Characteristics of extreme precipitation in the Vosges Mountains region (north-eastern France)

2.1 Study area

The study area (Figure 1) comprises of VG and covers Alsace, major part of Lorraine, and some parts of the Franche-Comté regions, north-eastern France. VG culminating at the Grand Ballon (1424 m a.s.l.) are characterized by hilly foreland, relatively gentle western slopes, and steep eastern slopes dipping to the Upper Rhine Plain at an altitude of 200 m a.s.l. (Gley, 1867; Alsatia, 1932; Ernst, 1988; Sell, 1998). Despite various microclimates, the temperate oceanic climate dominates at the western part and near the ridge of VG, and the temperate climate with continental features prevails in the Upper Rhine Plain (Sell, 1998; Météo-France, 2008).

The spatial distribution of precipitation is correlative to altitude and the prevailing westerlies from the Atlantic Ocean. The major precipitation differences are due to the almost perpendicular orientation of the mountain ridge to the dominant airflow direction (Sell, 1998; Météo-France, 2008). During 1960–2013, the highest mean annual precipitation total of 2329 mm was recorded at the Sewen-Lac Alfeld weather station (620 m a.s.l.) in the southern Vosges, 903 mm was recorded at the Rovillé weather station (278 m a.s.l.) on the windward side, and 599 mm at the Colmar–Mayenheim rain gauge in Upper Rhine River Plain due to rain shadow in the lee (Minářová et al., 2017).

2.2 Precipitation data set

Daily precipitation non-homogenized totals and metadata from 168 weather stations of the VG were analysed. The data covers the period 1960–2013 and were provided by Météo-France national meteorological network. Due to the fact that some of the time series included large discontinuities, only that data which covered more than half of the study period (i.e. 27 years) was analysed further. This criterion was met by data from 84 weather stations, and the remaining data were used for checking the interim results. The missing values in the new data set of time series from the 84 selected stations were not filled in by interpolation or extrapolation, and the data set was found sufficient for the subsequent analyses. Figure 2 shows the substantial progressive increase of daily data availability for the meteorological stations with time either due to the increase in the number of weather stations or due to the availability of digital precipitation records. The availability of daily precipitation totals has increased from 50–60% in 1960 to 90–100% in 1980 and onwards. Taking into consideration the spatial distribution of the selected 84 stations, the study area VG was adjusted in order to avoid the extrapolation of resulting spatial outputs (Figure 1).

The RHtests_dlyPrcp R-package (Wang et al., 2010; Wang and Feng, 2013), designed for testing the daily precipitation totals, was used to test the homogeneity of time series. The package is accessible at http://etccdi.pacificclimate.org/software.shtml. The computation includes the metadata of weather stations. In our case, 0.4 mm was selected as a suitable value for the error of data measurement in the test based on the estimated maximum error of the different rain gauges used for data measurement in our study area, despite the commonly used value of 0.2 mm for such analyses, which is also suggested by the WMO (World Meteorological Organization, 2008). Lower values (0.1, 0.2, and 0.3 mm) would have produced similar results, as documented by Minářová et al. (2017).

The test highlighted two non-homogenous time series recorded at the Aillevillers and Foucogney meteorological stations, which were adjusted according to the homogenization technique described by Wang et al. (2010) and Wang and Feng (2013). The mean of adjusted daily precipitation totals from both the stations is negligibly lower (in order of 10⁻² mm) than the equivalent of its raw data.

The non-zero daily precipitation totals were studied further, and the 1–10 days precipitation totals were assessed using the event-adjusted evaluation technique (Section 2.3) in order to select the EPEs. The limit of 10 days in the VG area was set based on the characteristics of mean precipitation in VG and the Czech Republic, and is in good agreement with the hydrological studies in the areas. For instance, van Pelt et al. (2014) stated that 10-day precipitation events in particular tend to result in flooding in the Upper Rhine River catchment.

