(a) Rainfall disaggregation

The multiplicative random cascade model (Müller and Haberlandt, 2015) is applied for the disaggregation of time series of the daily stations. A general scheme of this model is shown in Fig. 3. One coarse time step is divided into b finer time steps of equal length. The branching number b determines the number of finer time steps and is in the first disaggregation time step b=3 and in all following disaggregation steps down to 1 h resolution b=2. The cascade model is microcanonical, so the rainfall amount of each time step is conserved exactly. A re-aggregation of the disaggregated time series yields the observed time series used for the disaggregation. Since the focus of this study is not on the disaggregation itself, the interested reader is referred to Müller and Haberlandt (2015) for a more detailed explanation. However, the main results are a slight underestimation of dry spell duration (relative error of −6 %), percentage of dry intervals (−3 %), wet spell duration (−12 %) and amount (−9 %), while average intensity is slightly overestimated (4 %). While the autocorrelation function also shows underestimations, the extreme values are represented well.

Figure 3General disaggregation scheme of the applied multiplicative cascade model (values inside the boxes represent rainfall amount, and a blue or white box color indicates wet or dry time steps, respectively).

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(b) Bivariate characteristics

For the definition of spatial consistence applied in this study, the bivariate rainfall characteristics follow the ones used by Haberlandt et al. (2008) and are briefly described in the following.

The probability of occurrence P_k,l describes the probability of rainfall occurrence at the same time at two stations k and l:

where n is the total number of non-missing observation hours at both stations, z_i is the rainfall intensity and the number of simultaneous rainfall occurrence at both stations is represented by n₁₁.

Pearson's coefficient of correlation ρ describes the relationship between simultaneously occurring rainfall at two stations k and l as a measure of the linear relation between both rainfall time series (Eq. 2). Breinl et al. (2014) used this coefficient before for multisite rainfall generation:

Müller and Haberlandt (2015) found an intensity dependency for Pearson's coefficient of correlation and distinguished between ρ(k≤4 mm) and ρ(k>4 mm), which is adopted here.

The continuity ratio C_k, l compares the expected rainfall amount at one station for times with and without rain at the neighboring station (E is the expectation operator):

These characteristics are distance-dependent and prescribed values can be estimated as functions of the separation distance between two stations from observed data (see regression lines in Fig. 4 for each characteristic).

(c) Implementation of spatial consistence

As mentioned before, the disaggregation of single time series is a point process with no surrounding stations taken into account. Input rainfall products for the rainfall–runoff models consisting of just the disaggregated time series without subsequent steps to implement spatial consistence are referred to as V1 (no implementation of spatial consistence). Two methods for the implementation of spatial consistence, and resulting in the rainfall products V2 and V3, are applied in this study.

The first method, resulting in V2, is based on simulated annealing (Aarts and Korst, 1965; Kirkpatrick et al., 1983), a nonlinear optimization method from the group of resampling algorithms. The aim of simulated annealing is to modify the disaggregated time series and in doing so minimize an objective function including the deviations between the observed bivariate rainfall characteristics and those from the disaggregated time series. Relative diurnal cycles are swapped without changing the structure of the time series or the absolute daily totals of rainfall amounts. The interested reader is referred to Müller and Haberlandt (2015) for further details.

Figure 4Bivariate spatial rainfall characteristics of V1, V2 and V3 in comparison to observations for the Pionierbrücke catchment (for one realization, black circles represent observations – for details the reader is referred to Müller and Haberlandt, 2015).

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The second method, resulting in rainfall product V3, is a more pragmatic solution. It was introduced by Haberlandt and Radtke (2014) and is also based on the time series of V1 that is already disaggregated. For each day, the station with the highest rainfall amount is identified. The relative diurnal cycle of this station is transferred to all other stations for this day. This parallelization is carried out for all days of the disaggregated time series. The varying diurnal distributions of rainfall at each station without spatial patterns, leading to an underestimation of spatial consistence, are transformed instead to a simultaneous occurrence of rainfall at all stations with an overestimation of spatial consistence.

Both methods are compared against using the disaggregated time series without any subsequent steps. For analyses and discussion of the impacts of these methods, the designations listed in the summarizing Table 4 are used.

Table 4Short characterization of the three rainfall products.

