In choosing the soil erosion model to be applied, it is always necessary to know the availability of input data and the type of output data needed [14]. Modelling cannot be an alternative to measurement and monitoring but might be a powerful tool for simulation and prediction of the soil erosion potential [15]. Soil erosion models can be understood as a virtual laboratory which brings together all data, observations and knowledge of different fields [11].

This research aims to apply USLE-type model such as USLE, MUSLE or RUSLE. Presently, USLE is the most widely accepted method for estimating the annual soil loss because of less data demand [8]. It has been originally proposed in 1958 and actually modified to its present form in 1978 [16]. The empirical equation of this model is as follows:

A = K ∗ R ∗ ( L S ) ∗ C ∗ P (1)

where the involved parameters are: Average annual soil loss A (t/ha * year); Soil

erodibility or K-factor (t * ha * hr/ha * MJ * mm); Rainfall erosivity or R-factor (MJ * mm/ha * hr * yr); Slope length factor or LS-factor (Dimensionless); Crop management factor or C-factor (Dimensionless); Erosion-control practice factor or P-factor (Dimensionless). As described in the next section, the designed Soil Loss and Crop Yield (SOLCY) model is based on the USLE concept.

From the Center of GIS of University of Rwanda (UR-CGIS) parcels data, a parcels map and an Excel sheet containing 259,673 parcels each of them having a specific location (District, Sector, Cell and Village) were first made available. Each parcel is characterised by a Universal Parcel Identification code abbreviated as parcel UPI. The shapes of parcels are diversified: triangular, square, rectangular and many irregular shapes.

At the large-scale application of USLE model, the catchment area has been discretized into a series of 259,673 parcels having relatively independent land use. Figure 2 shows diversified processes used by Soil and Crop Yield (SOLCY) mode designed to predict soil loss and correlatively estimated crop yields.

The collected data from Rwanda Meteorological Agency were used to generate rain fall distribution ranges in Sebeya catchment using ArcGIS software tool. Knowing the minimum and maximum values for each range, the reading of average annually precipitation on each parcel in Sebeya catchment was made possible by overlapping the administrative boundaries map and the parcels map of Sebeya catchment on the obtained rainfall map.

In this present study, a shapefile of soil was obtained from the Center of GIS of University of Rwanda (UR-CGIS) to produce soil texture map. To determine the soil texture for each parcel within Sebeya catchment, the administrative boundaries map and Sebeya parcels map were overlapped to soil texture map.

The Digital Elevation Model data collected from the Center of GIS of University of Rwanda (UR-CGIS) were used to generate the slope range map of Sebeya catchment. In order to detect the slope range of each parcel, the administrative boundaries map and parcels map were overlapped on the slope range map of Sebeya catchment.

Terraces are more favorable in agricultural land with steep slopes ranging from 16% to 40% while progressive terraces and contour bunds are suited to flat areas of slopes less than 16% [17]. Erosion control measures in Sebeya catchment were proposed based on land slope range [17].

The data collected at UR-CGIS revealed that Sebeya catchment is characterized by seven land cover types: Built-up areas, closed agriculture, forest plantation, irrigation, natural forest, open agriculture and open land. In the determination of the type of land cover and land use on each parcel within Sebeya catchment, the administrative boundaries map and Sebeya parcels map were overlapped on the land cover map.

R-factor is the long term annual average of the product of rainfall kinetic energy (KE) in MJ∙ha⁻¹ and the maximum rainfall intensity in 30 minutes (I₃₀) in mm∙h⁻¹ [19]. The rainfall factor can vary from year to year, so an average over a number of years is usually used [20]. The average precipitation on each parcel was calculated based on the minimum and maximum precipitations found on the rainfall range map as:

P ¯ = ( P min + P max ) / 2 (2)

The observed average rainfall values were varying from 1132.5 mm to 1568 mm and were inserted in the SOLCY Excel spreadsheet with respect to each parcel UPI. Due to lack of rainfall kinetic energy and intensity data, which is the case of this study, many researchers [21] have predicted the erosivity factor R by using the following regression equation:

R = 81.5 + 0.38 P ¯ (3)

where: R = rainfall erosivity in MJ∙mm∙h⁻¹∙year⁻¹; P ¯ = average precipitation on the parcel in mm.

