Catena 145 (2016) 39–46

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Comparison between the USLE, the USLE-M and replicate plots to modelrainfall erosion on bare fallow areas

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

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

http://dx.doi.org/10.1016/j.catena.2016.05.0170341-8162/© 2016 Elsevier B.V. All rights reserved.

a b s t r a c t

a r t i c l e i n f o

Article history:Received 13 January 2016Received in revised form 11 May 2016Accepted 20 May 2016Available online 28 May 2016

It has been proposed that the best physical model of erosion from a plot is provided by a replicate plot (Nearing,1998). Event data from paired bare fallow plots in the USLE database were used to examine the abilities of rep-licate plots, the USLE and the USLE-M to model event erosion on bare fallow plots. The Nash-Sutcliffe efficiencyfactor as applied to logarithmic transforms of the data was used to evaluate the overall performance ofmodels ata number of locations. The value of this efficiency factor is influenced by both systematic and stochastic differ-ences between the pairs. Systematic differences are the result of systematic differences in event runoff orevent sediment concentration or both, and the degree of the impact of them varies as the regression coefficientfor the relationship between the soil losses from the pairs varies from the value of 1.0. Inmost cases the replicatemodel performed better than the USLE-M thatmodelled event soil loss as a product of observed event runoff andevent sediment concentration directly related to the EI30 index. Generally, failure of replicates to match runoffwas compensated by the ability of the replicated to determine sediment concentrations better than the USLE-M.

© 2016 Elsevier B.V. All rights reserved.

Keywords:USLE databaseSoil loss predictionUSLE/RUSLEUSLE-MRunoff

1. Introduction

The Universal Soil Loss Equation (USLE; Wischmeier and Smith,1965, 1978) and subsequent revisions (eg RUSLE; Renard et al., 1997)and refinements, have provided a model for predicting soil erosionloss that has been used rightly and wrongly throughout the world.The USLE operatesmathematically in two steps. The first step is the pre-diction of long term (~20 years) average annual soil loss from the unitplot (A1), a bare fallow area 22.1 m long on a 9% slope gradient, interms of a rainfall runoff factor (R) and a soil dependent factor (K).

A1 ¼ RK ð1Þ

where A1 has units of mass per unit area, R is the long term product ofstorm kinetic energy (E) and the maximum 30-minute intensity (EI30),and K is the loss of soil per unit of R. In order to predict soil lossesfrom areaswhich differ from the unit plot, A1 is multiplied in the secondstep by factors that account for slope length (L), slope gradient (S), cropand crop management (C) and soil conservation practice (P).

A ¼ A1 LSC P ð2Þ

where L= S= C= P=1.0 for the unit plot. Eq. (1) provides themeans

of taking account of spatial variations in climate and soil. Consequently,the unit plot is the primary physical model on which the USLE model-ling approach is based. However, it has been proposed that the bestphysical model of erosion from a plot is provided by a replicate plot(Nearing, 1998). The USLE data base contains data from replicatedbare fallow plots installed at a number of locations. The objective ofwork reported here is to examine the concept that “the best physicalmodel of erosion froma plot is provided by a replicate plot” by analyzingevent data from individual pairs of replicated bare fallow plotscontained in the USLE data base and compare the result with the abilityof the USLE/RUSLE and the USLE-M (Kinnell and Risse, 1998) to modelevent soil losses on bare fallow areas.

1.1. Measures of model effectiveness

Replicated plots show “random” (stochastic) variations in soil lossesbetween them (Wendt et al., 1986) at the event scale that tend to benormally distributed (Nearing, 1998). The primary issue that concernedNearing was the observation that the coefficients of variation werehigher for small soil losses than high soil losses so that he perceivedthat the observation that models like the USLE and WEPP (Flanaganand Nearing, 1995) tended to over predict small soil losses and underpredict large soil losses (Tiwari et al., 2000) was a mathematical phe-nomenon rather than a function of any bias inherent in the modelsthemselves. Subsequently, Nearing et al. (1999) examined data from

40 P.I.A. Kinnell / Catena 145 (2016) 39–46

replicated plot using a relative difference term described by.

