Published online 11 April 2005
Published in Soil Sci Soc Am J 69:674-680 (2005)
DOI: 10.2136/sssaj2004.0047
© 2005 Soil Science Society of America
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Soil & Water Management & Conservation
ALTERNATIVE APPROACHES FOR DETERMINING THE USLE-M SLOPE LENGTH FACTOR FOR GRID CELLS
Peter I. A. Kinnell*
School of Resource, Environmental and Heritage Sci., Univ. of Canberra, Canberra, ACT 2601, Australia
* Corresponding author (peter.kinnell{at}canberra.edu.au)
ABSTRACT
Erosion within a grid cell depends on cell size and the size of the area that is upslope of the cell. The factor accounting for slope length when the USLE-M is applied to predicting erosion in grid cells needs to (a) equal the universal soil loss equation (USLE) slope length factor for the cell when the area above the cell is completely pervious and (b) be directly related to the Desmet and Govers slope length factor when runoff is generated uniformly over the whole area. The procedure for calculating the slope length factor proposed previously fails to meet the second criteria. Alternative approaches which consider the contribution of the upslope area to the determination of the slope length factor to vary when the ratio of the runoff coefficient of the upslope area varies from that of the cell are described and shown to meet both criteria. The approach where the slope length factor is based on the ratio of the runoff coefficients for the upslope area and the upslope area plus the cell produces slope length factor values that lie within theoretically acceptable boundaries.
Abbreviations: RUSLE, revised universal soil loss equation • USLE, universal soil loss equation
KINNELL AND RISSE (1998) showed that the ability of the USLE (Wischmeier and Smith, 1978) to account for event soil loss could be improved by multiplying the USLE event erosivity factor by the runoff coefficient for the event (QCe, the amount of runoff per unit quantity of rain on an area). The version of the USLE using this modification of the USLE erosivity factor is called the USLE-M. While the USLE-M is an empirical model, the event erosivity factor has some physical basis. It was developed from the concept that erosion is directly related to the product of runoff volume and sediment concentration and the suggestion that the sediment concentration for an event is directly related to (a) the average amount of rainfall for kinetic energy per unit quantity of rain and (b) effect of rainfall intensity which could be accounted for by I30, the maximum 30-min rainfall intensity.
As shown by Kinnell and Risse (1998), the modification of the USLE event erosivity factor used in the USLE-M requires changes to be made to all USLE factors that influence runoff when the new erosivity index is used. In modeling erosion within catchments or watersheds, it is common to represent the catchment or watershed by a grid of square cells in which factors like slope gradient, soil erodibility, and crop management are uniform in any given cell. In the case of applying the USLE-M to such grid cells where the event erosivity factor is given by the product of the runoff ratio (QRe, the volume of the water flowing out of the cell during the event divided by the volume of rain falling on the cell) and EI30 (the product of event kinetic energy and the maximum 30-min intensity), Kinnell (2001) proposed that the slope length factor for a cell with coordinates i,j that applies during Event e (LUMe.i,j) could be expressed by
 | [1] |
where
 | [2] |
and Ai,j-in is the area (m2) contributing to flow into the cell with coordinates i,j, D is the length (m) of the sides of the grid cell, xi,j a factor that is dependent on flow direction relative to grid cell orientation and m is the slope length exponent defined for use with the RUSLE (Renard et al., 1997). The variable m varies with slope gradient. The terms (Ai,j-in + D2)m+1, Ai,j-inm+1, Dm+2, xmi,j, and (22.13)m result from a slope length factor for applying the USLE to grid cells developed by Desmet and Govers (1996) on which Eq. [1] is based. QCe.i,j-all is the runoff coefficient for the whole area that contributes to runoff out of the cell during an event (hence the use of the subscript e) and is given by Qe.i,j-all/Be where Qe.i,j-all is the runoff per unit area from the area that includes the cell and Be as rainfall amount per unit area during the event. Likewise, QCe.i,j-in is the runoff coefficient for the area upslope of the cell and is given by Qe.i,j-in/Be where Qe.i,j-in is runoff per unit area for the area upslope of the cell. The QRe.i,j-cell is the runoff ratio for the cell. In the case of runoff coefficients (QCe), Qe, and Be apply to the same area and so take on values from 0 to 1 when runoff is generated by infiltration excess. However, the runoff ratio for a cell (QRe.i,j-cell) is given by the volume of runoff discharged from the cell per unit volume of rain falling on that cell. The volume of runoff discharged from the cell is given by Qe.i,j-in Ai,j-in + Qe.i,j-cell D2 and because a large proportion of that volume may come from upslope, QRe.i,j-cell can take on values much >1.