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(a) HBV-IWW including calibration procedure

The HBV-IWW model is based on the HBV model that was originally developed at the Swedish Meteorological and Hydrological Institute (SMHI) in the early 1970s (Bergström, 1976) and was modified by Wallner et al. (2013). HBV-IWW, denoted HBV for simplification, is a conceptual model, whereby runoff generation and runoff transformation are represented by simple relationships between storage and effective precipitation, or runoff (see flowchart of the model in Fig. S1 in the Supplement). For the spatial discretization of the study areas, subcatchments (see Fig. 2) with an approx. area of 20 km² are applied. It could be questioned whether a rainfall–runoff model with subcatchments is useful for the validation of the spatial consistence of rainfall. A daily station covers an area of 65 km² on average in Germany (Müller, 2016). This spatial resolution is not increased by the cascade model in this study, since only a temporal disaggregation is applied. Also, no additional information is gained by a model with higher spatial resolution. So the only disadvantage could be a sort of numerical diffusion due to the spatial resolution. However, since subcatchments of this size are used throughout a number of studies, the HBV with this spatial resolution represents the state of the art and is applied for the current study.

Table 5Calibration and validation period for all catchments.

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For the estimation of the areal rainfall of each subcatchment, a two-step approach was chosen. First, rainfall is interpolated with a nearest neighbor approach on a raster basis with cell widths of 1 km. In the second step, areal rainfall for each subcatchment is calculated through the arithmetic mean of all raster cells within the subcatchment. If the areal rainfall of a subcatchment is dominated by one station, it could be questioned whether areal rainfall intensities should be reduced (by, e.g., areal reduction factors; Sivapalan and Blöschl, 1998; Veneziano and Langousis; 2005; Wright et al., 2013) to avoid an overestimation (e.g., Peleg et al., 2018). Since underestimations also occur in the continuous simulation if this station was not in the center of the storm, no areal reduction was carried out.

Snow accumulation and snowmelt are based on a threshold temperature and the degree day method. After snow storage, all precipitation and snowmelt enters the soil storage where actual evaporation is considered. Depending on the state of the soil storage, water is released to the upper groundwater layer from where surface runoff and interflow can occur. Both are controlled by a storage coefficient. Water from the upper groundwater layer can also percolate to the lower groundwater layer. The outflow from the latter represents the baseflow component. Surface runoff, interflow and baseflow are finally summarized and transformed via a triangular unit hydrograph. River routing is carried out via the Muskingum method. Further details about the model parameters can be found in Wallner et al. (2013) and in Table S2 in the Supplement.

For the calibration, the following runoff statistics are used: quantiles of the distribution functions fitted to the extreme values of (i) summer (Extr-Su, May to October) and (ii) winter (Extr-Wi, November to April), (iii) quantiles of the flow duration curve (FDC) and (iv) monthly averages (Q-mon). The calibration is carried out for each rainfall product separately, but for all 10 realizations at the same time (resulting in 1 parameter set for 10 realizations) The calibration procedure is also illustrated in Fig. S1.

For Extr-Su and Extr-Wi, a two-parametric Gumbel distribution is fitted to the annual series of extreme values. L moments are used for parameter estimation to reduce the sensitivity against outliers (Hosking and Wallis, 1997). Although extreme values only occur in a few time steps, their reproduction in the discharge time series is the main aim of the simulation on an hourly basis. However, since the extreme values only represent a small fraction of the discharge time series, FDC and Q-mon are also used to represent the more frequent discharge values. Q-mon accounts for the temporal dependency on the interannual variation of the discharge. The analyses of FDC and Q-mon allow no direct validation of the rainfall products, but enable an overall plausible simulation of rainfall–runoff processes. Hence, FDC and Q-mon are calculated from averaged daily discharge values in order to reduce computation time. For the goodness-of-fit analyses of simulated (Sim) and observed (Obs) statistics, the Nash–Sutcliffe-efficiency, NSE (Nash and Sutcliffe, 1970), is used. A perfect fit would result in NSE = 1, while assuming the average of the observed data for all time steps would result in NSE = 0. The equation for the NSE is given in Eq. (4) and the corresponding quantiles for Extr-Su, Extr-Wi and FDC and months for the Q-mon, respectively, are given in Eq. (5).

The goodness-of-fit values of all runoff statistics are summarized in the objective function O_stat, which should be minimized during the calibration:

For the optimization, simulated annealing is used. The parameters modified during the optimization with the corresponding ranges are given in Table S2. The periods for calibration and validation are listed in Table 5 for each catchment.

