1 INTRODUCTION

Numerical weather prediction (NWP) models exhibit insufficient forecast skill for extreme precipitation events (EPEs), particularly due to the high spatial and temporal variability of precipitation (Sukovich et al., 2014). However, accurate forecasts of precipitation, particularly extreme events, are essential for mitigating the impact of possible precipitation extremes as well as subsequent floods, which can cause serious damage to society and ecosystems.

Our focus is large-scale extreme precipitation in central Europe, where events can be connected to synoptic-scale processes of moisture transport mainly from the Atlantic and the Mediterranean (Sodemann and Zubler, 2010; Hofstätter et al., 2017), which distinguish two main types of large-scale extreme precipitation in central Europe. However, when going into details, some studies suggest larger variability of synoptic-scale patterns that induce extreme precipitation and floods in the area (Jacobeit et al., 2006; Wypych et al., 2018).

The connection between hydrometeorological extremes in central Europe and synoptic types of subjective, mixed or objective classifications is relatively well discussed in the literature (e.g. Ustrnul and Czekierda, 2001; Jacobeit et al., 2006). In connection with extreme precipitation and floods, the classifications are often qualitative, that is, the circulation patterns are classified by similar fields of sea-level pressure or geopotential height (Mastrantonas et al., 2021), backward air trajectories (Seibert et al., 2007), cyclone tracks (Hofstätter et al., 2017), etc. However, when associated with extreme events, Müller and Kašpar (2010) emphasized the need to add a quantitative aspect to the classification process, which allows expressing the extremity of circulation conditions at different places.

Such a quantitative evaluation of the synoptic–dynamical conditions can be based on anomalies of some meteorological variables, including for example, potential vorticity, total column water vapour, vertical velocity, moisture flux, or a combination of several anomalous variables. The transport of moisture is often considered essential for the emergence of hydrometeorological extremes in central Europe. A close relationship between extreme floods and intense moisture transport has been demonstrated in the Alps by Froidevaux and Martius (2016), who examined the vertically integrated water vapour transport (IVT), and by Müller and Kašpar (2011), who focused on the moisture flux at single pressure levels. Outside central Europe, the importance of IVT as a predictor of heavy precipitation was highlighted by Grazzini et al. (2020) over northern Italy and, on a larger European scale, by Lavers et al. (2014; 2016).

However, extreme precipitation always requires moisture supply supported by a sustained lifting of air, which has been considered in our previous study (Gvoždíková and Müller, 2021), where we examined the moisture flux anomalies in the area of extra high upward vertical velocity. Results showed that these moisture flux anomalies might be a suitable indicator of EPEs, especially events with dominating northern moisture flux (i.e. travelling towards central Europe from the north; Nf). The EPEs that belong to the Nf type represented a large group of EPEs that affect the eastern parts of central Europe in the warm half-year and that cause severe floods in the Oder, Elbe and Danube basins (Gvoždíková and Müller, 2017). On the other hand, there were several EPEs with dominating western moisture flux (i.e. travelling from the west; Wf), which occurred only from September to March (Gvoždíková and Müller, 2021) and mostly affected the western half of central Europe. However, the correspondence between western moisture flux anomalies and Wf EPEs was not as strong due to cases when high values of moisture flux from the west occurred due to extra high wind velocities despite rather small amounts of air moisture.

Given that the moisture flux is related to a larger-scale circulation, we can assume that its predictability will be better than in the case of precipitation, which is driven by more complex small-scale atmospheric processes (Lavers et al., 2016). There is already evidence from previous studies that IVT is more predictable than precipitation (Lavers et al., 2014). The finding was based on the comparison of the degree of linear association between forecasted fields and observed fields measured by the Pearson correlation coefficient. Since the correlation coefficient is a dimensionless value, it may be useful for comparing the quality of forecasts for different atmospheric variables.