Rdiff ¼ M2–M1ð Þ= M2 þM1ð Þ ð3Þ

where M1 and M2 are the paired losses from two replicate plots. Theproperties of Rdiff are that its value may vary between −1 and +1 andwhenM1 =M2 have a value of zero (Nearing et al., 1999). In their anal-ysis, Nearing et al. computed two values of Rdiff. For each pair of plots, Aand B, the first Rdiff valuewas calculated using the soil loss from plot A asM1 and the soil loss from plot B asM2. The second value was calculatedusing the soil loss from plot B asM1 and the soil loss from plot A asM2.The valueswere then plotted against the respective values ofM1. Conse-quently, for every paired loss, there are two values of Rdiff of equal abso-lute value but one is positive and the other is negative, and the negativevalue is always plotted against the lower of the two soil losses and thepositive value always plotted against the higher of the two soil losses.Nearing (2000) proposed that

Rdiff :lower ¼ 0:236 log10 Mð Þ–0:641 ð4aÞ

Rdiff :upper ¼ −0:179 log10 Mð Þ þ 0:416 ð4bÞ

provided the lower and upper boundaries for the 95% occurrence inter-val for Rdiff for the soil losses on both bare fallow and vegetated plots inUSDA repository.

Eqs. (4a) and (4b) result from the analysis of event soil losses from alarge number of replicate plots that included not just bare fallow butalso cropped plots. Fig. 1A shows the values of Rdiff obtained for bare fal-low plots 1-8 and 1-18 at Presque Isle, ME for the 85 events when bothplots produced soil loss. 6% of the Rdiff fell beyond the limits set byEqs. (4a) and (4b). In contrast, for bare fallow plots 1-8 and 1-3 at thesame location, 27% of the Rdiff values for 82 erosion events that producedsoil loss on both plots fell beyond the limits set by Eqs. (4a) and (4b)(Fig. 1B). Although, this comparison obtained using the Nearing's ap-proach indicates that plots 1-8 and 1-18 were better at modelling the

Fig. 1. Rdiff values for (A) replicate bare fallow plots 18 and 8, and (B) for replicate barefallow plots 3 and 8, at Presque Isle calculated using Eq. (3) together with the upper andlower limits determined by Eqs. (4a) and (4b).

soil losses from each other than plots1-8 and 1-3, that approachdoes not reflect the fact that the average absolute difference fromthe observed values when plots 1-8 and 1-18 were considered(0.324 t ha−1) was more than half that when plots 1-8 and 1-3 wereconsidered (0.687 t h−1). Also, the method does not facilitate compari-sons to bemadewhen all the Rdiff values fall within the upper and lowerlimits. Consequently, the approach proposed by Nearing (2000) doesnot provide a usable index for evaluating the capacity an individualplot to act as a model of a replicate.

Recently, Bagarello et al. (2015) suggested that the relationship be-tween the absolute differences between observed or measured(M) and modelled or predicted (P) values as expressed by.

│P–M│ ¼ aMb1 ð5Þ

where a and b1 are empirical coefficients, is sufficient to establish the ac-curacy level of the predictions. However, Eq. (5) cannot be evaluated ifany of the absolute differences in the data set equal zero. Consequently,because there are a number of events where event soil loss from thereplicate bare fallow in the USLE database were the same, Eq. (5) isnot suitable for evaluating the capacity of a bare fallow plot to act asmodel of a replicate in the USLE database.

It follows from above that the methodologies adopted by Nearing(2000) and Bagarello et al. (2015) are not suited to evaluating the ca-pacity of replicates to model soil loss from individual bare fallow plotsin the USLE database. Often, the Nash-Sutcliffe model efficiency index((Nash and Sutcliffe, 1970) is applied to determine how effective amodel is in predicting observed values. The index,

NSE ¼ 1−

NXYo−Ymð Þ2

n ¼ 1NX

Yo−Moð Þ2n ¼ 1

ð6Þ

where Yo is the observed value, Ym is the modelled value, and Mo is themean of Yo, provides a comparison between the ability of themodel andusing the mean of the observed values to predict the observed values.Positive values indicate that using the model is better using the meanwhereas negative values indicate that using the model is worse usingthemean. A value of 1.0 is produced by the perfect model. Consequent-ly, as demonstrated here in this paper, the fundamental approach un-derlying the Nash-Sutcliffe model efficiency index provides a methodthat is well suited to evaluating the capacity of replicates, the USLEand the USLE-M to model soil loss from individual plots.

