1 Introduction

Soil erosion by water is one of the most important environmental problems as it leads to the loss of fertile top soil for agriculture and causes sedimentation and eutrophication of surface water bodies. The mountainous torrential Lukovska River catchment (114.31 km2) in the south part of Serbia is affected with a serious problem of soil erosion. Steep slopes of the terrain together with intensive deforestation, weak shallow soil and cracked, unstable and tectonic damaged geologically formations make this catchment prone to serious erosion. A specific hydraulic and hydrological regime of torrents is manifested in occurences of torrential waves, that can be intense but short in duration and they can transport a heavy load of sediment. It was noticed that a few number of flash floods caused by a few number of intensive rain events account for over 80 % of the total soil loss and the total annual sediment transport (Petković and Kostadinov 1989).

Sediment yield predictions and spatial and quantitative assessment of soil erosion on a catchment scale with the aid of erosion models are particularly important for the identification of erosion-vulnerable regions for the implementation of a suitable erosion control measures. Since surface erosion have large spatial variability in a catchment due to spatial heterogeneity of physical characteristics of a catchment, the Geographic Information Systems (GIS) can be utilized to handle large spatially distributed model parameters. The empirical Universal Soil Loss Equation (USLE) (Wischmeier and Smith 1978) is frequently coupled to GIS because of its simple structure and ease of application (Sanjay et al. 2001; Fistikoglu and Harmancioglu 2002; Pandey et al. 2007; Bhattarai and Dutta 2007; Dabral et al. 2008; Mishra and Deng 2009; Manoj and Debjyoti 2010; Jain and Das 2010). Physically based models can represent erosional and sedimentation processes in basins more realistically than empirical models because they consider the governing physical laws between various hydrological processes related to soil erosion and sediment transport. Many physically based models have been developed in the past such as: ANSWERS (Beasley et al. 1980), SHETRAN (Bathurst et al. 1995), WEPP (Nearing et al. 1995) or AGNPS (Young et al. 1989).

The purpose of this study is to present a physically based methodology for estimating soil erosion and sediment yield during isolated intensive storm events on the example of the mountainous torrential Lukovska River catchment. The applied methodology implies the coupling of the SHETRAN physically based distributed hydrological modelling system with GIS. SHETRAN (Ewen et al. 2000) was developed at the Water Resource Systems Research Unit (WRSRU), University of Newcastle upon Tyne. It is capable of simulating water flow, sediment transport, and contaminant migration. SHETRAN can be used to display the unsteady and non-uniform movement of torrential waves realistically owing to the use of a diffusive wave approximation to the Saint–Venant equations (Saint-Venant 1871) for describing overland and channel flows and the use of the unsteady equation of sediment continuity for describing sediment transport; it can be applied to individual rain events and at the river basins of different size. Although the capability of the SHETRAN model to simulate water and sediment discharge was tested in a number of publications (Bathurst et al. 2006; Elliott et al. 2012; Zhang et al. 2013), its capability for predicting erosion rates and its spatial distribution within a catchment were tested only in a few applications (Banis et al. 2004).

2 Methodology

The model of soil erosion was established by the calibration and the validation of the SHETRAN modelling system, while the ArcGIS software ArcView 9.3 was used both for the preparing of the input data related to the physical characteristics of the basin for each grid cell and for the displaying and visualisation of the results of soil erosion and sediment transport.

The evaluation of the model performance is based on simulations of runoff hydrographs and the graphs of sediment concentrations at the basin outlet. The two statistical parameters were used to compare the simulated values with the observed, including: the relative mean square error (RMSE) and the coefficient of determination (CD) (Legates and McCabe 1999):

$$ \mathrm{RMSE}=\frac{100}{\overline{O}}\sqrt{\frac{{\displaystyle \sum_{i=1}^n{\left({O}_i-{P}_i\right)}^2}}{n}} $$
(1)
$$ \mathrm{CD}=\frac{{\displaystyle \sum_{i=1}^n{\left({O}_i-\overline{O}\right)}^2}}{{\displaystyle \sum_{i=1}^n{\left({P}_i-\overline{O}\right)}^2}} $$
(2)

where O is observed discharge (m3 s−1), P is simulated discharge (m3 s−1), \( \overline{\mathrm{O}} \) is the average value of the observations and n is the total number of observations. The RMSE value indicates the extent to which the simulations are overestimating or underestimating observed values, expressed as a percentage of the average value of the observations. The CD statistics describes the ratio of the scatter of the simulated to that of the observed values. A CD value of one indicates that the simulated and observed values match perfectly.

