A hydrological impact analysis concerns the study of the consequences of certain scenarios on one or more variables or fluxes in the hydrological cycle. In such an exercise, discharge is often considered, as floods originating from extremely high discharges often cause damage. Investigating the impact of extreme discharges generally requires long time series of precipitation and evapotranspiration to be used to force a rainfall-runoff model. However, such kinds of data may not be available and one should resort to stochastically generated time series, even though the impact of using such data on the overall discharge, and especially on the extreme discharge events, is not well studied. In this paper, stochastically generated rainfall and corresponding evapotranspiration time series, generated by means of vine copulas, are used to force a simple conceptual hydrological model. The results obtained are comparable to the modelled discharge using observed forcing data. Yet, uncertainties in the modelled discharge increase with an increasing number of stochastically generated time series used. Notwithstanding this finding, it can be concluded that using a coupled stochastic rainfall–evapotranspiration model has great potential for hydrological impact analysis.

Precipitation is the most important variable in the terrestrial hydrological
cycle that determines soil moisture and discharge from a watershed. As such,
it also impacts water management where generally the occurrences of extreme
events, e.g. storms or droughts which have very low frequencies, are of
concern. Hence, very long time series of precipitation are needed. Because
this kind of data is not always available, one may consider using a
stochastically generated rainfall time series

Besides precipitation, the water balance is also highly influenced by the
amount of water that is lost due to evapotranspiration. An accurate
estimation of evapotranspiration is essential for hydrological and
agricultural designs, irrigation plans and for water distribution management

Many modelling approaches exist for simulating catchment discharge. The
simplest models are the conceptual models in which several linear (or nonlinear)
reservoirs are put in series and/or parallel. Well-known examples of such
conceptual models are the following: the Hydrologiska Byräns Vattenbalansavdelning
model

Section

This study uses observed time series measured in the climatological park of
the Royal Meteorological Institute (RMI) at Uccle, near Brussels, Belgium.
The data include time series of observed precipitation (mm) from 1898 to 2002,
and mean daily temperature

In order to use the above-described data to fit copulas, the data should be
independent and identically distributed (iid), indicating that the
distribution of the data should not change with time. To this end, the time
series is split into monthly series to which a vine copula model can be
fitted. Hence, for each month a different model will be obtained. However,
the data distributions can also change within the monthly series, i.e. a
within-month trend may exist. Therefore, the daily distributions, each
containing 72 observations, were compared within each month by means of an
ANOVA test when distributions were homoscedastic, a Welch ANOVA
test

General model structure of the PDM (adapted from Moore, 2007).

The PDM is a lumped rainfall-runoff model which basically conceptualizes the
absorption capacity of soil in the catchment as a collection of three
different storages (

In this study, PDM is calibrated for the Grote Nete catchment using the
particle swarm optimization algorithm (PSO;

A copula is a multivariate function that describes the dependence structure
between random variables, independently of their marginal distributions

The use of copulas allows us to decompose the construction of a joint
distribution function into two independent steps, i.e. the modelling of the
dependence structure and the modelling of the marginal distribution functions

Examples of four-dimensional vine copulas:

A flexible construction method for high-dimensional copulas, known as the
vine copula construction, has been introduced in the work of

There is, however, a large number of possibilities for the construction of
vine copulas

Construction of C-vine copula

In order to generate stochastic time series of evapotranspiration, we make
use of the vine-copula-based approach proposed in the work of

Further, the White goodness-of-fit test

Bivariate copula families selected by AIC for

Comparison between the probability density functions of
evapotranspiration of observed and simulated values: Uccle is in red and the
ensemble of 50 time series simulated using the C-vine copula

The construction of

Comparison between Kendall's tau for the relations of

Temperature data are required for the stochastic modelling of
evapotranspiration. However, in situations where no long-term time series of
temperature is available, it is necessary to use a stochastically generated
temperature time series. We use a similar approach as

Bivariate copula families selected by AIC for

The construction procedure of

Comparison between the probability density functions of the monthly
mean

Comparison between the return periods of monthly extremes of the
observed and simulated temperature values: Uccle is in red and the ensemble of 50
time series simulated using the C-vine copula

To assess the performance of the model, the statistics of 50 stochastic time
series of temperature using the observed daily precipitation from 1931 to
2002 are compared to those of the observations. The empirical probability
density functions of the monthly mean temperature for each of the simulated
72 year time series are shown in Fig.

Optimal parameter set for the (monthly) MBL (modified Bartlett–Lewis) model.

Comparison between observed and simulated precipitation data for the mean, variance, autocovariance and zero-depth probability (ZDP): Uccle is shown by the blue triangles and the ensemble of 50 simulated time series by the MBL model is shown by the box plots.

