During wet periods, when the depth of available water exceeds the maximum available soil storage capacity (Sb, in mm), the actual evapotranspiration is equal to the potential evapotranspiration (Ep, in mm day-1). When S(t) is lower than Sb, Ea is assumed to decrease linearly with moisture content as follows (Steenhuis and van der Molen, 1986): Ea=Ep(S(t)Sb),Sb=Dϕ, where D is the soil depth [mm] and ϕ is the soil porosity (-).

Subsurface runoff, Qss [mm day-1], occurs only when the storage depth exceeds the field storage capacity (Sf, in mm). It is calculated as the difference between the storage and the field storage capacity, divided by the response time (Tr) of the catchment with respect to subsurface flow (Jothityangkoon et al., 2001): Qss=S(t)-SfTrwhenS(t)>Sf,Qss=0whenS(t)≤Sf. The field storage capacity of the soil reservoir, Sf [mm], is calculated using Sf=FcD, where Fc (-) is the field capacity of the soil (dimensionless).

The catchment response time is the time taken by the excess water in the soil to be released from the soil and drained out from the catchment. This response time depends on the properties of the soil and the topography of the system, and the subsurface flow velocity (Vb, in mm day-1) can be expressed as Vb=LTr, where L is the average slope length of the catchment [mm]. From Darcy's law in saturated soils, Vb is also given as Vb=Ksi, where Ks is the saturated hydraulic conductivity of the soil [mm day-1] and i is the hydraulic gradient, which is approximated by the average slope gradient (G) of the catchment.

Brooks et al. (2004) analyzed the variability of saturated hydraulic conductivity with depth and found large Ks values near the surface or root zone layer and the transmissivity that decreases exponentially with depth. Accordingly, a variation is made between the upper soil layer (which affects interflow) and deep soil layer (percolation to groundwater) hydraulic conductivities. The permeability (K, in mm day-1) of the upper soil layer for the interflow under different soil water conditions is modeled as K=Ks,u(1-e-βS(t)Sb), where β is a dimensionless parameter and Ks,u [mm day-1] is the saturated hydraulic conductivity of the upper soil layer, both of which are to be calibrated.

The response time (Tr) in Eq. () is hence approximated from Eqs. (), () and () as Tr=LGK, where L and K are as defined in Eqs. () and () and G is average slope gradient of the catchment.

The deep percolation or recharge to groundwater (R, in mm day-1) under varying soil water content conditions is modeled as R=Ks,e(1-e-γS(t)Sb), where γ a dimensionless parameter, and Ks,e [mm day-1] is the saturated hydraulic conductivity of the deep soil layer, which is to be estimated from the aquifer properties of the catchments. This equation is identical to Eq. (); therefore in both cases it is assumed that conductivities vary exponentially under varying soil water content conditions but with different magnitudes.

Saturated excess runoff or surface runoff (Qse1, in mm day-1) is calculated as the depth of water that exceeds the total water storage in the soil reservoir at each time step (Jothityangkoon et al., 2001; Krasnostein and Oldham, 2004). Qse1=S(t)-SbΔtwhenS(t)>SbQse1=0whenS(t)≤Sb

Field visits on the Upper Blue Nile Basin (including the study catchments) revealed the existence of exposed surfaces that cause strong runoff response. These are areas with little or no soil cover and bedrock outcropping in some parts of the catchment as well as soils with well-developed tillage pans (Temesgen et al., 2012a, b) (Fig. 3). Hence, runoff from these almost impermeable areas is modeled as infiltration excess (Hortonian flow) runoff with a very small amount of retention before runoff occurs (Steenhuis et al., 2009). The surface runoff from these areas (QSe2, in mm day-1) is calculated as QSe2=P-EpwhenP>EpQSe2=0whenP≤Ep, where P and Ep [mm day-1] are as defined above. The impermeable portion of the catchment area (Ar, in km2) is modeled from the total catchment area (At, in km2) as Ar=λAt, where λ is the fraction of impermeable surface within the catchment.

