In this study, we applied the same hydrological model as Merz2011, which is a semi-distributed conceptual model that follows the structure of the Hydrologiska Byråns Vattenbalansavdelning (HBV; Bergström, 1995). The model equations can be found in Parajka et al. (2007). The model parameters are listed in Table 1. The model operates on a daily time step, and the spatial discretisation is based on 200 m elevation bands. Precipitation is partitioned into snow, rain or mixed precipitation based on air temperature, using a lower and an upper threshold temperature Ts and Tr. A snow-correction factor (SCF) corrects the stronger undercatch of the precipitation gauges during snowfall. Snowmelt is calculated using a temperature-index approach based on the degree-day factor (DDF) and the melt temperature (Tm). Actual evaporation (Esim) is estimated as a function of Eref and soil moisture. It equals Eref if soil moisture is above a calibrated threshold (LP). Below this threshold, it linearly decreases to zero at a soil moisture level of zero. The fraction of the sum of rain and snowmelt that results in discharge is calculated as a non-linear function of soil moisture. This involves the parameters (FC), the maximum soil moisture storage and the non-linearity parameter B, where a larger B is associated with a smaller fraction of direct run-off and vice versa. The run-off module consists of a hillslope component and a river-routing component. The hillslope component is represented by two linear stores that are connected through a constant percolation rate Cp. Fast run-off is generated if the state of the upper zone store is above a threshold (LSUZ), using a fast storage coefficient K0. Medium and slow run-off components are calculated as outflow from the upper and lower zone store, using the storage coefficients K1 and K2. In the river-routing component, run-off routing in streams is simulated using a triangular transfer function involving the parameters CR and Bmax⁡.

Despite being technically external to the applied HBV model, the estimation of Eref is considered part of the hydrological model rather than part of the input data since it is calculated and not available as measured data. Eref is computed on a 1 km2 grid and aggregated to elevation zones for each catchment, as used in the hydrological model. For the baseline model, Eref was derived based on a modified Blaney–Criddle method (DVWK, 1996), following Merz2011, and denoted as E0 as follows: 2E0=-1.55+0.96⋅8.128+0.457⋅T⋅S0⋅100Syear, where T is the mean daily air temperature at 2 m height (∘C), S0 is the potential daily sunshine duration (h), and Syear is the mean yearly sum of the potential sunshine duration (h).

In order to consider interannual variations in global radiation and vapour pressure deficit in addition to air temperature, we calculated Eref using the Penman–Monteith equation for well-watered short grass vegetation (Allen et al., 1998), and denoted as E1 as follows: 3E1=0.408⋅Δ⋅Rn-G+γ⋅185400(T+273)⋅ra⋅es-eaΔ+γ⋅1+rsra, where Rn is the net radiation at the crop surface (MJ m-2 d-1), G is the soil heat flux density (MJ m-2 d-1), ra is the aerodynamic resistance (s m-1), rs is the surface resistance (s m-1), es is the saturation vapour pressure (kPa), ea is the actual vapour pressure (kPa), Δ is the slope of the vapour pressure curve (kPa ∘C-1), and γ is the psychrometric constant (kPa ∘C-1). According to the reference conditions of a vegetated surface with a height of 0.12 m, rs=70 s m-1 and ra=208/u2, where u2 is the wind speed at 2 m height (m s-1), which was derived from the wind speed at 10 m height based on a logarithmic wind speed profile (Allen et al., 1998). The ground heat flux was neglected. The vapour pressure deficit es-ea was calculated as the average of the vapour pressure deficit at the minimum air temperature (using relative humidity at 07:00 LT) and at the maximum air temperature (using relative humidity at 14:00 LT). Rn was estimated from global radiation (Rs; MJ m-2 d-1), albedo (α; set to 0.23), and net long-wave radiation (Rnl; MJ m-2 d-1) as follows: 4Rn=(1-α)⋅Rs+Rnl, where Rnl was estimated according to Allen et al. (1998) based on minimum and maximum air temperature, clear-sky solar radiation, measured Rs, and the mean daily vapour pressure.

