In climates with a warm and a cold season, low flows are generated by different processes, which violates the homogeneity assumption of extreme value statistics. In this second part of a two-part series, we extend the mixed probability estimator of the companion paper

In seasonal climates with a warm and a cold season, low flows are generated by different processes so that the annual extreme series will be a mixture of summer and winter low-flow events. This can violate the basic assumption of extreme value statistics and give rise to inaccurate conclusions. In the first part of the study

In this second part of the study, we address the problem when summer and winter minima are not completely independent. This can play a role in low-flow statistics, where the events have a long timescale and may last for several months or even some years

The use of multivariate distributions for extremes dates back to

Copula models have recently received increasing attention in hydrology. Most examples are for flood frequency analysis but some examples for droughts and low flows also exist. A good overview of copula theory with respect to hydrological applications is given in

In a low-flow hydrologic context, bivariate analysis of volume and duration of below-threshold events was performed by

While copula approaches have been used to combine either different event characteristics (e.g. volume and duration) in a local analysis or the same event characteristic at multiple sites, we are not aware of any study that has used copulas to account for seasonal dependence in a mixed distribution approach to modelling the frequency of extreme events. In this second paper, we propose such a mixed copula estimator based on annual and seasonal minima series to conduct low-flow frequency analysis for mixed seasonal regimes. The aims of the paper are as follows:

to formulate an enhanced mixed distribution approach for minima that incorporates seasonal correlation,

to review its behaviour and plausibility for archetypal catchments,

to evaluate its possible performance gain compared to the mixed probability estimator,

to give a recommendation of which estimator to use under specific hydrological conditions.

Let us consider the case where the annual minima series is composed of events that arise from different processes occurring in the summer and winter season. Here, the annual minima series AM can be viewed as the minima of the annual summer minima AM

The multiplication rule is just a special case of a more general problem of frequency analysis where we are concerned with finding the joint occurrence probability of events in two variables. This can be solved by bivariate frequency analysis. Here, a bivariate probability model is used to estimate the joint probability that

The approach requires the choice of a copula model. As we are modelling extreme values, a Gumbel–Hougaard copula may be an adequate choice. The model can be written as

The model can now be used to estimate the occurrence probability of a low-flow event by inserting its magnitude

In the same way as for the theoretical estimator, we can also define an empirical estimator that generalizes the mixed probability approach to the case of seasonal correlation.

Overview of the example gauges used for model demonstration. Shown are the station identifier (ID), catchment area (AREA), Spearman's correlation between summer and winter event series (

Let

Like any empirical probability estimator,

As in the companion paper we use archetypal examples to demonstrate the behaviour of the model. The examples used now differ not only in terms of the strength of seasonality, but also in terms of seasonal correlation, ranging from insignificant correlations to highly correlated summer and winter events. An overview is given in Table

The first example is the Ebensee gauge at river Langbathbach, representing the case of almost no seasonality (

Probability plots of summer (red), winter (blue), annual (grey), mixed (green), and mixed copula (magenta) distributions for archetypal cases:

The second example is gauge Weg at river Isen situated in low, hilly terrain in Bavaria, Germany, representing the case of weak summer seasonality (

The third example is gauge Trausdorf an der Wulka at river Wulka situated in the eastern foreland of the Alps in Burgenland, Austria. It represents the case of a medium-sized lowland catchment (

The fourth and final example is gauge Schalklhof at river Schalklbach representing a medium-sized alpine catchment (

On the whole, the examples suggest that there is little difference between the two mixed estimators in the lower part of the distribution, so using the copula-based approach should have little effect on the estimated return period of an event. Seasonal correlation is observed to affect mainly the upper part of the distribution, which is less relevant for low-flow analysis. Although the differences between the two mixed probability estimators tend to be small, they increase with increasing correlation and may become relevant for highly correlated cases. In the following, we assess the possible gains of the mixed copula estimator based on a comprehensive pan-European data set.

