Although the consequences of floods are strongly related to their peak discharges, a statistical classification of flood events that only depends on these peaks may not be sufficient for flood risk assessments. In many cases, the flood risk depends on a number of event characteristics. In case of an extreme flood, the whole river basin may be affected instead of a single watershed, and there will be superposition of peak discharges from adjoining catchments. These peaks differ in size and timing according to the spatial distribution of precipitation and watershed-specific processes of flood formation. Thus, the spatial characteristics of flood events should be considered as stochastic processes. Hence, there is a need for a multivariate statistical approach that represents the spatial interdependencies between floods from different watersheds and their coincidences. This paper addresses the question how these spatial interdependencies can be quantified. Each flood event is not only assessed with regard to its local conditions but also according to its spatio-temporal pattern within the river basin. In this paper we characterise the coincidence of floods by trivariate Joe-copula and pair-copulas. Their ability to link the marginal distributions of the variates while maintaining their dependence structure characterizes them as an adequate method. The results indicate that the trivariate copula model is able to represent the multivariate probabilities of the occurrence of simultaneous flood peaks well. It is suggested that the approach of this paper is very useful for the risk-based design of retention basins as it accounts for the complex spatio-temporal interactions of floods.

The design of flood retention basins for small catchments usually focuses on
the runoff of the main channel. The relevant analyses involve both the peak
runoff and the corresponding flood volume. Both characteristics are
important parameters, which decide if a flood detention measure is
sufficient to meet the required protection targets. Consequently, both
variables need to be examined together. The use of bivariate copulas offers
one possibility to do this (e.g. Favre et al., 2004; De Michele et al., 2005).
However, the design of technical flood detention is getting more
complex with increasing area of the watershed and the related branching of
the river network. Because of the superposition of flood peaks of different
sub-basins or adjoining catchments, the risk of flooding and overloading of
storage (-systems) may increase downstream which increases the complexity of
the analysis. In addition, the distribution of precipitation and the
watershed-specific processes have a huge impact on the resulting flood
waves. The associated probability of the peak coincidence can be quantified
by multivariate statistics. The more tributaries the river network consists
of, the more variates have to be taken into account by the models. Copula
models are suitable to account for a number of variates. Copulas in the
context of flood coincidence have been used by Wang et al. (2009). They
utilized bivariate Frank-copulas to generate pairs of peak discharges of
nearby gauging stations. Kao and Chang (2012) demonstrated the influence of
dams on the time series of runoff of coincidence-affected sites using
bivariate Gauss-copulas. Chen et al. (2012) presented copula models of
higher dimensions. They chose the 4-D-Gumbel-copula to estimate the
probability of simultaneous floods at the Yangtze River in China and the
Colorado River in the United States. In addition to the peaks, they modelled
the timing of the flood events. Ghizzoni et al. (2010, 2012) go a step
further concerning the dimensionality. Applying the

This paper presents a case study that adopts copulas to represent the superposed peak discharges of three adjoining catchments. We use trivariate Archimedean copulas as well as pair-copulas to estimate the multivariate return periods of historical flood events, and compare the different copula types. The results suggest that joint return periods are indeed able to represent the spatio-temporal flood patterns within the river basin in a meaningful way.

The conventional flood statistics analyses the univariate probability that
the peak value

Composition of Pair-Copulas.

Another possibility to construct copula models of higher dimensions is the
use of pair-copulas (Joe, 1996; Aas et al., 2009). They are based on conditional
bivariate copulas, which can be coupled by the concept of vines (Bedford and
Cooke, 2002). Figure 1 shows the popular D-Vine. It is easy to imagine how a
D-Vine can represent a network of three joining rivers. According to Joe (1996)
the marginal conditional distribution functions are derived as

Position of gauging stations Wechselburg, Lichtenwalde and Nossen in the Mulde catchment.

Flood events in large river basins are composed of the contributions of a
number of tributaries. The main intention of this case study is to estimate
the variation of multivariate probabilities of these flood peaks and to
compare the results of the two differing copula models. The case study
illustrates the methodology for the Mulde catchment in eastern Germany,
where the three streams Zwickauer Mulde, Zschopau and Freiburger Mulde merge
(Fig. 2). We analysed the time series from the respective gauging stations
Wechselburg (area: 2107 km

Bivariate correlation coefficient of the coinciding peak values at the gauging stations Wechselburg, Lichtenwalde and Nossen.