The introduction of a deep groundwater storage (Fig. 2b) helps to improve low-flow predictions. This baseflow reservoir is assumed to act as a nonlinear reservoir (Wittenberg, 1999) and its outflow, Q2 [mm day-1], and storage, Sg [mm], are related as Q2=Sg(t)k1Δt, where k1 is a dimensionless model parameter. The water balance of the slow-reacting reservoir (groundwater reservoir) is given by Sg(t)=Sg(t-Δt)+(R-Q2)Δt, where Sg(t) [mm] is the groundwater storage at the given time step, Sg(t-Δt) [mm] is the previous time step groundwater storage and R [mm day-1] is the deep percolation, as given by Eq. ().

In total the model has seven parameters: i.

Parameters for the recharge (α1 and α2): in the three slope classifications, α1 is to consider for the recharge from the steep slope into the medium slope surface and α2 is for the recharge from the medium slope surface into the flat surface. There is no parameter for the steep slope surface since there is no surface that recharges it. Therefore, there are two parameters for the three slope classifications.

The total river discharge (Qt, in mm day-1) at the outlet of the catchments is given by Qt=Qss+Qse1+QSe2+Q2.

Steenhuis et al. (2009) found that overland flow in the Blue Nile Basin is generated from saturated areas in the relatively flatter areas and from bedrock areas, while in the rest of the catchment all the rainfall infiltrates and is lost subsequently as evaporation, interflow or baseflow. Topographical processes have been found to be the dominant factors in affecting runoff in the Blue Nile Basin (Bayabil et al., 2010). We used topography of catchments as the main criterion to divide the catchment into different runoff production surfaces. Based on slope criteria (FAO, 2006), each study catchment was divided into three sub-catchments as steep (slope gradient > 30 %), hilly or medium (slope gradient between 8 and 30 %) and flat (slope gradient < 8 %) to consider spatial variability in catchment properties and runoff generation mechanisms (Fig. 4).

The 30 m×30 m resolution global digital elevation model (GDEM) was used to define the topography (downloaded from the ASTER website, http://earthexplorer.usgs.gov/). The GDEM (Fig. 1b) was used to delineate and calculate the average slope gradient and average slope length of the catchments (topography-related inputs to the model).

The model requires data on depth, porosity and field capacity of the soils. Soil depth and soil types data (Figs. 5 and 6) were obtained from the Abay River basin integrated master plan study BCEOM (1998).

In this modeling philosophy, the soil depth is meant to represent the depth of water stored in the topmost layer (root zone) of the soil (Fig. 2). The porosity and field capacity of the soils were derived from the soil texture based on the work of McWorter and Sunada (1997). From this, we determined the soil textures of the study catchments (Table 1). The saturated hydraulic conductivity for the deep percolation (Eq. 12) was estimated using ranges of conductivities given by Domenico and Schwartz (1990) for the saturated hydraulic conductivities of a deep soil layer (colluvial mantle on top of the igneous rock). A summary of the topographic, soil and saturated hydraulic conductivity data for the study catchments is provided in Table 1.

Daily precipitation is the key input meteorological data for the model. Other meteorological data like minimum and maximum air temperature, humidity, wind speed and duration of sunshine hours were also used to calculate the potential evapotranspiration, the other input variable to the model. All weather data were obtained from the Ethiopian National Meteorological Agency (NMA) for 13 stations located within and around the catchments (www.ethiomet.gov.et). The location map of the rain gauge stations used for this study is depicted in Fig. 7. The data for most of the stations are consistent and continuous, particularly for the first-class stations like Dangila, Adet and Debretabor. However, we encountered gaps in some stations like Sekela station for some periods in the year. In such instances, only the rainfall data from the other stations were considered. Most of the rainfall stations in Gilgel Abay catchment are installed at the water divides, and there is no station in the middle of the catchment. In this regard, the Gumara catchment has a higher density of rainfall stations. The areal rainfall distribution over the catchments was calculated using the Thiessen polygon method, and the potential evapotranspiration was calculated using the FAO Penman–Monteith method (Allen et al., 1998).

