Spatial patterns and time distribution of central European extreme precipitation events between 1961 and 2013

2.1 Searching for extreme precipitation events

Unlike the precipitation totals, the return periods are suitable for the comparison of regions with different precipitation climatologies. We estimate the return periods from series of 1–10-day precipitation totals; the evaluation of multi-day precipitation totals allows us to optimize the duration t of the event. The generalized extreme value (GEV) distribution is fitted to the annual maxima of the totals and the return periods are calculated from that distribution. We set the maximum considered return period to a value of 1,000 years. The GEV distribution is frequently used in climatological or hydrological applications (Wilks, 2006), and it was found to be the most suitable distribution for the modelling of precipitation extremes in many regions of Czechia (Kyselý and Picek, 2007) and elsewhere in Europe (Fowler and Kilsby, 2003). We can estimate the GEV parameters by the methods of L-moments or maximum likelihood, which was applied in this study. The point return periods have to be interpolated, as the WEI works with a regular grid instead of station data. Using linear kriging, we interpolated rather the common logarithms of the return periods due to the exponential nature of the GEV (Müller and Kašpar, 2014). The resulting grid has a horizontal resolution of 10 km.

The calculation of the WEI is made by maximizing the variable E _ta. According to Müller and Kašpar (2014), E _ta is the product of the common logarithm of the spatial geometric mean G _ta of the return periods N _ti and the radius R of a circle whose area a is equal to the area summed up according to the number i of cells included in the geometric mean,

(1)

The cells enter Equation 1 sequentially in descending order (from the highest to the lowest return period). The variable E _ta increases when accumulating cells of high return periods, but it starts to decrease when the continuously expanding area does not compensate for the mean return period decrease (Figure 2). The inflexion point corresponds to the value of the WEI and relates to a certain size of the affected area a. Equation 1 can be applied to the return periods of each day. However, we did not calculate E _ta for days with return periods shorter than 2 years, due to the elimination of rarer precipitation episodes and to save the processing time.

Within a single event, the WEI changes when separately considering return periods of 1–10-day precipitation totals (t increasing from 1 to 10 in Equation 1). If the WEI continuously increases upon extending the event duration, the maximum value will be considered as the final absolute WEI (WEI_abs) describing the EPE with the affected area a and duration t. An example is shown for the May/June 2013 event in Figure 2, where E _ta curves are displayed up to 4-day durations of the event. In this case, the WEI_abs was reached in the fourth day, with an affected area of approximately 177,100 km². We again exclude multi-day values of the WEI when the single-day WEI is missing due to low return periods of the precipitation totals. The events are labelled according to their first day, which is also crucial for a seasonal classification of the events.

2.2 Comparing the events

In this study, the WEI is calculated on 3 different spatial levels: the study area as a whole, 5 basins, and 20 subcatchments (Figure 1). However, the values are hard to compare even on a single sublevel due to the various extents of the basins and the subcatchments. The standardization of the WEI is needed with respect to the highest theoretically possible WEI in the given area (WEI_max). The potential value of the WEI_max counts with t-day return periods of 1,000 years over all cells within the basin or the subcatchment of the surface area a. After basic operations transforming Equation 1, WEI_max is expressed as 3 times the radius R as follows:

(2)

The standardized value of the WEI (the relative WEI, WEI_rel) corresponds to the ratio of the absolute and maximum WEIs of the same event,

(3)

The difference between WEI_abs and WEI_max is demonstrated for three precipitation events by diagrams in Figure 3, where one can compare how the WEI_rel was derived for central Europe and the individual basins. A specific difference between the values of WEI_rel and WEI_abs can be found in the example of Figure 2, with E _ta and WEI (both absolute and relative) displayed on the spatial levels of central Europe and one subcatchment. To unify the results, we use only WEI_rel for all spatial levels. We primarily select 53 maximum central European events detected at the first spatial level—one event per year on average. They are analysed with respect to the areal precipitation extremity when we mainly focus on comparing the temporal and spatial distributions of the events. The method of directional statistics (Black and Werritty, 1997) is used for displaying the events’ seasonality in a circle representing 1 year. The method transforms the day of the EPE occurrence into directional vectors in that circle.