2.3 Event-adjusted evaluation technique of weather and climate extremes

The event-adjusted evaluation method of weather and climate extremes (Müller and Kaspar, 2014) was applied in order to obtain a data set of EPEs and to perform comparison among events. This technique introduces the WEI, which quantifies the extremity of an event on the basis of three parameters, i.e. rarity, spatial extent, and duration of an event; all varying and combined in one single index. In the first step, the return periods of precipitation are estimated at individual sites for various time windows separately. Then the resulting point return period data are interpolated spatially, and in the third step that area and time window is identified in which the event has the maximum extremity, which is termed as WEI.

The technique starts by assessment of rarity, which is based on return period estimates of 1 to x-day precipitation totals (1–10 days in our case) at rain gauges individually. The return period was estimated using three-parametric GEV distribution that is widely used for analysis of heavy rainfall. The three parameters of the GEV were calculated based on precipitation annual maxima values by means of L-moments (Hosking and Wallis, 2005). Since such local analysis may create variations in the estimates of GEV parameters and high quantiles, the maximum return period estimate was set to 1000 years. To express the spatial aspect of weather extremity, the maximum return period estimates from individual gauges are not considered. Instead, the resulting rain gauge return period estimates from the gauges were expressed in their common logarithmic equivalents that were interpolated using ordinary kriging interpolation method into a regular grid of 2 × 2 km resolution. The interpolated logarithmic values were transformed back to return period estimates N during t days (i.e. 1–10 days) at grid points i. The values of grid points (N_ti) were sorted in decreasing order, since the area affected by an EPE can be discontinuous. The analysis starts at the grid point with the highest value of return period estimate N, and other grid points are added one by one according to the decreasing value of return period estimates, i.e. the area a increases with each addition of the grid point. The spatial geometric mean G_ta is calculated step-by-step for n grid points.

The WEI is defined based on the spatial geometric mean as follows (Müller and Kaspar, 2014):

(1)

where N_ti is the return period estimate in years at a grid point i for t days, and a is the area in km² comprising n grid points. The resulting E_ta is the indicator of extremity of a weather/climate event, and it corresponds to the multiplication of a common logarithm of the spatial geometric mean G_ta of return period estimates N_ti by the radius of a circle R in km whose area is equal to that delimited by the spatial geometric mean G_ta.

The Equation 1 implies that the maximum value of E_ta is considered. It corresponds to the inflection point of its curve which represents an optimized combination between rarity and affected area. In fact, at the beginning pixels of high return period estimates are accumulated and the area and E_ta increase inflection point of E_ta, when it starts decreasing since newly accumulated pixels are of low return period estimates and the decrease of the return period estimates prevails over the increase in the accumulated area a.

The final WEI corresponds to the first maximal E_ta among non-zero E_ta values computed for 1–10 days (t) overlapping events, starting from the duration of 1 day. All the 1-day E_ta values included in an event longer than one day have also to be non-zero values so that the daily precipitation totals within the event are all considerable as sufficiently significant, i.e. as extreme. For further details about the computation and reasons of WEI, we refer the reader to (Müller and Kaspar, 2014).

In contradiction to the widely used approaches for evaluating precipitation extremes (annual block maxima or peaks over threshold), the WEI consists of areal assessment of events – it enables to optimize and delimit the area affected by the extreme precipitation within a wider precipitation field.

Based on the highest WEI independent values (irrespective of their 1–10 days duration), we selected and further examined the first 54 EPEs in this study; one EPE per year of the study period.

2.4 Other data sets

Two catalogues of the weather types were used to analyse the synoptic conditions during the EPEs; a manual ‘Grosswetterlagen’ catalogue (GWLc, Werner and Gerstengarbe, 2010) and an automated SynopVisGWL-catalogue (SVGc, James, 2007; James, 2015; personal communication). Subsequently, a weather type was assigned to each EPE. For EPEs lasting longer than one day, the most frequent weather type during such EPEs was taken into consideration. If the weather types were of similar frequency during an EPE, the weather type assigned to the day of the highest 1-day E_ta value was considered.

Since the GWLc provides qualitative rather than quantitative information about synoptic situation during EPEs, the ERA-40 gridded reanalysis (2.5°a horizontal resolution) daily data (Uppala et al., 2005) provided by ECMWF for the study area (5°–10°E, 47.5°–50°N) at two isobaric levels (500 and 850 hPa) at 12 UTC were used to quantify synoptic conditions during EPEs that occurred during 1960–2010. The velocity of meridional and zonal airflow components was derived to provide information about wind direction during EPEs. Meridional and zonal flux of specific humidity was calculated since it was suggested as one of predictors of extreme large-scale precipitation by Müller et al. (2009).