Applicable for areas where annual precipitation ranges from 340 and 3500 mm, Equation (3) was selected because its range of validity includes all precipitation events of this study. By inserting all calculated values for all parcels in the SOLCY Excel sheet, the R-factor in Sebeya catchment ranges from 511.85 to 677.34 MJ∙mm∙ha⁻¹∙h⁻¹∙year⁻¹.

The soil erodibility factor (K) indicates the susceptibility of soil to erosion where it reflects the effect of soil properties and soil profile characteristics on soil loss [8]. The soil erodibility is the resistance of the soil to both detachment and transportation. It can be determined through the measurement of soil loss from a standard runoff plot (called “Standard USLE Plot”). A “Standard USLE Plot” is a field experimental plot having 9% slope along 22.13 m length which is kept fallow (bare soil) with periodic tillage [22]. In this case, LS = C = P = 1, the soil loss becomes a function of R and K. Then from a known R and soil loss measurement, K can be estimated. The soil erodibility factor (K) rates from 0 to 1, where 0 indicates soils with the least susceptibility to erosion and 1 for soils that are highly susceptible to soil erosion by water. From literature, different researchers have identified the variability of the soil erodibility values with the soil texture (Table 1). The soil erodibility factor is determined by soil texture and organic matter content [23]. The determination of K-factors for all parcels within Sebeya catchment was determined by reading in Table 1 and inserted in the SOLCY Excel sheet.

The topographic factor LS is defined as the ratio of soil loss under the given conditions to that at the site with the “standard” slope steepness of 9% and the slope length of 22.13 m [29]. The topographic erosivity factor LS accounts for the effect of slope length (L) and slope steepness (S) on soil erosion. The steeper and longer the slope is, the higher is the risk of soil erosion. A higher slope gradient creates a higher flow velocity which causes more detachment and transport of soil particles [30]. In 1978, Wischmeier and Smith defined the slope length (L) as: “the distance from the point of origin of the surface flow to the point where each slope gradient (S) decreases enough for the beginning of deposition or when the flow comes to concentrate in a defined channel”. Further researches extend the LS-factor to topographically complex units using a method that incorporates contributing area and flow accumulation [31]. Generally, to come up with the result of a slope length factor of each parcel, the following equation has been found appropriate to be used [32]: For the slope less than 20%:

L S = ( L ) 0.5 × ( 0.0138 + 0.00965 S + 0.00138 S 2 ) (4a)

where: L = slope length (m) and S = slope gradient (%).

And when the slope is greater than 20%:

L S = ( L / 22.2 ) 0.6 × ( S / 9 ) 1.4 (4b)

Another approximation to this process was to take the maximum slope of each slope range as the slope steepness while the largest path of travel P/2 was attributed to the slope length (L) for each parcel in Sebeya catchment where P is the parcel perimeter. Finally, Equation (4) was applied to estimate the LS-factors for all parcels of Sebeya catchment and the obtained values were inserted in the SOLCY Excel sheet. For all parcels, the minimum and maximum observed slope lengths (and slope length factors) are 0.014 m (0.032) and 24707.00 m (559.798) respectively while the following Table 2 shows the percentages of areas covered in each slope range.

*: weighted average.

The cropping-management factor can vary according to farming practices. In USLE-type equations, C-factor measures the combined effects of all interrelated cover and management measures. Its value includes the effects of crop cover, crop sequence, and length of growing season, tillage practices, residue management and the expected time distribution of erosive rainstorms [23]. It is expressed as the ratio of the soil loss from a cropped land under specified field conditions and the soil loss from a continuously clean-tilled bare soil surface over a given period of time. One way to reduce soil loss is to choose a crop that provides cover during the year when rainfall is most erosive.