The functionality of the presented methodology in regard to erosion predictions was also validated by comparing the simulated soil losses with the observed soil losses in mountainous regions in Europe. In this context, the value of 1 t/ha/year is generally considered to be the maximum allowable value of the erosion rate. However, the soil losses of 10 to 40 t/ha in individual storms, that may happen once every 2 or 3 years, are observed regularly in Europe, with losses of more than 100 t/ha in extreme events (Morgan et al. 1998).

2.1 The SHETRAN Modelling System

In this study the SHETRAN water flow and sediment transport component were used. The SHETRAN water flow component accounts for interception, evapotranspiration, snowmelt, overland and channel flow, and unsaturated and saturated zone flow (Ewen et al. 2000).

Since erosion and sediment transport are the focus of this paper, the sediment transport component will be described in more detail. The sediment transport component (Bathurst et al. 1995; Wicks and Bathurst 1996) accounts for soil erosion by raindrop impact, leaf drip impact and overland flow, and transport by overland flow and channel flow. Modelling of soil erosion and sediment transport are based on the measured rainfall data and the simulated overland and channel flow data. The rate of soil erosion by raindrop and leaf drip impact is determined with the following empirical equation:

$$ {\mathrm{D}}_{\mathrm{r}}={\mathrm{k}}_{\mathrm{r}}{\mathrm{F}}_{\mathrm{w}}\left(1-{\mathrm{C}}_{\mathrm{g}}-{\mathrm{C}}_{\mathrm{C}}\right)\left({\mathrm{M}}_{\mathrm{r}}+{\mathrm{M}}_{\mathrm{d}}\right) $$
(3)

where Dr (kgm−2 s−1) is the rate of soil detachment; kr (J−1) is the raindrop impact soil erodibility coefficient; FW (−) is a correction factor representing shielding of the soil from detachment when there is surface water on the surface of the grid square; Cg is the proportion of soil covered by ground vegetation; CC is the proportion of ground covered by a canopy; Mr (kg2s−3) is the momentum-squared for rain per unit time per unit area; Md (kg2s−3) is the momentum-squared for leaf drip per unit time per unit area. The detachment rate decreases when surface water and/or vegetation cover is present on soil surface.

The rate of erosion by overland flow is calculated using:

$$ {\mathrm{D}}_{\mathrm{q}}=\left\{\begin{array}{c}\hfill {\mathrm{k}}_{\mathrm{f}}\left({1-\mathrm{C}}_{\mathrm{r}}\right)\left(\frac{\uptau}{\uptau_{\mathrm{c}}}-1\right)\kern0.6em \mathrm{if}\;\uptau >{\uptau}_{\mathrm{c}}\hfill \\ {}\hfill 0,\kern0.6em \mathrm{otherwise}\hfill \end{array}\right. $$
(4)

where Dq (kg m−2 s−1) is the rate of detachment of soil per unit area, kf (kg m−2 s−1) is the overland flow erodibility, τc(N m−2) is the critical shear stress for initiation of soil particle motion, and τ (N m−2) is the overland flow shear stress.

Transport of the eroded material by overland flow is calculated using numerical solutions of the two-dimensional physically based conservative transport equation (Wicks and Bathurst 1996):

$$ \frac{\partial (hc)}{\partial t}+\left(1-\uplambda \right)\frac{\partial z}{\partial t}+\frac{\partial {g}_x}{\partial x}+\frac{\partial {g}_y}{\partial_y}=0 $$
(5)

where h (m) is the water depth (m), c (m3m−3) is the sediment concentration, λ is the soil surface porosity, z (m) is the depth of loose soil, t (s) is time, gx and gy (m3s−1 m−1) are the sediment transport rates in the x and y directions respectively. Estimates of the values of the erodibility coefficients are generally made through model calibration (Wicks and Bathurst 1996).