In the context of this paper, the runoff ratio for the cell (QRe.i,j-cell,) is determined by dividing the volume of the water flowing out of the cell during the event by the volume of rain falling on the cell. Because the volume of runoff from the cell comes from both the cell and the upslope area, the runoff ratio can have values greater than 1.0. In contrast, the runoff coefficient for a cell (QCe.i,j-cell), is given by the volume of water discharged from the cell that is derived from rain falling on the cell divided by the volume of rain falling on the cell. For runoff produced by infiltration access, QCe.i,j-cell, will not exceed 1.0.
As noted above, the USLE-M was developed from the concept that erosion is directly related to the product of runoff volume and sediment concentration and the suggestion that the sediment concentration is directly related to (a) the average amount of rainfall for kinetic energy per unit quantity of rain and (b) effect of rainfall intensity which could be accounted for by I30. As a consequence of this, LUMe.i,j accounts for the effect of upslope area on the sediment concentration associated with erosion in the cell. In terms of the USLE-M, it follows from Desmet and Govers (1996) that when runoff is generated uniformly over an area, event soil loss from a cell with coordinates i,j (Ye.i,j) for bare fallow (CUMe.i,j = 1.0) on a 9% slope (S = 1.0) and cultivation up and down the slope (PUMe.i,j = 1.0) is given by
 | [3a] |
where KUMe.i,j is the soil erodibility in the cell during the event. Because the term Qe.i,j-all (Ai,j-in + D2)/Be (Ai,j-in + D2) equals QCe.i,j-all, Eq. [3a] can be written as
 | [3b] |
Given that the erosivity index for the cell is given by the product of EI30 and QRe.i,j-cell (Kinnell, 2001), it follows from Eq. [3] that
 | [4] |
when runoff is generated uniformly over an area. Equation [1] is a modification of Eq. [4]. In Eq. [1], QCe.i,j-all Am+1i,j-in has been replaced by QCe.i,j-in Am+1i,j-in to account for the fact that the runoff coefficient for the area including the cell may be different to that of the upslope area. F appears in Eq. [1] because without F, Eq. [1] overpredicts the slope length effect when QCe.i,j-in = 0 (Kinnell, 2001). When QCe.i,j-in = 0, LUMe.i,j should equal (D/22.13)m.
Figure 1 shows how LUMe.i,j values given by Eq. [1] and [4] vary with the total area contributing to runoff out of a cell as that total area increases when D = 30 m, m = 0.5 and both QCe.i,j-in and QCe.i,j-cell, the runoff coefficient for the cell, are equal to 0.6. While, without F, Eq. [1] overpredicts the slope length effect when QCe.i,j-all = 0, it can be seen from Fig. 1 that, when runoff is generated uniformly over the area, Eq. [1] does not give the same result as Eq. [4] when, in reality, it should. Consequently, a new approach needs to developed to overcome this problem. Two alternatives are considered here.
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Fig. 1. LUMe.i,j for a 900-m2 outlet cell calculated by Eq. [1] and [4] in relation to the total area in the catchment or watershed when m = 0.5, QCe.i,j-cell = 0.6, and QCe.i,j-in = 0.6.
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Theory
As noted above, Eq. [4] is valid when runoff is generated uniformly over the area. Logically, Eq. [4] is not valid when the runoff coefficient for the cell (QCe.i,j-cell) differs markedly from that of the upslope area (QCe.i,j-in). As noted earlier, the value for LUMe.i,j when QCe.i,j-in = 0 must be given by (D/22.13)m. As can be seen from Fig. 2, Eq. [1] produces this value but Eq. [4] produces a value that is less than required and the values LUMe.i,j for Eq. [4] do not vary with QCe.i,j-in.
As with Eq. [1], alternative approaches for determining LUMe.i,j can be based on a modification of the Desmet and Govers (1996) equation. The Desmet and Govers equation extends the equation for the slope length factor for Segment i developed for the USLE and the RUSLE (Renard et al., 1997)
 | [5] |
where 
i is the length of slope to the bottom of the segment and i–1 is the length of slope to the top of the segment, to grid cells by replacing i and i–1 with the respective contributing areas (Ai,j-in + D2 and Ai,j-in) divided by cell width (D) to give
 | [6] |
For a rectangular area one cell wide, the value of Ai,j-in/D is the same as i–1 and the value of (Ai,j-in + D2)/D is the same as i so that Eq. [5] and [6] produce the same slope length factor value with a rectangular area one cell wide.