In order to consider, additionally, changes in the vegetation dynamics, we calculated Eref using a variable surface resistance based on changes in a satellite-based vegetation index (E2). We used observed 15 d maximum value composite data of the Normalized Difference Vegetation Index (NDVI) at a resolution of 8 km from the Advanced Very High Resolution Radiometer (AVHRR) from Tucker et al. (2005). For each point in time of this biweekly series, we aggregated the NDVI data to 200 m elevation zones based on the NDVI data for a rectangle around Austria. As the NDVI data are only available starting from July 1981, we applied the data of July 1981–June 1982 for 1976–1981 where the NDVI data were not available. We used the parameterisation from Sellers et al. (1996) to estimate a variable rs from the NDVI data. This involved estimating the fraction of photosynthetically active radiation (FPAR) from transformed NDVI data (Eq. 5; Sellers et al., 1996), estimating the leaf area index (LAI) from the FPAR data (Eq. 6; Sellers et al., 1996), and estimating rs from the LAI data (Eq. 7; Allen et al., 1998). 5FPAR=S-Smin⁡Smax⁡-Smin⁡⋅FPARmax⁡-FPARmin⁡+FPARmin⁡, where S is a transformed NDVI value (1+NDVI)/(1-NDVI) and Smin⁡ and Smax⁡ are the 5 % and 98 % quantiles of S for a given land cover class. 6LAI=LAImax⁡⋅log⁡1-FPARlog⁡1-FPARmax⁡, where LAImax⁡ is the maximum LAI of a land cover class. In Eqs. (5) and (6), we applied the following coefficients for grassland: NDVImin⁡=0.039, NDVImax⁡=0.674, FPARmin⁡=0.001, FPARmax⁡=0.95, and LAImax⁡=5 (Sellers et al., 1996). 7rs=rl⋅LAI⋅0.5-1, where rl is the leaf surface resistance. We applied a value of rl=100 s m-1 for well-watered grass (Allen et al., 1998). Since the satellite-based LAI values derived in this way are often lower than the value of 2.88, which is assumed in the Penman–Monteith equation for well-watered short grass by Allen et al. (1998), E2 generally resulted in lower annual Eref than E0 or E1. In order to avoid water balance problems in the hydrological model, E2 was multiplied with the annual average ratio of E2 to E0 that was averaged over all catchments with a value of 1.2. Such an adjustment of Eref may be justified based on the fact that our study catchments are dominated by forests, and the maximum possible evaporation under well-watered conditions (Emax⁡) of forests is typically higher than Eref that assumes short grass. For example, analyses from non-weighable lysimeters suggest Emax⁡ to be 20 %–30 % higher for sites with pine forests at typical stand ages of 80–100 years compared to sites with grass (ATV-DVWK, 2002).

The objective function applied for the model calibration consisted of three parts. An average of the Nash–Sutcliffe efficiency of linear and logarithmic discharge values (fQ) was applied in order to achieve a balanced model performance for high and low flows. In order to keep the volume bias low, the absolute value of the relative volume bias (fbias) was added as a penalty. Furthermore, a penalty for model parameters that deviate from an a priori distribution (fbeta) was added. The penalty function fbeta is based on a beta distribution for each parameter, as described in Merz2011. The a priori distributions for the model parameters were applied since, on the basis of the literature and previous applications of the model, we believe to have more information on the likely parameter values than just the parameter range. Including this criterion in the objective function has very little influence on the difference between simulated and observed discharge trends (Sect. S1). The objectives were combined in the following way: 8f1=w1⋅1-fQ+w2⋅fbias+w3⋅fbeta, by setting the weights to w1=0.8, w2=1, and w3=0.2.

In order to test whether including annually aggregated discharge data in the objective function improves the model performance under transient climate conditions, we additionally applied a modified objective function as follows: 9f2=w1⋅1-fQ+w2⋅fbias+w3⋅fbeta+w4⋅1-fannual, where fannual is the Nash–Sutcliffe efficiency calculated for discharge data aggregated to hydrological years. The weights were set to w1=0.4, w2=1, w3=0.1, and w4=0.5.