For the verification of spatial fields, which are typically produced by numerical NWP models, anomaly correlation is one of the most common traditional measures of forecast accuracy (Wilks, 2011; Zhang et al., 2019). Anomaly correlation is utilized as one of the main verification measures at the European Centre for Medium-range Weather Forecasts (ECMWF) (Owens and Hewson, 2018), although it is not directly employed for precipitation. There may be a problem with forecasts that indicate smaller-scale precipitation patterns that are somewhat misplaced and are penalized twice for showing precipitation at the wrong location and not showing precipitation at the correct location (Ebert, 2008). Therefore, new verification methods that account for spatial translation and suppress the double penalty issue were developed (e.g. Gilleland et al., 2009). However, this issue is more problematic for high-resolution forecasts. Another solution may be to describe precipitation fields as individual events, characterised by for example, a daily value of extremity index, and thus to verify precipitation as a categorical event – it means whether or not it has exceeded a given threshold (Rossa et al., 2008).

Given that some EPEs were directly associated with moisture flux anomalies in our previous study (Gvoždíková and Müller, 2021), it is also possible to use the combination of vertical velocity and moisture flux – the accumulated ascending moisture flux – as a predictor of heavy precipitation. The aim of this study was to determine practical predictability (Lorenz, 2006) and forecast skill of mean daily accumulated ascending moisture flux and compare it with the forecast skill of one-day precipitation extremity, which are both derived from ERA-Interim reanalysis and forecasts. The focus is on selected EPEs from 1979 to 2013, but the non-extreme precipitation days are also evaluated to detect possible false-alarm events. We hypothesize that forecasts of the accumulated ascending moisture flux could support extreme precipitation warnings even at longer lead times and improve preparedness for these extremes.

2 DATA AND METHODS

2.1 Study area and reference precipitation events

Our study area is defined by five main river basins in central Europe – Rhine, Weser/Ems, Elbe, Danube up to Bratislava, and Oder (Figure 1). In this area, we detected a set of EPEs, which could later be divided into two main types according to the area of occurrence, seasonal distribution, and the direction of moisture flux that caused the event.

The examined extreme precipitation events (Table 1) were always large-scale extremes that were selected by the value of the weather extremity index (WEI: Müller and Kašpar, 2014), combining the information about precipitation intensity using return periods of precipitation totals, the area affected by precipitation and the duration of the event. The return periods are estimated from series of 1- to 10-day precipitation totals at rain-gauges within the study area and then interpolated into a regular grid with a horizontal resolution of 10 km. The WEI is equal to the maximum value of the variable E_ta (described in detail by Müller and Kašpar (2014)), which is calculated for the day-by-day expanding time window (t from 1 to 10 days) as the product of the radius of a circle (equivalent to the enlarging area a) and the mean common logarithm of the return periods N of precipitation totals within the considered area:

$$E_{\mathit{ta}}=\frac{{\sum }_{i=1}^n\log (N_{\mathit{ti}})}{n}\frac{\sqrt{a}}{\sqrt{\pi }}.$$ (1)

TABLE 1. Extreme precipitation events of Nf (left and central panel) and Wf (right panel) moisture flux variants in central Europe from 1979 to 2013 ranked by precipitation extremeness (WEI)

EPEs ranking	Precipitation event	Duration (d)	Moisture flux variant	EPEs ranking	Precipitation event	Duration (d)	Moisture flux variant	EPEs ranking	Precipitation event	Duration (d)	Moisture flux variant
1	17 Jul 1981	4	Nf	13	6 Aug 2002	2	Nf	10	20 Oct 1986	3	Wf
2	4 Jul 1997	5	Nf	14	16 Jul 2002	2	Nf	16	19 Dec 1993	2	Wf
3	30 May 2013	4	Nf	15	8 Jul 1996	1	Nf	17	19 Mar 2002	4	Wf
4	11 Aug 2002	2	Nf	18	31 Jul 1991	3	Nf	20	27 Oct 1998	2	Wf
5	6 Aug 1985	4	Nf	19	17 Jun 1979	1	Nf	23	13 Feb 1990	3	Wf
6	1 Aug 1983	5	Nf	21	22 Jul 2010	3	Nf	24	6 Oct 1982	2	Wf
7	3 Aug 2006	6	Nf	22	24 Jun 2013	2	Nf	26	21 Dec 1991	2	Wf
8	5 Sep 2007	2	Nf	25	12 Apr 1994	2	Nf	28	12 Sep 1998	4	Wf
9	25 Sep 2010	3	Nf	27	27 May 2007	3	Nf	29	28 Dec 1986	3	Wf
11	20 Jul 2011	3	Nf	30	16 May 2010	3	Nf	32	25 Feb 1997	1	Wf
12	15 Jul 1997	7	Nf	31	8 Jun 1990	3	Nf

The final list of central European EPEs, based on the size of the WEI and published by Gvoždíková et al. (2019), contains events with different sizes of affected area and different durations, mostly up to 5 days. However, the course of multiday events can always be analysed by the one-day E_ta.