Equation [5] is based on a mass balance approach where soil passing into the segment from upslope varies directly with erosion in the upslope area and the length of upslope area (i–1) and the soil passing out of the segement varies directly with erosion in the area that includes the segment and the length of slope to the bottom of the segment (i). According to Kinnell (2001), applying this approach to the USLE-M in grid cells gives
 | [7] |
Equation [7] equals Eq. [4] when QCe.i,j-in = QCe.i,j-all and consequently, meets the criteria for applying the USLE-M when runoff is produced uniformly over the whole area and when Ai,j-in = 0 but not when QCe.i,j-in = 0 when Ai,j-in > 0. (Kinnell, 2001) introduced the factor F to overcome this problem. However, in effect, Ai,j-in should equal zero when QCe.i,j-in = 0 and the failure of Eq. [7] to meet the criteria for the QCe.i,j-in = 0 results from the failure to set Ai,j-in = 0 when QCe.i,j-in = 0.
In an area where runoff is produced uniformly, the volume of water flowing across a unit width of a boundary is directly related to the upslope area divided by the width of the boundary. Using the contributing area divided by the cell width approach when the area is not rectangular assumes that the effect of the above cell area on the slope length factor is dependent on the volume of water inflow per unit cell width rather than the length of flow in the above cell area. It follows from this that if QCe.i,j-in is less than QCe.i,j-cell, then the effective value of Ai,j-in (Ai,j-in-eff) should be less than Ai,j-in in the context of determining LUMe.i,j (Fig. 3). Likewise, if QCe.i,j-in is greater than QCe.i,j-cell, then Ai,j-in-eff should be greater than Ai,j-in. Under these circumstances,
 | [8] |
where
 | [9] |
In the context of the combination of Eq. [8] and [9], the primary criteria for Ai,j-in.eff are that it equal (a) Ai,j-in when QCe.i,j-in = QCe.i,j-cell and (b) zero when QCe.i,j-in = 0. In addition, it can be argued that if QCe.i,j-in is half QCe.i,j-cell, then Ai,j-in.eff should equal half Ai,j-in and so on. These conditions are met by
 | [10] |
In effect, Eq. [10] considers that runoff entering a cell is produced from an area whose size is that required to produce that runoff given a runoff coefficient equal to that of the cell so that QCe.i,j-eff takes on a value equal to QCe.i,j-cell. Thus, the combination of Eq. [8], [9], and [10] calculates LUMe.i,j on the basis of a total area whose runoff coefficient is equal to that of cell i,j. In this way, the approach to determining LUMe.i,j. maintains the perception of runoff being produced uniformly over the whole area.
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Fig. 3. A diagrammatic representation of the sizes of the areas relevant to the calculation of the slope length factor in a rectangular watershed one cell width wide.
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Figure 4 shows how LUMe.i,j values produced by Eq. [8] vary with QCe.i,j-in when Eq. [10] is used in comparison with the values produced by Eq. [1] and [4]. It can be seen that the approach has positive features in terms that LUMe.i,j varies non linearly from the value given by (D/22.13)m when QCe.i,j-in = 0 to the value given by Eq. [4] when QCe.i,j-in = QCe.i,j-cell. The approach also produces LUMe.i,j values that are between those produced by Eq. [1] and [4] when QCe.i,j-in > QCe.i,j-cell and QCe.i,j-cell = 0.6.
As noted above, the approach maintains the perception that runoff is produced uniformly over the whole area used to determine LUMe.i,j and assumes that the runoff coefficient for that area is that of the cell. Thus LUMe.i,j = 0 when QCe.i,j-cell = 0 despite the fact that erosion can occur when QCe.i,j-cell = 0. QCe.i,j-cell = 0 can occur when the cell has an infiltration rate that is higher than the upslope area and runoff from upslope still occurs even though the rainfall rate is less than the infiltration rate for the cell. Also, Eq. [10] will produce very large values of Ai,j-in.eff and low values of LUMe.i,j whenever QCe.i,j-cell is very much less than QCe.i,j-in. Thus, the effect on LUMe.i,j produced by Eq. [8] under these conditions may generate an unrealistic erosion result. For example, in the case of the newly cultivated area cited above, the soil will be in a highly erodible state and the combination of raindrop impact on the soil surface and runoff from upslope may in fact cause severe erosion to occur in the cell while LUMe.i,j may be very low or zero. Consequently, because cell erosion is directly related to LUMe.i,j, the combination of Eq. [8], [9], and [10] cannot be applied indiscriminately when QCe.i,j-in > QCe.i,j-cell.