(b) WaSiM

WaSiM (Schulla, 1997, 2015) is a physically based and distributed hydrological model which has been designed to study climate change and land use change impacts on the water balance and floods in mesoscale catchments (e.g., Niehoff et al., 2002; Bormann and Elfert, 2010). WaSiM was formerly known as WaSiM-ETH, but has since been renamed (Schulla, 2015), and hence the new abbreviation is used throughout the paper. WaSiM is flexible regarding the resolution of spatial input data. In general, elevation, land use and soil data need to be prepared as gridded raster datasets. The spatial resolution of WaSiM applications covers several scales ranging from tens of meters to a few kilometers. For this study a spatial resolution of 150 m×150 m was chosen.

For the areal rainfall estimation, a combined inverse distance weighting and elevation-dependent regression approach is applied. This approach does not only account for a horizontal interpolation but also addresses the typically observed increase in precipitation with increasing elevation, which proves helpful given that the catchment spans an altitudinal range of several hundred meters.

A set of alternative hydrological process representations for each of the following sub-models is included in the model in order to cover different user needs and meteorological data requirements: (i) evapotranspiration, (ii) snow, (iii) interception and (iv) soil water. This list is not exhaustive since other processes can also be addressed using the model. Here, only the processes utilized in this study are described. Potential evapotranspiration is computed using the Penman–Monteith approach (e.g., Monteith, 1965), taking look-up tables of parameters defined for different land use classes into account. Seasonal snow cover dynamics is simulated using a temperature threshold for phase partitioning and a temperature index model for snowmelt calculations. A bucket approach is applied to consider interception of rainwater. The soil water dynamics including actual evapotranspiration, infiltration, lateral outflow (interflow) and percolation is simulated in a numerical scheme which is based on the Richards equation. The lowermost nodes in each grid cell, which are subject to saturation, represent the groundwater storage in the model. A linear storage approach is applied here to simulate the outflow from the groundwater.

Since WaSiM is more complex than HBV with respect to computational needs, a different strategy for model calibration was chosen. As the number of both adjustable parameters and iterations is limited due to limited computational resources, a lexicographical approach was set up for model calibration (Gelleszun et al., 2017). In this way, the optimization of parameters is divided into subsequent steps that are associated with different processes. In a first step, the parameters of the soil water balance and runoff generation (i.e., recession of hydraulic conductivity along the soil profile and the flow density) have been calibrated through maximizing NSE. Then, the baseflow recession is improved through minimizing the root mean square error of the lowermost part of the flow duration curve (two parameters). Both calibration steps have been performed using hourly meteorological time series and observed discharge time series from the period 2009–2012. As highly resolved meteorological observations are only available from 2000 onwards, an additional calibration step has been carried out using disaggregated rainfall time series in order to better match the long-term water balance characteristics through slightly modifying canopy resistance parameters of the evapotranspiration model. Without these pre-calibration steps an underestimation of the mean discharge and hence the water balance was identified. An incorrect representation of the water balance introduces other uncertainty sources, which hence superpose the effects of the different versions of spatial rainfall. However, this pre-calibration was only focused on the water balance itself and not on the objectives used in Eq. (6).

(a) HBV simulation results with calibration using three rain gauges as input

The parameterization was carried out by a split sampling technique with a calibration and validation period for each catchment. The results for Reckershausen, Pionierbrücke and Tetendorf are shown in Figs. 6, 8 and 9 for the calibration period. For Reckershausen, only results using three rain gauges as input are shown here. For Extr-Su and Extr-Wi, flood quantiles are shown for a return period of 100 years. However, the extrapolation is limited by the length of the simulated runoff time series. As per Maniak (2005), a maximum return period of 3 times the runoff time series length should be used to avoid statistical uncertainties that are too high, caused by extrapolation. This results in 75 years for Pionierbrücke, 21 years for Tetendorf and 45 years for Reckershausen. The discussion of the results is limited to these and more frequent return periods. For a quantitative analysis, NSE values for all criteria and for each catchment are given in Table 6. As mentioned before, NSE values are based on a few supporting points (see Eq. 5). Also, theoretical Gumbel distribution functions with two parameters are compared, which can be similar although the population of each distribution function used is different. Hence, values of 0.99 or even 1.00 can be achieved. On the other hand, small deviations from the observations can lead to even negative NSE values (see, e.g., the discussion of the simulation results for Reckershausen).

