Journal of Hydrology 519 (2014) 3328–3337

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Journal of Hydrology

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Applying the RUSLE and the USLE-M on hillslopes where runoffproduction during an erosion event is spatially variable

http://dx.doi.org/10.1016/j.jhydrol.2014.10.0160022-1694/� 2014 Elsevier B.V. All rights reserved.

E-mail address: peter.kinnell@canberra.edu.au

P.I.A. KinnellInstitute of Applied Ecology, University of Canberra, Australia

a r t i c l e i n f o

Article history:Received 11 December 2013Received in revised form 2 October 2014Accepted 4 October 2014Available online 22 October 2014This manuscript was handled byKonstantine P. Georgakakos, Editor-in-Chief

Keywords:Soil loss predictionNon-uniform runoff productionCropped hillslopes

s u m m a r y

The assumption that runoff is produced uniformly over the eroding area underlies the traditional use ofUniversal Soil Loss Equation (USLE) and the revised version of it, the RUSLE. However, although the appli-cation of the USLE/RUSLE to segments on one dimensional hillslopes and cells on two-dimensional hill-slopes is based on the assumption that each segment or cell is spatially uniform, factors such as soilinfiltration, and hence runoff production, may vary spatially between segments or cells. Results fromequations that focus on taking account of spatially variable runoff when applying the USLE/RUSLE andthe USLE-M, the modification of the USLE/RUSLE that replaces the EI30 index by the product of EI30

and the runoff ratio, in hillslopes during erosion events where runoff is not produced uniformly werecompared on a hypothetical a 300 m long one-dimensional hillslope with a spatially uniform gradient.Results were produced for situations where all the hillslope was tilled bare fallow and where half ofthe hillslope was cropped with corn and half was tilled bare fallow. Given that the erosive stress withina segment or cell depends on the volume of surface water flowing through the segment or cell, soil losscan be expected to increase not only with distance from the point where runoff begins but also directlywith runoff when it varies about the average for the slope containing the segment or cell. The latter effectwas achieved when soil loss was predicted using the USLE-M but not when the USLE/RUSLE slope lengthfactor for a segment using an effective upslope length that varies with the ratio of the upslope runoff coef-ficient and the runoff coefficient for the slope to the bottom of the segment or cell was used. The USLE-Malso predicted deposition to occur in a segment containing corn when an area with tilled bare fallow soilexisted immediately upslope of it because the USLE-M models erosion on runoff and soil loss plots as atransport limited system. In a comparison of the USLE-M and RUSLE2, the form of the RUSLE that uses adaily time step in modeling rainfall erosion on one-dimensional hillslopes in the USA, on a 300 m long 9%hillslope where management changed from bare fallow to corn midway down the slope, the USLE-M pre-dicted greater deposition in the bottom segment than predicted by RUSLE2. In addition, the USLE-Mapproach predicted that the deposition that occurred when the slope gradient changed from 9% to4.5% midway down the slope was much greater than the amount predicted using RUSLE2.

� 2014 Elsevier B.V. All rights reserved.

1. Introduction C is the crop factor, and P is the soil conservation practice factor.

The Universal Soil Loss Equation (Wischmeier and Smith, 1965,1978) was developed to predict the long term (�20 years) averageannual soil loss from field size areas using 6 factors focusing on theimpact of climate, soil, topography, cropping and soil conservationpractices;

A ¼ R K L S C P ð1Þ

where A is the average annual soil loss (weight per unit area), R isthe rainfall-runoff ‘‘erosivity’’ factor (climate), K is the soil ‘‘erodibil-ity’’ factor, L is the slope length factor, S is the slope gradient factor,

Only R and K have units. R is defined as the long term averageannual value of the product of storm energy (E) and the maximum30-min rainfall intensity (I30). R does not include direct consider-ation of runoff but provisions were made in the USLE to determinea R equivalent for runoff that occurs from melting snow(Wischmeier and Smith, 1978). L = S = C = P = 1.0 for the ‘‘unit’’ plot,a bare fallow area 22.1 m long cultivated up and down a slope with9% slope gradient. As a result, the USLE operates mathematically intwo steps. In the first step, soil loss is predicted for the unit plot,

A1 ¼ R K ð2Þ

In the second step, the soil loss from the unit plot is multiplied byappropriate values of L, S, C and P to predict soil loss from areaswhich differ from the unit plot.

Nomenclature

SymbolsA long term average annual soil loss per unit area (mass/

area)A1 long term average annual soil oil loss per unit area

(mass/area) from the ‘‘unit’’ plotAi long term average annual soil oil loss per unit area

(mass/area) from segment iAe.1 soil loss per unit area (mass/area) from the unit plot

during a rainfall eventC crop and crop management factor in the USLE/RUSLECe crop and crop management factor for a rainfall eventCi crop and crop management factor for segment iCUM crop and crop management for the USLE-M when pre-

dicting long term soil lossCUMe crop and crop management for the USLE-M for an eventD size of a cellE total storm kinetic energy(EI30)e the product of the total storm kinetic energy (E) and the

maximum 30-min rainfall intensity (I30) for a rainfallevent

I30 maximum 30-min rainfall intensityK soil erodibility factor in the USLE/RUSLEKe soil erodibility factor for a rainfall eventKi soil erodibility factor for segment iKUM soil erodibility factor in the USLE-M when predicting

long term soil lossKUMe soil erodibility factor in the USLE-for an eventk1 a coefficient in the regression equation between Qs.slope.i

and ki

L slope length factor in the USLE/RUSLELe slope length factor for an eventLi slope length factor for segment iLi,j slope length factor for cell i,jLslope.i slope length factor for a slope to the bottom boundary of

segment iLS the product of L and S referred to as the USLE/RUSLE

topographic factorLSp the USLE/RUSLE topographic factor associated unit

stream power theorym power involved in the calculation of the USLE/RUSLE

slope length factorP soil conservation protection factor in the USLE/RUSLEPe soil conservation protection factor for a rainfall event