2.3 Spatial similarity of events

The EPEs seems to repeat over time in accordance with recurring atmospheric circulation conditions (Hofstätter et al., 2017; Jacobeit et al., 2017). We expect a similarity in the spatial distribution of the precipitation and employ a clustering method to check the similarity and to find groups of analogous EPEs within the 53 maximum events.

As we do not know the number of the groups and their initial membership in advance, we use an agglomerative hierarchical clustering procedure, which merges the closest single- or multiple-member groups hierarchically according to a chosen measure of distance. Finally, only one group remains to contain all members (Wilks, 2006). The method seems useful for finding regional spatial patterns of weather and climate extremes (Gibson et al., 2017). Differences appear in defining an intergroup dissimilarity, which is commonly based on the Euclidean distances between points or clusters (Wilks, 2006). Among the various approaches for hierarchical clustering, Ward's method seems to be the most accurate (Gong and Richman, 1995), as it minimizes the within-group sum of the squared distances over all created groups.

Using Ward's method, we want to cluster the EPEs with respect to the spatial distribution of the precipitation extremity. The aim is to minimize the sum of the squared distances between the merging EPEs and the centroids of the resulting groups. The degree of similarity is assessed based on the subcatchment WEI_rel (sWEI_rel) during the 53 maximum events. For each subcatchment s, the value of the sWEI_rel is standardized and expressed as a proportion (pWEI_rel) of all sWEI_rel values summed for an individual event e as follows:

(4)

The standardization of the sWEI_rel seems to be appropriate due to differences in the variable means and variances (Gong and Richman, 1995). The final distances are determined from a numeric matrix with rows representing individual EPEs and columns indicating dimensions, that is, values of the pWEI_rel during the 53 maximum events. This means that the closest events form clusters in 20-dimensional space (as the number of the subcatchments). A final number of five groups is then set in to keep a reasonable quantity of events in each cluster. At the previous hierarchical level, one group would remain with only a single event.

3.1 Extreme precipitation events between 1961 and 2013

The calculation of the WEI_rel for the whole study area determines a ranking of the EPEs; the 53 maximum central European events are presented in Table 2. The methodology allows us to detect EPEs lasting up to 10 days, but in fact, the events are much shorter, usually between 2 and 4 days. Events longer than 5 days are very rare, with a maximum detected duration of 8 days. Generally, the duration of the EPEs slightly increases with the increasing WEI_rel, but the seven most highly ranked EPEs still last only up to 5 days. Moreover, there are two time-concentrated precipitation events among them (August 2002 and 1978), with only 2-day durations. In 1978, the event was associated with a high precipitation intensity (24-hr precipitation over 100 mm) mainly in northern Bohemia and the eastern part of Germany (Marx, 1980). The 2002 EPE was accompanied by even heavier precipitation, with a maximum of 312 mm within 24 hr measured at Zinnwald (Germany) on 12 August (Ulbrich et al., 2003).

To demonstrate the spatial patterns of the precipitation during the events, Table 2 also presents the 10 maximum values of the WEI_rel within the five river basins constituting the study area (Figure 1). Apart from only one Weser/Ems event, all other basin-related maximum events belong to the 53 central European EPEs. Eight of the top 10 central European EPEs are at the same time among the 10 maximum events in two or even three river basins (the latter case corresponds to the No. 1 EPE in July 1981). However, the representation of the basin-related maxima among the central European EPEs is very unequal. While the maximum events of the Elbe, Danube, and Oder basins are almost always highly ranked according to the total WEI_rel, the top 10 events of the Rhine and Weser/Ems basins do not appear very often among the highly ranked events. This particularly applies to the rather small Weser/Ems basin, whose precipitation extremes are isolated from the others, apart from three events, which coincide with the Rhine events. Nevertheless, the maximum events from July 2002 in the Weser/Ems basin and October 1986 in the Rhine basin did not have large responses in the other basin. Both events affected quite large areas of their own basins (82% in the Weser/Ems and 66% in Rhine basin), while the mean return periods were just 4 and 7 years, respectively. On the other hand, the Elbe, Danube, and Oder maximum events had mean return periods between 25 and 28 years, with the maximum in the Oder basin during the August 1985 event. The affected area reached 55% of the basin extent, as did the maximum Danube event in July 1981. In the case of the Elbe maximum event in August 2002, the affected area was slightly smaller, with 51% of the Elbe basin covered (see Figure 3 for a comparison of the 1981 and 2002 events in individual basins). Although the percentages are lower than for the Weser/Ems and Rhine basins, the maximum events of the Danube, Elbe, and Oder basins are always highly ranked in the other basins (Table 2). These facts suggest a significant difference in the nature of the maximum events between the Rhine and Weser/Ems EPEs on one hand and the EPEs of the other three river basins on the other hand.