According to our previous study (Gvoždíková and Müller, 2021), EPEs from 1979 to 2013 consisted of three types of event of different moisture flux conditions that prevail during their lifetimes. A geometric mean of relative moisture flux values (separately from each of the four cardinal directions) was used to derive a single-value representation of the moisture flux over multiple grid points. The geometric mean specified the following division: (a) events with dominating moisture flux from the northern sector (Nf), (b) events with dominating moisture flux from the western sector (Wf), and (c) low-flux events (Ot). Considering that the geometric mean was taken only over grid points of extra high upward vertical velocity, no Sf events or Ef events were detected (i.e. events with dominating southern moisture fluxes or eastern moisture fluxes, respectively). As a result, we identified 22 Nf events, 10 Wf events, and 11 Ot events. In this study, we focused on two of these three types: EPEs of Nf and Wf moisture flux variants (a total of 32 EPEs, Table 1), which completely differ in their origin.

3 RELATION BETWEEN EPES AND DAILY ACCUMULATED ASCENDING MOISTURE FLUX

Due to the position in midlatitudes, western directions of moisture flux prevail in the study area, which also applies to the values of T^*_d in general, both for their frequency and magnitude (left diagram in Figure 2). To evaluate the representation of EPEs among high T^*_d values, we separately defined sectors within the diagram for Nf and Wf events so that Nf and Wf sectors comprise maximum T^*_d values during all EPEs of the respective type. Each sector is characterised by a range of angles and a threshold of T^*_d (Table 2 and Figure 2a), which give the T^*_d values hereafter referred to as extreme T^*_d. The agreement between extreme T^*_d values and EPEs of the given type was tested for the reanalysis and forecasts within the sectors in two ways: (a) percentage of the extreme T^*_d values associated with EPEs in the total number of extreme T^*_d values, and (b) number of the EPEs with extreme T^*_d values at least on 1 day (in the case of the reanalysis, it equals the number of all EPEs, i.e. 22 Nf and 10 Wf events).

The results for the reanalysis of T^*_d are depicted on the far left in Figure 3a. Almost half of the extreme T^*_d values in sector Nf are associated with Nf EPEs. Regarding the Wf precipitation events, only a minority (15%) of the extreme T^*_d values in the Wf sector were associated with EPEs. A substantial portion of extreme T^*_d values in sector Wf, which were not accompanied by EPEs, was recorded on days when strong moisture fluxes were generated by extra high wind velocity combined with rather low values of air humidity.

In general, extreme T^*_d values became more frequent when computed from the forecast outputs. As a result, the proportion of extreme T^*_d associated with EPEs in the total number of extreme T^*_d forecasts generally decreased (Figure 3a) compared to the reanalysis. The decrease was more pronounced for the Nf sector, in which the proportion of extreme T^*_d related to Nf EPEs decreased to 26% already for 1-day forecast.

The situation was rather different for the forecasts of moisture fluxes from the western direction. The proportion of extreme T^*_d values associated with Wf EPEs to the total number of measurements even increased slightly within the first 2 days of the forecast and approached the values of the Nf sector (Figure 3a). However, the number of included EPEs in the Wf sector decreased a little faster with longer forecast lead times.

An interesting comparison is offered by Figure 3b, which displays the same as Figure 3a, but for moisture flux accumulated over all grid points (T_d) and not only over points with extra high upward vertical velocity. The T_d threshold (Table 2) was calculated in the same way as for T^*_d (i.e. the smallest of maximum daily T_d associated with EPEs). There was no significant decline with longer forecast lead times, which could indicate that forecasted vertical velocity values negatively affect the forecasts of T^*_d. However, despite the better correspondence of reanalysis and forecasts in Figure 3b, extreme T_d is less related to the EPEs than extreme T^*_d, as the percentage of the extreme T_d values associated with EPEs was very low in both sectors.