Pi soil conservation protection factor for segment iQc runoff coefficient in the area (the runoff coefficient ratio

of the volume of runoff generated by rain falling on anarea to the volume of the rain falling on that area. Ithas a maximum value of 1.0)

Qc.up runoff coefficient in the area upslope of a segment orcell

Qc.1 runoff coefficient for the unit plotQc.C runoff coefficient for a cropped areaQCe.i,j.eff effective runoff coefficient when the effective upslope

area of a cell or segment varies when runoff is not pro-duced uniformly over the hillslope

Qc.seg runoff coefficient for a segment on a hillslopeQe volume of runoffQe.kslope volume of runoff from a slope of length kslope

Qe.1 volume of runoff from the unit plotQc.all runoff coefficient in the area upslope of the bottom

boundary of a segment or cellQR the runoff ratio for an area. (the runoff ratio is the ratio

of the sum of volume of runoff generated by rain fallingon the area and the volume of water entering the areafrom elsewhere to the volume of the rain falling on thatarea. It has a value that can exceed 1.0)

QRe1 runoff ratio for an event on the unit plotQR1e.slope.i runoff ratio for the tilled bare fallow area down to the

bottom of the segment i or an eventQReC runoff ratio for an event on a cropped areaQReC runoff ratio for an event on a cropped area down to the

bottom of segment iQRe.kslope runoff ratio for a slope with a length of kslope

Qse.slope.i sediment discharge per unit width of the bottomboundary of segment i for an event

R rainfall runoff erosivity factor in the USLE/RUSLES slope gradient factor in the USLE/RUSLESi slope gradient factor for segment iki slope length for a slope to the bottom of segment iki-1 slope length for a slope to the bottom of segment i � 1ki-1.eff effective slope length that varies from ki�1 when runoff

is not generated uniformlyki.seg length of slope of segment ikslope length of a slope from which soil is lostvupslope area upslope of a cellvupslope.j,j area upslope of cell with coordinates i, j

P.I.A. Kinnell / Journal of Hydrology 519 (2014) 3328–3337 3329

A ¼ A1 L S C P ð3Þ

In the subsequent revision, the Revised Universal Soil Loss Equation(RUSLE, Renard et al., 1997), the fundamental mathematical struc-ture and operation depicted by Eqs. (1)–(3) were retained but pro-cedures for determining some of the factor values were changed.

Originally developed as tool to support management decisionsdirected at the conservation of soil on agricultural land in theUSA, the USLE/RUSLE has been widely used in many other partsof the world. In the RUSLE, provision was made to determine soilloss in segments on one-dimensional hillslopes enabling the RUSLEto be applied on the non-uniform one-dimensional hillslopes. TheUSLE/RUSLE has also been used to predict soil loss on non-uniformtwo-dimensional hillslopes in models designed to predict theeffect of landuse on water quality (e.g. AGNPS; Bingner andTheurer, 2001). In many models used to predict erosion in water-sheds or catchments, grid cells are used and assumed to be uniformin soil, slope gradient, and management, enabling the USLE/RUSLEto be used to predict the soil lost from individual cells depending

on their position in the landscape using the L factor. In some cases,the L factor is calculated using a one-dimensional slope whoselength equals the longest flow pathway on the hillslope (Hickeyet al., 1994). Although this approach is consistent with the stipula-tion that the slope length is the distance from the point where run-off begins to the point where deposition occurs or runoff enters awell defined channel (Wischmeier and Smith, 1965, 1978), itignores the fact that the slope length factor is an empirical factorthat accounts to the effects of runoff and sediment concentrationon soil loss. A more appropriate approach is to use a modificationof the RUSLE slope length factor for segmented one-dimensionalslopes such as that developed by Desmet and Govers (1996).

2. The RUSLE slope length factor for segmented one-dimensional slopes

A one-dimensional hillslope can be divided into a number ofplanar segments and the RUSLE applied to predicting the soil loss

3330 P.I.A. Kinnell / Journal of Hydrology 519 (2014) 3328–3337

from each segment. The average annual soil loss (mass/area/year)for the ith segment (Ai) on a one-dimensional hillslope is given by

Ai ¼ RKi Ci Pi Si Li ð4Þ

where R is the rainfall erosivity factor as used in Eqs. (1) and (2), Ki

is the soil erodibility factor for the segment, Ci the crop and cropmanagement factor for the segment, Pi is the soil conservation prac-tice factor, Si is the slope gradient factor for the segment and Li is theslope length factor for the segment. As a general rule the value of Ris common to all segments while all other factors may vary betweensegments. Li is the only one that is inherently dependent on theposition of the segment relative to the point where runoff begins.

