The model calculates rainfall excess at a fine temporal resolution (minutes to hours) using a modified curve number approach. The original version of the MCST model simulates single rainfall events but is converted into a continuous simulation model as follows. The input of the model is a continuous rainfall series with a time resolution of 10 min. A rainfall–runoff event is identified as a period (i) in which rainfall depth exceeds 2 mm in 24 h (< 1 % of total runoff excluded) and (ii) which is separated by at least 72 h without rainfall. Accordingly, a rainfall–runoff event is not necessarily defined by a single hydrograph, but might contain multiple runoff peaks. A moving window of 24 h is used to estimate cumulative rainfall (Pi,cum) and cumulative abstractions for each time step i (i.e. initial abstraction (Ia,cum) and continuing abstraction (Fa,cum) of the curve number method). The excess rainfall hyetograph (Ri) at time step i is calculated as Ri=Ri,cum-Ri-1,cum and Ri,cum=Pi,cum-Ia,cum-Fa,cumIf, where Pi,cum (mm) is the cumulative excess rainfall during the last 24 h.

If is a scaling factor for rainfall intensity which is calculated as If=INmax10100.9, where INmax10 is the maximum 10 min rainfall intensity (mm h-1).

Flow discharges for each grid cell and time step are calculated by numerically solving the kinematic wave equations (Van Oost et al., 2004). For sheet flow, cross-sectional flow area is calculated assuming a homogeneous flow depth for each raster cell, while for concentrated flow, a relationship between discharge and cross-sectional flow area is used (Govers, 1992). To distinguish between sheet and concentrated flow, a critical shear velocity of 3.5 cm s-1 for rill initiation, based on flume experiments conducted by Govers (1985), is used. The model keeps track of changes in the pattern of concentrated flow and rill network development. Finally, sediment movement is described by utilizing an event-based steady-state sediment continuity equation proposed by Yu et al. (1997).

Experimental research has shown that the Hairsine–Rose model provides a physically based description of sediment transport and deposition for multiple sediment classes that differ in terms of settling velocities (Beuselinck et al., 2002a, b). Transport of soil by overland flow is characterized by simultaneous re-entrainment and deposition (i.e. temporary settlement) of sediments: di-rri=αiCivsi-αiHFgδiδi-ρΩ-ΩcrDMdiMdt, where di is the mass rate of deposition per unit area of size class i (kg s-1 m-2), rri is the rate of sediment re-entrainment for settling velocity class i (kg s-1 m-2), Ci is the mean sediment concentration (settling velocity class i; kg m-3), αi is the ratio of the sediment class concentration of flow related to the local sediment class concentration of the parent material, vsi is the settling velocity of sediment size class i (m s-1), H is the fractional shielding of the soil by the deposited layer, F is the fraction of stream power used for re-entrainment, g is gravity (m s-2), δi is the sediment density of settling velocity class i (kg m-3), ρ is the water density (kg m-3), Ω is the stream power (W m-2), Ωcr is the critical stream power (W m-2), D is the depth of the water flow (m), Mdi is the mass of sediment class i in the deposited layer (kg m-2), Mdt is the total mass of the deposited layer per unit area (kg m-2).

If the local stream power (Ω) is less than a critical threshold (Ωcr), re-entrainment does not occur and deposition of size class i is a function of its specific settling velocity (Beuselinck et al., 1999; Hairsine et al., 2002). If the local stream power exceeds this threshold value, a shielding factor H is calculated to decide whether net erosion or deposition occurs (Hairsine and Rose, 1992a): H=δ-ρg∑viCiδFΩ-Ωcr. If H≥ 1, then net deposition, characterized by steady state flow and re-entrainment of previously deposited sediment, occurs. If H < 1, net erosion occurs, and soil detachment is modelled as Dr=aSserQde+bSseiDir=bI2Sf, where Dr and Dir are the rill detachment rate and the interrill sediment transport to the rill (kg m-2 s-1), respectively, a is the rill erodibility factor, b is the interrill erodibility factor, Q is the rill discharge (m3 s-1), S is the local slope gradient, I is the maximum 10 min rainfall intensity, Sf is a slope factor and ser, de and sei are calibration exponents.