4 RELATION BETWEEN EPES AND ONE-DAY PRECIPITATION EXTREMITY

According to the previous Section 3, there exists a good connection between extreme T^*_d and EPEs mainly of Nf type. This connection is even stronger than between EPEs and one-day precipitation extremity expressed by E_ta calculated from the reanalysis (Figures 2c and 3c). It is not possible to directly connect one-day E_ta with different types of EPE without knowing the direction of moisture flux. Therefore, each value of one-day E_ta is characterised by the direction vector inherited from T^*_d calculation. It is then possible to treat the one-day E_ta correspondingly to T^*_d and to determine the threshold for extreme one-day precipitation for both types of EPE. The individual thresholds calculated as the smallest of maximum one-day E_ta associated with EPEs are listed in Table 2.

The actual proportion of extreme one-day precipitation related to Nf EPEs in the total number of extreme precipitation days was only 34% according to the data from reanalysis (Figure 3c). However, the proportion rises to 62% when we consider one-day E_ta calculated from rain-gauge data (not shown). A general overestimation of the 1-day E_ta in the case of reanalysis is also evident from the comparison of Figure 2b,c, where the 1-day E_ta from rain-gauge data and reanalysis, respectively, are displayed with the same direction vector as in Figure 2a.

The proportion of extreme one-day precipitation associated with EPEs in the total number of forecasted extreme precipitation days was also lower than for extreme T^*_d up to the 3-day forecast. The proportions approached those of T^*_d in the case of longer lead times. This applies to both the Nf and Wf sectors.

5 PREDICTABILITY OF CONSIDERED VARIABLES

5.1 Predictability of accumulated ascending moisture flux

Section 3 indicated that the vertical velocity may reduce the quality of the accumulated ascending moisture flux forecast. To confirm the presumption, we decided to compare the accuracy of the T^*_d and T_d forecasts (Figure 4) using two selected scores: hit rate and false-alarm ratio (refer to Section 2.5). The thresholds for the YES events correspond to the Nf/Wf sector thresholds in Table 2.

The poor forecast of T^*_d in sector Nf was caused by a large number of false alarms, which is demonstrated by a high false-alarm ratio (Figure 4b). Even on the first forecast day, false alarms almost doubled the number of hits (Table 3a). For sector Wf, the forecast was better according to a lower false-alarm ratio (Figure 4b), but a slightly higher number of missed events (and lower hit rate) had the opposite effect and worsened T^*_d forecasts (Figure 4a, Table 3b).

TABLE 3. Contingency tables for verification of one-day forecasts of (a,b) accumulated ascending moisture flux (T^*_d), (c,d) accumulated moisture flux (T_d), and (e,f,g) one-day precipitation extremity expressed by E_ta from 1979 to 2018

(a)			(b)
T*d Nf sector	Reanalysis		T*d Wf sector	Reanalysis
	Yes	No		Yes	No
1-Day forecast			1-Day forecast
Yes	74	121	Yes	36	24
No	27	14,379	No	36	14,505

(c)			(d)
Td Nf sector	Reanalysis		Td Wf sector	Reanalysis
	Yes	No		Yes	No
1-Day forecast			1-Day forecast
Yes	1,077	219	Yes	1,445	235
No	147	13,156	No	155	12,764

(e)			(f)
Eta Nf sector	Reanalysis		Eta Wf sector	Reanalysis
	Yes	No		Yes	No
1-Day forecast			1-Day forecast
Yes	79	156	Yes	22	80
No	35	14,331	No	32	14,467

(g)
Eta all range	Reanalysis
	Yes	No
1-Day forecast
Yes	217	322
No	103	13,959

• Note: Except for (g), tables are divided according to the sector in which we evaluate the occurrence of daily events: (a,c,e) Nf sector and (b,d,f) Wf sector. For sector thresholds, refer to Table 2.

In contrast, the daily forecast of T_d seems to be quite accurate. Both the hit rate and the false-alarm ratio showed much better results than when we considered the condition of extra high upward vertical velocity (Figure 4). However, as Figure 3b shows, only a small part of extreme T_d values in the defined sectors were associated with EPEs. Therefore, it is not possible to use it as a supporting forecasting tool for extreme precipitation warnings. Simultaneously, working with T^*_d is also problematic due to the lower forecast skill.