In determining the value of Li, the ith segment is considered tobe on a planar hillslope that has the same slope gradient as the seg-ment (Fig. 1). A number of lesser hillslopes exists on a planar hill-slope and L factor values can be calculated for each one from

Lslope:i ¼ ðki=22:1Þm ð5Þ

where ki is the horizontal distance in metres from the point whererunoff begins to the end of the lesser slope that is terminated by thebottom of the ith segment. In the USLE, m varies between 0.2 asslope gradient increases from less than 1% to reach a maximumvalue of 0.6 for slopes with gradients greater than 10%(Wischmeier and Smith, 1965, 1978), where as m is set to valueswhich depend on the ratio of rill to interrill erosion in the RUSLE(Renard et al. (1997)). The amount of soil discharged from each hill-slope is directly related to the product of the length of the hillslope(ki) and (ki/22.1)m so that

Q s:slope:i ¼ k1kiðki=22:1Þm ¼ k1ðkmþ1i =22:1mÞ ð6Þ

where k1 is an empirical coefficient that varies with the climate,soil, slope gradient and the crop and its management on the slope.As a result, the amount of soil entering a segment is given by Eq. (6)when ki is set to the slope length to the top of the segment whereas,the amount of soil leaving the segment is given by Eq. (6) when ki isset to the slope length to the bottom of the segment. Consequently,in the RUSLE, the L factor for the ith segment is given by

Li ¼kmþ1

i � kmþ1i�1

� �

ðki � ki�1Þ22:1m� � ð7Þ

where ki is the horizontal distance in metres from top of the slope tothe bottom end of the segment being considered and ki�1 is the dis-tance from top of the slope to the bottom of the previous segment(Fig. 1). The length of the segment (kseg.i) is given by ki � ki�1. Asindicated by Fig. 1, Eq. (7) takes no account of variations in slopegradient above segment i.

Fig. 1. Schematic representation of a hillslope in the context of calculating the slopelength factor for segment i. The solid lined profile represents the existing profile ofthe hillslope while the gray triangle outlined by dotted lined represents thehillslope profile that is perceived to exist in the context of calculating the slopelength factor for segment i using Eq. (7).

Although it was formally included in the RUSLE (Renard et al.,1997), Eq. (7) was originally developed by Foster andWischmeier (1974). Eq. (7) conforms to the stipulation that theslope length is the distance from the point where runoff beginsto the point where deposition occurs or runoff enters a welldefined channel (Wischmeier and Smith, 1965) only if runoffbegins at the top of the slope. Eq. (7) can be used on slopes whererunoff begins elsewhere if ki is the horizontal distance in metresfrom point where runoff begins to the bottom end of the segmentbeing considered and ki�1 is the distance from point where runoffbegins to the bottom of the previous segment.

In the environment where the USLE was developed, the mass ofsoil lost from a hillslope of length ki varies with the volume ofwater discharged (Qw.slope.i) and the sediment concentration(cs.slope.i, mass of sediment per unit quantity of water);

Qs�slope:i ¼ Q w slope:ics slope:i ð8Þ

As noted above, traditionally the USLE/RUSLE model operates on theassumption that runoff is generated uniformly over the erodingsurface so that Qw.slope is given by the product of the runoff amount(volume of water discharged per unit area), the width of the one-dimensional slope and ki. Consequently, it follows from Eqs. (6) and(8) that, in terms of the USLE/RUSLE model, cs.slope.i is directly relatedto (ki/22.1)m. Also, in the USLE/RUSLE model, Qw.slope.i is directlyrelated ki, so that, when all segments have the same k1,cs.slope.i is related to ratio of Qw.slope.i to the volume of water dischargedper unit width of the boundary at the bottom of a 22.1 m slope.

3. The slope length factor for using the USLE/RUSLE in 2dimensional space

Although the L factor is an empirical factor, arguably it doeshave a physical basis (Moore and Burch, 1986). As indicated above,in effect, ki is a surrogate for the volume of water discharged over aunit width of the downslope boundary. A grid cell representationof a one-dimensional hillslope produces a straight line of cellswhich are analogous to segments in a one-dimensional hillslopeto which Eq. (7) can be applied. The amount of soil lost from a seg-ment or cell on such a hillslope is dependent on the volume ofwater flowing through the segment which, inturn, is influencedby the volume of water entering the segment from upslope. Theactual shape of that upslope area is irrelevant but its capacity togenerate the volume of water entering the segment varies withits area (vupslope). Consequently, on a two-dimensional hillslopewhere runoff production is spatially uniform, the effective slopelength to the top of a cell of size D is given by dividing the upslopearea by the width of the boundary over which the water flows.

k ¼ vupslope=D ð9Þ

Fig. 2. Schematic representation of one-dimensional and two-dimensional hill-slopes in the context of calculating the slope length factor for a cell.

Fig. 3. The effect of QC.upslope on the ratio of QC.upslope to QC.slope when the upslopelength is 3 times that of the segment and QC.seg is 0.5.

Fig. 4. The effect of variations in QC(seg) about the value of 0.5 on (A) L factor for7.5 m long segments and (B) the QC.upslope to QC.slope ratio with distance down a300 m long one-dimensional hillslope when m = 0.4.

P.I.A. Kinnell / Journal of Hydrology 519 (2014) 3328–3337 3331

(Fig. 2). As a result, the L factor for a cell with coordinates i,j and acell of size D is given by

Li;j ¼vupslope:i;j þ D2� �mþ1

� vmþ1upslope:i;j

Dmþ2ð22:1Þmð10Þ

Eq. (10) follows from Eq. (7) by replacing ki�1 by vupslope.j,j/D and theterm ki � ki�1 by D. Eq. (10) is based on the premise that, like on aone-dimensional hillslope, all flow enters the cell across a side ofcell, not a corner, and leaves the cell over the opposite side as indi-cated in Fig. 2. A variant of Eq. (10) that provides for flows leaving acell by a corner by multiplying the value obtained by Eq. (10) by0.707 was developed independently by Desmet and Govers (1996)but that ‘‘correction’’ involves a departure from the concept thatthe USLE applies to a rectangular or square surface that does nothave a cross slope. The gray rectangle in Fig. 2 provides a schematicfor the linkage of cells when Eq. (10) is applied to the two-dimen-sional flow system shown to the left of it.