The cartographical outputs were constructed in Esri's ArcGIS 10.3 software using a high resolution (100 × 100 m) global multi-resolution topography model obtained from GeoMapApp (http://www.marine-geo.org/tools/GMRTMapTool/) as base map.

2.5 Analysis approach

The three strongest EPEs and the EPE that affected the largest area in the VG were described in detail, i.e. their synoptic situation was analysed in conjunction with the precipitation totals and river discharges. The synoptic situation was described mostly based on National Centers for Environmental Prediction/National Center for Atmospheric Research (NCEP/NCAR) reanalysis data (Kalnay et al., 1996), and the data from Koblenz Global Runoff Data Centre (GRDC) was used to examine the river discharges.

The seasonality of EPEs was analysed according to the occurrence of the first day of event in meteorological seasons (e.g. spring for 1 March to 31 May), and a division between summer half-year (SHY) (from April to September) events and winter half-year (WHY) events (from October to March) was also derived from the first day of event. No influence of the selection of first day of event compared to the second, third until the last day was detected in the conducted sensitivity analysis. In order to shorten the terms, summer (warm) half-year events and winter (cold) half-year events are written as SHY events (SHY EPEs) and WHY events (WHY EPEs), respectively.

The resulting duration of events served to divide the EPEs between short and long. Various characteristics of short/long EPEs and SHY/WHY EPEs were studied: affected area, extremity (expressed by WEI), inter-annual changes, and synoptic conditions. The relationship between duration, affected area, and extremity was expressed through correlation coefficient at 1 and 5% p-value levels, and the covariance was also computed.

The inter-annual changes were examined using simple linear regression for different durations. The synoptic conditions were analysed based on the two GWLc and values of synoptic variables (Section 2.4). For the later, the daily means of the derived synoptic variables (meridional and zonal airflow components and meridional and zonal flux of specific humidity) were calculated in VG (i.e. six grid points), and the highest absolute values (i.e. minimum or maximum) of variables during EPEs were assigned to each EPE following Müller et al. (2009) and Kašpar and Müller (2014), who suggested that the anomalies are essential for heavy rainfall. We are aware that the non-availability of quantitative variables during 2011–2013 may influence our results. However, following Zolina et al. (2005, 2013) we consider the influence less significant, since 3 years represent less than 6% of the study period.

3.1 The three strongest EPEs

The maximum EPE (WEI = 120) started on 11 November 1996 and lasted 2 days (Table 1, Figure 3(a)). On 11 November, the highest daily precipitation total was recorded at the Bains rain gauge station (67.3 mm). On 12 November, even 68.6 mm total was measured at the Terre-Natale station situated not far away from the Bains station (the position of both stations is westward from the southern VG). The study area was under the influence of a stationary cold front separating warm and moist air over western Mediterranean and Central Europe from the cold air which earlier penetrated along the West-European coast up to Portugal. A strong temperature gradient in lower troposphere positioned below the front side of an upper-level trough remained for both days over the VG region, as it is obvious from NCEP/NCAR reanalysis data (Kalnay et al., 1996). As a result, heavy precipitation occurred mostly in the southwestern (SW) part of the VG and was not related to orography, which is rather typical for stationary cold front. Subsequent to this EPE, according to data from GRDC, a very strong increase in discharge generated a heavy flood on 14 and 15 November at the Moselle River with mean daily discharges of 1350 m³ s⁻¹ recorded at the hydrological station in Cochem. Flooding was documented in the Saône River Basin on 13 November in the villages of Monthureux-sur-Saône and Bourbéville, where house-marks can still be found (EPTB, n.d.).