The final division of the basins into the smaller subcatchments (Figure 1) enables the analysis of even detailed spatial patterns of the events, presented in Figure 4b. It reveals a large variability of the sWEI_rel for the individual EPEs. Nevertheless, some EPEs seem to be very similar from the viewpoint of their spatial patterns. To study these relations objectively, a cluster analysis (section 2.3) was performed using the sWEI_rel values as the criteria.

3.2 Clusters of events with respect to their spatial patterns

Figure 4a presents the results of the clustering process. Sharp differences are evident between the two main groups of events, which were generated at the next to last stage of the agglomerative clustering. The “Western” (W-CE) and “Eastern” (E-CE) groups of the central European events roughly refer to a border of the main drainage basins. The upper (Alpine) Rhine catchment (Rhine-a subcatchment) is an exception, as it belongs more frequently to the E-CE group otherwise limited to the Elbe, Danube and Oder events. On the other hand, the Saale catchment (Elbe-c) often stays with the W-CE group, which may also apply to the upper non-alpine Danube catchment (Danube-a) in some cases. However, the spatial variability of the precipitation extremes is still high within each of these groups (Figure 4b). Therefore, a finer division of the EPEs was employed with respect to the clustering dendrogram.

The W-CE events can be clearly divided into two groups: (a) EPEs affecting mainly the northwestern part of the study area (NW cluster)—the Weser, Ems, lower Rhine, and middle and lower Elbe and (b) EPEs of the non-alpine Rhine (R cluster), with only a small contribution of the upper Danube catchment. The group of the E-CE events consists of three distinct groups: (c) the Danube (without the Morava catchment) and upper Rhine events (RD), (d) Elbe-Danube events (ED), in which the Elbe is represented mainly by its Czech and Saxon parts, and (e) Oder events (O), when adjacent parts of the Elbe basin and the Morava catchment can also be hit. These five clusters were further considered. In Figure 5, each cluster is represented by the maximum EPE, when both the return periods and t-day precipitation totals are depicted and compared with the median return periods derived from all events merged in the respective cluster. It demonstrates the fact that the maximum events were often anomalous in rarity and extent, too.

Within the ED cluster, there are 6 of the top 10 central European EPEs, including the absolute maximum EPE of July 1981 and the more recent events of August 2002 and May 2013 (Table 2). The core area of the cluster is spread from the Alpine tributaries of the Danube (Inn, Enns, etc.) over Bohemia (the western part of Czechia) into Saxony, but individual events also affected parts of the Oder and Upper Rhine basins, as it was, for example, in July 1981 (Figure 5b). As a result, the cluster contains 6 of the 10 maximum events in both the Elbe and Danube basins, but only 2 events and 1 event of the 10 maxima in the Oder and Rhine basins, respectively (Table 2).

The other four central European EPEs of the top 10 belong to the O cluster. Three of these events were very similar with respect to the spatial pattern of precipitation when they also affected the lower part of the Danube and the Czech part of the Elbe River basins (Figure 4b), as it was, for example, at the beginning of July 1997. Nevertheless, the core area of the cluster shifts to the northwest (Figure 5c). Apart from 8 of the 10 maximum events in the Oder River basin, the cluster also contains four and three of such maxima in the Elbe and Danube River basins, respectively (Table 2).