Nevertheless, the forecasts of T^*_d during the five largest EPEs seem quite stable even if vertical velocity is included. There was consistency in forecasting extreme T^*_d, especially in the case of days with Nf extreme precipitation events. For the five largest EPEs, the model began to forecast extreme T^*_d 6 days in advance (we take into account the forecast for the same day with a possible one-day shift; Figure 5). Except for EPEs, there were only a few cases when forecasts continuously indicated extreme T^*_d for at least six consecutive days. Apart from the largest Nf events, these cases included the 4th largest Wf event (Table 1), several less extreme spring events and two February precipitation events (some of them outside the largest EPEs in Table 1). Only one case in April remained without a precipitation response.

5.2 Predictability of one-day precipitation extremity

As in the case of T^*_d, poor predictability appears for one-day precipitation extremity expressed by E_ta. Both hit rate and false-alarm ratio show even worse forecast skill than for T^*_d, (compare Figures 4 and 6) with a very high number of false alarms in both Nf and Wf sectors (Table 3e,f). It seems that especially during western moisture flux situations, the one-day precipitation extremity is highly overestimated (Figure 6b), although part of the error may be related to prediction of moisture flux direction inherited from T^*_d. However, the presence of the same error in T^*_d as well as in one-day precipitation extremity allows us to compare the predictability of both. Nevertheless, if we only focus on extreme precipitation, regardless of direction angles, the predictability is slightly better, but still it only improves to the level of T^*_d forecast in Nf sector (compare Figures 4 and 6).

5.3 Comparison of forecast skill of vertical velocity and moisture fluxes

Considering the previous findings of this study, it can be assumed that the vertical velocities negatively affect the quality of the T^*_d forecast. Therefore, we examined individual variables and assessed their forecast skill by calculating anomaly correlations (refer to Section 2.5) between the grid points of the daily pairs of reanalysis/forecast fields. On average, the vertical velocity field forecast skill appears to be much worse than the forecast skill of moisture flux (Figure 7a). Both zonal and meridional horizontal moisture fluxes showed very similar results (compare Figure 7c,d) and even in the 5-day forecast, their AC were approximately 0.6 compared to 0.4 for vertical velocity. The difference in AC corresponds to an increased forecast horizon of moisture fluxes over vertical velocity of 2–3 days. However, in the case of the vertical velocity, a significant difference occurred between AC for the Nf events and those for the Wf events, with much better results for the western moisture flux situations (Figure 7b).

High AC of moisture fluxes and less accurate results for the vertical velocity suggest an explanation for a much worse agreement between EPEs and forecasted T^*_d than in the reanalysis (Figure 3a). Simultaneously, this finding agrees with the greater accuracy of the forecasts of T_d (Table 3c,d, Figure 4), which are independent of the condition of extra high upward vertical velocity.

Figure 8 shows that within the study area, the ME of vertical velocity is mostly negative, indicating larger forecasted ascending motions mainly on the northern side of the Alps and in the eastern part of central Europe. This is a possible explanation for higher AC of vertical velocity in the case of Wf events (Figure 7b) and the large number of false alarms of extreme T^*_d in the Nf sector (Figure 4b). In the case of other variables, including precipitation, the MEs are positive, indicating an overestimation of precipitation and western and southern moisture fluxes.

6 DISCUSSION

Moisture flux forecasting can support the prediction of precipitation extremes, but first, it needs to be reasonably connected to these extremes, and second, the quality of its predictability must be sufficient (Froidevaux and Martius, 2016).