The second EPE of nearly the same magnitude (WEI = 119, Table 1) started on 12 September 1986 and lasted 5 days. It affected larger area as compared to the 1996 EPE (68% of the study area, Figure 3(b)). Badonviller weather station situated west–north-west of the Middle VG recorded 61.2 mm on 14 September. As in the case of the 1996 EPE, a stationary cold front prevailed over the region. In 1986, westwards of the front, a trough was present at higher altitudes, and there was an advection of warm and moist air in the foreground of the front. Then shallow lows or frontal waves passed at the front interface and resulted in heavy precipitation in the region. The discharge significantly increased from 12 to 18 September at the Meuse River, Saar River, and mainly Moselle River, where the mean daily discharge increased from 33 to 681 m³ s⁻¹ in Perl, and from 86 to 927 m³ s⁻¹ in Cochem (GRDC).

The third EPE (WEI = 116) occurred on 17 September 2006 (Table 1). Among the three strongest EPEs it was the most recent, and due to its very short (1-day) duration it affected the least part of the study area (35%, Figure 3(c)). The highest daily rainfall total of 142.0 mm was recorded at the Padoux rain gauge (343 m a.s.l.) situated southeast of Nancy and north of Épinal. No strong pressure gradient was influencing the area that day, and according to the SVGc (James, 2007), the synoptic situation was classified to be low over British Isles. Nevertheless, a shallow trough was also situated over Germany, Alps, and northern Italy. The shallow low was present in the early morning of 17 September, and moved towards southeast during the day. The combined influence of shallow low pressure, dominant eastern airflow, and divergence at 300 hPa level in the study area suggests favourable conditions for an EPE. This can also be supported by 90% relative humidity at 700 hPa and intense vertical movements. In addition, convection might have played a role because such precipitation occurs frequently in autumn when the eastern airflow prevails in Central Europe (Tolasz et al., 2007). No orographical effect seems to occur in the third EPE as for the other two strongest EPEs. Though discharges at the Moselle River were not as high as in previous two cases, the increase was more rapid: between 17 and 19 September, mean daily discharge increased from 27 to 426 m³ s⁻¹ and from 67 to 538 m³ s⁻¹ in Perl and in Cochem, respectively (GRDC). A house-mark in the village of Darney demarcates local flooding in the Saône River Basin (EPTB, n.d.).

Overall, the three heaviest EPEs were of similar WEI magnitude, and affected the study area according to their duration, i.e. shorter EPE affected smaller part of the VG. The return period estimates of the three strongest EPEs were very short (if detectable by the WEI) in the VG, whereas the longest return period levels were mostly detected on the windward Lorraine side. This may suggest comparatively lower orographical influence during the events. The two strongest EPEs were not related to the expected strong zonal circulation but to stationary cold fronts.

3.2 Seasonal distribution of EPEs

The seasonal distribution of 54 EPEs in meteorological seasons (Figure 4) shows that the EPEs occurred in all seasons (9 EPEs in spring and winter, 15 EPEs in summer) but most frequently in autumn (21 EPEs). The autumnal predominance of EPEs matches with the seasonality of mean precipitation in the study area only on the windward side of the VG, where the autumn is the most humid season (Section 2.1). The seasonality can also be documented by similar representation of SHY and WHY EPEs with 30 SHY EPEs found in the data set of 54 EPEs (Table S1, Supporting information).

The seasonal distribution suggests that the EPEs can occur irrespective of the mean precipitation season, which is in good agreement with Minářová et al. (2017). This may also be valid for the strongest EPEs as well since the ten strongest EPEs also occurred in all seasons (Table 1). However, the strongest EPEs (WEI value higher than 100) occurred mainly in autumn and spring, which is in contradiction to Minářová et al. (2017), who found the strongest events in peak summer. This might be related to the difference between station-to-station approach used in Minářová et al. (2017), which enables detection of even very local (peak summer) convective storms. The areal assessment by WEI in this study produced more reliable results for the area of interest.