4. Predicting the effect of slope length on soil loss from slopeswhere runoff is not produced uniformly

As noted above, the L factor for a segment or cell is based on theassumption that runoff is produced uniformly over the erodingarea. However, Wischmeier et al. (1958) noted that there werecases where the USLE/RUSLE L factor does not apply in the USAbecause a tilled bare fallow area did not produce runoff uniformly.Other cases have been reported outside the USA. For example, in anarea where tilled bare fallow runoff and soil loss plots of variouslengths were established at Sparacia, Sicily, runoff decreased withplot length (Bagarello et al., 2011; Kinnell, 2012). Also, runoff is notproduced uniformly on hillslopes where soils and/or cropping arenot spatially uniform. Under these circumstances, the flow througha segment or cell differs from that which would occur if runoff wasproduced uniformly. Kinnell (2007) proposed that the value of thelength of the upslope area could be adjusted to account for thiseffect so that Eq. (7) becomes

Li ¼ki�1:eff þ ki:seg� �mþ1 � kmþ1

i�1:eff

ki:seg22:1m ð11Þ

where ki:seg ¼ ki � ki�1 and, Eq. (10) becomes

Li;j ¼vupslope:i;j:eff þ D2� �mþ1

� vmþ1upslope:i;j:eff

Dmþ2ð22:1Þmð12Þ

where the values of ki�1.eff and vupslope.i,j.eff are respectively greaterthan ki�1 and vupslope.i,j when the runoff coefficient (ratio of the vol-ume of runoff generated by rain falling on an area to the volume ofthe rain falling on that area) in the upslope area (Qc.upslope) is greaterthat the runoff coefficient for the area including the segment or cell(Qc.slope), and vice versa when the runoff coefficient in the upslopearea is less than that of the area including the segment or cell. Kin-nell proposed that this could be achieved by

ki�1:eff ¼ ki�1Q c:upslope=Q c:slope ð13Þ

and

vupslope:i;j:eff ¼ vupslope:i;jQ c::upslope=Q c::slope ð14Þ

Fig. 3 shows how the ratio of Qc..upslope to Qc..slope varies with Qc.ups-

lope when Qc.seg is 0.5 and the upslope area is 3 times that of the seg-ment being considered.

Kinnell (2007) demonstrated the spatial variation in Li,j

produced by using Eq. (12) with Eq. (14) over a 1.5 ha area whererunoff from Bermuda Grass passed on to an area subjected to awheat–clover–cotton rotation using 25 m size cells. Either Eq.

(11) with Eq. (13) or (12) with Eq. (14) can be applied on one-dimensional hillslopes because D equals ki.seg. Also, one-dimen-sional hillslopes provide a way of demonstrating Eq. (12) in asituation where the effect of individual cells on the L factor for a cellis seen more easily than when converging and diverging flows occuron two-dimensional hillslopes. Fig. 4A shows how the slope lengthfactor values produced by Eq. (11) vary in comparison to the Li val-ues produced by the approach used in the RUSLE (Eq. (7)) for 7.5 mlong segments or cells on a 300 m long tilled bare fallow one-dimensional hillslope with a uniform slope gradient wherem = 0.4 and the average runoff coefficient for the hillslope is 0.5.

It can be seen from Fig. 4B that the Li values produced by adjust-ing the upslope area using the Eq. (13) or (14) vary inversely withthe QC.seg because the effect of any change in QC is applied to QC.slope

first as distance down the slope increases. Also, deviations from the

Fig. 5. The effect of variations in QC(seg) about the value of 0.5 on segment soil lossfrom for 7.5 m long segments on a 300 m long one-dimensional hillslope predictedby Eq. (21) and the combination of Eqs. (11) and (13) when m = 0.4,K = 0.0619 t h MJ�1 mm�1 and KUM = 0.1268 t h MJ�1 mm�1 when EI30 = 350 -MJ mm ha�1 h�1. The spatial variations in QC(seg) on a segment by segment basisare the same as shown in Fig. 4.

3332 P.I.A. Kinnell / Journal of Hydrology 519 (2014) 3328–3337

slope length factor values produced by the RUSLE approach (Eq.(7)) diminish with distance down the slope because of the dimin-ishing effect individual values of QC.seg have on the ratio betweenQC.up and QC.slope as distance increases down the slope (Fig. 4B). Thissimply reflects the mathematical fact that as the number of valuesin a set of values increases, the effect of an individual value on thetotal sum diminishes. The net result is that the L factor values pro-duced by the combination of Eqs. (11) and (13) show little differ-ence from those produced by the RUSLE approach as the distancedown the slope increases (Fig. 4A).

In the USLE/RUSLE model (Eq. (1)), K is given by

K ¼

e ¼ N

RAe:1

e ¼ 1e ¼ N

RðEI30Þee ¼ 1

ð15Þ

where Ae.1 is the event soil loss from the unit plot which, as notedearlier is a tilled bare fallow area 22.1 m long slope inclined at 9%with cultivation up and down the slope, E is the kinetic energy ofthe rain event and I30 is the maximum 30-min rainfall intensityfor that event. Although not designed to predict event erosion (Ae)well, it follows from Eqs. (1) and (15) that

Ae ¼ ðEI30ÞeKe L S Ce Pe ð16Þ

where Ke, Ce and Pe are factor values for the effects of soil, crops andmanagement and soil conservation practice during an event andthat may differ respectively from K, C, and P. It has been shown(Kinnell and Risse, 1998) that replacing the EI30 index by the prod-uct of EI30 and the runoff ratio (ratio of volume of runoff dischargedfrom an area to the volume of rainfall falling on that area) for theunit plot (QR1) to give

Ae ¼ ðQ R1EI30ÞeKUMeL S Ce Pe ð17Þ

improves the prediction of event erosion at a number of geographiclocations in the USA and elsewhere if runoff is know or predictedwell. The USLE-M is the name given to the model where the EI30

index is replaced by the QREI30 index (Kinnell and Risse, 1998). QR,the ratio of volume of runoff discharged from an area to the volumeof rainfall falling on that area, may differ from QC, the runoff coeffi-cient, the ratio of volume of runoff produced by rain falling on anarea divided to the volume of rain falling on that same area, becauseQC has a maximum value of 1.0 but QR > 1.0 will occur in the situa-tion where a plot has exfiltration. QR for a segment may also exceed1.0 as the result of runoff entering the segment from upslope.