The model is evaluated based on a pan-European data set. We use the data selection of

The study area covers large parts of western, central, and northern Europe and represents a great diversity of hydrological regimes. The diversity is clearly reflected by the low-flow seasonality patterns in Fig.

Location of gauges and main features of the study: seasonal dependency measured by Spearman's

As the copula family is a priori unknown, we screened a number of alternative families and compared them using a copula information criterion. Here we used the function xvCopula of the R package copula to compute the cross-validated log likelihood of

A similar procedure as in the companion paper is used for model evaluation, which is briefly described below. Again, we assume that the mixed copula estimator is superior to the simpler models, since it is a valid generalization of the annual and the mixed probability estimator (Sect. 2.1). From this point of view, we call the change in performance of the superior model compared to the inferior model a

In the first analysis, we assess the performance at the continental scale to learn about the relevance of the enhanced estimator in different regions. The evaluation focuses on the change in return period when using a seasonal probability estimator instead of the conventional annual estimator. This change, or

Consistent with

In the second analysis, we assess the gain of the mixed copula estimator over the mixed probability estimator. For this direct comparison of the two mixed distribution approaches we use their respective deviation

In the third analysis we want to gain deeper insight into the conditions under which the mixed copula approach improves the mixed probability estimator. For this purpose the influencing variables accounting for the difference in both estimators are examined using multiple regression. The predictors considered include the catchment area (AREA), the baseflow index (BFI), and the recession constant as defined in

Overview of hydrologic predictors used in the regression model.

Expecting the estimator to also depend on the strength of seasonality, we include the two seasonality indices described in Sect.

All predictors have been screened for possible nonlinear correlations using the same procedure as in the companion paper, with a significant difference between linear and rank correlation used as a symptom of nonlinearity. From this preliminary assessment, we have found log-linear relationships of the catchment area and recession constant and decided to use the (decadic) log-transformed variables

To test their significance, we use the common Student's

The data analysis was performed in R

We first assess the performance gain of the mixed copula estimator compared to the annual estimator based on the non-cumulative and cumulative distributions of relative deviances across Europe (Fig.

Uncertainty of the annual probability estimator compared to the mixed copula estimator. Full line with blue shaded area refers to the 100-year event. For comparison the 20-year event (dotted line) and the 2-year event (dashed line) are shown. Single outliers

Relative deviation (rd) of the mixed copula estimator from the annual probability estimator for a moderate low-flow event with

The companion paper has inferred from the Austrian study area that mixed distribution approaches should be relevant for a wide range of regimes, and it is now interesting to assess this finding at the pan-European scale. Figure

Table

Relative deviation

Relative absolute deviation (rad) of the mixed copula estimator (

In the second assessment we evaluate the relative difference in the two mixed distribution approaches in greater detail. Figure

Relative deviation (

Interestingly, the spatial patterns of Fig.

In a deeper analysis we assess the variables controlling this additional performance gain at low return periods (

We first analyse the predictors as main effects. This is a simplified assessment as interdependence of the predictors is not covered, and latent interactions may be misclassified as main effects. Nevertheless, the assessment is useful to give the first indication of variables that have some predictive value. The resulting model was carefully checked for overfitting and was found to be well determined. The summary statistics presented in Table

Multiple regression summary statistics (main effect model) of the relative deviation (

The same procedure was subsequently used to fit a model with main effects and two-fold interaction terms. Care had to be taken for possible overfitting, as the full model had high VIF values (up to 59) for all predictors. The model was therefore subjected to a stepwise variable elimination, which reduced the model by most of the insignificant predictors, thereby resulting in VIFs that were considered acceptable. As the fit improved and overall significances did not change, we found the full model well justified. The model has an

Multiple regression summary statistics (main effects and two-way interaction term model) of relative deviation (

Figure

Term plot of multiple regression (main effects and two-way interaction term model) of relative deviation (

It is interesting, finally, to assess the spatial patterns of the effects at the pan-European scale (Fig.

