Many hydrological studies are devoted to the identification of events that are expected to occur on average within a certain time span. While this topic is well established in the univariate case, recent advances focus on a multivariate characterization of events based on copulas. Following a previous study, we show how the definition of the survival Kendall return period fits into the set of multivariate return periods.

Moreover, we preliminary investigate the ability of the multivariate return period definitions to select maximal events from a time series. Starting from a rich simulated data set, we show how similar the selection of events from a data set is. It can be deduced from the study and theoretically underpinned that the strength of correlation in the sample influences the differences between the selection of maximal events.

Studying extremes in hydrological multivariate time series often aims at getting an estimate of the size of events to be expected in a period of 10, 50 or 100 years. This information is relevant for the construction of many hydrological structures such as dams and dykes. As most of these natural events are characterized by several variables (e.g. peak discharge, volume, duration, …) and several locations, it is important to understand their dependence structure and which constellations result in an extreme event. Copulas allow to flexibly model the dependence between the variables and add different marginal distribution functions to build a probabilistic multivariate model. The natural ordering in univariate time series does not extend to the multivariate case calling for different tools to identify multivariate extremes.

In a previous study

Currently, multivariate maxima are often selected based on a single driving variable (e.g. peak discharge) and the associated variables (e.g. volume and duration) are studied in a multivariate setting. However, this does not a priori reflect the joint extreme characteristic that is the actual focus of such a study. Different notions of maximality can be defined following the above return period definitions. These allow to calculate the empirical joint extremeness and to select the maxima of multivariate time series.

In this paper, we will only briefly quote the key concepts. The interested
reader is referred to the predecessor of this paper,

The driving tool underlying the multivariate return period definitions are
copulas. Copulas are multivariate distribution functions defined on the unit
hypercube. Based on Sklar's Theorem

In order to extend our previous study, we use the same data (simulated using
the COSMO4SUB model,

Figure

Comparison of different MRP definitions for a return period of 10
years (compare Figure 6 in

In the previous section and study, the annual maxima were selected based on
the maximum peak discharge and the volumes were only the corresponding, but
not necessarily maximal ones. An alternate approach can be taken either based
on the empirical copula or the adoption of multivariate distributions. For
these, the same MRP definitions can be applied as quoted above and the
largest values per year can be selected. In the following, we will follow
this avenue and investigate the differences between these approaches where
the copula

Comparison of the annual maximum value for the four different definitions. Left: based on 500 years of simulated rainfall data. Right: moderately correlated sample of a Gumbel copula.

For ease of notation, we will stick to bivariate events. We say that an event

To study the impact of the aforementioned definitions, we use a second run of 500 simulated years of 5.625 min resolution discharge data that were aggregated to separate rainfall events. This is different from the previous data set where only annual maxima have been used. This second data set contains 12 466 events. In order to reduce the effect of autocorrelation within this simulation, we only consider a random subset of 50 % of the data (autocorrelation plots indicate an uncorrelated time series, not shown here). We do not fit any parametric family, and solely use the empirical definitions of the above equations.

In our simulated data set, the largest event in a year often is the same for
all four definitions. This is not too surprising, considering that there are
on average less than 25 rainfall events in each year. What remains different,
is how extreme the event is for each of the four notions. The left plot of
Fig.

Identifying the single rainfall events and looking into the marginal distributions, visually reveals identical histograms for peak discharge as well as for volume for the four bivariate and respective univariate maxima selections. An overlay shows only very little variations for discharge values and volumes. Larger values of the margins tend to even better coincide.

As this data set follows a very strong correlation, we draw a sample of a
Gumbel copula with a moderate Kendall's tau of

The SKRP yields the most reliable separation into safe (sub-critical) and dangerous (super-critical) events. Nevertheless, the selection of a single design event, as often required by subsequent studies, remains an open question. Here, we selected the most probable bivariate event, but any event along the critical layer separates the sub- and super-critical regions.

The differences between the four definitions of maximality were minor in the simulated rainfall time series, but this is also due to the very strong correlation. This strong dependence causes the copula to be close to the upper Fréchet-Hoeffding bound where all four definitions coincide. If all points lie close to the diagonal, there is no difference whether the critical layer follows the OR definition (enclosing the lower left rectangle), the SKRP definition (very sharply bend contour lines enclosing the lower left region), the KRP definition (very sharply bend contour lines excluding the top right region) or the AND definition (excluding the top right rectangle).

The temporal structure was not changed, only the dependence structure to investigate the effect. A less dry study area with much more rainfall events in a year will further influence the selection. However, the large extreme values appear to be extreme in each of the definitions.

Here we use the raw definitions of multivariate return periods, but an
alternative would be to investigate derived measures.

The investigated data set features a very long time series. Shorter time series might be more sensitive to changes of the maximum selection regime applied, as few events might have a strong influence on the selection of the marginal distributions. The influence of these outer properties needs to be further investigated. An avenue of future research is to consider the joint extremeness for the selection of extremes to be fed into a peak over threshold approach.