## 2 Materials and Methods
### RUSLE Model
#### 2.3.2 Implementation of RUSLE Model Rainfall Erosivity (R) Factor
In this study, the LS calculation was executed using the original equation proposed by [PERSON] and [PERSON] [32], which incorporates a multiple flow algorithm and contributes to a precise estimation of flow accumulation. Many researchers worldwide use algorithms proposed by [PERSON] and [PERSON] [32] based on the DEM resolution and unit contributing area (UCA). These algorithms allow the model to be adapted to varied reliefs and include the influence of flow concentration. The calculation methodology for the LS-factor was applied to each pixel in the DEM. The L factor is expressed according to the equation proposed by [PERSON] and [PERSON] [32]. \[L_{i,j-in}=\frac{\left(A_{i,j-in}+D^{2}\right)^{m+1}-A_{i,j-in}^{m+1}}{D^{m+2} \times\chi_{i,j}^{m}\times 22.13^{m}} \tag{10}\]  
where _Li,j-in_ is the slope length for grid cell (_i,j_), _Ai,j-in_ is the contributing area at the inlet of the grid cell with coordinates (_i,j_) (m\({}^{2}\)), \(D\) is the grid cell size (m), \(m\) is the length exponent \begin{table}
\begin{tabular}{c c c c} \hline
**Structure Class** & **Description** & **Granular Size** & **European Soil Database** \\ \hline
1 & Very fine granular & 1–2 mm & Good \\ \hline
2 & Fine granular & 2–5 mm & Normal \\ \hline
3 & Medium or coarse granular & 5–10 mm & Poor \\ \hline
4 & Blocky, platy, or massive & \textgreater{}10 mm & Humic \\ \hline \end{tabular}
\end{table} 
Table 2: Soil structure according to the European Soil Database. \begin{table}
\begin{tabular}{c c c} \hline
**Permeability Class** & **Description** & **Texture** \\ \hline
1 & Fast and very fast & Sand \\
2 & Moderate fast & Loamy sand, sandy loam \\
3 & Moderate & Loam, silty loam \\
4 & Moderate slow & Sandy clay loam, clay loam \\
5 & Slow & Silty clay loam, sand clay \\
6 & Very slow & Silty clay, clay \\ \hline \end{tabular}
\end{table} 
Table 3: Soil permeability. of the USLE L-factor, _xi,j_ is the summation of the sine and cosine of the aspect direction (\(\alpha_{ij}\)) of the grid pixel \(x_{i,j}=\sin\alpha_{i,j}+\cos\alpha_{i,j}\), and 22.13 is the length of the USLE unit plot in meters. In attempts to calculate the upslope contributing area (_Ai,j-in_) for each pixel, several algorithms have been developed and are reported in the literature. The exponent (\(m\)) was implemented according to the algorithm proposed by [PERSON] et al. [33], where the slope length is a function of the rill-to-interrill (\(\beta\)) erosion ratio, as expressed in the following equation:  
\[m=\beta/(\beta+1) \tag{11}\]  
where \(\beta\) varies according to the slope gradient [33]. The \(\beta\) value is obtained as:  
\[\beta=\frac{\frac{\sin\theta}{0.0896}}{\left[0.56+3(\sin\theta)^{0.8}\right]} \tag{12}\]  
where \(\theta\) is the slope in degrees. The calculation of the S-factor proposed by [PERSON] and [PERSON] [24] was adjusted in the RUSLE model for a better representation of the slope steepness factor, which takes into account the ratio of the rill and interrill erosion:  
\[S=10.8\,\sin\theta+0.03,\,\,\theta\,\,<\,\,9\% \tag{13}\]  
\[S=16.8\,\sin\theta-0.5,\,\,\theta\,\,\geq\,\,9\% \tag{14}\]  
where \(\theta\) is the slope angle.