Allocation of SWC measures as decision variable
The decision variable of the optimization is a list containing the sub-watershed identifier and the decision, and whether or not SWC measures are applied (Fig.
2). The length of the list depends on the number of sub-watersheds that varies per study area depending on the digital elevation model and the size of the area.
The study areas are separated into sub-watersheds with a watershed delineation algorithm. The selected watershed delineation is performed with a multiple flow direction model (Holmgren,
1994) using the A
T least-cost path search algorithm (Ehlschlager,
1989). In addition, a basin threshold parameter serves to control the minimum inflow area for sub-watersheds.
The placements of the bench terraces within each selected sub-watershed for conservation depend on the slope and the depth of workable soil. The distance between the planned bench terraces should be 2.5 times the depth of workable soil (Hurni et al.,
2016). The distance between bench terraces becomes smaller with higher steepness levels and shallower soil profile depths.
Objective functions
Soil loss estimation
We use the empirical-based Revised Universal Soil Loss Equation (RUSLE) (Renard,
1997) to estimate the soil loss of protected and unprotected sub-watersheds. Even though the RUSLE only accounts for soil loss through sheet and rill erosion and not erosion types like gully erosion or dispersive soils (Rowlands,
2019), it belongs to the most widely applied methods to estimate soil loss rates (Ganasri and Ramesh,
2016). It is computed with
$$A=R\cdot K\cdot L\cdot S\cdot C\cdot P$$
(1)
where
A: the estimated average annual soil loss and temporal average soil loss per unit of area in t ha−1 yr−1,
R: Rainfall-runoff erosivity factor in MJ mm ha−1 h−1 yr−1,
K: Soil erodibility factor in (t ha h) (ha MJ mm)−1,
L: Slope length factor in m,
S: Slope steepness factor in radians,
C: Cover management factor (unitless),
P: Support practice factor (unitless).
The computations for the single factors of the RUSLE with study area specific parameter settings are explained in Appendix A.
Labor requirement estimation
We use empirical values of the labor requirements (Table
1) measured in person days from Tenge et al. (
2005) for different slopes and soil types. The soil types are categorized into stable and unstable soil, where clay soil is considered stable, and loam and sand are considered unstable (Tenge et al.,
2005). Soil is classified as clayey soil if the clay content is above 40%, or if the clay content is above 35% as long as the sand content is below 45% (García-Gaines and Frankenstein,
2015). The labor requirement map depends on the slope, clay, and sand content rasters (Table
1). The total labor requirement is computed for all cells with planned SWC measures:
$$L{D}_{total}=A\mathop{\sum }\limits_{n=1}^{N}l{d}_{n}$$
(2)
where
Table 1
Labor requirement estimation in labor days per hectare for building bench terraces per slope and soil classes
Stable soil | 66 | 148 | 237 | 354 | 427 |
Unstable soil | 92 | 205 | 328 | 491 | 592 |
LDtotal: total labor days
n: current cell of labor requirement raster
N: number of cells where SWC measures are applied
A: Cell size in ha
ldn: labor days per ha
Translation of labor requirements and soil losses to monetary units
Labor and soil losses are associated with estimated costs measurable in monetary units. We use estimated labor costs in US Dollars based on daily wages of 4.32 US Dollars after Bachewe et al. (
2016). The monetary loss associated with soil loss is based on yield loss estimates, assuming an estimated yield loss of 0.74% per mm of eroded soil (Rickson,
2020). The yield loss in percent is one component of the monetary loss estimate. The second component is market shares of agricultural products, crop yields and prices derived from official statistics Central Statistical Agency Ethiopia (
2020). We consider the total area used for a crop for the market share and use the cereals teff, sorghum and maize. These make up 88.52% of the total market share. For example, in the Ethiopian region Meher, 16.5% of the area is used to grow teff with a retail price of 750 US Dollars per ton in 2018 (United States Department of Agriculture Foreign Agricultural Services,
2019). In combination with crop productivity of 16.38 quintals (1 quintal = 100 kg), we obtain the expected monetary unit per ha. Both components and the total area size in hectares lead to the total estimated monetary loss in US Dollars per year. The estimate considers expected yield and soil losses for the coming 10 years.