This paper proposes a systematic assessment of the performance of an analytical modeling framework for streamflow probability distributions for a set of 25 Swiss catchments. These catchments show a wide range of hydroclimatic regimes, including namely snow-influenced streamflows. The model parameters are calculated from a spatially averaged gridded daily precipitation data set and from observed daily discharge time series, both in a forward estimation mode (direct parameter calculation from observed data) and in an inverse estimation mode (maximum likelihood estimation). The performance of the linear and the nonlinear model versions is assessed in terms of reproducing observed flow duration curves and their natural variability. Overall, the nonlinear model version outperforms the linear model for all regimes, but the linear model shows a notable performance increase with catchment elevation. More importantly, the obtained results demonstrate that the analytical model performs well for summer discharge for all analyzed streamflow regimes, ranging from rainfall-driven regimes with summer low flow to snow and glacier regimes with summer high flow. These results suggest that the model's encoding of discharge-generating events based on stochastic soil moisture dynamics is more flexible than previously thought. As shown in this paper, the presence of snowmelt or ice melt is accommodated by a relative increase in the discharge-generating frequency, a key parameter of the model. Explicit quantification of this frequency increase as a function of mean catchment meteorological conditions is left for future research.

Knowledge of the availability and variability of daily discharges in a given
stream section proves useful for many engineering applications (e.g., the
design of hydropower plants or water supply systems) and for studies
about stream ecology alterations and sediment transport or about water
quality and allocation

In hydrology, the probability distribution of daily discharges is
traditionally not represented as a probability density function
(pdf), but in
terms of flow duration curves (FDCs) that assign an exceedance probability
to each discharge value

Different methods exist to estimate FDCs (ie. to estimate their shape), the
most straightforward method being the assignment of empirical probabilities
to observed ranked data (yielding empirical FDCs;

An important category of FDC models are process-based models that combine
climate controls and catchment characteristics to estimate the shape of FDCs.
Such models describe the shape of FDCs either based on long-term simulations
of the system behavior or based on a direct parameterization of the FDC shape
as a function of key hydrological controls. One such model is the model
developed by

This modeling framework has been applied successfully for a range of case
studies in Italy

The original model framework was developed for rainfall-driven catchments
that show a linear recession behavior. Besides the aforementioned extension
to seasonally dry climates, the framework has namely been extended to
nonlinear recessions

In the previous applications of the model, the focus was generally on the
study of signatures of discharge regimes under different climates and
landscape conditions

The objective of this research is to assess and compare the performance of
the model in its linear and nonlinear forms for summer streamflows for a
range of Alpine discharge regimes. The selected set of case studies covers
all Swiss catchments that have a natural (unperturbed) discharge regime and
long-term discharge monitoring. Compared to existing studies

The paper is organized as follows: Sect. 2 provides a description of the analytical model, together with the methods adopted in this paper to estimate the model parameters and to assess the model performance, followed by a presentation of the Swiss case studies (Sect. 3). The obtained results for the linear and nonlinear model versions (Sect. 4) are discussed in Sect. 5 with a particular focus on the model performance under different hydrological regimes. The conclusions are summarized in Sect. 6.

Here, we first give a short overview of the used analytic modeling framework, followed by the two different methods adopted for parameter
estimation and for model performance assessment. All methods are applied only
to the summer season (1 June to 31 August; see also Sect.

Sketch of the adopted workflow for model parameter estimation and performance assessment.

The analytical modeling framework of

Annual cycle of discharge and air temperature for three selected catchments representing three different hydrologic regimes (pluvial, snow-dominated and glacier). The mean monthly values computed over the entire observation period for each catchment are shown (see Table 1).

It is assumed here that discharge

The overall rainfall forcing

Location of the case study catchments in Switzerland. The six biogeographical
regions of Switzerland

As discussed in detail by

Nonlinear storage–discharge relationships at the catchment scale are commonly
observed

Examples of the temporal variation of the model parameters over the
course of a year. The parameters are calculated for 90-day intervals beginning
at the calendar day for which the value is plotted; for a given time window,
the data points corresponding to this window in all available civil years are
pooled together. (

We use the term “forward parameter estimation” to emphasize that the
parameters are estimated directly from observed data, without calibration.
This method is generally used in the context of this modeling framework for
the estimation of the parameters related to the stochastic inputs
(

The computation of the precipitation parameters first involves the
computation of a reference catchment-scale precipitation time series (here
obtained from gridded data; see Sect. 3). Then interception losses (

Temporal variation of the residence time (

Instead of correcting the frequency of precipitation events

The recession parameter for the linear model is calculated directly from
observed daily discharge based on a classical Brutsaert–Nieber recession
analysis

Difference between

To objectively compare the potential of the linear and the nonlinear model
formulations to capture observed flow-duration curves, the recession
parameters for the linear model (

To objectively compare different models, we propose using the
Kolmogorov–Smirnov distance between the cdfs corresponding to different
models

Modeled cdfs with forward and inverse parameter estimation for the three selected catchments. The shaded area is located between the cdf envelopes and represents the natural variability of the daily discharges.

This comparison of the cdfs overcomes an important limitation inherent in the
comparison of analytic pdfs and empirical pdfs. In fact, the choice of the
number of classes for the calculation of the empirical pdf from observed data

Since the nonlinear model formulation has an additional parameter, the linear
and the nonlinear models are also compared based on the Akaike information
criterion

Based on the above criterion, we measure the relative performance increase
from the linear to the nonlinear model as follows:

As Fig.