Effect size of multiple regression (main effects and two-way interaction term model) of relative deviation (

This second paper of a two-part series presents the mixed copula estimator for low-flow frequency analysis. The method provides an extension of the mixed distribution approach of the companion paper

As a starting point, the distributional characteristics of the mixed copula estimator were assessed by frequency plots of archetypal examples representing contrasting low-flow regimes. Differences in the mixed estimator were always observed in the upper part of the distribution, which, however, is less relevant for low-flow frequency analysis. At severe low-flow events, the differences are very small. For mild to moderate events that fall between these cases, the differences between the two mixed distribution approaches are determined by the interaction between seasonality and seasonal correlation.

In the subsequent step, the performance gain was evaluated based on a larger pan-European data set. The patterns match well with the findings from the exemplary catchments. There is generally little difference between the mixed distribution approaches for severe low-flow events. However, for mild low-flow events the differences are large. For the

We then assessed the performance gain of the mixed copula approach over the mixed distribution approach in greater detail. The analysis shows that the gain for the

Multiple regression analysis suggests that the performance gain of the mixed copula over the mixed distribution estimator can be well explained by the seasonal correlation and catchment-related characteristics. We can interpret the model such that the copula estimator is indeed effective in reducing the underperformance of the mixed probability estimator caused by seasonal correlation. However, the impact of this correlation (and thus the potential of the mixed copula estimator to improve the estimate) is modulated by interactions with seasonality and storage. Effect maps show the strong antagonistic nature of seasonality and seasonal correlation at the European scale, with weak seasonality leading to a high potential for correction and strong seasonal correlation reinforcing the need to take this potential into account. These results are well in line with findings from the exemplary catchments and generalize them to a wide range of European regimes. Storage and catchment size only have a minor effect on the performance gain of the mixed copula estimator and appear to play a subordinate role. This should also be the case for other catchment characteristics, such as soil properties, vegetation, and terrain, which tend to have more influence on surface processes and fast components of the water balance than on long-term storage and thus less influence on the redistribution of water over time.

Although the differences found are high on relative terms, they mostly occur at low return periods. This raises the question of whether it is practical to use the more complex copula-based estimator or whether the simpler mixed distribution approach will suffice. Some indications were already given in the Results section, where regions with large differences and hence a stronger correction were identified. These relate to mid-mountain regions in cold and temperate climates where rivers have strongly mixed low-flow regimes. In these regions, the use of the mixed copula estimator always appears to be beneficial.

Implications of estimation uncertainty for drought event mapping, illustrated for the European cases of 2003

Another aspect of practical relevance is the impact of uncertainties on a Europe-wide drought mapping, which is essential for the assessment of major events. Such mapping is often based on return periods that are traditionally obtained by the annual minima approach. This was also the case in our own study of the European drought of 2015

Despite all the favourable features of the mixed copula estimator, it should be emphasized again that the difference between the two mixed distribution approaches is not large and mainly concerns the low return periods. It is therefore unlikely that severe events in their entirety, i.e. in terms of their spatial extent and severity, will be misclassified by the mixture distribution estimator. Differences are more likely at the regional level where drought conditions may be overlooked. Such regional view, however, is vital for drought monitoring, where the statistically more reasonable concept of the mixed copula estimator should allow for a more accurate assessment of drought severity.

Overall, we conclude that the two mixed probability estimators are quite similar, and both are conceptually more adequate than the annual minima approach. In regions with strong seasonal correlation, the mixed copula estimator appears most appropriate and should be preferred over the mixed distribution approach.

Details on streamflow data collection are given in the cited source:

The author has declared that there are no competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Data provision by the Hydrographical Service of Austria (HZB) was highly appreciated. I am grateful to Tobias Gauster for providing a dedicated R environment for European drought mapping. This research is a contribution to the UNESCO IHP-VIII FRIEND-Water programme. I would like to thank the editor and two reviewers for their valuable comments on the manuscript.

This research has been supported by the Climate and Energy Fund under the programme ACRP (grant no. C265154).

This paper was edited by Manuela Irene Brunner and reviewed by Poulomi Ganguli and one anonymous referee.