3 Study Area

The torrential Lukovska River catchment (Fig. 1) is situated in the south part of Serbia, in the region of a temperate continental climate. The Lukovska River is a tributary of the Toplica River, which belongs to the catchment of the South Morava. Altitudes in the Lukovska River catchment vary from 515 meters above sea level in the lower part of the basin to 1652 meters above sea level in the source areas of the basin. Once in every 10 years floods in the Lukovska River flooding zones provide for a flow of approximately 100 m3/s, whereas low flows constantly provide for 99 % flow of about 0.1 m3/s, so that the ratio between small and large water flows is approximately 1:1000, which shows the torrential character of streams.

Fig. 1
figure 1

The Lukovska River catchment with river system and the rain gauging and hydrological stations in the catchment

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From the studied catchment, the two pedological types of soil were singled out: humus-siliceous soil on schists (82 %) and eutric brown soil on flysch (18 %) (Fig. 2). Both types of the soil are of clay-loamy texture, according to USDA Soil Classification System (Baldwin et al. 1938). However, the humus-siliceous soil on schists is a shallow and poorly developed soil. The eutric brown soil on flysch is morphologically more developed than the humus-siliceous soil on schists because it was developed by further pedogenesis of the humus-siliceous soil on schists.

Fig. 2
figure 2

The digitized soil map of the Lukovska River catchment up to “Merćez” water level monitoring station (w.l.m.s.)

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The dominant geological types in the catchment, which define hydrogeological behaviour of the whole catchment, are flysch sediments (shales, marls, marly limestones, sandstones and siltstones) (84 %) (Fig. 3). Due to cracked and tectonical damage of surface parts of the terrain, the whole study area is of low to medium waterpermeability to depths of 10–20 m below the groundwater level. The water table is at the depth of about 4 m below the ground surface (Energoprojekt 1986).

Fig. 3
figure 3

The digitized geological map of the Lukovska River catchment up to “Merćez” w.l.m.s

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There are seven different vegetation types in the Lukovska River catchment (Fig. 4): deciduous forests (76 %), coniferous forests (2.22 %), mixed forests (0.87 %), transition area between forest and brush (9.83 %), agricultural land (6.86 %) and natural grasslands (4.21 %). Forests are often degraded and of low quality. Arable land is situated in valleys of streams.

Fig. 4
figure 4

The digitized vegetation map of the Lukovska River catchment up to “Merćez” w.l.m.s

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4 Material

The basic input data for the model are the maximum hourly rainfall in the catchment during the analysed rain events. The Hydrometeorological Observation System consists of a single climatological station (Kuršumlia), 5 rain gauging stations (Selova, Lukovo, Stave, Blaževo and Gornja Mirnica) and a single hydrological station (Merćez) (Fig. 1). Daily values of precipitation are measured at the rain gauging stations in the catchment while hourly values of precipitation and of other meteorological data are measured only at the climatological station in Kuršumlia. The hourly values of runoff and the maximum daily values of sediment concentration are measured at the hydrological station “Mećez”. It was assumed that there was an even distribution of rainfall at the same time over the whole surface of the basin, the intensity of which equals the values of average hourly heights of rainfall. Average hourly heights of rainfall were determined approximately basing on the hourly heights of precipitation measured at the climatological station in Kuršumlia and the maximum daily rainfall for the whole catchment. The maximum daily rainfall for the whole catchment were determined based on the maximum daily rainfall from 5 rain gauging stations in the catchment by application of the method of Thiessen polygons (Thiessen 1911). The model was calibrated for the rain event in 1986 and validated for the rain events in 1974, 1976 and 1979. The duration of simulation period and of the analysed rain events are presented in Table 1. Total amounts of precipitation fallen on the ground during the analysed rain events and the measured maximum daily values of sediment concentration which was used as a basis for calibration and validation of the sedimentation section of the model are also presented in this Table.

Table 1 Total amounts of precipitation (Ptot) fallen on the ground and the measured maximum daily values of sediment concentration (Cmax) during the analysed rain events
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The all remaining input data, the corresponding input parameters and the source of these data are summarised in Table 2. The preliminary values of the calibrated parameters were determined from the literature (Table 2), while their finally values were determined in the process of the model calibration and validation.

Table 2 Input data and parameters used in the model
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5 Establishment of the Model

The model calibration and validation were performed in particular for hydrological and for sedimentation section of the model on the basis of the measured streamflow hydrographs and measured maximim daily concentrations in a flow at the catchment outlet respectively. The adopted grid cell size in this study was the area of the size of 500 m * 500 m. The average elevation of each grid square was obtained by transforming the obtained 25-m DEM into a 500-m DEM using the ArcView 9.3 tools.