Table 6NSE values for all catchments and all criteria for calibration (Cal) and validation (Val) periods.

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Figure 7Runoff simulation results with HBV for Reckershausen, validation period.

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Figure 8Runoff simulation results with HBV for Pionierbrücke, calibration period.

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For Reckershausen, the Extr-Su and Extr-Wi are similar to those from observations (Fig. 6). While for summer all observed flood quantiles are within the range of Extr-Su ($\mathrm{0.99}\le \text{NSE}\le \mathrm{1.00}$), for Extr-Wi a slight overestimation occurs for V2 and V3.

For the validation period, flood quantiles for both Extr-Su and Extr-Wi are overestimated. The overestimation is higher in winter (approx. 20 m³ s⁻¹ for HQ₅₀) than in summer (approx. 10 m³ s⁻¹). One possible cause can be the higher yearly maximums in the calibration period. It is assumed that parameters, calibrated to achieve high floods, tend to generate larger discharges even if lower yearly maxima are observed. This is also indicated by the results for FDC and Q-mon. Although both are represented well in the calibration period ($\mathrm{0.88}\le {\text{NSE}}_{\mathrm{FDC}}\le \mathrm{0.90}$, $\mathrm{0.96}\le {\text{NSE}}_{\mathrm{Q}\text{-}\mathrm{mon}}\le \mathrm{0.99}$), both criteria are overestimated in the validation period ($\mathrm{0.57}\le {\text{NSE}}_{\mathrm{FDC}}\le \mathrm{0.63}$, $\mathrm{0.81}\le {\text{NSE}}_{\mathrm{Q}\text{-}\mathrm{mon}}\le \mathrm{0.89}$). In the validation period the range, and hence the uncertainty, for both Extr-Su and Extr-Wi, is smaller for V2 and V3 in comparison to V1.

Figure 9Runoff simulation results with HBV for Tetendorf, calibration period.

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The simulation results of Extr-Su of the validation period for the Reckershausen catchment show the sensitivity of the NSE as a goodness-of-fit criterion. V1 and V3 lead to positive NSE values (0.60 and 0.31), while V2 leads to a negative value of NSE $=-\mathrm{0.05}$. However, from a visual inspection (see Fig. 7), differences between all three approaches are small and less intense as one might expect from the NSE value itself. The high sensitivity of the NSE makes a direct interpretation of its values more difficult (Schaefli and Gupta, 2007; Criss and Winston, 2008). However, for the calibration process, a high sensitivity leads to an improvement of the simulation results.

Values for the objective function are given in Table 7. For Reckershausen, the objective function values are very similar for V1, V2 and V3 for both calibration and validation periods, especially by taking into account that the value for the objective function depends on four NSE values.

For Pionierbrücke it should be mentioned that at points during the calibration (see the FDC in Fig. 8) and validation periods, a simulated discharge of Q=0 m³ s⁻¹ was obtained. Zero discharge implies that all storages have been emptied. This only occurs for Pionierbrücke and is due to the very steep conditions in the mountainous catchment (see Fig. 1) and hence the low soil depth and storage capacity. In the observed time series the minimum value is Q=0.1 m³ s⁻¹. The underestimation is caused by the selection of criteria selected for the objective function used for calibration as well. The main aim is to represent the extreme flows, while the shapes of the intra-annual cycle of monthly average discharges and of the FDC are only implemented to achieve an overall realistic mean discharge behavior. For the FDC, four quantiles greater than 0.5 and only two quantiles smaller than 0.5 are used. Smaller quantiles are not of interest in these simulations, since discharge values in that range belong to dry periods with low flows, for which daily values of rainfall are sufficient for simulations and hence no rainfall disaggregation would be necessary. For the FDC, V3 leads to a slightly better fit to observations for non-exceedance probabilities smaller than 35 %, but to a worse fit between 35 % and 60 % non-exceedance probability. However, FDC is underestimated, independent of the applied rainfall product, for non-exceedance probabilities higher than 60 %. The underestimation identified by the FDC can also be identified for Q-mon in winter and in the underestimation of the Extr-Su and Extr-Wi. The results for the validation period are very similar and not shown here.