3.3 Duration, affected area, and extremity of EPEs

3.3.1 Duration

The maximal duration of 54 EPEs was 5 days, i.e. 6–10 days EPEs did not occur (Figure 5). 1- and 2-days EPEs were the most frequent (26 and 19 EPEs, respectively). The short duration of EPEs is against our expectation, which was based on the general behaviour of precipitation in the VG area where precipitation lasts rather longer on average (Parlow, 1996; Minářová, 2013). The short duration may be explained by leeward convection, which is generally short lasting (Houze, 2014) and has been documented in the leeward side of the VG by the Convective and Orographically induced Precipitation Study campaign (Planche et al., 2013). Nevertheless, since the leeward convection mostly occurs in summer in Europe (Barry, 2008), short duration of EPEs in the VG area, which occurred mostly in autumn, is more likely related to rapid changes in precipitation activity during precipitation event in the area. In fact, the event-adjusted method enables to distinguish the most anomalous 1- to x-day EPE within a more continuous precipitation period, which also suggests that there can be more episodes of heavy rainfall within a precipitation sequence but separated by no or less extreme precipitation resulting in the decrease of the E_ta. Moreover, since the WEI tends to increase with increasing duration of event (and area) by definition, the short duration of EPEs found in VG suggests its plausibility.

Given that the study was limited to precipitation totals available only at daily resolution, return levels at a resolution of 3- or 1-h were not computable. This limitation hindered any comparison of 1- to 3-h return levels with the 1-day EPEs, which may categorize the 1-day EPEs into stratiform and convective.

We propose to consider 1–2 days EPEs as short EPEs, and 3–5 days EPEs as long EPEs because 1–2 days EPEs evince much higher frequency clearly differentiating them from 3 to 5 days EPEs (Figure 5). This division is maintained hereafter.

3.3.2 Extremity and affected area

The more frequent short EPEs were of similar range (WEI values 28–120) as compared to long EPEs (35–119) although less extreme in general (Figure 6). It may imply that the long EPEs are more severe. However, the ten strongest EPEs comprise only two long events (Table 1), suggesting that the relationship between duration and extremity of EPEs is more complicated.

Figure 6 also shows that short EPEs tend to affect smaller areas. Most commonly they affected 17–39% of the study area. It is in good agreement with the expectations since the short lasting heavy rainfall events affect smaller area as compared to the long lasting events due to likely restricted time for changes in circulation patterns (Houze, 2014). However, the area affected by short EPEs (6–72% of the VG area) is similar to that by long EPEs (16–75%). The reason for the similarity could be that the data set of long EPEs was too small (only 9 of the 54 EPEs were long) to show substantial differences from short EPEs.

The correlation coefficients calculated between pairs of variables [duration, extremity (WEI), and size of the area affected by 54 EPEs] showed that the pairs of variables are positively correlated (99% probability, except for the pair duration-extremity, where it was significant at the confidence level of 95%). The covariance was higher for the pair affected area-extremity (cov = 215.4, r = 0.45) than for duration-size of the affected area (cov = 6.5, r = 0.43). The stronger positive correlation between the size of the affected area and extremity is natural due to the definition of the WEI value, which increases with the size of the area.

The correlation coefficients were also calculated for the same variables for short and long EPEs, and SHY and WHY EPEs, separately. The short EPEs showed no correlation between the duration and extremity of events, and duration and size of the affected area of events; only the extremity and size of the affected area were positively correlated (r = 0.35 at α = 5%). The long EPEs exhibited the same results between three pairs of variables as the short EPEs (r = 0.68 at α = 5% for the pair extremity-affected area). The SHY EPEs showed positive correlation between all the variables (r = 0.49 for duration-extremity, r = 0.55 for duration-size of the affected area, and r = 0.58 for extremity-size of the affected area at α = 1%), whereas for WHY EPEs no correlation between the variables was found. The results thus suggest that except the natural positive correlation between the affected area and extremity of EPEs following the definition of WEI, the relationships between the variables are not straightforward and the positive correlations are mostly due to the SHY EPEs.

3.4 Inter-annual changes in EPEs

The inter-annual changes of maximal annual WEI values of events show that the extremity (WEI) of events was lower at the beginning of the study period and got higher mostly since 1980 (Figure 7). The lower extremity at the beginning of the study period might be connected with lesser availability of data (Figure 2) and limitations of the available instruments in measuring heavy rainfall (e.g. gauge overflow or wind influence on unshielded gauges). The EPEs being stronger since 1980 is in good agreement with the Beck's (2011) findings and with the findings of IPCC (2014), which showed likely increase in intensity of EPEs in Europe.