Rill erosion is considered to be unselective, i.e. the sediment particle size distribution of the eroded material equals the distribution of the source material at the source location. In contrast, interrill erosion is simulated as a selective process: assuming steady state flow conditions, Eq. (4) is used to estimate the particle size distribution of the sediment detached by interrill erosion and Eq. (7) is used to estimate the transport for sediment delivered to the rill network (or that leaves a grid cell when there is no incised rill). This approach is consistent with empirical observations showing that the enrichment of finer sediment particles and SOC in suspended sediment is mainly controlled by the transport capacity of the flow (Schiettecatte et al., 2008a). To represent the amount of primary particles vs. soil aggregates of suspended sediments, the model interpolates the settling velocity for each particle class and grid cell according to the proportion of particles detached by interrill or rill erosion.

The MCST model keeps track of spatio-temporal changes in particle size distribution of the eroded and deposited topsoil sediment within 10 different size fractions. However, the particle size distribution is spatially homogeneous and constant throughout the 100-year modelling period.

For our modelling-based analysis, we combined data from different sources into a virtual catchment data set: (i) All basic data (i.e. digital elevation model, soils) were taken from a small first-order catchment in central Belgium, located about 15 km southwest of Leuven. The site has a size of 3 ha with a mean and maximum slope of 7 and 14 %, respectively. The catchment consists of diverging convex hillslopes and a central concavity where ephemeral gullying and sediment deposition are frequently observed (Fig. 1; Desmet and Govers, 1997). Soils in the catchment are loess-derived, silty-loamy Luvisols, with a clay, silt and sand content of 14, 82, and 4 %, respectively (Desmet and Govers, 1997). (ii) For the 100-year modelling period, high-resolution rainfall data (1898–1997; 10 min intervals), measured in Ukkel (Brussels-Capital Region), were used (Fig. 1; Verstraeten et al., 2006).

Modelling parameters were derived from earlier studies. (i) We assumed continuous maize cropping, where monthly curve number values range between 83 and 89 (Van Oost et al., 2004) to account for seasonal changes in crop cover and soil crusting. This range resulted in runoff volumes that are consistent with field observations (Gillijns et al., 2005). (ii) Two annual tillage operations are assumed to erase the network of rills and ephemeral gullies which may have evolved during preceding erosion events. Apart from removing rills, tillage erosion is not taken into account. (iii) The rill and interrill erodibility parameter values, as well as the slope and discharge exponents (Eqs. 6 and 7), were assumed to be constant over time and space. Therefore, spatio-temporal variability of soil moisture is not accounted for. The parameter values are taken from flume and plot-scale experiments, conducted using soils from the Belgium loess belt (Table 1; Van Oost et al., 2004). With these parameters, MCST has already shown to be able to predict the spatial patterns and rates of sediment detachment and transport in the test catchment (Van Oost et al., 2004).

In simulation studies, the particle size distribution is typically derived from dispersed sediment samples and therefore reflects the settling velocities of the primary particles of the sediment. However, sediment transport and deposition can also occur in the form of aggregates, particularly for fine-textured soils, as is the case in our study area (Beuselinck et al., 2000c). Therefore, we considered the particle size distributions of both aggregated soil and primary particles in our simulations. We considered two erosion scenarios. In erosion scenario 1, both detachment by rill and interrill erosion leads to aggregate breakdown and soil is transported and deposited following the settling velocity classes of primary particles (Fig. 2). Furthermore, particulate organic matter (POM) is an individual free floating particle class. In erosion scenario 2, interrill erosion still breaks down aggregates and transports primary particles. In contrast, detachment by rill erosion does not lead to aggregate breakdown and entrains aggregated soil, following the settling velocity classes of aggregated soil (Fig. 2). For aggregated soil POM is assumed to be encapsulated in soil aggregates and is not treated as an individual class. Following detachment, the model simulates the transport and deposition of primary particles or aggregated soil based on the erosion type of detachment that they underwent. The particle size distributions of primary particles and aggregated soil were taken from direct measurements (n= 81) in the Belgian loess belt conducted by Beuselinck et al. (1999). The grain size distribution for aggregated soil represents the relative difference between fully dispersed and non-dispersed soil in 10 different diameter classes. For these classes, the corresponding settling velocities were calculated according to the model of Dietrich (1982), using a density of 2.6 and 1.3 kg m-3 for primary particles and aggregates, respectively. The density of primary particles is assumed to be close to quartz, whereas a pore space of 50 % is assumed for aggregates. The settling velocity distributions (Fig. 2) show that the aggregated sediments are dominated by fractions with settling velocities between 10-4 and 10-3 m s-1, i.e. silt-sized particles. In contrast, erosion scenario 1, which solely considers primary particles, shows very low settling velocities relative to aggregated sediments (Fig. 2). This results from differences in particle size between the two fractions: aggregated soils contain fewer clay and silt-sized particles, because particles of this size tend to be occluded in aggregates. As a result, aggregates have larger particle sizes and faster settling velocities.