In addition, to assess the performance difference between different models,
the obtained models are compared to the natural variability of the observed
discharge cdfs. Therefore, an empirical long-term cdf is constructed,
obtained by ranking the observed data in ascending order and dividing the
rank numbers by the total sample size. Furthermore, to assess the natural
yearly variability, individual cdfs are constructed for each summer season of
each civil year

Performance of the linear model and nonlinear model as a function of mean
catchment elevation. The performance measure shown,

In this paper, we analyze 25 Swiss catchments with areas
ranging from 1.05 to 377 km

Characteristics of Swiss case study catchments as given in the
FOEN database, including the FOEN identification code (ID), the catchment
name, the Swiss coordinates of the gauging station, the drainage area, the mean
elevation of the catchment and the gauging station elevation, the percentage of
glacier-cover of the catchment, the mean annual precipitation, the mean annual
temperature and the period of data acquisition. The 16 regime classes of

The average precipitation at the country scale is around 1300 mm yr

Relative increase in the performance of the nonlinear model with respect to
the linear model (as measured by

Most Swiss discharge regimes show a strong seasonality

It is noteworthy that surface runoff processes definitely can play a certain
role in extreme events in all regions of Switzerland

Besides observed daily discharge, the model requires catchment-scale daily
precipitation as input. Most of the previous applications of the models used
precipitation from one or several meteorological stations as input

Corresponding catchment-scale average precipitation time series per case
study catchment are obtained by averaging the daily precipitation time series
of all pixels contained in the catchment (a list of pixels per catchment is
included in the Supplement). In addition, we also used the
corresponding gridded temperature data set

Before estimating rainfall frequency (

To gain further insights into the hydrological processes underlying the
different regimes, Fig.

The inverse of the linear recession coefficient

Parameter values and performance indicators for all the catchments for
summer with linear model and forward estimation, summer linear model and inverse
estimation, summer nonlinear model and forward estimation, winter nonlinear model
and inverse estimation and winter linear model and forward estimation.

All estimated parameters for both forward and inverse estimations are
summarized in Table

The exceedance of

The cdfs obtained from all estimated parameters are presented in Fig.

Overall, there is a strong increasing trend of the linear model performance
with mean catchment elevation (Fig.

The inverse estimation of the model parameters improves the results
significantly, but the

The results obtained from inverse parameter estimation for the nonlinear
model are very good (Fig.

It is noteworthy that the two catchments for which the performance increase in the nonlinear model over the linear models exceeds 20 % are the two catchments that have a strongly karst-influenced regime (Scheulte at Vicques and Venoge at Ecublens).

As for the linear model, the forward estimation mode gives worse results
than the inverse estimation mode. For some catchments (i.e., Murg-Wängi,
Gürbe, Sense, Ilfis, and Grosstalbach), the forward estimation mode gives
nevertheless very good results with

Our results show that analytical modeling framework for
streamflow distributions proposed by

In forward estimation mode, the linear model outperforms the nonlinear model for catchments with summer high flows; the nonlinear model outperforms the linear model for catchments with rainfall-driven regimes (i.e., summer low flows). This results from the fact that for regimes with summer high flow, the linear model overestimates the recession timescale (resulting in a underestimation of the discharge variance). For regimes with summer low flow, the linear model in exchange underestimates the recession timescale. Given that the nonlinear model yields longer recessions, the nonlinear model shows accordingly a better performance for regimes with summer low flow.

The comparison between the forward and inverse estimation methods shows a
clear underestimation of

Overall, good model performance in many different catchments with
different regimes indicates that the modeling framework is suitable for the
prediction of FDCs in Switzerland. A more detailed temporal model
validation

Recomputing their results with their model parameters yields slightly
different

The most remarkable result of the presented analysis is the fact that the
modeling framework is applicable in its original formulation to catchments
where summer flow is influenced by snow processes. The additional source of
water from snow or ice melt is accommodated by increasing the frequency

Furthermore, the increase in the discharge-producing frequency to account for snow or
ice melt is also coherent with the original description of the
analytic modeling framework, which incorporates losses as a decrease in the
discharge-producing frequency. This type of behavior can be identified in
previous studies. For the Sitter at
Appenzell,

As can be seen in Fig.

It should be kept in mind here that for the present study,

Besides the important result that the model is applicable to snow-influenced
catchments, additional insights can be obtained from the highlighted model
performance trends with mean catchment elevation (Figs.

This application of the analytic framework of

The detailed comparison between the performance of the linear and the nonlinear model formulation shows that the description of Swiss summer flows strongly benefits from using a nonlinear storage–discharge relationship, in particular for catchments with summer low flow and for the karst catchments. In general, the linear model performance increases for increasing total summer flows or, equivalently, for catchments with higher mean elevation. Future work will focus on improving the model parameter estimation directly from observed data (without parameter optimization), which is a precondition for parameter regionalization. Better insights into the physical drivers of the different parameters will also open new potential for extending the model framework to all four seasons for all Swiss streamflow regimes.

The meteorological data used are not currently freely
available; they require a license from MeteoSwiss. The discharge data are
available upon request from the Swiss Federal Office for the Environment
at

The supplement related to this article is available online at:

The authors declare that they have no conflict of interest.

The work of the first author is funded by the Portuguese Science and Technology Foundation (FCT), grant no. PD/BD/52663/2014. The work of Bettina Schaefli is funded by the Swiss National Science Foundation (SNSF), grant no. PP00P2_157611. We thank the editor, Fabrizio Fenicia, and two anonymous reviewers for their detailed comments on an earlier version of this manuscript. Edited by: Fabrizio Fenicia Reviewed by: two anonymous referees