Table 3 indicates various C-factors for 13 main crops selected with high suitability of growing in Sebeya catchment [33]. All C-factors for 13 main cultivated crops in Sebeya catchment were inserted in SOLCY Excel sheet for all 195,533 arable parcels.

The P-factor represents the effect of various support practices such as contour farming, terracing and strip cropping for arresting soil erosion being taken up in the area [9] [27]. It is the ratio of soil loss from a land where conservation practices (like: contouring, strip cropping, terracing, etc.) are adopted to soil loss from a land where soil erosion control measures were not implemented. The value of P-factor ranges from 0 to 1. The value of P-factor for a water body is 0 while the value of P-factor is taken as 1 for no conservation practice [34]. From literature, Table 4 indicates typical values of P-factors of various soil erosion control measures.

Various combinations of soil erosion control measures to be implemented in Sebeya catchment have been proposed based on the land slope range [17]. As simulated by SOLCY model, Table 5 summarizes various combinations of soil erosion control measures applied on each parcel of Sebeya catchment to predict soil loss and crop yields. It is recommended that biological measures such as planting trees and grasses as well as lime and compost applications can be undertaken. The first step was to assign a slope range to each parcel before calculating P-factor corresponding to each proposed combination of the soil

erosion measures. The product of 2 or more P-factors gives a P-factor which represents the effects of the combined soil erosion control measures to be implemented. Table 4 gives all P-factors for all soil erosion control measures to be used in calculating the P-factors corresponding to various combinations of soil erosion control measures simulated in SOLCY model and presented in Table 5.

In many developed countries, the best management practices (BMPs) for soil erosion management have been technologically implemented. These techniques helped greatly to meet the tolerable soil loss criterion [45]. T-value is a concept useful to judge if a soil has a potential risk of erosion, productivity loss and off-site damages as a river or reservoir sedimentation [46] [47]. The generally accepted maximum limit of soil loss or T-value is 11.5 t∙ha⁻¹∙year⁻¹ [16] [44] [48]. The aim of soil conservation strategies is to limit rilling using a threshold soil loss value. But the occurrence of soil loss equal to or lower than 11.5 t∙ha⁻¹∙year⁻¹ does not ensure absence of rills. The procedure used in assigning T-value has relied on multiple judgments of various researchers.

In 2017, the National Institute of Statistics of Rwanda (NISR) 3 recognized agricultural seasons in Rwanda as: Season A (September-February); Season B (March-June) and Season C (July-September). Table 6 gives a typical crop rotation simulated for soil loss prediction in Sebeya catchment with SOLCY model.

SOLCY model is a USLE-type model at each parcel level:

A = R ∗ K ∗ L S ∗ C ∗ P

where R, K, LS are the already determined USLE parameters and now considered as physical constants in nature at each parcel. The only remaining variable is C-factor which will change according to the type of the seasonally cropping patterns. In fact, the simulated soil erosion control measures presented in Table 5 are assumed constant for any crop at any slope range.

The first part in developing this SOLCY model was to simulate how soil loss can be reduced if suitable combinations of soil erosion control measures are applied specifically to well-known site conditions at all parcels in Sebeya catchment. Comparatively by cultivating any type of the 13 main crops on any parcel, each annually average soil loss which can be predicted with SOLCY model will lay between 2 extreme values: minimum and maximum soil loss as A_min < A < A_max. The 1st step of simulation was done with the typical crop rotation described in Table 6, the 2nd step simulated 3 seasonally crops which have a minimum C-factor of C = 0.15 and the 3rd step simulated 3 seasonally crops which have a maximum C-factor of C = 0.55. Referring to the combinations of soil erosion control measures defined in Table 5, the results for these 3 steps are presented in Table 7.