No significant increasing trend was identified in the frequency of EPEs unlike reported by IPCC (2014), which might be related to regional differences that have been suggested in the report. We tested 1, 3, 6, 18, and 27 years equally long-time intervals (divisible of 54 years study period) since the trend analysis can be influenced by the selected number of time intervals. All resulted in insignificant linear trends at α = 0.01, α = 0.05, and α = 0.10 except for three equal time slices (i.e. 18-years) that showed an increasing trend in frequency of EPEs in VG at α = 0.10. The trend analysis can also be influenced by the trend curve. Nevertheless, the other trend curves such as exponential, polynomial of second degree and logarithmic also resulted in insignificant trends for the analysed time slices and α.

Figure 8(a) shows that the short and long EPEs were the most frequent during 1980–1990 and no long EPE occurred during 1961–1977 and since 2005. Both the long and short EPEs experienced an insignificant trend during the period (at α = 0.01, α = 0.05, and α = 0.10). The increase in numbers in short events in the latest period correspond with the climate projections by Klimaveränderung und Konsequenzen für die Wasserwirtschaft (KLIWA) (Söder et al., 2009) who predicted increase in frequency of very short heavy rainfall.

The SHY EPEs were less frequent during 1990–2000 and the WHY EPEs were the most frequent during 1978–2002; only two WHY EPEs occurred out of the period 1978–2002 (Figure 8(b)). The trends in SHY and WHY EPEs were both insignificant (at α = 0.01, α = 0.05, and α = 0.10), suggesting their difficult prediction. A similar regional study has been performed for the period 1931–2010 by KLIWA (climate change and its consequences for water management) in southern Germany (KLIWA, 2011). In five regions of the Rhine River Basin situated close to the VG area (from Basel to the tributary basin of Schwarzbach), the authors found increasing trends (significance lower than 80%) in 1-day maximum regional precipitation for both the summer (May–October) and winter (November–April) halves of the hydrological year.

In order to minimize the influence of trend analysis related to the arbitrary number of time slices, the SHY and WHY EPEs were also studied for 2, 3, 6, 9, and 18 equally long-time slices (not depicted). The results confirmed the general trend that was found in Figure 8, i.e. higher representation of SHY EPEs at the beginning and in the end of the study period, interrupted by a period of preponderance of WHY events.

3.5 Synoptic conditions of EPEs

Figure 9 displays the weather types that occurred during EPEs (the abbreviations are explained in Table 2). It shows that the west cyclonic weather type (Wz) prevailed during all 54 and 27 first EPEs for both the GWLc and SVGc. It is in good agreement with REKLIP (1995), where it was found that the precipitation in VG is often related to Wz. According to GWLc, the other most frequent weather types during the 54 EPEs were low over Central Europe (TM), trough over Western Europe (TrM) and south-shifted westerly circulation (WS). The TM weather type can cause precipitation on the eastern side of the VG and in the Upper Rhine River Plain when northern or northeastern airflow prevails in the area (REKLIP, 1995). According to SVGc, the north-west cyclonic (NWz) and south-shifted westerly (WS) weather types were among the most frequent during the 54 EPEs (i.e. after Wz). Two more weather types related to EPEs were found in SVGc contrary to the GWLc, although both catalogues include equal number of types.

Although Wz prevailed during the 54 EPEs, it was not related to any of the five strongest EPEs for both the GWLc and SVGc, and to any of the first 11 EPEs for SVGc (Figure 9). This suggests that Wz is not the prevailing synoptic pattern during the very EPEs in VG and that there is a discrepancy between the most frequent weather types during EPEs and the ones producing strongest EPEs, although we are aware about the low number of representatives for the strongest EPEs.

Among the five strongest EPEs no weather type dominated, i.e. the synoptic pattern was diverse. It is in accordance with the general conjecture about various characteristics of the strong EPEs or their strong sensitivity on selected data set (Stephenson et al., 2008).