In a further model variant, we tested whether snow data improve the model performance under transient climate conditions. The snow-related part of the objective function aimed at minimising the number of days with poor snow cover simulations and was defined following Parajka et al. (2007). Observed snow cover was derived from maps of interpolated snow depth. An elevation zone was considered as snow covered if the average interpolated snow depth was greater than 0.5 mm and snow free otherwise. In the model, an elevation zone was considered snow covered if the simulated snow water equivalent was greater than 0.1 mm and snow free otherwise. If the difference between simulated and observed snow cover on a particular day was greater than 50 % of the catchment area, it was considered to be a day with poor snow cover simulations. The snow-related part of the objective function fsnow was defined as the ratio of the number of days with poor snow cover simulation and the number of days with observed snow cover. The overall objective function was then defined as follows: 10f3=w1⋅1-fQ+w2⋅fbias+w3⋅fbeta+w4⋅fsnow. The weights were set to w1=0.7, w2=1, w3=0.1, and w4=0.2, following Parajka et al. (2007).

The objective function was minimised automatically with the shuffled complex evolution algorithm (SCE-UA; Duan et al., 1992), a global optimisation method based on the simplex downhill search scheme (Nelder and Mead, 1965). The calibration included 11 parameters. The upper and lower bounds and two further parameters of the beta distribution for each parameter were selected following Merz2011 (Table 1). Four parameters that showed little sensitivity were preset to the following values: TR=2 ∘C, TS=0 ∘C, Cr=25 d2 mm-1, and Bmax⁡=10. As the focus of this study was on calibrating the model many times for different calibration periods, catchments and model variants, characterising parameter uncertainties was beyond the scope of this study. For the baseline model, we used seven consecutive 5-year calibration periods without temporal overlap (based on hydrological years), during 1978–2012. Each simulation was started with an additional 22-month warm-up period. As a modification, we also tested using a 25-year period as calibration period (1978–2002).

Model performance was evaluated using the relative bias in discharge volume and the Nash–Sutcliffe efficiency (NSE). The relative bias in discharge volume was calculated as follows: 11bias=∑t=1nQsim,t-∑t=1nQobs,t/∑t=1nQobs,t, where Qsim,t and Qobs,t are, respectively, the simulated and observed discharge on day t, and n is the number of time steps.

In order to focus on the change in discharge under transient climate conditions, we used the difference between simulated and observed discharge trends as an additional criterion. Good performance in the calibration period but inability to estimate the changes in the observed discharge resulting from the climatic changes indicates problems under transient climate conditions. Trends were evaluated over the entire study period (1978–2013). Trend significance was assessed by the nonparametric Mann–Kendall test (Mann, 1945; Kendall, 1975), and lag-1 serial correlation was removed by applying the trend-free prewhitening technique (Yue et al., 2002). Trend slopes were estimated by the Sen's slope estimator (Sen, 1968). Uncertainties of the trend slope were estimated using a bootstrapping approach. For this purpose, 1000 samples of size N were drawn, with replacement, from the record of length N years, and the Sen's slope was calculated for each of the 1000 samples. Then, the standard deviation was determined. The trends and the standard deviations were first derived for each catchment and then averaged over the catchments to determine average trends and their uncertainties over a number of catchments.

We compiled possible explanations for the expected divergence between the observed and simulated changes in discharge based on the frameworks suggested by Westra et al. (2014) and Fowler et al. (2018) and the discussion in Coron et al. (2014). The working hypotheses are grouped into (1) data problems, (2) problems related to the model calibration, and (3) problems of the model structure (see Table 2). In a first analysis, the hypotheses were evaluated based on process understanding and literature. During this process, a number of the working hypotheses were rejected or assessed as being unlikely to cause the differences between the observed and simulated discharge changes. Other hypotheses were evaluated using simulations with modifications of the model (Table 3).