The implementation of SOC characteristics in the model is based on a SOC fractionation study that was carried out with similar soils (Luvisols) from the Belgian loess belt (Doetterl et al., 2012). In the study of Doetterl et al. (2012), soil samples were taken at 11 locations along a topographic gradient, from non-eroded to eroding and depositional sites. The results showed that 85 % (±10 %) of the total SOC was associated with the mineral fraction (clay and silt size), while the remaining 15 % (±3 %) was POM. To our knowledge, no detailed information is available on the allocation of SOC in particle size fractions from 2 to 63 µm. In terms of simplicity, and given the constraints imposed by the model structure, we considered two types of SOC for both primary particles and aggregated soil: (i) mineral-bound SOC, which represents 90 % of the total and is associated with the finest sediment class (< 2 µm) and (ii) a POM fraction, which represents 10 % of total SOC and is considered a separate class in the model, with a particle size of 250 µm and a density of 1000 kg m-3. Hence, SOC is represented in different particle classes but the model does not account for geochemical processes.

We evaluated the performance of the model by comparing the predicted characteristics with those that were continuously observed in the Kinderveld and Ganspoel (Table 2) agricultural catchments for two observation periods of 3 years each (6 years total observation; Van Oost et al., 2005b). The two catchments are situated approx. 15 km from our study site and are larger but very similar to our site in terms of soil properties and geomorphology. We were unable to directly apply our model to these two agricultural catchments, as our model has high data requirements, which could not be met due to large uncertainties in input data, or in some cases because the data were simply not available. Rather than providing an evaluation on an event-basis, we evaluated the model performance by looking at the characteristics of sediment and carbon delivery, in response to a range of erosion event-sizes. This provides a first, but stringent, test of model structure and assumptions.

For an analysis of event-based recurrence intervals, we follow the rainfall event definition given in Sect. 2.1 (72 h window). Thereby, some events may contain multiple runoff peaks. The recurrence interval (T) is related to the frequency (P) with which soil erosion (SL) exceeds the value X: T=1PSL≥X. The recurrence interval is expressed in years when T is multiplied by the number of modelled years.

To calculate the frequency of exceedance, monthly soil erosion values were ranked in increasing order, and a rank m is given to each modelled soil erosion event. The exceedance probability for event m is given by: PSL≥Xm=mn+1, where n is the total number of events during the period.

Application of the rainfall/runoff model over a period of 100 years resulted in 792 individual rainfall/runoff events. The temporal variability of rainfall events is relatively low, as more than 70 % of total rainfall is associated with events with a recurrence interval of less than 1 year. Extreme rainfall events do occur, but their relative contribution to total rainfall is limited (i.e. events with a recurrence interval ≥ 2 years contribute less than 18 % of total rainfall). The model simulates that, integrated over the period of simulation, about 10 % of the total rainfall does result in surface runoff. This is consistent with field observations in the study area, where an average of 8 % was reported by Steegen et al. (2000, 2001). In contrast, the simulated temporal variability in runoff is high, and events with a larger recurrence interval, i.e. ≥ 2 years, make up more than 36 % of total runoff. The variability in runoff is higher than that of rainfall because it is controlled by multiple factors, including rainfall amount and intensity, vegetation characteristics, soil surface conditions and the presence and/or absence of a rill/ephemeral gully network at the beginning of an event.

