# LS Factor & RUSLE Equation

Instruction document by my former Boise State University students Jim Pelton, Eli Frazier, Erin Pikcilingis:
LS Factor in RUSLE with ArcGIS 10.docx

Instructions and worked example by my former Boise State University students Richard Martinez & Keisha Tomaszewski:
RUSLE Soil Loss Project_Martinez and Tomaszewski.pptx

“There are still a lot of concepts and assumptions embedded in erosion estimation models that need to be verified” – C. Wu, 2000

The Revised Universal Soil Loss Equation can be used to estimate soil loss from agricultural fields and certain rangeland areas. RUSLE has limited application in steep, complex mountainous terrain. RUSLE generates the value A, the average loss in tons over a “field slope”, a concept somewhat at odds with distributive nature of DEMs. DEM data was not used during development of the model, rather DEM data was substituted in later.

The RUSLE equation accounts for:
– Rainfall
– Soil Erodibility
– Slope Length
– Slope Steepness
– Cover Management Type
– Conservation Practice

The RUSLE equation is:

A = R x K x L x S x C x P

Of all the inputs, the LS factor is perhaps the most confusing.

(Area/22.1)^m * (sin(t)/0.09)^n * (m+1)

Constants “m” and “n” have been empirically derived by others. There is a range for each. The idea is that you select values that best suit your particular landscape you are working (climate, soils, etc.). See instruction doc by Pelton et al. and references therein.

L = (λ / 22.13)m

22.13 meters = 72.6 feet
λ = length of slope in the field (or data layer)

Values for L
0.2 on uniform, < 1% slopes
0.3 for 1-3% slopes
0.4 for 3.5-4.5% slopes
0.5 for >5% slopes

The m Exponent
m = exponent representing the Rill-to-Interrill Ratio

m = β / (1+β)
or
m = (sinθ / 0.896) / (3(sinθ)0.8 + 0.56)
or
m = (Gr/Gi) / (1 + Gr/Gi) = Gr/(Gr + Gi)
or
m = Gr/G

Where,
G = Gr + Gi
Gr = sediment load contributed from Rill erosion
Gi = sediment load contributed from Interrill erosion

Values for Exponent m
0.02 – 0.05 for low β slopes
0.04 – 0.71 for moderate β slopes
0.07 – 0.83 for high β slopes

Refs:
Renard et al. (1997) USDA-ARS Agricultural Handbook 703 PDF
Foster et al. (1977) Transactions of the Amer. Society of Agricultural Engineers, v. 20
Foster (1994) Journal of Soil and Water Conservation v.49
McCool et al. (1989) Transactions of the Amer. Society of Agricultural Engineers, v. 32
McCool (1994) Journal of Soil and Water Conservation v.49
Moore and Wilson (1992) Journal of Soil and Water Conservation v.47
Moore and Wilson (1994) Journal of Soil and Water Conservation v.49
Wischmeier and Smith (1978) USDA-ARS Agricultural Handbook 537
Wu, C-C., 2000, L Factor for short, steep slopes, in Laflen, J.M et al. (editors), Soil Erosion and Dryland Farming, CRC Press, p. 591-600

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Length of Overland Flow can be calculated in several ways, including the list below.

A. Field measurements.
B. Regional estimation method (Troeh et al. 1991).
C. Unit stream power method (Mitasova 1993, Mitasova et al. 1996, Moore and Wilson, 1992, Moore and Burch 1986).
D. Upslope contributing area proxy method (Desmet and Govers 1995, 1996).
E. Slope length from high points (ridges/peaks) and the flow direction raster (Hickey 2000).
F. Cell-by-cell Non-cumulative Slope Length (NCSL) method: FlowAcc * SR, where FlowAcc = Flow accumulation raster (orientation of maximum downhill slope), SR = Cell size of raster (i.e., 30 x 30m).
G. TIN analysis (Cowen 1993).
H. Average length of overland flow by the equation:

Average Length of Overland Flow = 1 / (2Dd √1 – (Sc / Sg)2 or 1 / 2Dd

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RUSLE Equation Notes:
The LS factor in the Revised Universal Soil Loss Equation is:
(Area/22.1)^m * (sin(t)/0.09)^n * (m+1)

m+1 is the mysterious 1.4, where m=0.4. Careful though, m=0.4 isn’t always appropriate.

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Jeffrey Evans, ESRI Forum:
The original USLE assumed little slope curvature and no deposition. To account for flow convergence in complex terrain, modifications were made to the LS factor with an equation that incorporates flow accumulation. Because they are now used interchangeably, when you see reference to slope-length most likely it is in reference to the LS factor, as calculated below.

To calculate the LS factor for the RUSLE equation, first calculate flow accumulation (flowacc) and slope in degrees (slope). Then a bit of map algebra in the raster calculator yields LS.

Power(flowacc * cell resolution / 22.1, 0.4) * Power(Sin(slope * 0.01745) / 0.09, 1.4) * 1.4

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Van Remortel et al. articles on SL in GIS:
PDF Computers & Geosciences
PDF ResearchGate/Geography v30
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Robert Hickey article (2000):
http://www.onlinegeographer.com/slope/hickey_slope_length.pdf
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Camposano thesis (2004):
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How to Lesson from CSU:
http://gisedu.colostate.edu/webcontent/nr505/ethiopia/group4/GIS%20Analyses.html
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Wiki: USLE:
http://en.wikipedia.org/wiki/Universal_Soil_Loss_Equation
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From Da Ouyang and Jon Bartholic:

A detailed description of these factors is available from the web site. Among these factors, terrain factor – “L” is most difficult to compute. Fortunately, the soil loss equation is much less sensitive to L factor than another terrain factor – S factor which can easily be computed from the digital elevation model (DEM).

L factor and S factor are usually considered together to combine the effect of slope and slope-length, which basically reflects the terrain on a given site. For this project, an approach developed by Moore and Burch (1985) is used to compute LS factor. They developed an equation to compute length-slope factor:

LS = (As / 22.13) ^m * (sin β / 0.0896) ^n (2)

where:
m = 0.4 – 0.6 and n = 1.2 – 1.3.
LS = computed LS factor.
As = specific catchment area, i.e. the upslope contributing area per unit width of contour (or rill), in m‘ / m. It is calculated in Are/Info using the function called “flow accumulation” multiply by the squared cell size and divided by the cell size.
As = calculated flow accumulation * 30 * 30 / 30 (for cell size = 30 m).
β = slope angle in degrees. It is calculated in Arc/Info using the function called “slope” with option “percent rise” which is 100 times Tan β. Then β is calculated using “Atan” function in Arc/Info.
Tan β = slope (in percent rise) / 100
β = Atan (Tan β)
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Kim & Julien, PDF
NRCS RUSLE for Western States (press release) LINK
Renard et al. (1997) Predicting Soil Erosion by Water, USDA Agricultural Handbook, #703
Cowen (1993) Proceedings 13th ESRI User Conference
Desmet and Govers (1995) Catena 25
Desmet and Govers (1996) Journal of Soil and Water Conservation 51
Hickey, R. (2000) Cartography 29
Mitasova (1993) GRASSclippings 7
Mitasova et al. (1996) International Journal of GIS 10
Moore and Burch (1986) Soil Science Society of America Journal 50
Moore and Wilson (1992) Journal of Soil and Water Conservation 47
Renard et al. (1997) USDA Agricultural Handbook #703
Troeh et al. (1991) Soil and Water Conservation (2nd edition), Chapter 6
Wischmeier and Smith (1978) Predicting rainfall erosion losses-A guide to conservation planning: Agricultural Handbook #537