Starting from July 2011, water level was measured at the Wanzaye station (11.788073∘ N, 37.678266∘ E) on the Gumara River and from December 2011 at the Picolo station (11.367088∘ N, 37.037497∘ E) on the Gilgel Abay River. The water level measurements were made using mini-divers, automatic water level recorders (every 10 min), and manual readings from a staff gauge (three times a day, at 07:00, 13:00 and 18:00), following the procedures described by Amanuel et al. (2013).

Discharges were computed from the water levels using rating curves (Eqs. 21 and 22) for each station. The rating curves (Fig. 8) were calibrated based on detailed surveys of the cross sections of the rivers and measurements of flow velocity at different flow stages, using the following commonly used expression: Q=ahb, where a and b are fitting parameters and Q [m3 s-1] and h [m] are discharge and water level, respectively. The resulting rating curve equation for the Gumara catchment at the gauging station (Wanzaye station) is Q=44.1h1.965(R2=0.997,n=12), and that of the Gilgel Abay catchment at the Picolo station is Q=70.39h2.105(R2=0.985,n=14).

Compared to the discharge data that have been gathered in the past, the discharge data that are acquired for this study are of superior quality, since a high time resolution during the measurement has been used. This minimizes the risk of missed peaks, particularly during the night. Furthermore, frequent supervision was also conducted during the data collection campaign. Hence, these data were used for the model calibration. Discharge data collected before December 2011 were obtained for nearby stations from the Hydrology Department of the Ministry of Water Resources of Ethiopia, which has a long data record (since 1960) for these stations. However, the latter measurements were made using staff gauge readings twice a day, with many data gaps and discontinuities, particularly at the end of the observation window. The discharge data from 2000 to 2005 are relatively better and are used to validate the model.

The 2012 discharge data for Dirma catchment (outlet at 12.427194∘ N, 37.326209∘ E), collected in the same way as those of Gilgel Abay and Gumara, were used to assess the transferability of the model parameters.

SWAT is a basin-scale and continuous-time model used to simulate the quality and quantity of surface and ground water and predict the environmental impact of land use, land management practices and climate change (Arnold et al., 1998). The hydrological model is based on the water balance equation SWt=SW0+∑i=1t(Ri-Qi-ETi-Pi-QRi)Δt, where SWt is the soil water content at time t [mm]; SW0 is the initial soil water content [mm]; Δt is the time step (day) and Ri, Qi, ETi, Pi and QRi are the daily amounts of precipitation, runoff, evapotranspiration, percolation and return flow [mm day-1], respectively.

In SWAT, a watershed is divided into homogenous hydrologic response units (HRUs) based on elevation, soil, management and land use, whereby a distributed parameter such as hydraulic conductivity is potentially defined for each HRU. Hence, an analyst is confronted with the difficult task of collecting or estimating a large number of input parameters, which are usually not available for regions like the Upper Blue Nile Basin. Details of the model can be accessed at the SWAT website (http://swatmodel.tamu.edu).

Automatic calibration and validation of the model was made using SWAT-CUP. It is an interface that has been developed for SWAT automatic calibration and model uncertainty analysis (Abbaspour et al., 2007). R2 and NSE were used as objective functions during the calibration process of the search for the optimal value.

This model is a lumped conceptual type and is characterized by three reservoirs as described by Fenicia et al. (2008): the unsaturated soil reservoir (UR), the fast-reacting reservoir (FR) and the slow-reacting reservoir (SR). The model has eight parameters: a shape parameter for runoff generation β [-], the maximum UR storage Sfc [mm], the runoff partitioning coefficient D [-], the maximum percolation rate Pmax [mm h-1], the threshold for potential evaporation Lp [-], the lag times of the transfer functions Nlag [h], and the timescales of FR and SR: Kf [h] and Ks [h]. Details of the model and the various equations of the model can be found in Fenicia et al. (2008).