Data problems

Discharge data can be misleading if they are influenced by abstractions or streamflow diversions. For example, a general increase in water abstractions would reduce a positive streamflow trend. However, our study includes only catchments that were classified as devoid of substantial anthropogenic influences (Viglione et al., 2013) and any existing streamflow diversions were introduced before the beginning of our study period (BMLFUW, 2015). Changes in water abstractions due to irrigation are not believed to be a major cause of the deviations between simulated and observed discharges as only about 3 % of the arable land in Austria is irrigated (FAO, 2016), the fraction of arable land is small in most of the study catchments (median 5 %, see Sect. 2.1), and the study catchments have only a little overlap with those regions where irrigation is most relevant. These are small areas east, southeast and northwest of Vienna, where estimated average irrigation amounts of the agricultural areas exceed 10 mm yr-1 (BMLFUW, 2011). Erroneous trends in the discharge data could be caused by systematic trending errors of the rating curve. However, it seems unlikely that the discharge data of a large number of catchments are afflicted by systematic trends in the same direction. Problems in the discharge data were thus assumed unlikely to be a relevant cause of the differences between simulated and observed discharge trends.

Inhomogeneities of the precipitation data would result in biased estimates of the precipitation trends. A problem that would affect a large number of catchments is a varying number of precipitation stations included for generating the gridded precipitation data set. The precipitation data set used by Merz2011 was based on all available stations and included ∼800 stations at the end of the 1970s and ∼1050 stations around the year 2000 (Fig. S2 in the Supplement). The effect of the changes in the number of stations on the trends in the water balance components was analysed by simulations with a precipitation data set based on all available stations (P0) and simulations with a precipitation data set based on a constant number of stations (P1). Changes in the gauge undercatch error due to changes in climate would also affect a large number of catchments. An increase in precipitation intensity and a decrease in the snow-to-rain ratio are expected to result in a higher catch ratio, meaning that the precipitation increase is lower than perceived by the observed data. The effect of neglecting the systematic precipitation error was estimated by simulations with a precipitation data set that is based on a constant number of stations that was corrected for the systematic gauge undercatch considering the influence of the precipitation type and daily precipitation intensity on the catch ratio (precipitation data set P2).

Similar to the precipitation data set, the air temperature data set in the baseline model was based on a variable station network, though the number of air temperature stations varies much less than the number of precipitation stations (Fig. S2). We investigated the effect of the changes in the number of air temperature stations by simulations with air temperature data sets based on all available stations (T0) or a constant number of stations (T1).

Driving the hydrological model with a precipitation data set based on a variable number of precipitation stations may influence the estimated trend of precipitation and thus the trend of simulated discharge. In order to quantify this effect, we performed model simulations with a precipitation data set based on a constant number of stations (P1) in comparison with the baseline precipitation data set P0 that uses a variable number of stations. This reduced the gap between simulated and observed discharge from 95±50 to 55±47 mm yr-1 per 35 years (Table 4); i.e., a reduction of 39±26 mm yr-1 per 35 years (Table 5). The reduced gap between simulated and observed discharge is consistent with the difference in the trends in the precipitation data sets. The baseline precipitation data set P0 suggests a precipitation increase of, on average, 159±89 mm yr-1 per 35 years, whereas the precipitation data set P1 results in an increase of 121±89 mm yr-1 per 35 years (Fig. 4a). Better model performance with respect to changes in streamflow volume is also reflected by smaller increases in bias during evaluation in the different subperiods (Fig. 5a).

Changes in the snow-to-rain ratio and in the precipitation intensity may affect the undercatch error and thus the precipitation trend. Figure 4c–e shows that, over the study period, the snow-to-rain ratio decreased and the daily precipitation intensity increased, whereas the number of precipitation days remained relatively stable. In the precipitation data sets P0 and P1, the precipitation undercatch error is neglected. In order to estimate the magnitude of the effect of changes in air temperature and precipitation intensity on changes of the undercatch error, we performed simulations with a precipitation data set that was corrected for undercatch accounting for daily precipitation intensity and precipitation type, which was estimated based on air temperature (precipitation data set P2). Precipitation data set P2 generally exhibits higher precipitation and, with an average trend of 120±93 mm yr-1 per 35 years, a similar absolute and a lower relative precipitation increase over time when compared to the precipitation data set P1 (Fig. 4a). Simulations with precipitation data set P2 resulted in a gap between simulated and observed discharge trends of 48±47 mm yr-1 per 35 years (Table 4); i.e., a reduction of 47±28 mm yr-1 per 35 years compared to the baseline model V0 that uses precipitation data set P0 (Table 5). Comparing model variants V2 to V0, strong reductions of the differences between simulated and observed discharge trends particularly occurred in catchments where the differences between simulated and observed discharge trends were large (Figs. S6d and 2c). The tendency to further reduce the gap compared to simulations with the precipitation data set P1 of 8±9 mm yr-1 per 35 years was not significant.