In the USLE-M, KUM is given by

KUM ¼

e ¼ NPAe:1

e ¼ 1e ¼ NPðQR1EI30Þee ¼ 1

ð18Þ

KUMe is usually greater than Ke because QR1 is usually less than 1.0.As a general rule, the values for Ke and KUMe for individual stormsare not known so that the long term average values (K, KUM) areused here instead. The L, S, Ce and Pe values that can be used to pre-dict event erosion using Eq. (16) can also be applied in Eq. (17)because both equations operate by initially predicting soil loss fromthe unit plot (Ae.1 = (EI30)e Ke, Ae.1 = (QR1 EI30)e KUMe) before usingthose factor values to predict soil loss for other situations assumingthat runoff is generated uniformly over the eroding area. However,the QREI30 index is based on the concept that soil loss is directly

related the product of runoff (Qe) and sediment concentration withsediment concentration varying directly with the product of thekinetic energy per unit quantity of rain and I30 (Kinnell and Risse,1998; Kinnell, 2012). In contrast, the USLE/RUSLE operates withsediment concentration varying directly with the product of thekinetic energy of the rainstorm and I30 divided by runoff amount,not rainfall amount

When runoff is produced uniformly over the eroding area, the Li

factor values determined by Eq. (7) can be combined with theproduct of (QR1 EI30)e and KUM to predict soil loss from segmentson a tilled bare fallow 9% slope with cultivation up and down theslope,

Ae:i ¼ ðQ R1EI30ÞeKUMðkmþ1

i � kmþ1i�1 Þ

ki:seg22:1m ; S ¼ C ¼ P ¼ 1:0 ð19Þ

However, with the USLE-M, the soil loss from a slope varies directlywith the runoff ratio for the slope as a whole and is not affected byhow runoff production varies spatially upslope of the downslopeboundary. Consequently, the amount of soil discharged over thebottom of a segment during an event (Qse.slope.i) on a tilled bare fal-low slope with cultivation up and down the slope is given by

Qse:slope:i ¼ Q R1e:slope:iEI30KUMkmþ1i =ð22:1Þm; S ¼ C ¼ P ¼ 1:0 ð20Þ

where QR1e.slope.i is the runoff ratio for the tilled bare fallow areadown to the bottom of the segment. When expressed as the massof soil per unit area, the soil loss in a segment is given by the differ-ence between the result produced by Eq. (20) for the 9% hillslopeending at the bottom of the segment and the result produced byEq. (20) for the 9% hillslope ending at the bottom of the previoussegment divided by the length of the segment. The result of thatmathematical process can be expressed by

Ae:i ¼ EI30KUMQR1e:slope:ik

mþ1i � QR1e:i � 1kmþ1

i�1

ki:segð22:1Þm;

S ¼ C ¼ P ¼ 1:0 ð21Þ

Fig. 5 shows the results produced by Eq. (21) and the combinationof Eqs. (11) and (13) for a storm that fell at Bethany, MO, on tilledbare fallow with K = 0.0619 t h MJ�1 mm�1 and KUM = 0.1268 -t h MJ�1 mm�1 when EI30 = 350 MJ mm ha�1 h�1 on 31 August1931 with the same hypothetical segment runoff ratios and thevalue of m (m = 0.4) used for Fig. 4. In this case, the effect of the var-iation in runoff on the soil loss predicted in each segment by Eq.

P.I.A. Kinnell / Journal of Hydrology 519 (2014) 3328–3337 3333

(21) is of greater magnitude and opposite to that obtained usingEqs. (11) and (13).

5. Discussion

Although spatial variations in the runoff ratio can occur on asegment by segment basis, there are situations where variationsin the runoff coefficient may operate at a larger scale. For example,the moisture status on a hillslope may be such that an appreciablestep change in the runoff coefficient occurs part way down theslope. As noted above, the soil loss predicted for a slope usingthe USLE-M varies directly with the runoff ratio for the slope as awhole and consequently, despite the soil loss from individual seg-ments varying spatially, the amount of soil material dischargedover the bottom boundary of slope given by Eq. (20) is determinedby the runoff ratio for the slope that end at the bottom of segment i(Fig. 6A). The soil loss from each segment given by Eq. (21) isshown in Fig. 6B and is directly related to the difference betweenthe amounts of soil material discharged over the top and bottomboundaries of each segment calculated using Eq. (20). The reduc-tion in soil loss when QR1e decreases from 0.8 to 0.2 reflects the sit-uation where the increase in discharge of water between the top ofthe segment and the bottom of the segment has a limited capacityto transport material detached in the segment. Conversely, theincrease in discharge of water when QR1e increases between 0.2and 0.8 reflects the increase transport capacity that takes placein that segment.