Calibration of this model was made using the PSO technique, following similar procedures of the Wase–Tana model calibration algorithm. The same objective function, RMSE, is also used in the search for the optimal value.

This hydrological model (the Wase–Tana model) is based on the generation of direct runoff from saturated and impermeable (degraded surfaces and rock outcrops with little or no soil cover) areas, interflow from the soil storage in the root zone layer and baseflow from the deeper layer as groundwater storage. The understanding of the relative importance of these processes on the hydrological response of each catchment is still unknown. The mean annual surface runoff (Qse, sum of Qse1 and QSe2), interflow or subsurface flow (Qss) and baseflow (Q2) components of the total daily hydrograph computed by the model for the calibration and validation periods are given in Table 4.

The total mean annual runoff generated by the model is in line with the observations for both catchments in the calibration period (Table 4), while an appreciable difference is noticed in the values for the Gilgel Abay catchment in the validation period. One of the problems in accurate modeling of the discharge is that precipitation measurements do not cover well the catchments. This is particularly the case for the Gilgel Abay catchment, where the rainfall stations are poorly distributed as most of the meteorological stations lie near the water divides. The calibration results are better, since the data from the recently established precipitation stations (e.g., Durbetie) could be used. There are also doubts about the representativeness of the discharge data used for the validation of the model, because the water level measurements were made manually and twice daily (in the morning and late afternoon), leading to the possibility of missing flash floods at other moments of the day as the stream discharge is very variable. This can be clearly seen from the mean annual observed flows during the calibration and validation periods for Gilgel Abay. The mean annual observed flow in the validation period was found to be much smaller than the corresponding flow during the calibration period (Table 4). The closer total mean annual runoff values and the better model performance indicators for the Gumara catchment during the calibration period suggest that the model can perform satisfactorily with better input discharge and precipitation data.

From PBIAS results (Table 3), the FlexB model showed overestimated bias and the SWAT model behaved the opposite for both catchments during the calibration period.

Despite the variations in mean annual runoff generated by the Wase–Tana model, the partitioning of the total runoff into the different components (Table 4) in each period is almost identical for each catchment, as expected. About 65 % of the runoff appears in the form of interflow for the Gumara catchment, and baseflow takes the larger proportion for Gilgel Abay catchment (44–48 %). Uhlenbrook et al. (2010) found the baseflow to be about 32 % from similar model study results for Gilgel Abay catchment. Vogel and Kroll (1992) have shown that baseflow is a function of catchment area, and geomorphological, geological and hydrogeological parameters of the catchment have a linear incidence on the discharges. The difference between the baseflow of the two catchments is high, despite their comparable catchment sizes, suggesting rather the different structure, functioning and hydrodynamic properties of the two catchments. Hence, the model results reveal that the groundwater in the Gilgel Abay catchment receives more recharge and makes a greater contribution to the river flow. This is in line with Kebede (2013) and Poppe et al. (2013), who showed that the largest part of the Gilgel Abay catchment consists of pumice stones and fractured quaternary basalts with a high infiltration capacity and hydraulic properties, which clarifies the large groundwater potential. In line with this, several large springs exist in the catchment, including one that is used as a source of water supply for the city of Bahir Dar (Fig. 13).

The other interesting result is that direct runoff is the smallest fraction of the total runoff for both catchments (18–19 % for Gumara and 20 % for Gilgel Abay) and almost all peak flow incidences are associated with direct runoff. More than 90 % of this direct runoff is found to be from the relatively impermeable (degraded areas, plough pans or rock outcrops with little or no soil cover) surfaces. The calibrated result shows that this type of runoff production area covers 15 % of the Gumara and 17 % of the Gilgel Abay catchments, respectively. In a similar study, Steenhuis et al. (2009) mention that the rock outcrops occupy 20 % of the total catchment area in the Abay (Blue Nile) catchment at the Ethiopia–Sudan border upstream of the Rosaries Dam, which is very similar to the result of Gilgel Abay catchment in this study.