In order to investigate the possible effect of changes in the station network for air temperature data, we performed simulations with gridded air temperature data based on stations with a complete record over the study period (T1), compared to simulations with a gridded data set based on all available air temperature series (T0). This showed virtually no differences in discharge trends between the two variants (Table 4). The small effect of varying the air temperature data set can be explained by the fact that changes in the station network were only small (Fig. S2), and the two data sets result in very similar changes over time (Fig. 4b).

In order to evaluate whether the calibration period was too short, we increased the calibration period from 5 years (1978–1982) to 25 years (1978–2002; model variant V4). This resulted in an average discharge trend of 113±82 mm yr-1 per 35 years over 1978–2013 (Table 4), and thus there was virtually no effect compared to the baseline model.

Changing the objective function by including annually aggregated discharge data (model variant V5) led to an average discharge trend of 115±83 mm yr-1 per 35 years over 1978–2013 (Table 4), and thus there was no improvement in the simulation of the long-term discharge trends either.

Including a snow-related criterion into the objective function (model variant V6) improved the model performance with respect to snow without deteriorating the model performance for discharge (Table S1). The performance of the model compared to observed snow cover derived from interpolated snow depth was comparable to Parajka et al. (2007), when considering the same set of catchments. Model performance with respect to long-term trends was not improved, with an average gap between simulated and observed discharge trends of 95±50 mm yr-1 per 35 years over 1978–2013 (Table 4).

To estimate the effect of using a simplified versus a more physically based equation for estimating Eref, we compared simulations with Eref estimated by the Blaney–Criddle method (simulation V0) to simulations with Eref estimated by the Penman–Monteith method (model variant V7). The results showed only negligible differences between the two model variants in terms of simulated discharge trends (Table 4). This is consistent with small differences between the trends in Eref estimated by the two different methods, with average trends of 70±13 mm yr-1 per 35 years for E0 (Blaney–Criddle) and 71±17 mm yr-1 per 35 years for E1 (Penman–Monteith; Fig. 6).

In order to consider changes in the vegetation dynamics, we estimated changes in surface resistance based on changes in a satellite-based vegetation index for the calculation of Eref. Accounting for vegetation dynamics in the calculation of Eref increased trends in Esim to 88±16 mm yr-1 per 35 years (model variant V8), compared to 52±13 mm yr-1 per 35 years in the baseline model V0 (Table 4). This reduced the gap between simulated and observed discharge trends from 95±50 to 58±49 mm yr-1 per 35 years (Table 4); i.e., a reduction of 36±9 mm yr-1. Increased trends in Esim are consistent with Eref trends that increased from 70±13 mm yr-1 per 35 years in the baseline model V0 to 110±17 mm yr-1 per 35 years in model variant V8 (Fig. 6). Accounting for vegetation dynamics had a rather consistent effect on the discharge trends throughout the catchments (Fig. S6b and e). In order to evaluate the effect of combining the model modifications that had a considerable effect on the gap between trends in observed and simulated discharge, we combined the use of the precipitation data set P2 (model variant V2) and the consideration of vegetation dynamics in the calculation of Eref (model variant V8) as model variant V9. Compared to the baseline model, the differences in the trends between simulated and observed discharge were reduced by 90±31 mm yr-1 per 35 years in this model variant so that the differences largely disappeared (Table 4). Bias values in the evaluation period for variant V9 show only little variation between subperiods S2 and S6, but some variation remains when transferring models from subperiods S1 or S7 to subperiods S2 or S6 or vice versa (Fig. 5h). Bias values in the evaluation period were reduced from -0.13–0.18 in the baseline model to -0.03–0.10 in model variant V9. When comparing model variant V9 and the baseline V0, the differences in the trends of simulated and observed discharge were reduced in most catchments, with stronger reductions in catchments that showed higher differences in the trends of simulated and observed discharge in the baseline model (Fig. S6f).