain param-eters: rainfall erosivity, soil erodibility, crop factor, bt coeffi-cient, or runoff coefficient. For each basin, the measured eventrainfall erosivity values Rt were found to be lognormally dis-tributed (Fig. 9) with mean m and standard deviation s listedin Table 7.

The comparison between the empirical cumulative fre-quency distribution of crop factor of each event Ct, calculatedby Cinnirella et al. (1998), and a lognormal distribution (LN2)having mean value and standard deviation equal to the samplevalues (Table 7) is shown in Fig. 10.

For determining the frequency distribution of bt coeffi-ciency, for each basin having known values of Kb and Cb, thesediment yield Yt and the rainfall erosivity Rt measured foreach event t were first used to calculate by (10) the corre-sponding bt coefficient.

For basin W1 bt is distributed according to a beta distribution,whereas for basins W2 and W3 the Gauss law agrees with theempirical frequency distribution of bt variable (Fig. 4).

The soil erodibility factor was treated as having only knowl-edge uncertainty representing the range of possible valuesavailable from the literature. The knowledge uncertainty, dueto the difficulty of establishing an appropriate value for use inthis study’s model for the examined soil type, was consideredby a uniform distribution that describes K values ranging from

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 topo-graphic factor was treated as a constant deterministic value ofeach morphological unit, under the assumption that the lengthsand the slope of the units are controlled.

Each Monte Carlo simulation was carried out drawing atrandom a value from each of the distributions of rainfall ero-sivity, soil erodibility, crop factor, and bt coefficient. Thesevalues were then used with the constant topographic factor ofeach morphological unit as input to the model [(10)], whoseoutput Yc,t represented one iteration of the simulated scenario.The resampling of rainfall erosivity, soil erodibility, crop fac-tor, and bt was repeated 15,000 times, resulting in 15,000 es-timates of the output basin sediment yield Yc,t. For each basin,the 15,000 values of Yc,t were used to define the theoreticalprobability distribution P(Yc,t) to compare with the empiricalfrequency distribution F(Ym,t) of the measured values Ym,t ofthe basin sediment yield. For each basin and for the case oftopographic factors calculated by (6), the agreement betweenthe distributions F(Ym,t) and P(Yc,t) is shown in Fig. 11. Themean value m and the standard deviation s estimate of thesediment yield (shown as an example for basin W1 in Fig. 12)were essentially constant for iterations >3,000. The agreementbetween the frequency distribution of the measured sedimentyield values and the generated ones remained acceptable whenthe resampling was carried out using a constant bt value equalto bm.

The Monte Carlo technique was also applied to resamplingof runoff coefficient RCt values, which are distributed accord-ing to a beta distribution (Table 7) and calculating bt by (13).The agreement between the generated frequency distributionP(Yc,t), with the topographic factors calculated by the expres-sion of Moore and Burch (1986) and bt estimated by (13), andthe measured F(Ym,t) distribution (Fig. 13), shows the abilityof the model to reproduce the empirical distribution of basinsediment yield.

The parameter uncertainty analysis was also developed atannual scale, even though the basins have slightly differentrecording periods (for basins W1 and W3 the recording periodis 13 years, and for basin W2 the recording period is equal to14 years). At the annual scale, for each basin, the rainfall ero-sivity factor is lognormally distributed, and the normal distri-bution agrees with the empirical frequency distribution of thebt coefficient. Table 8, which lists the statistical parameters,shows a good agreement between the mean value of the twodistributions, whereas basin W1 underestimates the medianvalue m and basin W3 underestimates the standard deviations. Finally, for basin W1 the theoretical distribution is moreskewed than the empirical one, and for basins W2 and W3 theempirical values of G are greater than the theoretical ones.

FIG. 11. Comparison, for Each Basin, between Measured Sed-iment Yield Frequency Distribution and That Generated byMonte Carlo Technique Using Distribution of bt Coefficient

In conclusion, the parameter uncertainty analysis showedthat the SEDD model, even if the bt coefficient is estimatedby (13), is able to generate a theoretical distribution functionof basin sediment yield, which has a satisfactory agreementwith the cumulative distribution function of the measuredvalue.