2. Data source and analysis

Kinnell and Risse (1998) used bare fallow plot data held in an ar-chive of data originally used by Wischmeier and Smith in developingthe USLE to provide metric values of K at 14 locations in the USA. Themajority of the plots had slope lengths of 22.1 m and slope gradientsvaried from 3 to 19% (Kinnell, 1998). 11 locations had replicated barefallowplots. Data from the same archive is used here. TheUSLE databasecurrently available online (http://topsoil.nserl.purdue.edu/usle/) pro-vides more extensive runoff and soil loss data at some locations butlacks data at others (e.g. Morris (MN), LaCrosse (WI), Madison (WI),Guthrie (OK), Castana (IA)). Also, the current data base lacks informa-tion about the treatments applied to the plots.

The data for the pairs of replicated plots were sorted to remove asmall number of events where data were recorded for one plot butnot for the other. The model efficiency was then calculated usingthe Nash-Sutcliffe index (Nash and Sutcliffe, 1970) applied to log

Fig. 2. The relationship between NSE(ln) and b in Eq. (8).

41P.I.A. Kinnell / Catena 145 (2016) 39–46

transforms of the observed and modelled values,

NSE lnð Þ ¼ 1−

NXlnYo− lnYm1ð Þ2

n ¼ 1NX

lnYo−M lnoð Þ2n ¼ 1

ð7Þ

where Yo is the observed value, Ym is themodelled value, andMlno is themean of the log transforms of Yo. Eq. (7) is used instead of the standardNash-Sutcliffe index (Eq. (6)) in order to give more emphasis to differ-ences between low observed andmodelled values than provided by thestandard index (Krause et al., 2005) because the objective of the analy-sis is to examine the capacity of a model to predict all event soil losseswell without favoring those events which have a major influence onthe long term average annual values that are used traditionally inassessing erosion for management purposes. The NSE(ln)values for thereplicate pairs at the 11 locations in the USA and replicates at Canberra,Australia (Kinnell, 1983, 1985) are presented in Table 1.

TheNSE(ln) values at the 12 locations vary from aminimumof 0.592(Plots 1-3 and 1-8, Presque Island) to 0.993 (Plots 3 and 1, and 3 and 2,Canberra). The values of b for the equation.

Ae:principal ¼ bAe replicate ð8Þ

where Ae.principal is event soil on the principal plot and Ae replicate is eventsoil loss on the replicate, are also presented in Table 1. At most of the lo-cations, the event soil losses from the paired plots were highly correlat-ed. However, generally, the NSE(ln) values for the pairs tended todecline linearly as b varied from 1.0 (Fig. 2) but some variation in thevalues occurred that was not directly associated with variations in b.Two types of variation affect NSE(ln) values, systematic variation andstochastic variation. The trend for NSE(ln) values for the pairs to varyas b varies from 1 is associated with systematic variation being domi-nant, whereas variations in NSE(ln) that show no relationship with bare associated with stochastic variation being dominant.

Table 1NSE(ln) values for replicate model applied to bare fallow plots together with values of bobtained for Eq. (8).

Location Obs Principal Replicate NSE(ln) b R2

Presque Isle 82 Plot 1-3 Plot 1-8 0.592 0.596 0.881Presque Isle 85 Plot 1-8 Plot 1-18 0.838 0.957 0.960Marcellus 65 Plot 1-2 Plot 1-3 0.957 0.996 0.965Marcellus 65 Plot 1-3 Plot 1-2 0.952 0.979 0.966Morris 74 Plot 1-5 Plot 1-10 0.706 0.876 0.920Morris 74 Plot 1-10 Plot 1-13 0.636 1.022 0.966Morris 74 Plot 1-13 Plot 1-5 0.737 0.953 0.968Castana 116 Plot 1-3 Plot 1-4 0.829 0.934 0.928Castana 116 Plot 1-4 Plot 1-3 0.878 1.021 0.925Bethany 135 Plot 1-9 Plot 1-10 0.772 1.247 0.882Bethany 135 Plot 1-10 Plot 1-9 0.765 0.759 0.893McCredie 124 Plot 2-1 Plot 2-18 0.763 0.934 0.962McCredie 124 Plot 2-18 Plot 2-1 0.750 1.039 0.962LaCrosse 97 Plot 1-8 Plot 1-9 0.828 1.181 0.941LaCrosse 97 Plot 1-9 Plot 1-8 0.832 0.813 0.941Holly Springs 187 Plot 3-5 Plot 3-7 0.826 1.162 0.885Holly Springs 187 Plot 3-7 Plot 3-5 0.843 0.784 0.883Watkinsville 111 Plot 2-47 Plot 2-64 0.722 1.322 0.827Watkinsville 111 Plot 2-64 Plot 2-47 0.765 0.665 0.827Madison 66 Plot 1-5 Plot 1-12 0.893 0.881 0.735Madison 66 Plot 1-12 Plot 1-5 0.886 0.888 0.850Tifton 103 Plot 1-2 Plot 2-4 0.820 1.095 0.850Tifton 103 Plot 2-4 Plot 1-2 0.838 0.832 0.852Canberra 50 Plot 2 Plot 1 0.884 0.901 0.991Canberra 50 Plot 3 Plot 1 0.993 0.993 0.993Canberra 50 Plot 3 Plot 2 0.993 1.100 0.987