Fig. 6. The effect of variations in QC(seg) about the average value of 0.5 on (A) soilmaterial discharged from for 7.5 m long segments on a 300 m long one-dimensionalhillslope predicted by Eq. (20) and (B) soil loss from segments predicted by Eq. (21)when m = 0.4, K = 0.0619 t h MJ�1 mm�1 and KUM = 0.1268 t h MJ�1 mm�1 whenEI30 = 350 MJ mm ha�1 h�1. The spatial variations in QC(seg) on a segment bysegment basis are the same as shown in Figs. 4 and 5.

The responses shown in Fig. 6 are indicative of modeling atransport limited erosion system. In the USLE-M, soil loss is relatedto the factors that are directly linked to runoff producing eventsand it is well known that erosion is limited either by detachmentand transport processes. If raindrops have sufficient kinetic energyto cause detachment on tilled bare soil areas, all rainfall will pro-duce soil material that is available for transport from the soil sur-face by runoff when it occurs. The transport processes involved intransporting coarse particles in rain-impacted flows are known tolimit the transport rates of these particles (Kinnell, 2005a,b) so thatnot all the material detached in sheet and interrill erosion areasprior to and during runoff can be transported over the downslopeboundary during a runoff event. However, rilling and the flushingof some soil material from the soil surface by changes from trans-port by raindrop impact induced bedload motion to flow drivenbed load motion may produce temporal variations in soil loss overand above those caused by the mechanisms that constrain the lossof soil much of the time. These variations in detachment and trans-port processes are to some considerable extent responsible for dif-ferences between observed soil losses and those predicted by theUSLE-M.

The range in QC(seg) used above is extreme and unnatural but thehypothetical situations selected were selected to demonstrate theway that the various equations respond to spatial variations inthe production of runoff on a hillslope. Step changes in runoff coef-ficients of great magnitude part the way down a hillslope are unli-kely to occur with tilled bare fallow on a spatially uniform soil butmay occur when crop and management vary spatially on a hill-slope. For example, Kinnell and Risse (1998) observed that cornat a number of locations in the USA had associated runoff coeffi-cients that were about half those for tilled bare fallow. If runofffrom a cropped 9% slope is used to determine values of the QREI30

index then Eq. (20) is replaced by

Qse:slope:i¼QRCe:slope:iEI30KUMCUMe:ikmþ1i =ð22:1Þm; S¼ P¼1:0 ð22Þ

where QRCe.slope.i is the actual runoff ratio for the rain falling on thehillslope down to the bottom of segment i, CUMe.i is the crop andcrop management factor for the segment. CUMe.i differs from Ce.i

because QRCe is being used instead of QR1e so that the C factor doesnot have to account for the effect of crops and crop management onvariations in both the runoff and sediment concentration (Kinnelland Risse, 1998). As noted previously, the soil loss from each seg-ment on the slope is given by the difference in Qse.slope.i betweenadjacent segments divided by the length of the segment and thisresults in

Ae:i¼ EI30KUMQ RCe:slope:iCUMe:ik

mþ1i �Q RCe:slope:i�1CUMe:i�1k

mþ1i�1

ki:segð22:1Þm;

S¼ P¼1:0 ð23Þ

Fig. 7 shows the result of applying Eq. (22) to a 9% 300 m long slopefor the hypothetical cases where corn changes to tilled bare fallowmid-way down the slope (Fig. 7A) and the converse (Fig. 7B) for anevent that occurred on 31 August 1931 at Bethany, MO in the USA.For tilled bare fallow at Bethany, QC = 0.5 (Kinnell and Risse, 1998),and as noted above, Qc for corn is about half that for tilled bare fol-low, so that for corn at Bethany, QC = 0.25. The actual CUMe values forcorn during the event at Bethany are unknown but the long termaverage annual value (CUM) can serve as an example. For corn atBethany, CUM = 0.674 (Kinnell and Risse, 1998). The case wherethe same step change in QCseg associated with the change betweencorn (QC = 0.25) and bare fallow (QC = 0.5) occurs in a completelytilled bare fallow slope is provided as a comparison.

Fig. 7. The effect of step changes in QC(seg) at 150 m from 0.25 to 0.5 (A) and from0.25 to 0.5 (B) on soil material discharged from 7.5 m long segments on a 300 mlong one-dimensional hillslope predicted by Eq. (23) when m = 0.4, KUM = 0.1268 -t h MJ�1 mm�1 when EI30 = 350 MJ mm ha�1 h�1, QC(bare fallow) = 0.5, QC(corn) = 0.25.CUM (corn) = 0.0674.

3334 P.I.A. Kinnell / Journal of Hydrology 519 (2014) 3328–3337

In the case where tilled bare fallow lies below the cropped area,Fig. 7A indicates that a major increase in soil loss occurs in the seg-ment where the increase in QCseg occurs. That major increase is fol-lowed by increases in segment soil losses that result in the amountof soil material discharged at the bottom of the 300 m slope beingsame as if there had been tilled bare fallow in the top part of theslope but the same step change in QC had occurred. The increasein sediment discharge associated with the change from corn totilled bare fallow at 150 m gave a soil loss in the first bare fallowsegment of 209 t ha�1 where as 58 t ha�1 occurred in the next seg-ment. In the case of corn below tilled bare fallow on the slope,Fig. 7B indicates that a considerable amount of soil was depositedin the segment where the decrease in QCseg occurs followed byincreases in segment soil losses that lead to the amount of soilmaterial discharged over the bottom end of the 300 m slope beingappreciably higher than if corn had been cropped over the wholearea. This results from the fact that runoff ratios for those segmentsof corn are higher than when corn covers the whole slope. Thedecrease in the sediment discharge associated with the stepchange from tilled bare fallow to corn deposited 275 t ha�1 in thefirst corn segment where as 29 t ha�1 of soil loss occurred in thenext corn segment. The values for the soil loss and deposit in thefirst segment following the crop change depend on the segmentsize used. If, for example, the segment size was 30 m, then the soilloss in the first segment containing tilled bare fallow below cornwould be 97 t ha�1, not 208 t ha�1. Similarly, the deposit in the firstcorn segment below tilled bare fallow would be 47 t ha�1, not275 t ha�1. This effect can be avoided by using routines that controlthe size of the area where deposition occurs independently of seg-ment size used to model erosion on the hillslope.