The 2nd part in developing SOLCY model was to apply successively a set of soil erosion control measures (Pi1, Pi2 and Pi3) as defined in Table 5 in order to predict the resulting soil loss and crop yield on each arable parcel in Sebeya catchment. Table 8 gives an estimate of the minimum and maximum crop yields for the 13 main cultivated crops in Sebeya catchment.

Many researchers have proposed linear relationships between soil loss and crop yields [50] [51]. In order to predict the crop yields, a linear relationship was assumed by considering the 1st point (x1 = the maximum soil loss; y1 = the minimum crop yield) and the 2nd point (x2 = the minimum soil loss; y2 = the maximum crop yield). From literature, the maximum soil loss has been fixed by previous researchers at 137 t/ha/yr [50] while the minimum soil loss can be taken as the maximum acceptable limit of soil loss or T-value of 11.5 t/ha/yr [44]. Having 2 points in plane XY Cartesian coordinates at this stage, a linear relationship becomes well defined for each crop in the form:

Y = a ∗ X + b (5)

where a & b are known constants for each crop yield prediction; X = A = soil loss and Y = crop yield.

The soil loss for each of the 13 crops simulated on each arable parcel of Sebeya catchment and the value of the crop yield correlatively predicted using the above linear relationship (Equation (5)) were already simulated for each of the combination of soil erosion control measures Pi1, Pi2 or Pi3 as defined in Table 5 and now available in SOLCY Excel spreadsheet. The results as shown in Table 9 were extracted from SOLCY Excel spreadsheet by using the INDEX-MATCH type lookup formula. Taking an example of the parcel of UPI = 30,305,034,908, the user has to type only the parcel UPI and the soil loss with its corresponding crop yield are extracted for all 13 main cultivated crops in Sebeya catchment to be printed in Table 9.

Farmers, District agriculture planners and technicians in Sebeya catchment can use SOLCY model. Most of the cases, an agriculture technician may be available at each sector and ready to facilitate farmers who will need SOLCY model guidance on how they can improve their farming systems. Also, researchers can develop similar models in other catchments based on SOLCY design concept. For 13 simulated crops, a District planner can take advantage of SOLCY to get advice in proposing a crop to be cultivated in a sector or a region based on its productivity and by suggesting BMPs for an appropriate farming system.

Initially, the estimation of soil loss using USLE model with GIS applications on Sebeya catchment was done at large-scale. In order to get more precise results and flexibility in assessing soil loss at parcel level, USLE model itself was limited to measure the soil loss at the outlet of the catchment but it was very difficult to print all the results of soil loss at the outlet of each of 259,673 parcels in Sebeya catchment. That’s why an improvement was proposed to apply analytically the USLE model on each parcel by simulating the cultivation of 13 main crops in Sebeya catchment with selected combinations of soil erosion control measures. A such designed SOLCY model is estimating the soil loss at each parcel and prints results of soil loss for all 259,673 parcels and results of the predicted crop yields for all 195,533 arable parcels of Sebeya catchment in an Excel sheet for each combination of soil erosion control measures.

By simulating the combination Pi1 of soil erosion control measures in each parcel (Table 5), the soil loss from the entire catchment by SOLCY model was 143.27 t/ha/yr compared to 130.72 t/ha/yr of soil loss using GIS. In order to judge the significance of the difference between the means of these 2 populations, a sample of 30 parcels was selected randomly from 259,673 parcels of Sebeya catchment. Table 10 shows the estimates of soil loss using GIS in comparison with the predicted soil loss by SOLCY model.

In order to assess the significance of the difference for 2 sets of results as in the case of Table 10, One-Way ANOVA technique using F-test is usually applied [55]. Using an Excel tool for a significance level α = 5%, the F-test results obtained and shown in Table 11 allow us to draw the conclusion that the two samples have been extracted from populations having the same mean.

Fertilizer has effects on soil loss and crop yield because it changes the organic matter of soil. However, its doses were not indicated in SOLCY model. Finally, SOLCY model is limited to Sebeya catchment but this study provides an appropriate methodology and procedures to be duplicated to other catchments.