CONCLUSIONS

For evaluating the quantity of sediment that is transferred,in a given time interval, from eroding sources through thehillslopes and the channel network to the basin outlet, a plotsoil erosion model can be coupled with a distributed mathe-matical function expressing the sediment transport efficiency.

At first, for a basin discretized into morphological units, adistributed model based on RUSLE, in which different ex-pressions of the erosivity and topographic factors are consid-

FIG. 12. Mean Value and Standard Deviation of GeneratedSediment Yield Sequences

FIG. 13. Comparison, for Each Basin, between Measured Sed-iment Yield Frequency Distribution and That Generated byMonte Carlo Technique, Using bm 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 SedimentYield at Annual Scale

Parameter(1)

EQ. (5)

b (Median Value)

W1(2)

W2(3)

W3(4)

b (from Theoretical Law)

W1(5)

W2(6)

W3(7)

EQ. (6)

b (Median Value)

W1(8)

W2(9)

W3(10)

b (from Theoretical Law)

W1(11)

W2(12)

W3(13)

NEmpirical 13 14 13 13 14 13 13 14 13 13 14 13Theoretical 12,886 13,006 12,886 12,886 13,006 12,886 12,886 13,006 12,886 12,886 13,006 12,886

mEmpirical 10.473 27.808 8.249 10.473 27.808 8.249 10.473 27.808 8.249 10.473 27.808 8.249Theoretical 8.170 23.916 5.698 11.782 25.084 6.129 8.398 24.234 5.641 11.403 24.392 6.393

mEmpirical 10.606 13.032 4.956 10.606 13.032 4.956 10.606 13.032 4.956 10.606 13.032 4.956Theoretical 4.348 12.793 3.323 5.693 12.346 3.076 4.469 12.794 3.290 5.684 11.764 3.056

sEmpirical 9.007 39.646 10.412 9.007 39.646 10.412 9.007 39.646 10.412 9.007 39.646 10.412Theoretical 9.459 27.581 6.042 14.809 31.095 7.447 9.696 28.120 5.943 13.841 30.386 7.900

COVEmpirical 0.860 1.426 1.262 0.860 1.426 1.262 0.860 1.426 1.262 0.860 1.426 1.262Theoretical 1.158 1.153 1.060 1.257 1.240 1.215 1.155 1.160 1.053 1.214 1.246 1.236

GEmpirical 1.224 2.597 2.941 1.224 2.597 2.941 1.224 2.597 2.941 1.224 2.597 2.941Theoretical 1.866 1.860 1.665 2.063 2.004 1.957 1.838 1.874 1.657 1.942 1.997 1.955

Note: COV = coefficient of variation.

ered, coupled with the sediment delivery ratio of each mor-phological unit was proposed. The model neglects the channelsediment delivery component and can be applied at mean an-nual temporal scale or at event scale for a small basin havingan ephemeral channel network and with no well-developedfloodplains. The analysis, carried out by the sediment yieldmeasurement at the event scale in three Calabrian experimentalbasin, initially used an erosivity factor that, as suggested inthe literature, linearly weights the rainfall factor of Wischmeierand 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 theselected equations for estimating the topographic factors andcan be simple obtained by using the rainfall erosivity factor.

At the event scale the model was calibrated using the first15 events of the recording period to estimate the bm coeffi-cient. A good agreement between measured and calculated sed-iment yield values was obtained validating the model by theremaining set of events for each basin. The analysis alsoshowed that this agreement is independent of the 15 eventsused for estimating the bm coefficient.

The median value bm of the bt coefficient estimated by allevents of the recording period was then correlated with theevent runoff coefficient [(13)] by taking into account that theefficiency of the sediment transport is linked to hillslope run-off. The analysis showed that (13) allows a reliable estimateof the bt coefficient because by using (13) a good agreementbetween measured and calculated sediment yield values wasobtained.

The applicability of the SEDD model was also tested at theannual scale, and the analysis showed that the reliability ofthe sediment delivery distributed approach, measured by thecorrelation between measured and calculated annual sedimentyield values, improves with the temporal scale (from the eventscale to the annual scale).