In the study area, erosion can be found in the mid-slopes, whereas a depositional area is located in the valley bottom (Fig. 3). Interrill erosion, modelled here as a function of slope and rainfall intensity, accounts for 14 % of total sediment mobilization over a 100-year period. Rainfall intensity is the main factor controlling interrill erosion and explains about 70 % (p < 0.001) of its variability. In contrast, incised (i.e. rill/ephemeral gully) erosion was modelled as a function of slope and discharge and is therefore mainly controlled by surface runoff (R2= 88 %; p < 0.001). The simulated relative contribution of interrill erosion depends on the suspended sediment concentration (SSC) and the sediment delivery ratio (SDR), which is the fraction of eroded soil that is transported to the catchment outlet. For events with low values for SSC and SDR, the contribution of interrill erosion can account for up to 100 % of total sediment mobilization, but this contribution declines rapidly with increasing SSC and SDR (Fig. 4). This reflects the role of event size, whereby only larger events produce significant amounts of concentrated erosion once the hydraulic threshold for rill initiation is exceeded. This large contribution of rill erosion for sediment delivery was also observed by Wang et al. (2010) in the Kinderveld and Ganspoel catchment. In a modelling study, Wilken et al. (2016) tested the effect of different rill initiation characteristics on carbon delivery in a catchment of similar loess-derived soils. The results showed that rill erosion widely controls sediment and carbon delivery in catchments with high connectivity.

We evaluated two erosion scenarios where different assumptions about particle size distribution are made: erosion scenario 1, where soil is transported and deposited as primary particles and POM is an individual class, and erosion scenario 2 where rill erosion detaches, transports, and deposits aggregated soil and POM is encapsulated in soil aggregates (see Sect. 2.3). The simulated long-term enrichment ratio of the deposits for the fine fraction (< 2 µm), which results from selective transport and deposition processes, was found to be 0.02 and 0.6 for erosion scenario 1 and 2, respectively. For erosion scenario 1, this implies that the deposition of clay particles and POM is virtually non-existent and also suggests a very efficient export of clay minerals and POM from first-order catchments. However, these results are not consistent with field observations. Data derived from the Belgian Soil Map (Baeyens, 1959) show only small differences between the primary particle size distributions of colluvial and non-eroded agricultural soils in the study area. The reported enrichment ratio for the clay fraction of colluvial soils is 0.8 (Baeyens, 1959). The colluvial sediment is thus only slightly depleted in clay when compared to the source material. Based on this analysis, we consider the results of the simulations for erosion scenario 1 to not be physically valid. In contrast, the results of erosion scenario 2 are qualitatively similar to the observations (Baeyens, 1959), which suggests that erosion scenario 2 more accurately depicts erosion processes in our study area. This implies that the assumptions made for erosion scenario 2 are appropriate, i.e. interrill erosion detaches and transports primary particles, whereas rill erosion is unselective and detaches and transports soil aggregates. The concept of particle size-selective interrill and non-selective rill erosion which detaches and entrains the entire soil matrix has been documented in numerous studies (Kuhn et al., 2010; Polyakov and Lal, 2004; Quinton et al., 2001; Schiettecatte et al., 2008a). Following non-selective splash erosion (Poesen and Savat, 1980; Poesen, 1985; Parsons et al., 1991), selectivity is caused by particle size specific deposition differences, where coarser and heavier particles settle out earlier than finer and lighter particles (Schiettecatte et al., 2008b). The model tends to slightly underestimate the deposition of the finest fractions (enrichment of colluvial sediments: 0.6 model versus 0.8 field observations).

The clay enrichment ratios at the outlet of the catchment (i.e. clay exported sediment/clay source material) for the simulated events range between 1 and 4.8 (Fig. 5a). These ratios are strongly related to the SSC: high enrichment ratios occur when SSC is low, i.e. during small events with a low recurrence interval. In contrast, low enrichment ratios (i.e. close to 1) are associated with events characterized by a high SSC. These findings are in line with other studies (Schiettecatte et al., 2008a, b; Wang et al., 2010) and emphasize the importance of event size. The contribution of interrill erosion is higher for small events and, since interrill erosion is modelled as the detachment and export of individual sediment particles, this results in a higher clay enrichment ratio. Vice versa, the contribution of interrill erosion is small for large events, resulting in enrichment ratios close to 1, since concentrated erosion is assumed to be unselective. The simulated range of enrichment ratios and the relationship between those ratios and SSC are both very similar to that which was observed at the Kinderveld and Ganspoel catchment (Fig. 5; Wang et al., 2010).

However, over a simulation period of 100 years, the flux-weighted predicted clay enrichment ratio in exported sediments was found to be 1.4, which is lower than the field-observed ratio of 1.5–2.6 for a 6-year period for the Ganspoel and Kinderveld catchments (Wang et al., 2010). We assume that this discrepancy results from difference in sedimentological connectivity, whereas a cascade of selective erosion and deposition processes in the larger catchments lead to stronger enrichment in the delivered fines. In contrast, an earlier study applying MCST in catchments of similar scale (0.7 and 3.7 ha; Fiener et al., 2008) showed a good representation (R2= 0.93) of the modelled transport of fines compared to 8 years of observations.

