# Spatial-Temporal Variability of Future Rainfall Erosivity and Its Impact on Soil Loss Risk in Kenya
## 2 Materials and Methods
### Methodological Framework
#### 2.4.5 Soil Erodibility Factor  
The soil erodibility (\(K\)) factor takes into account the inherent soil properties (including soil texture, organic matter content and the permeability) to quantify soil erodibility or susceptibility to disintegration due to surface water runoff action. This present study utilized the EPIC (erosion-productivity impact calculator) model, as contended by [PERSON] [71], to derive the \(K\) factor using the sand, organic, silt and sand soil fractions of the area, as compiled by the Africa Soil Information Service (AfSIS) [66]. \[K=F_{csand}\times F_{si-cl}\times F_{orgc}\times F_{hisand}\times 0.1317 \tag{20}\]  
\[F_{csand}=\left[0.2+0.3\;exp\left(-0.0256 SAN\left(1-\frac{ SIL}{100}\right)\right)\right] \tag{21}\]  
\[F_{si-cl}=\left[\frac{ SIL}{ CLA+ SIL}\right]^{0.3} \tag{22}\]  
\[F_{orgc}=\left[1.0-\frac{0.0256C}{C+exp(3.72-2.95C)}\right] \tag{23}\]  
\[F_{hisand}=\left[1.0-\frac{0.70\;SN1}{ SN1+exp(-5.51+22.9\; SN1)}\right] \tag{24}\]  
where \(SAN\), \(SIL\) and \(CLA\) are the percentage of sand, silt and clay content, respectively; \(C\) is the organic carbon content and \(SN1\) is the sand content subtracted from 1 and divided by 100. \(F_{csand}\) (Equation (4)) gives a low soil erodibility factor for soil with coarse sand and a high value for soil with little sand content. \(F_{si-cl}\) (Equation (5)) gives a low soil erodibility factor with a high clay to silt ratio; \(F_{orgc}\) (Equation (6)) is the factor that reduces soil erodibility for soil with high organic contents. \(F_{hisand}\) (Equation (7)) is the factor that reduces soil erodibility for soil with extremely high sand content. Multiplication by the constant value 0.1317 converts the values to International (\(SI\)) units. 2.4.6. Slope Length and Slope Steepness (LS) Factor  
The dimensionless \(LS\) geophysical factor is a product of two terrain derivatives (slope length (\(L\)) and slope steepness factor (\(S\))) that expresses the influence of the landscape on soil loss. [PERSON] [72] defined slope length as the distance between the point of origin of overland flow to the point where the surface runoff waters flow into a well defined channel. High slope lengths along steep slopes result in increased surface runoff and, thus, high erosion rates. A 30 m void-filled SRTM DEM was first pre-processed to correct for sink errors before estimating slope length and steepness attributes using the Spatial Analyst extension of ArcMap 10.2 (Environment Systems Research Institute (ESRI) Inc., Redlands, CA, USA). Equation (25) was used to compute the slope length factor, while the \(S\) factor was generated using the [PERSON] et al. [73] (1987) method (Equation (28)). \[L_{i,j}=\frac{\left(A_{i,j-in}+D^{2}\right)^{m+1}-A_{i,j-in}^{m+1}}{D^{m+2} \cdot x_{i,j}^{m}(22.13)^{m}} \tag{25}\]  
\[m=\frac{\beta}{1+\beta} \tag{26}\]  
\[\beta=\frac{sin\theta/0.0896}{3(sin\theta)^{0.8}+0.56} \tag{27}\]  
\[S_{i,j}=\left\{ \begin{array}{l}10.8 sin\;\theta_{i,j}+0.03,\;tan\theta_{i,j} <9\%\\ 16.8 sin\;\theta_{i,j}-0.50,\;tan\theta_{i,j}\geq 9\%\end{array} \right. \tag{28}\]  
where \(L_{i,j}\) = slope length factor for the grid cell with coordinates (\(i\),\(j\)); \(D\) = the grid cell size (\(m\)); \(X_{i,j}\) = \(sina_{i}+cosa_{i,j}\); \(a_{ij}\) = aspect direction for the grid cell with coordinates (\(i\),\(j\)); \(A_{i,j-in}\) = flow accumulation or contributing area at the inlet of a grid cell with coordinates (\(i\),\(j\)) (m\({}^{2}\)), \(\beta\) = the ratio of inter-rill erosion and \(\theta\) = the slope in degrees  
2.4.7. Cover Management Factor (C) Factor  
The \(C\) factor varies from 0 to 1 and models the conservative property of ground vegetation cover against surface water runoff. This implies that holding other parameters constant, thick canopies, e.g., forests, are more likely to prevent soil erosion better than bare or sparse cover; thus, they are assigned a low \(C\) factor value [74]. [PERSON] et al.