To represent the model limitations and uncertainties, the ef-fect of parameter uncertainty on calculated sediment yield wasalso studied. Monte Carlo simulations were performed usingpredetermined probability distributions of the input param-eters.

The resampling of the model factors produced 15,000 esti-mates of the output basin sediment yield Yc,t, which definedthe 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 measuredvalues Ym,t.

For each studied basin, the uncertainty analysis developedat event scale, showed that the SEDD model, even if the bt

coefficient is estimated by the runoff coefficient, is able togenerate a theoretical distribution of sediment yield P(Yc,t) hav-ing a satisfactory agreement with the cumulative distributionfunction of the measured values.

ACKNOWLEDGMENTS

Research is supported by a grant from the Ministero Universita eRicerca Scientifica e Tecnologica, Governo Italiano, ‘‘Produzione disedimenti a scala di bacino e tecniche di ricostruzione naturalistica delsistema alveo-versante’’ and quota 60%. Both authors set up the research,analyzed the results, and participated in writing the paper.

APPENDIX I. REFERENCES

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APPENDIX II. NOTATION

The following symbols are used in this paper:

A = parameter of beta distribution (Table 7);Ai = soil loss from each morphological unit;At = rainfall depth of each event t;

AW = basin area;a = numerical constant;B = parameter of beta distribution (Table 7);b = numerical constant;

Cb = cover and management factor of each basin;Ci = cover and management factor of each morphological

unit;Cm = mean suspension concentration;Ct = cover and management factor of each event;Dt = runoff depth of each event t;

D50 = median particle size;E = storm rainfall kinetic energy;

EFt = erosivity factor for given rainfall event t;EFt,i = erosivity factor for given temporal scale of each mor-

phological unit;fi = ratio of rill to interrill erosion for ith morphological unit;

G = skewness;Hm = mean altitude;I30 = maximum rainfall intensity measured over 30-min time

interval;Kb = soil erodibility factor of each basin;Ki = soil erodibility factor of each morphological unit;

Li Si = topographic factor of each morphological unit;lp,i = length of hydraulic path;mi = slope length exponent for ith morphological unit;Ne = number of events of each investigated basin;Np = number of morphological units localized along hydrau-

lic path;Nu = number of morphological units into which basin is di-

vided;

JOURNAL OF HYDROLOGIC ENGINEERING / OCTOBER 2000 / 421

Pi = support practice factor of each morphological unit;qp,t = peak flow rate of flood event t;

R = rainfall erosivity factor in the USLE;RCt = runoff coefficient of each event t;Rd,t = runoff factor of Williams of each event t;Rt = rainfall erosivity factor of each event t;r = coefficient of correlation;

SDRi = sediment delivery ratio of each morphological unit;SDRW = basin sediment delivery ratio;

SUi = area of each morphological unit;s = mean slope;

si, j = slope of each morphological unit i localized along hy-draulic path j;

sp,i = slope of hydraulic path;tp,i = travel time of each morphological unit;Vt = runoff volume of flood event t;

Yb,t = basin sediment yield for given event t;Yc,t = basin sediment yield calculated for given event t;Yi = sediment production of each morphological unit;

Yi,t = sediment yield of each morphological unit for givenevent t;

422 / JOURNAL OF HYDROLOGIC ENGINEERING / OCTOBER 2000

Ym,t = basin sediment yield measured for given event t;Zmax = maximum elevation;Zmin = minimum elevation;

a = parameter of beta distribution (Table 7);ai = slope angle of each morphological unit;b = coefficient;

bm = median yield of calculated bt values;bm,15 = bm value corresponding to first 15 historical events;

bt = value of b coefficient of given event t;g = parameter of beta distribution (Table 7);li = slope length of ith morphological unit;

li, j = length of each morphological unit i localized along hy-draulic path j;

ls,i = ratio between area of the morphological unit and widthmeasured along contour line;

m(bm)= mean value of 15,000bm values generated by MonteCarlo techniques;

s(bm) = standard deviation of 15,000bm values generated byMonte Carlo techniques; and

f = routing coefficient.