Through numerous simulations it was concluded that the size of the runoff and the sediment discharge in the catchment resulting from a given rainfall are the most dependant on the following parameters of the hydrological section of the model: the Strickler’s coefficients for overland flow and for river flow, the vertical saturated hydraulic conductivity of the subsurface soil and the horizontal saturated hydraulic conductivity in the saturated zone. The above mentioned parameters primarily affect the value of surface runoff, except the horizontal saturated hydraulic conductivity in the saturated zone that affects the values of base flows. The most significant parameters of the sedimentation section of the model determining the size of the sediment discharge and the sediment concentration in a flow are: the erodibility coefficient due to rain and the erodibility coefficient due to overland flow. The values of these parameters were determined through the process of calibration of the model, and checked through the process of the model validation. The values of the all other model parameters were fixed and adopted from the literature (Table 2).

The Strickler’s coefficient for overland flow and for streams, as well as the vertical saturated hydraulic conductivity of subsurface soil were adjusted to model the surface runoff. The horizontal saturated hydraulic conductivity in the saturated zone were used to adjust the base flows.

During the calibration and validation of the sedimentation part of the model, the Yalin (1963) and Engelund and Hansen (1967) equations were compared in order to determine the equation for calculating the sediment transport capacity of overland flow. The Engelund-Hansen and the Acker’s and White (1973) equations were compared in order to determine the equation for calculating the sediment transport capacity of channel flow. The equation of Engelund-Hansen was adopted to be an appropriate equation for calculation of the transport capacity for both the overland flow and channel flow because it produced realistically possible values of sediment concentration in the flow. The equation of Acker and White’s was not adopted because the negligible values of bed load sediment discharge were obtained by appliction of that equation. This result was not in accordance with the realistic conditions in the catchment. The calibrated parameters are shown in Table 3.

Table 3 Calibration intervals and the optimal values (in parenthesis) of calibrated parameters for the Lukovska River catchment
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6 Results and Discussion

The results of the model calibration and the validation are presented in Fig. 5, together with the correlative relations between the simulated and the registered streamflow hydrographs.

Fig. 5
figure 5

Predicted (Qmod) and registered (Qreg) hydrographs and predicted sediment concentrations c (Qmod) for the calibration and the validation rain events (P) (a). Correlative relationships between the registered and the predicted hydrographs for the calibration and the validation rain events (b)

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Basing on the predicted and the registered hydrographs as well as on the obtained correlative relationships and statistics CD and RMSE (Fig. 5) it can be concluded that a satisfactory agreement between the modelled and the registered runoff hydrographs was obtained. The shape of the hydrographs, the peak runoff ordinates and the time of their occurence were simulated appropriately during both the calibration and the validation periods. The largest discrepancy between the modelled and the observed hydrographs was noticed on the example of the wave in 1974 because this wave of water belonged to a medium and not to high flows. The predicted values of maximum sediment concentrations in flow corresponded entirely to their observed values for the all analysed waves.

The changes of intensities of erosion and deposition within the catchment and in the course of time for the torrential rain event in 1986 were estimated and presented in Figs. 6a–h. The total net erosion at the end of this rain event, obtained as the difference between the total soil loss and the amount of deposited sediments during the observed rain event is presented in Fig. 6i.

Fig. 6
figure 6

Spatial representation of intensities of erosion (−) and deposition (+) of sediments in the Lukovska River catchment: a) 16.07.1986. at 23.00 h (t = 10 h); b)17.07.1986. at 00.00 h (t = 11 h); c) 17.07.1986. at 01.00 h (t = 12 h); d)17.07.1986. at 02.00 h (t = 13 h); e) 17.07.1986. at 03.00 h (t = 14 h); f)17.07.1986. at 04.00 h (t = 15 h); g) 17.07.1986. at 05.00 h (t = 16 h); h)17.07.1986. at 06.00 h (t = 17 h) i) the total soil losses at the end of the rain event in 1986

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The results obtained for each grid cell were classified using the nine classes of erosion/deposition listed below: extreme erosion (> −30 tha−1), high erosion (−30/-10 tha−1), medium erosion (−10/-3 tha−1), low erosion (−1/-3 tha−1), stable (−1/1 tha−1), low deposition (1/3 tha−1), medium deposition (3/10 tha−1), high deposition (10/30 tha−1) and very high deposition (>30 tha−1) (Fig. 6). This classification has been derived by adopting the erosion risk classes used by INRA (The Institute National Recherche Agronomique) (Le Bissonnais et al. 2002) and by extending them to take deposition into account.