Comparisons between the probability density functions of mean,
variance, autocovariance and ZDP calculated for the observed and simulated
precipitation data for different aggregation levels for each year: Uccle is in
red and the 50 simulated time series by the MBL model is in grey. Densities are shown
for the

Comparisons between the return periods of extremes for the observed and simulated precipitation data at different aggregation levels: Uccle is in red and the ensemble of 50 simulated time series by the MBL model is in grey. Calculation of the extremes for a given return period on a time series that is based on concatenating the 50 simulated time series results in the blue line.

Different cases for discharge simulation.

In situations where no long time series of precipitation is available, one
can use a stochastic rainfall model. In this study, the modified
Bartlett–Lewis (MBL) model

Comparison between the probability density functions of the
reference discharge

Comparison between the probability density functions of observed
temperature time series (red) and the ensemble of simulated time series of
temperature values (grey) using the C-vine copula

Comparison between the probability density functions of observed
evapotranspiration time series (red) and the ensemble of simulated time
series of evapotranspiration (grey) using the C-vine copula

In Fig.

Figure

Figure

Comparison between the empirical return periods of annual extremes
of the observed and simulated discharge for all cases. Reference discharge

Comparison between the empirical return periods of annual extremes of the observed and simulated discharge for case 3 based on 50 time series of 3600 years of rainfall and corresponding evapotranspiration.

Comparison between the probability density functions of the mean of
discharge of the observed and simulated values in three cases. Reference
discharge

Root mean square difference (RMSD) for simulated discharge in different cases: case 1 (red), case 2 (blue) and case 3 (green). The RMSD is plotted against the cumulative relative frequency of the discharge given by the empirical cumulative distribution (ECDF) value.

The catchment discharge is calculated by the PDM that uses precipitation and
evapotranspiration data as inputs. In order to assess the impact of each
stochastic variable on the modelling of discharge, three cases have been
developed that can be compared to a reference situation
(cf. Fig.

The catchment discharge can be simulated by means of the PDM that uses
precipitation and evapotranspiration data. In case 1
(cf. Fig.

In case 2 (cf. Fig.

As shown in Sect.

This case accounts for a situation in which no time series (of sufficient
length) are available as shown in Fig.

First, the simulated time series of precipitation are used as inputs to the
C-vine copula

Compared to the simulated discharge of cases 1 and 2, even higher extreme
values are generated and the grey areas representing the ensemble of 125 000
time series are generally wider, indicating that mainly the stochastic
generation of precipitation has introduced considerable variations into the
discharge simulations. The top row of Fig.

In order to further investigate the quality of the simulated discharge for
all cases, Fig.

To account for the variations between the modelled and reference discharge,
the simulated discharge values are further evaluated using the root mean
square deviation (RMSD):

Figure

In water management, discharge is a very important variable which can be
simulated via a rainfall-runoff model using recorded precipitation and
evapotranspiration data. However, in situations that suffer from data
deficiency, one may consider using stochastically generated time series. In
this study, the impact of using the stochastically generated precipitation
and evapotranspiration on the simulation of the catchment discharge is
investigated. In order to assess the influence of each stochastic variable on
the discharge simulations, three different cases have been considered. In the
first case, it is assumed that insufficient evapotranspiration data would be
available, requiring stochastically generated evapotranspiration based on
observed precipitation and temperature data by means of a copula. In the
second case, where only precipitation data would be sufficiently available,
the temperature and evapotranspiration are each reproduced by vine copulas.
The third case addresses the situation where too short time series of
observations are available. In this case, the precipitation time series could
be generated using an MBL model calibrated to the
limited precipitation data available and then the time series of temperature
and evapotranspiration could be obtained using the copula-based models. In
all cases, C-vine copulas

With respect to extreme discharge, it was shown that the uncertainties encountered in case 3 are partly caused by the limited length of the time series used. The uncertainties in the predictions are highly reduced when input time series are used that are much longer than the maximum return period aimed at. As in this particular case, all forcing data are generated, the modeller is not restricted to the length of an observed time series, and can hence generate time series of whatever length as input to the hydrological model, taking into account that the longer the time series used, the more the uncertainty reduces at the expense of increasing runtime.

From this study, we may conclude that in situations that suffer from a lack of observations, one can rely on the stochastically generated series of precipitation, temperature and evapotranspiration to reproduce time series of discharge for water resource management. However, care should be taken as the modelled extreme discharges may experience the largest errors.

All data are property of the Royal Meteorological Institute (KMI) of Belgium and are obtained on an exclusive basis. We thank Patrick Willems for providing the evaporation data.

The authors declare that they have no conflict of interest.

The authors gratefully acknowledge the Vietnamese Government Scholarship (VGS), the King Baudouin Foundation (KBF) and the project G.0013.11N of the Research Foundation Flanders (FWO) for their partial financial support for this work. The historical Uccle series were provided by the Royal Meteorological Institute of Belgium. Edited by: Carlo De Michele Reviewed by: Thomas Nagler and Mojtaba Sadegh