Although based on widely used GWLc and automated SVGc, some rather unusual weather patterns such as anticyclonic weather are linked to EPEs, more frequently in the case of subjective GWLc. It may be connected to the fact that the weather types over Europe from GWLc and SVGc are assessed from the view of Central Europe, VG being situated at its most western part, and at large scale. It suggests that the results are catalogue-dependent and not so precise for very regional analyses. Thus the synoptic conditions during EPEs were also assessed quantitatively.

Figure 10 shows that the EPEs occurred mostly in strong SW airflow at 850 hPa and in western, SW and southern airflow at 500 hPa level. Similar findings can be found in REKLIP (1995), where the SW airflow was related to high precipitation totals (Rd > 100 mm) in VG. Analogous direction and strong values are also found for the flux of specific humidity. Figure 10 shows clearly that strong values of synoptic variables are frequently responsible for extreme precipitation. This corresponds to Müller and Kašpar (2010), who found that usually strong moisture fluxes accompany hydrometeorological extremes in this part of Europe. The strongest EPEs occurred when strongest values of variables were measured, which is especially true for the airflow at 500 hPa level. Other synoptic variables can be analysed such as vertical velocity and relative vorticity with respect to EPEs, while our analysis was restricted to the aforementioned accessible variables. Another quantitative approach introducing a Circulation Extremity Index proposed by Kašpar and Müller (2014) can provide the in-depth study of circulation causes of the EPEs and can be tested in future.

Two EPEs were missed in the quantitative analysis of synoptic variables since they occurred after 2010, i.e. beyond the available data set. However, these EPEs were not among the strongest (20th and 25th in 54 EPEs, Table S1) and they represented <4% of EPEs, thus their influence on the results was considered negligible and the results accurate (Zolina et al., 2005, 2013).

3.6 Comparison of WEI with standard indices

The ten strongest EPEs defined by WEI (Table 1) were compared with standard indices, i.e. exceeding a defined precipitation threshold value at a station (Štekl, 2001; Muluneh et al., 2016; Tošić et al., 2016; Wang et al., 2016a; Ngo-Duc et al., 2017), and mean areal precipitation totals MAP (e.g. Wang et al., 2000; Konrad, 2001). Since the EPEs in VG lasted 1–5 days and a fixed duration in both the methods is required, the 1-, 3-, and 5-days totals were considered. The threshold precipitation total was set to 100 mm for 1-day totals (according to REKLIP, 1995), 200 mm for 3-days totals and 300 mm for 5-days totals.

Figure 11 shows that the WEI values correspond with 1-day maximum precipitation totals if the EPE lasted 1-day (Table 1, e.g. EPE from 17 September 2006). The 3-day totals match with WEI values in most cases except some EPEs, for which a station-to-station approach may result in longer duration of such EPEs. The highest fluctuation as compared to WEI values is found for 5-day point maximum precipitation totals, which may be due to only one 5-day EPE found through WEI. Figure 11 also demonstrates that the fixed duration of EPEs can lead to some uncertainties. For instance, if 3- or 5-days totals are considered, the ninth strongest EPE from 1995 may be considered longer than it obviously was or not considered at all. Thus the major advantage of WEI is that it enables to adjust the duration for each EPE without any arbitrary criterion. Even the maximum allowed duration of precipitation totals does not influence the results of WEI if it is fixed sufficiently long, as in our case.

The 3-days MAP was best matching with decreasing WEI values, and the 1-day MAP showed most fluctuation as compared to WEI. More differences of 1-day MAP from WEI may be related to the fact that the short events generally affect much smaller area than the whole study region. In fact, the size of the area affected by EPEs was only 50% of the study area on average. Thus in comparison to MAP where a fixed area is needed, and to point specific totals, the adjustable size of the affected area by EPEs through WEI is another valuable advantage. The results are in good agreement with Müller et al. (2015b), who discussed the WEI with the standard indices for the Czech Republic.

Figure 11 shows that the ranking of EPEs also depends on the assessment method, however a comparison in the ranking of ten first EPEs based on each aforementioned method highlights that seven out of ten first EPEs based on WEI were also recorded by at least one of other method among the ten strongest. Thus the WEI method can be considered capable of providing relevant results, and its adjustable duration and size of the area affected by EPEs make it unique and simple tool for the analysis of weather and climate extremes.