An alternative to Eq. [10],
 | [11] |
also meets the primary criteria for Ai,j-in.eff being equal to (a) Ai,j-in when QCe.i,j-in = QCe.i,j-cell and (b) zero when QCe.i,j-in = 0. In the case of Eq. [11], as with Eq. [10], Ai,j-in.eff < Ai,j-in when QCe.i,j-in < QCe.i,j-cell and Ai,j-in.eff > Ai,j-in when QCe.i,j-in > QCe.i,j-cell but Ai,j-in.eff will not tend toward infinity when QCe.i,j-cell tends to zero and LUMe.i,j will not equal zero when QCe.i,j-cell = 0. Figure 5 shows how LUMe.i,j varies with QCe.i,j-in when Eq. [11] is used when QCe.i,j-cell = 0.6 in comparison with when Eq. [10] is used. In contrast to when Eq. [10] is used, with Eq. [11], LUMe.i,j lies close to the value for when QCe.i,j-in = QCe.i,j-cell except at low values of QCe.i,j-in.
As noted earlier, cell erosion is directly related to the product of QRe.i,j-cell and LUMe.i,j. Figure 6 shows the product of QRe.i,j-cell and LUMe,i,j produced using Eq. [10] and [11] when QCe.i,j-all is held constant. Erosion should not vary significantly when the flow of water over the downstream boundary does not vary and the flow of water over the downstream boundary does not vary with QCe.i,j-cell when QCe.i,j-all is held constant. Figure 6 shows that erosion predicted using Eq. [11] varies little with QCe.i,j-cell while that predicted using Eq. [10] is not close to reality except when QCe.i,j-cell is close to QCe.i,j-all.. Using Eq. [11] predicts erosion rates that are close to those predicted when LUMe.i,j is determined using Eq. [4] except at low values of QCe.i,j-in when QCe.i,j-all varies with QCe.i,j-in (Fig. 7). By design, using either Eq. [10] or [11] will predict the appropriate erosion rates when QCe.i,j-in = 0 and when QCe.i,j-in = QCe.i,j-cell. but Eq. [10] does not predict erosion values that lie within theoretically acceptable limits.
Discussion
Of the two approaches considered, the approach where Ai,j-in.eff varies with QCe.i,j-all (Eq. [11]) rather than QCe.i,j-cell (Eq. [10]) provides LUMe.i,j values which can be applied when QCe.i,j-cell = 0. Consequently, the combination of Eq. [8], [9], and [11] provides the more appropriate method for determining the slope length effect for use when the USLE-M is applied to grid cells.
In effect, Eq. [8] is the product of Li,j given by Eq. [6] with Ai,j-in replaced by Ai,j-in.eff,
 | [12] |
and the ratio of QCe.i.j-eff to QRe.i,j-cell. Figure 8 shows how the Li,j for the outlet 30-m cell varies with QCe.i,j-in when Eq. [12] is applied to the 0.9-ha total area considered here when Ai,j-in-eff is determined using Eq. [11]. In the context of Eq. [8] being the product of Li,j values determined by Eq. [12] and the ratio of QCe.i.j-eff to QRe.i,j-cell, LUMe.i,j values tend to be directly related to the Li,j values determined by Eq. [12] as QCe.i,j-in varies when Ai,j-in-eff is determined using Eq. [11] except when QCe.i,j-in tends toward zero because the ratio of QCe.i.j-eff to QRe.i,j-cell varies little with QCe.i,j-in except when QCe.i,j-in toward zero (Fig. 9).
It should be noted that the determination of LUMe.i,j does not consider the form of the hillslope profile in which a grid cell lies or the actual erosion that occurs in the upslope area. The approach considers only the slope gradient of the cell and the slope lengths of the cell and upslope areas. That is standard for determining what is commonly known as erosion in a cell or segment. However, the total mass of soil loss, which is calculated by multiplying the total area by the average erosion rate over that total area, may differ from the mass of soil actually passing across the downstream boundary of a cell. This is because deposition may occur in the cell, particularly when a hillslope is concave, and the USLE, RUSLE (as implemented in the computer program RUSLE 1) and USLE-M models do not account for deposition. In terms of looking at the impact of land management on the health of rivers and streams, sediment delivery to channels in a watershed or catchment is the ultimate focus of the modeling exercise and requires the use of a sediment transport model to control the movement of sediment when deposition occurs. The concept involved is illustrated in Fig. 10. This is recognized in the computer program RUSLE 2 (Foster et al., 2003) that produces two outputs— "erosion", which is the output that would occur if all the sediment available for transport is transported to the bottom of the hillslope, and "sediment delivery", which takes account of deposition on the amount of sediment discharged from the bottom of the hillslope. RUSLE 2 models erosion and sediment delivery in a one-dimensional system. The issue addressed here is "erosion" in a two-dimensional system.