3. The replicate plotmodel in comparison to theUSLE/RUSLE and theUSLE-M

Although not designed to predict event soil loss (Wischmeier, 1976),the linear relationship between Ae1 and EI30 described by.

Ae1 ¼ EI30K ð9Þ

Fig. 3. The relationships between NSE (ln) values and average runoff ratios for erosionsevents associated with the USLE, the USLE-M and the replicate plot models. NB. The twonegative NSE(ln) values associated for the USLE-M are for bare fallow plots at Tifton, GA.

Table 2NSE(ln) valueswhen USLE, USLE-Mand replicatemodelswere applied to bare fallow plotsin the USLE data base.

————NSE(ln)————

Location Obs Principal roffratio

Replicate USLE USLE-M Replicate

Presque Isle 82 Plot 1-3 0.254 Plot 1-8 0.091 0.693 0.592Presque Isle 85 Plot 1-8 0.343 Plot 1-18 0.137 0.772 0.838Marcellus 65 Plot 1-2 0.407 Plot 1-3 0.365 0.786 0.957Marcellus 65 Plot 1-3 0.410 Plot 1-2 0.366 0.771 0.952Morris 74 Plot 1-5 0.177 Plot 1-10 0.015 0.838 0.706Morris 74 Plot 1-10 0.204 Plot 1-13 0.133 0.784 0.636Morris 74 Plot 1-13 0.180 Plot 1-5 −0.078 0.758 0.737Castana 116 Plot 1-3 0.300 Plot 1-4 0.396 0.777 0.829Castana 116 Plot 1-4 0.283 Plot 1-3 0.307 0.809 0.878Bethany 135 Plot 1-9 0.465 Plot 1-10 0.498 0.761 0.772Bethany 135 Plot 1-10 0.425 Plot 1-9 0.396 0.716 0.765McCredie 124 Plot 2-1 0.216 Plot 2-18 0.332 0.842 0.763McCredie 124 Plot 2-18 0.260 Plot 2-1 0.058 0.548 0.750LaCrosse 97 Plot 1-8 0.437 Plot 1-9 0.566 0.854 0.828LaCrosse 97 Plot 1-9 0.432 Plot 1-8 0.547 0.853 0.832Holly Springs 187 Plot 3-5 0.662 Plot 3-7 0.520 0.645 0.826Holly Springs 187 Plot 3-7 0.639 Plot 3-5 0.491 0.704 0.843Watkinsville 111 Plot 2-47 0.459 Plot 2-64 0.443 0.594 0.722Watkinsville 111 Plot 2-64 0.364 Plot 2-47 0.354 0.453 0.765Madison 66 Plot 1-5 0.514 Plot 1-12 0.024 0.781 0.893Madison 66 Plot 1-12 0.513 Plot 1-5 0.058 0.751 0.886Tifton 103 Plot 1-2 0.342 Plot 2-4 0.263 −0.511 0.820Tifton 103 Plot 2-4 0.328 Plot 1-2 0.157 −0.346 0.838

42 P.I.A. Kinnell / Catena 145 (2016) 39–46

where Ae1 is event soil loss from the unit plot, underpins Eq. (1) as dem-onstrated in Fig. 3 of Wischmeier and Smith (1958). However, in manycases.