In the next step, marginal distribution functions were estimated for the three univariate samples. The peak discharges at Wechselburg and Lichtenwalde were described by a log-Weibull distribution, those at Nossen by a generalized Extreme Value Distribution.

Randomized trivariate samples of 1000 elements overlayed with the observed simultaneous peak values, 1st row: trivariate Joe-Copula, 2nd row: Pair-Copula (HQ stands for peak discharge).

The fitting of several trivariate Archimedean copulas via the pseudo-likelihood method (Genest et al., 1995) showed the best goodness-of-fit for the Gumbel-Hougaard and Joe copulas. Because of the better performance in the test of Genest and Rivest (1993), we finally chose the Joe-copula for the statistical model of coinciding flood events. The superposition of a copula-generated trivariate sample and the observed flood peaks in the first row of Fig. 3 shows that the choice of Joe-copula is justified. The scatter plot reproduces the shape of the measured data and their interdependencies. As can be seen from Eq. (4) we also need some bivariate copulas for estimating multivariate return periods. Therefore, we reduced the 3-D-model to the three possible bivariate cases. The type of copula and the parameterisation were retained for consistency.

The construction of the pair-copula followed the D-Vine composition. The
gauging station Lichtenwalde served as the linking variate (variable

Return periods based on trivariate Joe-Copula and Pair-Copula for the simultaneous peak discharges at the sites Wechselburg, Lichtenwalde und Nossen for selected flood events; the last column shows the univariate return periods at the gauging station Golzern based on the official flood statistics. The results of the multivariate model are highlighted in bold font.

Basin wide flood events always differ in the spatial distribution of the
runoff contributions. Therefore, the multivariate probabilities differ, even
if the runoff below the confluence (here gauging station Golzern) is similar
between flood events. Table 2 specifies the peak values of the last three
extreme flood events in the river basin. This shows the relations among the
events and, by including the catchment areas, the corresponding core area.
So the event in August 2002 had its focus especially in the eastern part of
the catchment whereas eight years later the focus was clearly in the western
part. Using both copula models we estimated the corresponding return
periods. In addition, we determined the univariate return periods of the
resulting runoff in Golzern by use of the official local gauge statistic.
The table indicates that, overall, the multivariate return periods are
higher than the univariate ones. This is because the copula models include
the probabilities of the individual catchments and their combination whereas
the univariate statistics only relates to the total runoff downstream of the
confluence. The spatial composition of the flood peaks are not part of the
univariate distribution function. The flood event of August 2010 is a case
in point. About 75 % of the total runoff originates in the
catchment of the Zwickauer Mulde. This spatial heterogeneity can not be
considered in the univariate flood statistic for the gauge Golzern where the
peak value was 697 m

This study shows that both multivariate copula approaches estimate very similar return periods. This indicates that both of them can be adopted for the multivariate statistical assessment of flood events in large river basins. Although the trivariate Joe-copula only has one parameter, it seems not to be worse than the pair-copula, at least not in this application. In addition, the effort of estimating the return periods via 3-D-Archimedean copula is minor. However, the pair-copula should provide better fits to the data because of its more detailed structure and because of considering conditional bivariate dependencies. The copula-generated random samples in Fig. 3 demonstrate that this is the case. The scatter plots generated by the pair-copula show a less distinctive variation in the lower range than the 3-D-Joe-copula.

The application of trivariate copula models shows that they are able to estimate the multivariate probabilities of the occurrence of simultaneous flood peaks. They quantify the dependencies of the variates among each other and, consequently, capture the probability of infrequent spatial combinations of extreme events in the resulting return periods. Because of this they provide an suitable instrument for the spatial assessment of flood events within a river basin. The generation of random samples from the copula models suggests that the pair-copula gives a better fit to the data than the trivariate Archimedean Copula because of its smaller variation. The shape of the trivariate distribution seems to be reproduced more realistically. Flood design applications may benefit from this property.