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Fig. 10. The approach used by Meyer and Wischmeier (1969) to simulate the processes of soil erosion by water.
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Conclusions
Kinnell (2001) proposed an equation (Eq. [1]) for determining the slope length factor for applying the USLE-M in grid-cells. This equation was a modification of the one proposed by Desmet and Govers (1996) for determining the slope length factor for applying the USLE/RUSLE in grid cells and considered that the effect of variations is runoff from the area upslope of a cell was dependent on the runoff coefficient of that upslope area. That equation has been found to be deficient in terms of determining the value of the slope length factor when runoff is uniformly produced over the whole area. A new method of determining the slope length factor has been developed (Eq. [8] with Eq. [9] and [11]) which has the required characteristics for applying the USLE-M to erosion in grid cells.
Symbols
i, length of slope to the bottom of the segment i.
i–1, length of slope to the top of the segment i.
Ai,j-in, area of upslope area of cell with coordinates i,j.
Ai,j-in.eff, effective area of upslope area when runoff not uniformally generated over the area contributing to runoff out of cell with coordinates i,j.
Be, amount of rain falling during an rainfall event.
CUMe.i,j, crop management factor for applying the USLE-M to a grid cell with coordinates i,j for a rainfall event.
E, event kinetic energy.
D, length of cell sides.
I30, maximum rainfall intensity measured using a 30-min time frame.
KUMe.i,j, USLE-M soil erodibility factor for cell with coordinates i,j for a rainfall event.
Li,j, slope length factor for applying the USLE to a grid cell with coordinates i,j.
LUMe.i,j, slope length factor for applying the USLE-M to a grid cell with coordinates i,j for a rainfall event.
m, slope length exponent defined for use with the RUSLE.
PUMe.i,j, support practice factor for applying the USLE-M to a grid cell with coordinates i,j for a rainfall event
QCe, Runoff coefficient volume of runoff per unit volume of rain for rain falling on the same area during a rainfall event.
Qe, runoff per unit area.
Qe.i,j-all, runoff per unit area from the area that includes the cell with coordinates i,j.
QCe.i,j-all, runoff coefficient for upslope area plus cell with coordinates i,j for a rainfall event.
QCe.i,j-cell, runoff coefficient for a cell with coordinates i,j for a rainfall event
QCe.i,j-eff, effective runoff coefficient for an area including cell with coordinates i,j for a rainfall event assuming that the area is given by the sum of Ai,j-in.eff and D2.
QCe.i,j-in, runoff coefficient for upslope area to cell with coordinates i,j for a rainfallevent.
QRe.i,j-cell, runoff ratio for cell with coordinates i,j–volume of runoff from cell per unit volume of rain falling on cell during a rainfall event. Volume of runoff from cell includes water running into the cell from upslope.
S, USLE slope gradient factor.
xi,j, coefficient that adjusts for width of flow at the center of the cell. It has a value of 1.0 when the flow is toward a side and 
2 when the flow is toward a corner.
Ye.i,j, soil loss from a cell with coordinates i,j.
Received for publication February 8, 2004.
REFERENCES
- Desmet, P.J.J., and G. Govers. 1996. A GIS procedure for automatically calculating the USLE LS factor on topographically complex landscape units. J. Soil Water Conserv. 51:427–433.
- Foster, G.R, D.C. Yoder, G.A. Weesis, D.C. McCool, K.C. McGregor, and R.D. Bingner. 2003. User's guide (Draft January 2003)–Revised Universal Soil Loss Equation Version 2 (RUSLE2). USDA-ARS, Washington, DC.
- Kinnell, P.I.A. 2001. Slope length factor for applying the USLE-M to erosion in grid cells. Soil Tillage Res. 58:11–17.
- Kinnell, P.I.A., and L.M. Risse. 1998. USLE-M: Empirical modeling rainfall erosion through runoff and sediment concentration. Soil Sci. Soc. Am. J. 62:1667–1672.[Abstract/Free Full Text]
- Meyer, L.D, and W.H. Wischeier. 1969. Mathematical simulation of the process of soil erosion by water. Trans. ASAE 12:754–758, 762.
- Renard, K.G., G.R. Foster, G.A. Weesies, D.K. McCool, and D.C. Yoder. 1997. Predicting soil erosion by water: A guide to conservation planning with the Revised Universal Soil Loss Equation (RUSLE). Agric. Handb. No. 703. USDA, U.S. Gov. Print. Office, Washington, DC.
- Wischmeier, W.C., and D.D. Smith. 1978. Predicting rainfall erosion losses–A guide to conservation planning. Agric. Handb. No. 537. USDA., Washington, DC.