The simulated enrichment of carbon is directly related to the preferential export of the clay fraction through by interrill erosion. The simulated carbon enrichment ratios are higher than the clay enrichment ratios previously discussed and range between 1 and 9 (Fig. 5b). Exported sediment is more enriched in carbon than it is in clay due to the fact that the clay fraction is itself enriched in carbon relative to the bulk soil. The simulated relationship between SSC and carbon enrichment is similar to what was found for clay enrichment, i.e. enrichment is higher when SSC is low. This is again consistent with the Kinderveld and Ganspoel field observations (Wang et al., 2010).

It should be noted that the enrichment of exported clay and carbon was simulated assuming that interrill erosion resulted in the detachment of primary particles while concentrated erosion resulted in the detachment of aggregates. Alternatively, clay and POM fractions could be considered as individual classes in the model. However, due to very low settling velocities, nearly the entire mobilized clay and POM fractions are exported from the catchment when this is simulated. This is not in line with field observations or with experiments that show that the transport of fine-textured sediments mainly occurs in the form of aggregates (Beuselinck et al., 2000c; Wang et al., 2013).

Based on the 100-year modelling period, we analysed the effect of event based frequency and magnitude of erosion on mobilization and delivery of bulk sediment, clay, and SOC (Fig. 6). We found that for within catchment erosion, a large number of relatively small events (recurrence interval < 1.5 years) account for about half of the cumulative erosion, while larger events (> 10 years recurrence) account for only about 15 %.

The SDR was 0.18 over the 100-year simulation period, while the mean erosion rate was 12.5 Mg ha-1 yr-1. Figure 6 clearly shows that larger events play a more important role in determining SDR than they do in determining soil erosion. Approximately 57 % of the total sediment delivery comes from events with a recurrence interval less than or equal to 10 years (Fig. 6). This is explained by the fact that sediment delivery is not linearly related to runoff amount: once the hydraulic threshold is exceeded (i.e. an extensive network for concentrated flow is established) the sedimentological connectivity is highly enhanced and SDRs can be very large. The simulation of hydrological and sedimentological connectivity requires the introduction of (i) differentiated hydrological behaviours for sheet and concentrated flow, (ii) rill/ephemeral gully network development tracking, and (iii) the rill/ephemeral gully network connectivity to the outlet of the catchment. Our simulations show that the highest export rates occurred when the rill/ephemeral gully network was already well established at the beginning of an event. The important role of a rill/ephemeral gully network for the catchment connectivity was also pointed out in other studies (López-Vincente et al., 2013, 2015). However, structures which interrupt the rill/ephemeral gully network potentially reduce the sedimentological connectivity to the outlet and reduce the SDR substantially (Wilken et al., 2016).

The importance of event size for simulating clay and SOC delivery is also shown in Fig. 5. Compared to bulk sediment, the delivery of clay and SOC is less driven by rare, large events, since small events with more interrill erosion already deliver relatively large amounts of clay and SOC. In general, the model results underline the importance of a more process-oriented analysis of SOC redistribution, as the effects of small erosion events, e.g. upon aquatic ecosystems, are underestimated when modelling only mean bulk erosion rates.

In order to qualitatively evaluate our predicted temporal patterns for sediment delivery, we compared our results to studies that continuously measured export from small catchments. In one such study, which was carried out in small agricultural catchments in the Belgian loess belt, Steegen et al. (2000) measured sediment delivery over a 3-year period in two first-order streams. The authors found that a single event contributed to more than 40 % of the total sediment delivery during the observation period and that the sum of rare and extreme events accounts for 46 %. Even more extreme results were reported from a small loess catchment (3.7 ha) in Southern Germany, where a short-term series of single runoff events accounted for up to 67 % of total sediment export (> 0.5 mm runoff) over an 8-year period (Fiener et al., 2008). Although a quantitative comparison of the model results with these empirical observations is not possible, as empirical observations in central Europe typically cover far fewer than 100 years, this analysis strongly indicates that the mechanisms incorporated into the MCST model (i) allow for a quantitative representation of the relative importance of both small and large events and (ii) account for event size related sediment and carbon delivery.