When the hillslope is completely cropped with a single crop, thecombination of Eq. (11) and the event form of Eq. (4) that gives

Ae:i ¼ EI30KCe:ki�1:eff þ ki:seg� �mþ1 � kþ1

i�1:eff

ki:segð22:1Þm; S ¼ P ¼ 1:0 ð24Þ

can be used to predict segment soil loss when using the runoffadjusted slope length approach to determining Li within the RUSLE.If all material eroded in a segment on a hillslope is transported fromthe end of the hillslope, then

Ae:i ¼ EI30KCe:iki�1:eff þ ki:seg� �mþ1 � kþ1

i�1:eff

ki:segð22:1Þm; S ¼ P ¼ 1:0 ð25Þ

where Ce.i is the crop factor for segment i, can be applied when crop-ping is spatially variable. Fig. 8 shows the soil losses and dischargesfrom the 7.5 m long segments obtained by using Eq. (25) for thesame situations where Eq. (22) was applied. As noted previously,the effect of a change in QC(seg) on Li values given by Eq. (11) isreduced as distance down the slope increases so that the stepchange at 150 m has little effect on the soil loss from the segmentwhere the change occurs, but the change in Ce.i results in soil lossesfrom the segments on the lower half of the slope that are identicalor nearly identical to those that would occur had the whole slopebeen either completely tilled bare fallow or completely croppedwith corn. In contrast to Eqs. (23) and (25) does not predict anydeposition when corn occurs below tilled bare fallow because Eq.(25) is based on the assumption that all eroded material passes tothe end of the slope. The sediment discharge for the case wheretilled bare fallow lies downslope of corn produced by using Eq.(25) (Fig. 8B) is 13,727 t m ha�1 (average 45.8 t ha�1 for wholeslope) which is slightly lower than that resulting from using Eq.(23) (13,837 t m ha�1(Fig. 7A, average 46.1 t ha�1 for whole slope).Conversely, the sediment discharge produced using Eq. (25) forthe case where corn lies below bare fallow is 10,874 t m ha�1 (aver-age 36.2 t ha�1 for whole slope) and higher than when Eq. (23) isused (9326 t m ha�1 (average 31.1 t ha�1 for whole slope). The ratiofor the soil loss for the whole slope for the corn below tilled barefallow to tilled bare below corn is 0.79 when Eq. (25) is used and0.67 when Eq. (23) is used.

In the RUSLE dedicated depositional routines are included todeal with the deposition that occurs in the case where corn liesbelow continuous tilled bare fallow. RUSLE2, the form of the RUSLEthat uses a daily time step in modeling rainfall erosion on one-dimensional hillslopes in the USA (Foster et al., 2001), has recentlybeen given a capacity to predict storm soil loss for a series of rep-resentative storms at a geographic location (Dabney et al., 2011).This enhancement makes it possible to predict soil loss for theserepresentative storms using the USLE-M within RUSLE2 (Kinnell,2014) and facilitated a comparison to be made between theUSLE-M and RUSLE2 on a 300 m long 9% hillslope where manage-ment changed from bare fallow to corn midway down the slope.Fig. 9A provides the comparison of the sediment discharges fromthe 150 m and 300 m slopes when the 300 m long 9% hillslope atBethnay, MO was cropped with corn with a yield of 112 bushesper acre was ploughed in spring. The USLE-M predicts sedimentdischarges that are close to those predicted by RUSLE2 whenKUM = 0.087 t h MJ�1 mm�1 and CUMe ranged from 0.09 to 0.77depending on when the storm occurred during the year. Ce forRUSLE2 ranged from 0.07 to 0.48.

The comparison between the sediment discharges from the300 m long slope where the management changes from bare fallowto corn midway down is shown in Fig. 9B. In RUSLE2, the value ofm used in the calculation of the slope length factor for corn differsfrom that used for bare fallow. Similarly, differences exist in thevalues of P used in the calculation of soil loss from corn and barefallow. The RUSLE2 values of m and P were also applied to the mod-

Fig. 8. The effect of step changes in QC(seg) at 150 m on (A) soil loss from 7.5 m longsegments and (B) soil material discharged from 7.5 m long segments on a 300 mlong one-dimensional hillslope predicted by Eq. (25) when m = 0.4,K = 0.634 t h MJ�1 mm�1 when EI30 = 350 MJ mm ha�1 h�1, QC(bare fallow) = 0.5,QC(corn) = 0.25. C(corn) = 0.337.

Fig. 9. Sediment discharges predicted by the USLE_ and RUSLE2 for a 300 m longhillslope split into two 150 m long segments at Bethnay, MO when the wholehillslope was cropped with spring ploughed corn (122 bushels per acre yield) andwhen the top segment was bare fallow above corn in the bottom segment.