The remaining three clusters contain fewer highly ranked EPEs. The RD cluster, the third one complementing the E-CE group of events, is rather small with only six EPEs. The core area of the cluster is limited to the Alps in both the Rhine and Danube River basins. Only its maximum event from March 2002, which substantially exceeded the core area (Figure 5e), was large enough that it belongs to the 10 maximum Danube events (Table 2). In contrast, the R cluster contains as many as 9 of the 10 maximum Rhine events (Table 2), when the core area of the cluster lies in the middle part of the Rhine basin (Figure 5d). However, the EPEs of the R cluster also often affected the Weser/Ems River basin (Figure 4b). As a result, the cluster contains 3 of the 10 maximum events there. The rest of them belong to the NW cluster (Table 2), with a rather small core area covering mainly the Weser and the adjacent parts of the Elbe River basin (Figure 5a).

3.3 Spatial extent of extreme precipitation events

The NW cluster generally includes less extensive events; the largest one did not exceed 100,000 km² (Figure 6a). It is comparable with the RD events; on the contrary, the extent of the events varies a lot for the other clusters. Especially in the case of the ED and O clusters, the events are distributed unevenly within Figure 6a, but they often have a larger extent and longer return periods of precipitation totals. This suggests the exceptional character of the highly ranked EPEs, which only arise from an interaction of high return periods and large extents of events. Within the top 10 EPEs, at least 15% of the areas were hit by precipitation of the 9-year average return periods. During the No. 1 case of July 1981, the average precipitation return period of 19 years affected approximately 30% of the study area. It resulted in a much higher WEI than for the No. 2 event in July 1997, which was characterized by a very high average return period (28 years) and the affected area reaching 17%. Finally, the No. 3 (May 2013) event was the most extensive, with 31% of the area hit, but the average return period was only approximately 11 years (see also Figure 3 for the comparison of the three maximum events). The remaining clusters contain events with lower WEI_rel values, which often result from a combination of high/low return periods and a local/large extent of the event (Figure 6a).

Although the affected area of the individual EPEs reaches up to 31% (Figure 6a) of the total area of central Europe (as defined in Figure 1), it is not always continuous, but pronounced relations in precipitation are evident between distant areas due to similar orientations of mountain ranges, which enhance precipitation on their windward side. As a result, precipitation often occurs simultaneously in the Elbe-a and Danube-b subcatchments (Figure 7), which are both located north of mountain ranges (the Bohemian Forest and the Alps, respectively) but are otherwise adjacent only marginally (Figure 1). However, the orography not only connects distant subcatchments but also divides the neighbouring ones—as it does, for example, in the case of the Elbe-b and Elbe-d subcatchments, which, despite their proximity, have a relatively low value of Spearman's correlation coefficient between the values of the sWEI_rel (Figure 7). The Ore Mountains between subcatchments acts as a barrier to airflow, causing the Elbe-b to lie in the rain shadow. The disparity is even greater between the Elbe-b and Elbe-c subcatchments (with a correlation coefficient close to 0). On the other hand, the low correlation between the Danube-c and Elbe-a is rather related to the narrower precipitation zone during the events caused by Vb cyclones (Müller et al., 2009).

High positive correlation values occur especially among the Rhine (except for the Rhine-a) and Weser/Ems or the Oder and Danube-c subcatchments (Figure 7), when they do not necessarily need to be adjacent. However, there are some other highly correlated neighbouring subcatchments within the Elbe (Elbe-a and Elbe-b; Elbe-c and Elbe-d), Danube (Danube-a and Danube-b; Danube-b and Danube-d) or two adjacent river basins (Danube-a and Rhine-a), etc.

Surprisingly, we also noticed many negative correlations, which means that the occurrence of precipitation in two subcatchments is often mutually exclusive. In this context, we can separate two groups of subcatchments, which have mostly negative correlations with the members of the other group. It roughly corresponds to the division of the EPEs into the two main clusters, W-CE and E-CE (Figure 4a). The strongest negative correlations appear between the Oder, Danube-c, Elbe-a, and the non-alpine Rhine and Weser/Ems subcatchments (Figure 7), although there exist negative correlations within individual basins, too: e.g., the Elbe-a and Elbe-c or the Danube-a and Danube-c, which is apparently connected to the rather narrow precipitation zone producing the EPE.