In contrast, for Tetendorf, FDC and Q-mon (except September and October) are overestimated by all rainfall products (Fig. 9). However, for Q-mon the shape of the intra-annual cycle is represented well. For the extreme values it should be mentioned again that the analyses are only valid for return periods more frequent than 21 years. For Extr-Su, underestimations occur for return periods more frequent than 5 years for all variants in the calibration period (less than 2 years in the validation period). For Extr-Wi, the median of V1 represents the observed values well, while for V2 and V3 the median leads to overestimations for return periods frequent than 5 years. However, observations are still in the range of the simulation results, whereby the range is wider for V1 and V3 in comparison to V2. In total, the resampling in V2 leads to a reduction of the overestimation of the observed summer extreme values, but to a stronger overestimation for winter extremes in comparison to V1 and V3.

Figure 10Bivariate spatial characteristics estimated for summer (S) and winter (W) seasons as well as over the whole year (Y).

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Since for Tetendorf seasonal differences regarding V2 were identified, the spatial rainfall characteristics of the objective function applied for the resampling process have been re-analyzed, differing between the summer and winter half years. The results regarding both periods as well as the estimation over the complete year are shown in Fig. 10 for all bivariate spatial rainfall characteristics based on all 24 hourly stations in Lower Saxony that have been used before for the estimation of these characteristics (Müller, 2016). For the continuity ratio, probability of occurrence and both volume classes of correlation coefficients, differences can be identified, based on the different geneses of rainfall in summer and winter. The probability of rainfall occurrence is lower in summer due to a higher number of convective rainfall events. However, the distance-dependent curve progression is very similar between the seasonal and annual estimated spatial characteristics. Since spatial characteristics are just moved closer to the regression line by V2 (without a perfect fit; see Fig. 4), an improvement of the spatial rainfall characteristics by introducing slightly different season-dependent regression lines cannot be expected and is hence not applied.

As main reasons for the seasonal differences, the short validation and calibration periods are considered. Short periods mean a small number of days with rain and hence a small number of relative diurnal cycles to swap during the resampling, limiting the ability of the algorithm to improve the spatial characteristics. The usage of time series of V2 as input for HBV and the additional short time for the calibration process lead to the seasonal differences.

For longer calibration and validation periods (Reckershausen and Pionierbrücke) the results for V1, V2 and V3 are very similar regarding the runoff statistics. An influence of the chosen method on the implementation of spatial consistence cannot be recognized.

Table 7O_stat values for all catchments and all criteria for calibration (Cal) and validation (Val) periods.

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Figure 11Runoff simulation results for V2 with three, five and eight rain gauges with HBV for Reckershausen, calibration period.

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(b) HBV simulation results' calibration using different numbers of rain gauges as input

A possible reason for the non-visible influence of the chosen method for the implementation of spatial consistence in the simulated runoff statistics is the low rain gauge network density. With a low network density, it is not possible to reflect the spatial rainfall variability, and hence the influence of V1, V2 and V3 cannot be identified. The influence of the spatial rainfall variability on the runoff can only be determined by rainfall–runoff simulations.

Therefore, for Reckershausen, different numbers of rain gauges are applied for the calculation of the areal rainfall used as input for HBV. Areal rainfall is estimated by three rain gauges (representing a network density of 0.9 gauges per 100 km²) as carried out in (a), five rain gauges (1.6 gauges per 100 km²) and eight rain gauges (2.5 gauges per 100 km²). The results are shown for V2 in Fig. 11 for the calibration and in Fig. 13 for the validation period. The results for V1 and V3 are very similar and not shown here. However, for a quantitative analysis the NSE and O_stat values are shown in Tables 8 and 9.

Figure 12Runoff simulation results for V2 with three, five and eight rain gauges with HBV for Reckershausen, validation period.

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Again, independent of the number of rain gauges used for the estimation of the areal rainfall, the results from the calibration period (Fig. 11) represent the observations better than those from the validation period (Fig. 12). In the validation period, Extr-Su and Extr-Wi are overestimated as well as the majority of Q-mon and the FDC. Minor differences can be identified between the different rain gauge network densities, but no general conclusion is possible; e.g., the overestimation of Extr-Wi in the calibration period is increasing with an increasing network density. However, in the validation period, the overestimation is decreasing with an increasing number of rain gauges from three to eight. Also for Q-mon or the FDC, no systematic improvement can be identified. This is an unexpected finding because with the additional information from the daily total rainfall amounts, an improvement of at least the continuum characteristics was expected. Also for the NSE and O_stat values, no systematical improvement can be identified: O_stat(V2, three rain gauges) = 0.03, O_stat(V2, five rain gauges) = 0.04 and O_stat(V2, eight rain gauges) = 0.03 (see Tables 8 and 9).