Ae1 ¼ QRe1EI30KUM ð10Þ

where QRe1 is the runoff ratio for the event on the unit plot and KUM isthe soil erodibility factor associated with using QRe1EI30 instead of EI30,predicts event soil losses better than Eq. (9) when runoff and rainfallamounts are known. Eq. (10) is the equation used to predict event soilloss from the unit plot in the USLE-M (Kinnell and Risse, 1998). TheQRe1EI30 results from the observation that sediment concentrations arebetter related to EI30 per unit rainfall amount than EI30 per unit runoffas implied by the USLE.

Although, from Eq. (9), K may be perceived as being the regressioncoefficient in the relationship between event soil loss and EI30, originallyin the USLE, K was calculated as the long term average annual soil lossdivided by the long term average annual value of EI30 for the events con-tributing to that loss,

K ¼

NXA1eð Þn

n ¼ 1NXEI30ð Þn

n ¼ 1

ð11Þ

where N is the total number of erosion events contributing to the longterm soil loss. However, in terms of comparing the abilities of theUSLE and the USLE-M with ability of a replicate plot to model the lossfrom the principal plot undertaken above,

AeBF ¼ EI30K LS ð12Þ

where AeBF is event soil loss from a bare fallow plot and.

AeBF ¼ QReBFEI30KUMLS ð13Þ

where AeBF is event soil loss from a bare fallow plot, QeRBF is the runoffratio for the event producing that loss and the product of L and S differsfrom 1.0. Consequently, the values of K L S and KUM L S were calculatedusing.

K LS ¼

NXAeBFð Þn

n ¼ 1NX

EI30ð Þnn ¼ 1

ð14Þ

and,

KUM LS ¼

NXAeBFð Þn

n ¼ 1NX

QReBFEI30ð Þnn ¼ 1

ð15Þ

where values of L and S are the same to each plot in the pair. UsingQReBF

in Eq. (15) instead of QRe1 follows from the assumption that any varia-tion in slope length from 22.1 m and slope gradient from 9% does nothave a significant influence on the volume of runoff produced per unitarea. The NSE(ln) values associated with the application ofEqs. (12)–(15) on the bare fallow plots at the 11 locations in the USAare presented in Table 2.

Fig. 3A shows the NSE(ln) values for the USLE and USLE-M modelsplotted against the average runoff ratios for erosion events on theplots at the 11 locations in the USA. NSE(ln) values for the USLE-M do

not show any appreciable relationship with the average runoff ratiofor erosion events but NSE(ln) values for the USLE tend to increasewith the average runoff ratio. This is because, as the average runoffratio for erosion events increases, the values of EI30 and the values ofQReBFEI30 move towards each other. When a replicate is used as amodel, NSE(ln) values tend to increase as the average runoff ratio in-crease at a lesser rate than for the USLE (Fig. 3B). In 16 of the 23 cases,using the replicate produces higherNSE(ln) values than those producedby the USLE-M (Table 2). However, the relative performance of theUSLE-M and the replicatemodel approach on individual plots is not par-ticularly obvious when the NSE(ln) values for the replicate and USLE-Mmodels are plotted together against the average runoff ratios for erosionevents on the plots at the 11 locations in the USA (Fig. 3C).

As noted above, the Nash-Sutcliffe index provides a comparison be-tween result produced by themodel and the result when event soil lossare predicted by the mean of the observed losses. It is a dedicated ver-sion of the Model Comparison Index (MCI) that compares one modelagainst another,

MCI ¼ 1−

NXYo−Ym1ð Þ2

n ¼ 1NX

Yo−Ym2ð Þ2n ¼ 1

ð16Þ

where Yo is the observed value, Ym1 is the value modelled by model 1and Ym2 is the value modelled by model 2. In the Nash-Sutcliffe index,themean of the observed soil losses is used as Ym2. Obviously, individualNash-Sutcliffe index values for models 1 and 2 can be compared as ameans of determiningwhichmodel is best but theMCI provides a directcomparison using a single value. The MCI value will vary from 1.0 ifmodel 1 is the perfect model to 0 when both models are equally effec-tive in their predictions. A value of 0.5 indicates that model 1 produceshalf the value of the residual sum of squares produced bymodel 2. Neg-ative values will be generated if model 1 is less effective in predictingthe observed values thanmodel 2. Consequently, theMCIprovides a sin-gle numerical indicator of whether model 1 is better or worse thanmodel 2.