3.4 Spatial patterns in the seasonal distribution of events

Central European EPEs are distributed through the whole year but with higher concentrations in the warm half-year (from May to October) and mainly in July and the first half of August (Figure 8a). This is especially true in the case of highly ranked EPEs: all of the top 10 EPEs occurred in the warm half-year, 7 of them in the second half of July or the first half of August (Table 2). In contrast, only seven cold half-year events (from November to April) were detected at all.

The cold half-year events are only limited to the NW, R and RD clusters (Figure 8a), when they have typically lower return periods, and at the same time, the affected area is comparable to others or even smaller (Figure 6b). That is why the resulting values of the WEI_rel are quite small for these events. The situation is mostly similar to that of events occurring in May with few exceptions. The May 2013 event had an abnormal spatial extent; it occurred at the end of May, but it remains exceptional when comparing it to June events (Figure 6b). In contrast, two other May EPEs (1970 and 1978) with WEI_rel values above 0.1 had limited spatial extents, but they deviated from the average due to high return periods; both were concentrated mainly in the Rhine basin (Figure 4b) and classified into the R cluster.

The summer EPEs (JJA) are quite varied; the combination of relatively long return periods and large spatial extents often results in high values of the WEI_rel. However, it applies mainly to July and August EPEs (Figure 6b), while June events have rather lower WEI_rel values, including also the June 1993 event, which is located in the upper left corner of Figure 6b, indicating its local character and very high average return periods. The summer EPEs form the majority of the ED and O events, which both have clear seasonal patterns—occurring from May to September and the beginning of August, respectively (Figure 8a). Generally, the affected area is roughly the same for the summer and the rest of the year, but the return periods of the precipitation totals during the summer events overtop the others (Figure 6b).

In the autumn (SON), lower return periods are evident, especially within October events, when the affected area is approximately equal to or greater than that in the winter and spring. The autumn EPEs almost exclusively belong to the R cluster except for three September EPEs (Figure 8a).

Among the top 10 EPEs in individual subcatchments, the number of autumn events decreases towards the east, when it is the highest in the lower Rhine River basin (Figure 8c). In contrast, the upper and middle Rhine is the only area with two or more winter EPEs among the top 10 in subcatchments. The event seasonality is quite various in the Rhine River basin as well as in the Danube basin (Figure 8c). This feature is not very obvious in the Elbe or Oder River basins, where the seasonal distribution of the top 10 EPEs is similar across all subcatchments and their respective basins, too (Figure 8b). In the case of the Oder basin, a high concentration of events in the summer months is more or less apparent at both spatial levels.

3.5 Spatial patterns in the inter-annual distribution of events

During the period between 1961 and 2013, the 53 maximum central European EPEs tend to occur in accumulations, whether lasting years or just a few weeks (Figure 9). The first 16 years of the study period (1961–1976) are characterized by a relatively low number of EPEs. Only six events were recognized, and all of them occurred in the warm half-year. These years are completely without greater events in the Danube and Weser/Ems basins, where EPEs first appeared in summer 1981. This year, we have recorded four central European events, including the absolute maximum event, over a relatively short period of 2 months (Figure 9b). Approximately a decade of wetter conditions ranges from 1977 to 1986, when we identified a large number of events with a high overall WEI_rel as well as the basins WEI_rel (Figure 9a).

This is in contrast with the 1990s, when many events occurred, but their WEI_rel values were much smaller; the two exceptions both appeared in July 1997 (Figure 9b). The period around the 1990s is, however, unique in the occurrence of cold half-year events, which only occurred between 1986 and 2002. This applies to the central European area and the individual basins, too. The period is somewhat longer in the case of the EPEs in the Rhine subcatchments (Figure 9a), but still, the cold half-year events occurred only from 1978 to 2003. The last period from 2002 to 2013 has, again, a large number of events with higher overall and basin WEI_rel although without any greater Rhine basin event. Many smaller accumulations of events occurred in this period: e.g., July/August 2002, August/September 2007 (Figure 9b) and May/June 2013.

An interesting shift in the seasonality is evident when looking at the top 10 EPEs in certain subcatchments. Only summer events occurred in the subcatchments (except for the Rhine-a to Rhine-d) until the mid-1980s (Figure 9a). In contrast, there has been no summer EPE in the Weser/Ems-a subcatchment since 1984 and in the Danube-a and Danube-b subcatchments since 1992.