Figures 5 and 6 show that the erosional and sediment transport processes follow the quickly dynamic of runoff. Intensities of erosion and sediment discharge increased during the time of runoff increasing and reached its highest values in the moment of the occuring of the maximum runoff in the catchment. After that the decrease of runoff caused the decrease of sediment transport capacity of flow and consequently the decrease of the erosion intensities and the increase of the sediment deposition on soil surface. It could be noticed that the influence of intensive storm event, the steep slopes of the terrain and the weak soils and rocks of low permeability resulted in intensive torrential wave of short duration containing lot of eroded sediments.

The analysis of the obtained spatially distributed net erosion values (Fig. 6i) indicate that approximately 88.3 km2 (69.1 %) of the total catchment area was affected by erosion during the analysed rain event in 1986 with the average soil loss of 2.92 t/ha. This result suggest that average annual soil losses should be within the estimated tolerable limits for soil formation of between 2 and 12 t/ha in mountainous regions in Europe (Trabucchi et al. 2012). The average rate of deposition is 1.56 t/ha. It was also estimated that nearly 50 % of the catchment is stable or affected by low erosion or low sedimentation. The classes of medium erosion and medium sedimentation have been observed for 23 % and 21 % of the catchment area, respectively. The erosion intensities are the highest in the zones of steep slopes, in the upper parts of the catchment, while the deposition occurrs mostly in the valleys of rivers. Higher values of sedimentation have been observed for 5.4 % of the catchment area, while only 0.58 % of the study area was affected by high erosion. The results indicate that some of the medium and high erosion/deposition zones could present potentially dangerous places on annual level. The impact of the catchment physical characteristics on the values of erosion was assessed by overlapping the maps of soil types, land use and slope with the net erosion map. The obtained maximum and mean net erosion values within different land covers and different soil types are shown in Table 4.

Table 4 Maximum and mean erosion intensities within different land use and soil type classes
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The results presented in Table 4 show that spatial variations of erosion in the study area appears to be dominated by slope and soil types, while the influence of soil cover is not significant. The mean soil loss for the humus siliceous soil on schist was 2.78 times higher than for the eutric brown soil on flysch. The maximum erosion value of 10.55 t/ha in the catchment was obtained for the area covered by deciduous forest which seems unlikely to occur. Beside that the forest areas were found to be more susceptible to soil losses by water than agricultural land and in the range with natural grasslands. The obtained results can be explained by the fact that degraded forests are often located on steep slopes with poorly developed humus-siliceous soil on schists.

Based on the analysis of spatial variations of erosion in this study, it can be concluded that the amount of soil loss and of sedimentation could be decreased by protecting the upper parts of the catchment from erosion.

7 Conclusions

In this study a physically based distributed soil erosion and sediment transport model has been developed through the joint application of the SHETRAN modelling system and the GIS. Taking into account that investigations of erosional and sediment transport processes in torrential basins in Serbia show that most erosion and sediment transport take place in two or three storms each year, the model was calibrated and validated to four representative intensive rain events on the example of the torrential Lukovska River catchment in Serbia.

In addition to the estimation of water and sediment discharge at the catchment outlet, the presented methodology enables estimation of spatial and temporal patterns of soil loss and deposition within a catchment. Basing on this model it is possible to evaluate the amount of soil losses and/or the amount of deposited sediments within a catchment during individual rain events. In that way, the areas of intensified erosional and sedimentation processes could be identified. These information are crucial for the implementation of erosion control measures. Since this approach considers both soil erosion and deposition, the results approximate reality to a greater degree compared to the models taking into account only erosion. Although the accuracy of the obtained soil loss and deposition values from this model is open to discussion due to the lack of measurement data, the obtained relative picture of existing conditions in any area of the catchment in regard of erosion and sedimentation could be very useful.