43P.I.A. Kinnell / Catena 145 (2016) 39–46

Fig. 4A shows theMCI values for the log transforms of the observedand predicted values obtained the 11 locations in the USA using.

MCI lnð Þ ¼ 1−

NXlnYo− lnYm1ð Þ2

n ¼ 1NX

lnYo− lnYm2ð Þ2n ¼ 1

ð17Þ

when the replicate model is model 1 and the USLE-M is model 2. As in-dicated in respect to using NSE(ln) values,MCI(ln) values are used herebecause the objective of the analysis is to examine the capacity of amodel to predict all event soil losses well not just those which have amajor influence on the long term average annual values that are usedtraditionally in assessing erosion for management purposes. The nega-tive values ofMCI(ln) shown in Fig. 4A indicate that theUSLE-Mpredictssoil losses better than using a replicate on 7 of the plots. As to be expect-ed from Fig. 3B,MCI(ln) valueswhen the replicatemodel ismodel 1 andtheUSLE ismodel 2 are all positive (Fig. 4B) and so indicate that the rep-licate model was always better than the USLE.

It should be noted that the purpose of the MCI index is to provide asingle measure that is relevant to assessing the relative capacities oftwo different models to predict observed losses. It is not a replacementfor the NSE index which provides a comparison between result pro-duced by the model and the result when event soil losses are predictedby the mean of the observed losses.

4. Discussion

As noted above, differences between observed and modelled valueswhen a replicate is used as the model may be systematic or stochastic(random) or both and the values of NSE(ln) are influenced by bothtypes despite plot management being by design uniform in space andtime. The NSE(ln) observed for the prediction of soil losses from plot1-3 by plot 1-2 at Marcellus (NSE(ln) = 0.952) was greater than thatfor Presque Isle plot 1-18 predicted by plot 1-8 (NSE(ln) = 0.838)

Fig. 4. Relationships between plot runoff ratio for erosion events andMCI(ln) for replicatemodel to (A)USLE-M and (B) USLE.

because there was more stochastic variation associated with the twoplots at Presque Isle (Fig. 5). In the case of the prediction of lossesfrom plot 1-8 by plot 1-3 at Presque Isle, the much lower NSE(ln)value (NSE(ln) = 0.592) resulted from both stochastic and systematicvariation resulting from b = 1.5 (Fig. 5) rather than b being close to1.0. The three plots near Canberra were established on a crusting soiland cultivated only once a year with chemicals used for weed control.This together with sheet erosion being the dominant form of erosion,produced less stochastic variation in the soil losses than was generallyassociated with the 23 plots considered in the USA.

Given that soil loss is directly related to the product of runoff andsediment concentration, the soil loss per unit runoff, both types of vari-ation in soil loss may result from stochastic and systematic variations inrespect to runoff or sediment concentration, or both (Fig. 5). As shownby Fig. 3B, NSE(ln) values for the replicate plot model tend to vary tosome degree with the average runoff ratio for erosion events. The likeli-hood that event runoff from the replicate will be close to that from theprincipal plot increases as the average runoff ratio for erosion events in-creases, so that the error generated by differences in runoff generationbetween the plot pairs decreases as the runoff ratio increases. The roleof event runoff in influencing the soil loss predicted by a replicate is il-lustrated by Fig. 6. Fig. 6A shows the relationship between event soillosses on plot 1-3 at Presque Isle and event soil losses on plot 1-8.Fig. 6B shows the relationship between event soil losses on plot 1-3 atPresque Isle and event soil losses calculated using the product of eventrunoff from plot 1-3 and event sediment concentration for plot 1-8.This resulted inNSE(ln) increasing for 0.592 to 0.799. This value is great-er than the 0.693 obtained when the USLE-Mwas used (Table 2). Obvi-ously, the capacity of the plot 1-8 to determine event sedimentconcentrations for plot 1-3 plot at Presque Isle is better than the USLE-M.

As noted above, in the USLE/RUSLEmodelling environment, the unitplot can be seen as being the physical model upon which the model isbuilt. Consequently, there is an expectation from Eqs. (1) and (2) thatthe two stepped approach adopted in the USLE/RUSLE model appliesto predicting event soil losses from a cropped area (AeC) so that.