Table 8NSE values for all catchments and all criteria for calibration (Cal) and validation (Val) periods.

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Table 9O_stat values for all catchments and all criteria for calibration (Cal) and validation (Val) periods.

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It can be summarized that the number of rain gauges has only a minor but no systematic influence on runoff statistics for the catchments used in this study. This contradicts conclusions from other studies. Seliga et al. (1992) recommend information every 5 km² (20 rain gauges per 100 km²) for spatial rainfall applications. So an improvement by an increasing station density up to this threshold should have been expected. For a French catchment with an area size of 71 km², Obled et al. (1994) investigated the influence of using 5 or 21 rain gauges, representing rain gauge network densities of 7 and 22 rain gauges per 100 km². With 21 rain gauges Obled et al. (1994) improved their results significantly. Nevertheless, they conclude that the improvement is based on the better estimation of the total rainfall amount, not on its spatial distribution. Xu et al. (2013) investigated the influence of station density on a Chinese catchment with an area size of 94 660 km² and daily rainfall time series; hence a direct comparison of network densities is not possible. Nevertheless, they point out that the distribution of rain gauges inside the catchment is of importance. A distribution covering regions with different rainfall behaviors in a catchment can lead to better simulation results with only a few rain gauges in comparison to a less efficiently distributed network with more rain gauges. In the current study, the rain gauges for each network density scenario have been selected in a way that covers the catchment area and its rainfall representatively (see Fig. 2). This could be one reason why an increase in rain gauge network density shows no systematic improvement in this study.

Figure 13Runoff simulation results with HBV without calibration for Reckershausen, validation period.

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(c1) HBV simulation results without calibration using three rain gauges as input

Another possible reason for the small differences between V1, V2 and V3 is the calibration of the rainfall–runoff model parameters for each of the rainfall products. Parameters are allowed to vary between V1, V2 and V3, and hence damp the effects of the different degrees of spatial consistence. To exclude the calibration as a possible reason for the damping behavior, a calibration with a neutral rainfall product offering the same spatial rainfall coverage without giving preference to one of the investigated versions would be recommended. This would enable a direct comparison between V1, V2 and V3 without re-calibration of the models. Since high-resolution time series do not exist with the required spatial network density, radar data could be a possible solution. However, radar time series are too short for model simulations and subsequent derived flood frequency analyses.

To avoid recalibrations, a pragmatic solution is chosen: for each parameter, the arithmetic mean of the upper and lower bound for each parameter (as described by Wallner et al. (2013); see also Table S2) is utilized to form what is called a “default” parameter set. The default parameter set is independent of calibration and therefore observed rainfall data, which in turn might have stronger similarities to a certain rainfall product, and hence might introduce biases in the comparison of rainfall products. In this way, we do not attempt to provide highest accuracy through utilizing the default parameter set. Instead, we intend to provide reliable first guesses that do not favor V1, V2 or V3. The application of a default parameter set includes some shortcomings, e.g., regarding the physical interpretability, but it enables a comparison of the rainfall products.

For the validation period, simulation results based on this default parameter set have been analyzed. Although a splitting in calibration and validation period is not necessary if no calibration is carried out, comparisons are possible between the simulation results with and without calibrated parameters. The results are shown in Fig. 13 for Reckershausen; results are similar for Pionerbrücke and Tetendorf. For a quantitative evaluation, NSE values for all catchments are provided in Table S6 and O_stat values in Table S7.

Figure 14Runoff simulation results with WaSiM without calibration for Pionierbrücke, calibration period.