P.I.A. Kinnell / Journal of Hydrology 519 (2014) 3328–3337 3335

eling of soil loss by the USLE-M and, as shown in Fig. 9B, RUSLE2predicts much higher discharges in the latter half of the year thanthe USLE-M. Fig. 9C shows the discharges of soil from the bare fal-low and the corn areas predicted by RUSLE2. Net erosion is pre-dicted to occur in the corn when the soil discharged from theupslope bare fallow area is less than that from the corn. Con-versely, net deposition is predicted to occur when the soil dis-charged from the upslope bare fallow area is more than thatfrom the corn. Overall, RUSLE2 predicts that the average annualsoil loss from the 300 m hillslope is 295.98 t ha�1 resulting from585.11 t ha�1 of erosion in the bare fallow area and 6.85 t ha�1

net erosion in the corn. As shown in Fig. 9D, net erosion is pre-dicted by the USLE-M to always occur when the top segment iscropped with corn but net deposition is predicted to occur in moststorms when the top segment is bare. A soil loss of 177.7 t ha�1 ispredicted by the USLE-M for the 300 m long hillslope when the topsegment is bare fallow but that soil loss does not result from neterosion in the corn.

Deposition may also occur when slope gradient decreases in thedownslope direction. Fundamentally, the soil loss or gain in a seg-ment is calculated by the difference between the sediment dis-charged for segment (Qse.slope.i) and the sediment discharged fromthe previous segment (Qse.slope.i�1) divided by the length of the seg-ment (ki.seg)

Ae:i ¼Q se:slope:i � Q se:slope:i�1

ki:segð26Þ

In the RUSLE, as indicated previously, a segment is consideredto be part of a hillslope whose slope is the same as the segment(Fig. 1) so that, for an erosion event, Qse.slope.i�1 is calculated using

Q se:slope:i�1 ¼ EI30 K kmþ1i�1 Si Ce:i Pe:i=ð22:1Þm ð27Þ

where Si is the slope gradient factor, Ce.i is the crop factor, and Pe.i isthe soil conservation practice factor for segment i. The soil dis-charged from the segment is calculated using

Fig. 10. Schematic representations of a hillslope in the context of calculating the effect of topography on the discharge of sediment into a segment using the RUSLE and USLE-M models. Note that both representations are for the same hillslope profile as shown in Fig. 1 but the gray shaded area in (A) represents the effective profile of the hillslopeabove segment i when the RUSLE is used whereas, in (B), it represents the effective hillslope profile above segment i when the USLE-M is used.

3336 P.I.A. Kinnell / Journal of Hydrology 519 (2014) 3328–3337

Q se:slope:i ¼ EI30 K kmþ1i Si Ce:i Pe:i=ð22:1Þm ð28Þ

In the case of the USLE-M, the slope gradient factor for the pre-vious segment (Si�1) used to calculate Qse.slope.i�1 from

Q se:slope:i�1 ¼ Q RCe:slope:i�1EI30 KUM kmþ1i�1 Si�1 CUMe:i�1 PUMe:i�1=ð22:1Þm

ð29Þ

where Si is the RUSLE slope gradient factor for segment i�1, andCUMe.i�1 is the crop factor, and PUMe.i�1 is the soil conservation prac-tice factor for the USLE-M for segment i�1. The respective hillslopesassociated for calculating Qse.slope.i�1 are depicted diagrammaticallyin Fig. 10. In the USLE-M, the soil discharged from the segment iscalculated using

Q se:slope:i ¼ QRCe:slope:iEI30 KUM kmþ1i Si CUMe:i PUMe:i=ð22:1Þm ð30Þ

Eq. (26) will always produce a positive result when Qse.slope.i�1 andQse.slope.i are calculated using Eqs. (27) and (28) but this is not thecase when Eqs. (29) and (30) are used. Consequently, dedicated rou-tines are used to deal with deposition on hillslope with non-uni-form topography when Qse.slope.i�1 is calculated using Eq. (27). Theratio for the soil loss from the bottom half of a 300 m long contin-uous bare fallow slope to soil loss from the top half produced byRUSLE2 when the slope changes from 9% to 4.5% midway downthe slope is 0.169. Using Eq. (27) for the case where the RUSLE isapplied without deposition being taken into account produces avalue of 0.845 where as Eq. (29) (USLE-M) produces a value of0.072, less than half the value for the ratio produced by RUSLE2.Further work is necessary in order to account for the differencesbetween RUSLE2 and the USLE-M in dealing with deposition onhillslopes.

6. Conclusion

The Universal Soil Loss Equation (USLE) is an empirical modeldeveloped from data obtained from more that 10,000 plot yearsof data collected on rainfall-runoff plots in the USA. The RUSLE,the revised version of the USLE, maintained the mathematicalstructure which was based on the assumption that runoff was pro-duced uniformly over the area where soil losses are being pre-dicted. That assumption has led to the development of equationsto predict the effect of slope length when hillslopes are dividedinto segments (Eq. (7)) and cells (Eq. (10)). However, the assump-tion that runoff production is spatially uniform is often inappropri-ate under natural conditions where infiltration is spatially variable.It has been shown here that using an upslope slope length that var-ies with the ratio of the upslope runoff coefficient to the runoff

coefficient for the area down to the downslope boundary of thesegment in modifications of the RUSLE approach produces onlyminor variations in soil loss from those predicted using the stan-dard RUSLE approach when runoff is spatially variable and thenumber of segments increases. In contrast, the USLE-M approachprovides predictions of soil loss that are influenced more stronglyby runoff when runoff varies in both space and time, so that anincrease in the runoff through a segment produces an increase insoil loss while a decrease in the runoff through a segment or cellproduces as decrease in soil loss. The hypothetical examples usedto demonstrate how the USLE-M model predicts spatial variationsin soil loss on the hillslopes where runoff production is spatiallyvariable produce results that are consistent with the notion thatsoil losses from segments on the hillslopes are dominated by atransport limited rather than a detachment limited erosion system.

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