AeC ¼ Ae1K LSCePe ð18Þ

where Ae1 is given by Eq. (9), event soil erodibility (Ke) remains constantand equal to K, and Ce and Pe are event values for the C and P factors.Under these circumstances, there should be a direct relationship be-tween the soil losses from a cropped plot and an adjacent bare fallowplot. Fig. 7A shows the relationship between the soil losses observedon plot 2-1 cropped with conventional corn at Clarinda, IA and thosepredicted from an adjacent bare fallow plot (plot 2-10) using the fort-nightly Ce values when Eq. (18) was applied at that location0.001 t ha−1 has been added to the observed soil losses so that valuespredicted when the observed values are zero can be displayed usinglog scales. Fig. 7A shows that the assumption that runoff occurs onboth the bare fallow plot and the cropped plot for every erosion eventthat occurs on the bare fallow plot is incorrect at Clarinda and almostcertainly at most other locations.

Eq. (18) operates irrespective of Ae1 being predicted by Eq. (9),which used the EI30 as the event erosivity index, or Eq. (10), whichuses the QRe1EI30 as the event erosivity index. However, the QREI30index approach provides themeans predicting event soil loss from veg-etated areas by using the runoff ratio for a cropped plot (QRC) ratherthan the runoff ratio for the unit plot (QR1),

AeC ¼ QReCEI30Ke;UMLSCe:UMPe:UM ð19Þ

whereKe,UM, Ce.UM, and Pe.UM are the event values for soil erodibility, cropand crop management, and soil conservation protection when QReCEI30is used as the event erosivity index. Procedures do not currently existto estimate Ke,UM, Ce.UM, and Pe.UM but they do exist to determine the

Fig. 5. Observed and predicted event soil loss, runoff and sediment concentration for selected replicate bare fallow plots at Marcellus and Presque Isle.

44 P.I.A. Kinnell / Catena 145 (2016) 39–46

long term average annual values KUM, CUM, and PUM (Kinnell and Risse,1998). Fig. 7B shows the relationship between the observed and pre-dicted event soil losses for the same plot with corn at Clarinda, IA asused in Fig. 7A when soil losses were predicted using.

AeC ¼ QReCEI30KUMLSCUMPUM ð20Þ

Fig. 6. Relationship between event soil loss fromplot 1-3 and Presque Island and (A) eventsoil loss fromplot 1.8 and (B) event soil loss calculated as the product of event runoff fromplot 1-3 and event sediment concentration for plot 1-8.

Unlike the USLE/RUSLE approach, Eq. (20) does not operated math-ematically in two steps despite KUM likeK being determined for unit plotconditions. The standard version of theMCI given by Eq. (16) produced

Fig. 7. Relationships between observed soil losses on plot 2-1 cropped with corn atClarinda, IA and soil losses predicted by (A) the RUSLE and (B) the USLE-M.

Table 3NSE(ln) values obtained for replicate mode at Presque Isle with various values of α added to data to enable logarithmic values to be obtained when no soil loss was measured.

————NSE(ln)———— Regression

Location Obs Principal Replicate USLE USLE-M Replicate b R2

Presque Isle (α = 0.1) 91 Plot 1-3 (7 × 0) Plot 1-8 (2 × 0) 0.106 0.731 0.259 0.583 0.854Presque Isle (α = 0.01) 91 Plot 1-3 (7 × 0) Plot 1-8 (2 × 0) 0.055 0.781 0.202 0.583 0.854Presque Isle (α = 0.001) 91 Plot 1-3 (7 × 0) Plot 1-8 (2 × 0) 0.035 0.843 0.041 0.583 0.854

45P.I.A. Kinnell / Catena 145 (2016) 39–46

a value of 0.456 when the USLE-M approach (Eq. (20)) was used asmodel 1 and the USLE/RUSLE unit plot approach (Eq. (18)) as model 2indicating that average mean square error for Eq. (20) was a littlemore than half that for Eq. (18). Consequently, the USLE-M approachadopted in Eq. (20) outperformed the two stepped mathematical ap-proach used by the USLE/RUSLE model (Eq. (18)) when runoff fromthe cropped area was known. In both cases, EI30 for an event is thesame, and the product of L, S and the relevant erodibility factorremained constant with time. Consequently, the differences in eventsoil losses for the corn at Clarinda produced by the twomodels are gen-erated by the product ofQRC and CUM varying between events in the caseof USLE-M (Eq. (20)) and Ce varying between events in the case of theRUSLE (Eq. (18)). For the data set where only event soil losses greaterthan zero from plot 2-1 were considered, the NSE(ln) value forEq. (20) was 0.605 whereas that for the replicate model using plot 2-2was 0.701. Arguably, the difference would have been less had it beenpossible to determine and use CUMe values instead of CUM.