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For Pionierbrücke and Tetendorf simulation results are worse without calibration (e.g., for Pionierbrücke, V1: $O_{\mathrm{stat},\mathrm{not}\mathrm{calibrated}}=\mathrm{1.14}$ and $O_{\mathrm{stat},\mathrm{calibrated}}=\mathrm{0.21}$). For Reckershausen a slight improvement can be identified without calibration. In the validation period, the calibrated parameters led to an overestimation of extreme values for both seasons as well as an overestimation of FDC and Q-mon (e.g., for V3: $O_{\mathrm{stat},\mathrm{not}\mathrm{calibrated}}=\mathrm{0.28}$ and $O_{\mathrm{stat},\mathrm{calibrated}}=\mathrm{0.40}$). For all catchments, Extr-Su is underestimated by every version of spatial consistence. Extr-Wi is also underestimated for Reckershausen and Pionierbrücke, but overestimated for Tetendorf. For all catchments, an intra-annual cycle of Q-mon can be identified. For Reckershausen, Q-mon is similar to observations, while for Pionierbrücke underestimations can be identified and for Tetendorf overestimations can be identified in winter. The FDC is not represented well for any of the catchments. However, the results based on the default parameter sets provide feasible estimates of the hydrological response of the catchments without calibration. In this way, the default parameter set provides a possible way to compare different rainfall products without favoring one of them. As the model parameters are not representing the real behavior of the catchments, this procedure is a pure relative comparison between the rainfall products (V1, V2, V3) and not valid for a comparison between the simulation results and observed data.

Although a default set of parameters has been applied, the differences in the simulation results between V1, V2 and V3 are still small. For Pionierbrücke, the values of the objective function show the same range without and with calibration (1.10 (V2) $\le O_{\mathrm{stat},\mathrm{not}\mathrm{calibrated}}=\le \mathrm{1.14}$ (V1) or 0.21 (V1) $\le O_{\mathrm{stat},\mathrm{calibrated}}\le \mathrm{0.23}$ (V2, V3)). The similarity of the simulation results exists even if the model parameters are not calibrated and a default parameter set is used.

(c2) WaSiM simulation results without calibration using three rain gauges as input

For the comparison of V1, V2 and V3, WaSiM (Schulla, 1997, 2015) is used as an additional rainfall–runoff model. The application of more than one model increases the reliability of the simulation results and excludes the possibility of being model-dependent. As far as possible, the same parameter values as in HBV in the uncalibrated case (c1) have been applied. The investigation with WaSiM is carried out only for the Pionierbrücke catchment, since here the highest differences in simulation results are expected due to the short reaction time of the catchment.

Figure 15Runoff simulation results with WaSiM without calibration for Pionierbrücke, validation period.

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The results are shown in Fig. 14 for the calibration period and Fig. 15 for the validation period, and a quantitative analysis is given in Table 10. For the calibration and the validation period, Extr-Su and Extr-Wi are simulated slightly higher with V2 and V3 in comparison to V1. In addition, the range for both criteria is higher for V2 and V3 in comparison to V1, whereby V2 leads to even wider ranges than V3 in some cases (e.g., Extr-Win the validation period). This is consistent with the areal rainfall extremes presented for Pionierbrücke in Fig. 5. In this context it should be repeated that a relative comparison is carried out and under- or overestimations are not points of interest. The NSE values for both Extr-Su and Extr-Wi are very similar for V2 and V3 (e.g., NSE${}_{\mathrm{Extr}\text{-}\mathrm{Wi},\mathrm{Cal},\mathrm{V}\mathrm{2}}=\mathrm{0.98}$ and NSE${}_{\mathrm{Extr}\text{-}\mathrm{Wi},\mathrm{Cal},\mathrm{V}\mathrm{3}}=\mathrm{0.99}$), but show differences to V1 (NSE${}_{\mathrm{Extr}\text{-}\mathrm{Wi},\mathrm{Cal},\mathrm{V}\mathrm{1}}=\mathrm{0.90}$). Hence, in WaSiM a slight effect of the spatial consistence of rainfall is visible from the simulation results. Possible reasons for the differences are the spatial resolution (150 m×150 m for each raster cell). However, for FDC and Q_mon, values for V1, V2 and V3 are again very similar. While for the calibration period the O_stat values are similar for all rainfall products, in the validation period the O_stat values for V2 and V3 ($O_{\mathrm{stat},\mathrm{Val},\mathrm{V}\mathrm{2}}=\mathrm{0.45}$ and $O_{\mathrm{stat},\mathrm{Val},\mathrm{V}\mathrm{3}}=\mathrm{0.46}$) are much closer to each other than to V1 ($O_{\mathrm{stat},\mathrm{Val},\mathrm{V}\mathrm{1}}=\mathrm{0.30}$).

Table 10NSE and O_stat values for Pionierbrücke without parameter calibration using WaSiM.

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