Soil Erosion in the NanoFASE Model
In order to quantify soil erosion in the model it is necessary to calculate the sediment yield (S_{y}). To do this, the MUSLE equation (equation 1) is a modification of the original USLE which included soil erodibility (K), cover and management (C), practice (P) topography (LS) and a coarse fragment factor (F).
\(S_{y}=(q_{surfQ_{peak}A})^{0.56}.K_{USLE}.C_{USLE}.P_{USLE}.LS_{USLE}.F_{USLE}\) [Equation 1] 
The modification involved representing the USLE rainfall erosivity factor (R) with surrogate information from runoff (q_{surf}Q_{peak} and A). Q_{surf}_{ }is the surface runoff (mm day1), Q_{peak} is the peak runoff discharge and A is the area for sediment yield calculation.
Alternatively, the rainfall erosivity term in R_{USLE} is defined in MJ mm ha^{1} hr^{1}. It represents the product of rainfall kinetic energy and the maximum rainfall in a 30 minute period (KE x I30). As highly dependent on detailed records I30 is not well defined but is widely used in model applications.
However, Davison et al. (2005) propose a power law relationship (KE = aPb) to define KE on a daily resolution from daily rainfall totals (Equation 2). Across the UK no significant geographic variation was found for this relationship (Davison et al. (2005).
\(KE_{DT}={a_{1}+a_{2}}\cos[D_{J}(\frac{2\pi }{365}+a_{3})]P^{b}\) [Equation 2] 
where: KE_{DT} is daily kinetic energy of rainfall as direct throughfall (J m2), P is daily rainfall (mm) and DJ is Julian Day; parameter values for UK: α_{1} = 6.608, α_{2} = 0.519, α_{3} = 2.727, b =1.204.
Total kinetic energy is the sum of the leaf drainage and direct throughfall components. Kinetic energy of leaf drainage is related to plant height (PH in m: leaf drainage from plants lower than 0.15 m contribute no kinetic energy) (Equation 3). The relative size of the components is defined by crop cover fraction (CC holding a value between 0 and 1) (Equation 4).
\(KE_{LD} = 15.8.PH^{0.5}5.87\) [Equation 3] 
\(KE = (1CC).KE_{DT}+CC.KE_{LD}\) [Equation 4] 
As the variation of rainfall erosivity across Europe (Panagos et al., 2015) strongly reflects hydroclimatic classification it was decided to estimate R based on the rainfall kinetic energy calculation (Equation 2) for which the fitting parameters (α,b) are related to the Koppen climatic classification.
Therefore the equation adopted here (Equation 5) is a modification of R_{USLE}.
\(S_{y}=KE.K_{MMF}.C_{USLE}. P_{USLE}.LS_{USLE}.F_{CFRG}\) [Equation 5] 
The soil erodibility parameter (K) adopted differs from those originally used in the USLE family of models (based on textural composition and organic matter content) by using the simpler formulation related to textural composition as used by Morgan and Duzant (2008). Units of K_{mmf} are g j_{1} whose value is determined by the particle size fractions (F_{clay}, F_{silt} and F_{sand} which hold values between 0 and 1) and coefficients related to those fractions (Equation 6).
\(K_{MMF}=F_{clay}.0.1+F_{silt}.0.5+F_{sand}.0.3 \) [Equation 6] 
The cover and management factor C_{USLE} is defined as the ratio of soil loss from land cropped under specified conditions to the corresponding loss from cleantilled, continuous fallow. Though data exists for temporallyaveraged C_{USLE} factors in Europe (Panagos, Borrelli, Meusburger, Alewell, et al., 2015), these do not take into account daily variation in plant cover, and daily C_{USLE} factors are calculated empirically (Neitsch et al., 2011):
\(C_{USLE}=exp((ln(0.8) ln(C_{min}))exp(0.00115\cdot r_{surf})+ln(C_{min}))\) [Equation 7] 
where r_{surf} is the crop residue on the soil surface [kg/ha] and C_{min} is a minimum C_{USLE} calculated as follows:
\(C_{min}=1.463\cdot ln(C_{av})+0.1034\) [Equation 8] 
where C_{av} is the average annual C_{USLE} factor.
The support practice factor P_{USLE} is defined as the ratio of soil loss with a specific support practice (e.g. contour ploughing) to the corresponding soil loss with upanddown slope culture. Spatiallyresolved data to 1km resolution are available for Europe (Panagos, Borrelli, Meusburger, van der Zanden, et al., 2015).
The topographic factor LS_{USLE} takes into account the steepness and slope of the hillslope on which erosion is occurring. It can be calculated from hillslope length and slope angle (Neitsch et al., 2011) and spatiallyresolved data are available at 25 and 100m resolution (Panagos, Borrelli, & Meusburger, 2015).
The coarse fragment factor F_{CFRG} takes into account the amount of rock in the soil profile, and is calculated as:
\(F_{CFRG} = exp(0.053\cdot rock)\) [Equation 9] 
where rock is the percentage of rock in the first soil layer. Thus, F_{CFRG }=1_{ } when no rock is present.

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Consult the NanoFASE Library to see abstracts of these deliverable reports: 
Davison PS, Withers PJA, Lord EI, Betson MJ, Strömqvist J. PSYCHIC – A processbased model of phosphorus and sediment mobilisation and delivery within agricultural catchments. Part 1: Model description and parameterisation. Journal of Hydrology. 2008 Feb; 350(34):290–302. Morgan R, Duzant J. Modified MMF (MorganMorganFinney) model for evaluating effects of crops and vegetation cover on soil erosion. Earth Surface Processes and Landforms. 2008; 32:90–106. Neitsch S, Arnold J, Kiniry J and Williams J. Soil and Water Assessment Tool – theoretical documentation. Texas Water Resources Institute Technical Report 406 Texas A&M University System, College Station Texas US. 2011. Panagos P, Ballabio C, Borrelli P, Meusburger K, Klik A, Rousseva S, et al. Rainfall erosivity in Europe. Science of the Total Environment. 2015 Apr 1; 511:801–814. Panagos, P., Borrelli, P., Meusburger, K., Alewell, C., Lugato, E., & Montanarella, L. (2015). Estimating the soil erosion covermanagement factor at the European scale. Land Use Policy, 48 (Supplement C), 3850. doi: 10.1016/j.landusepol.2015.05.021 Panagos, P., Borrelli, P., Meusburger, K., van der Zanden, E. H., Poesen, J., & Alewell, C. (2015). Modelling the effect of support practices (Pfactor) on the reduction of soil erosion by water at European scale. Environmental Science & Policy, 51 (Supplement C), 2334. doi: 10.1016/j.envsci.2015.03.012 Panagos, P., Borrelli, P., & Meusburger, K. (2015). A New European Slope Length and Steepness Factor (LSFactor) for Modeling Soil Erosion by Water. Geosciences, 5(2), 117126. doi: 10.3390/geosciences5020117 Мodelling: Science of the Total Environment, v. 650, p. 530540. 
Contact
Mike Hutchins
Centre for Ecology and Hydrology (CEH)