The issue of runoff occurring on one plot and not on another is notrestricted to the situation involving bare fallow and cropped plots. It

Fig. 8. Relationships between observed event soil losses and event soil l

also an issue with bare fallow replicates (Bagarello et al., 2015) but, asnoted earlier, relatively few events where erosion occurred on oneplot but not on another existed in the USLE plot data used here andthesewere removed from the data set analyzed so as to avoid themath-ematical problem associated with obtain logarithmic values when zerooccurs. Adding a small value α to both sets of data can overcome thismathematical problem but, as shown by Table 3, the value of NSE(ln)varies appreciably as the value of α declines so the approach cannotbe recommended.

As a general rule, the NSE(ln) values for the USLE-M are positive.However bare fallow plots at Tifton, GA produces negative values(Table 2, Fig. 3A and C). At Tifton, event sediment concentrations de-creasedwith event runoff (Fig. 8) but at all other locations, as illustratedby the data for plot 1-9 at Bethany (Fig. 8), event sediment concentra-tions did not vary appreciably with event runoff. The USLE-M operatesas a transport limitedmodel (Kinnell, 2015) but the decline in sedimentconcentration with runoff on the bare fallow plots at Tifton indicateserosion occurring under detachment limiting conditions. Why this isso has yet to be explained. Obviously, the advantage of using the

osses predicted by the USLE and the USLE-M at Bethany and Tifton.

46 P.I.A. Kinnell / Catena 145 (2016) 39–46

replicate model at Tifton is that the replicate model automatically dealswith this relationship whereas the USLE-M does not.

5. Conclusion

The objective here has been to examine the concept that “the bestphysical model of erosion from a plot is provided by a replicate plot”by analyzing event data from individual pairs of replicated bare fallowplots contained in the USLE data base. Although, as a general rule,there is a high correlation between the losses from the individualpairs, very few produce values near 1.0 for b in Eq. (8),

The deviation from b = 1.0 results from systematic errors whichmay be generated by differences in the capacity of the individual plotsto produce runoff or differences in event sediment concentration.Models like the USLE-M that explicitly consider runoff as a factor in pro-ducing soil loss have the potential to account to event soil loss from aplot better than theUSLEwhen runoff is known or predictedwell. How-ever, even though the runoff from the principal plot is not determineddirectly by the replicate plot model, the performance of that model isoften better than the USLE-M using observed runoff values. It would ap-pear that the lack of the ability to predict observed runoff is more thanoffset by the ability of the replicate to determine the sediment concen-trations on the principal plot. This result is understandable given thatthe concept that event sediment concentration is directly related toEI30 in the USLE-M results from nothing more than an empirical obser-vation that sediment concentrations are better related to EI30 per unitrainfall amount than EI30 per unit runoff as implied by the USLE(Kinnell and Risse, 1998).

The data sets used in the analysis preformed here did not includeevents where runoff occurred on one plot but not on another becauseNSE(ln) values cannot be determined in data sets when event soil lossesof zero occur. There were few cases where soil loss occurred on one plotbut not on another but failure to produce runoff on a replicate is an issuethatmay need to be considered in some experiments. Failure to producerunoff from one plot while runoff is produced from another is issue inthe USLE modelling system where the soil loss from the unit plot ismodified by L, S, C and P factors tomodel soil losses fromareas that differin some way from the unit plot (Eq. (18)). The strength of any model topredict event soil loss depends on its ability to account for the variationsin both event runoff and event sediment concentration that occur on thearea where the soil losses are being modelled. The weakness of theUSLE-M, and other event models like WEPP (Flanagan and Nearing,1995), is that when it comes to predicting event soil loss in practical

situations the actual event runoff values are unknown and have to bepredicted. That is usually difficult to do but that does not mean thatthe model is, in itself, a bad model.

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