The remaining direct runoff is generated from the flat slopes of the catchments as saturated excess runoff, probably near the valley bottoms. The hillslopes (medium and steep slope source areas in this paper) generated almost no direct runoff as saturated excess flow. Similar results were obtained by different researchers in the Blue Nile Basin, who identified hillslopes as main recharge areas (Steenhuis et al., 2009; Collick et al., 2009; Tilahun et al., 2013). Our results contribute to the debate on the relative importance of saturated excess runoff versus infiltration excess runoff (Hortonian overland flow) mechanisms in the Upper Blue Nile Basin, showing that the rainfall–runoff processes are better represented by the soil reservoir methodology. However, further research is necessary that involves rainfall intensity and event-based analysis of hydrographs.

The sensitivity analysis was performed on model parameters for Gumara catchment with respect to the RMSE.

The parameters β, α1 and γ show poor sensitivity for a wide range of values with respect to the local sensitivity analysis. The local sensitivity analysis shows the sensitivity of a variable to the changes in a parameter if all other parameters are kept constant at some value (optimal value in this case). An increase in the value of β beyond 1.4 showed almost no sensitivity, while the model efficiency decreased slightly after an increase in the value of γ from the optimum. This means that there is little confidence in the model's correspondence with these parameters and that the parameters can be reduced without appreciable impact on the model (Fenicia et al., 2008). k1, Ks,u and λ are very sensitive parameters in this model and the model performance drops abruptly if the parameters exceed a particular threshold value (Fig. 14).

The global sensitivity analysis (Fig. 15), however, shows interactions among all the input parameters of the model. Although global sensitivity analysis reveals details of the model behavior in a more general sense through random parameter sampling and that the parameters are all sensitive, the local sensitivity analysis indicates that moderate variations in the parameter values for some parameters can still drastically change the model performance.

The model parameter transferability to other ungauged catchments in the basin has been tested by analyzing the variability among the calibrated parameters of the two catchments. Table 2 shows that the calibrated parameters are nearly identical for both catchments, except for γ and λ, which are related to deep percolation and impermeable fraction of the catchment, respectively. As described above, they affect the baseflow and direct runoff contributions to the total river flow. However, we showed that the contributions of these components to the total runoff are relatively small and γ is poorly sensitive to a wide range of values. Thus the influence of these parameters is expected to be minimal. This is verified by generating flows using the average of the calibrated parameters of the two catchments and analyzing the effect on the model performance indicators (Table 5). The model performance obtained using the average model parameter values is similar to the results found using the optimal model parameters (Table 3). To further verify the adaptability of the average calibrated model parameter values outside the study catchments and see the impacts of scale, we applied the average parameter values to another catchment (Dirma catchment in the northern part of the Lake Tana sub-basin, Fig. 1) with an area of 162.6 km2. Encouraging model efficiency could be obtained, with NSE and R2 values of 0.58 and 0.6, respectively (Table 5). This is to be elaborated further in the future, involving more catchments and more years of data.

In general, transferability results showed good performance of the daily runoff model in the two study catchments and an average performance in the test catchment (Dirma catchment). This can be explained by the fact that effort was made to incorporate more knowledge in the model structure to increase model realism. We based our model strongly on the soil storage characterization of the soil reservoir in the rainfall–runoff process and representation of the maximum storage of the unsaturated reservoir at the catchment scale, which is closely linked to rooting depth and soil structure and strongly depends on the ecosystem. Transferability of the model has benefited from this in that we were able to derive most of the input data from the test catchments. The consideration of topography-driven landscape heterogeneity analysis and catchment information extraction based on topography (slope) for the model is another reason for the better performance of the model transferability. The role of topography in controlling hydrological processes and its linkage to geology, soil characteristics, land cover and climate through coevolution have been indicated in different studies (Sivapalan, 2009; Savenije, 2010; Gao et al., 2014). The results suggest the possibility of directly using the average model parameter values for other ungauged catchments in the basin, even though further tests on such catchments are still recommended. However, we believe that this is a useful result for operational management of water resources in this data-scarce region.