The close relationship between accumulated ascending moisture flux T^*_d and especially large EPEs of the northern moisture flux variant (Nf) was confirmed in this study. This result agrees with previous findings by Gvoždíková and Müller (2021), which were based on data from the recent ERA5 reanalysis. The agreement was even better using ERA5 data when expressed as the percentage of Nf event points to the total number of extreme T^*_d values, which accounted for 68% in the sector that corresponds to Nf (Table 2) compared to 47% using ERA-Interim. Higher data resolution (0.25° spatial grid) may explain this stronger relationship, as it captures local extremes of accumulated ascending moisture flux, or the stronger relationship may be related to the selected vertical velocity threshold. In our previous study (Gvoždíková and Müller, 2021), extra high upward vertical velocity corresponded to only 0.1% of the lowest reanalysis values, which was not possible in the case of ERA-Interim data, where some EPEs would not be recorded. Despite ERA5 outperforming the ERA-Interim reanalysis and exhibiting a closer connection between EPEs and extreme T^*_d, we could not use ERA5 due to the unavailability of the reforecasts.

In addition to the existing relation between EPEs and T^*_d at lower tropospheric isobaric levels, research by other authors often links extreme precipitation and floods to intense IVT, which usually occurs in atmospheric river regions. Thus, IVT had a significant effect on extreme events in the western United States (Konrad and Dettinger, 2017) or western Europe (Lavers and Villarini, 2013). According to Froidevaux and Martius (2016), IVT perpendicular to orography also caused major floods in the Swiss Alps, although it was not always associated with atmospheric rivers. Simultaneously, the authors were aware of the strong vertical wind shear situations, which can affect the IVT direction and reduce its magnitude. A significant change in the wind direction with altitude is often related to EPEs in the eastern part of central Europe (Bližňák et al., 2019), where precipitation extremes are triggered by Vb cyclones that originate in the Mediterranean region (Messmer et al., 2015).

Despite the obvious connection between extreme moisture flux and precipitation, few studies have compared their predictability. Lavers et al. (2014; 2016) focused on IVT and confirmed the higher predictability of moisture flux compared to precipitation. They explained this finding by a connection of moisture flux to synoptic-scale circulation, which is more predictable than microphysical phenomena, such as precipitation. Our study demonstrates good forecast skill for zonal and meridional horizontal moisture fluxes at the 850 hPa level up to 5-day forecast, when AC was still higher than 0.6. However, combined with vertical velocity, the accumulated ascending moisture flux is not an appropriate forecasting tool due to the worse forecast skill of vertical velocity (Figure 7).

Figure 8 indicates some changes in biases of the displayed variables depending on lead times. Therefore, the method could be further refined by using lead-time-dependent thresholds of the vertical velocity, which could improve the forecast performance of T^*_d. An improvement would also be possible in the E_ta forecast if we estimated parameters of GEV distribution from the forecasted time series. However, due to different mechanisms driving precipitation throughout the year, the biases could differ for each season (Argüeso et al., 2013). These possible variations make the use of lead-time-dependent thresholds difficult with an uncertain outcome. Other studies frequently use bias correction methods (e.g. Jabbari and Bae, 2020; Lavers et al., 2021), which are particularly useful for improving real-time hydrological modelling built on the precipitation forecast. To compare the forecast performance of different variables, we do not consider the bias correction necessary, but it may be the subject of further research.

7 CONCLUDING REMARKS

- The connection between extreme accumulated ascending moisture flux and EPEs exists mainly in the case of precipitation extremes with predominant northern moisture flux (Nf events). In the case of extreme T^*_d in the western moisture flux directions, the correspondence to Wf EPEs was less significant due to the larger proportion of non-precipitation events among these values caused by the high wind velocities with rather small air moisture.

- Forecasts of one-day precipitation extremity expressed by E_ta showed worse skill than forecasts of extreme T^*_d. However, forecasts of extreme T^*_d also do not provide sufficient reliability, because they produce a large number of false-alarm situations with northern moisture flux. This finding was connected to a poorer vertical velocity forecast during these situations. Therefore, improving the forecast skill of vertical velocity in NWP models could bring additional benefits in the forecasting of EPEs.

- Despite worse predictability of T^*_d, there was continuity in forecasting extreme T^*_d in the case of large Nf EPEs. For the five largest Nf events, extreme T^*_d values were forecasted at least 6 days in advance, while there was no such continuity for smaller or false-alarm events. We could use the continuity of high T^*_d forecasts as support for predicting large EPEs of the Nf type, such as in July 1981 and 1997, August 2002 or May/June 2013, which led to disastrous floods in central Europe. For less-extreme events, forecasts of accumulated ascending moisture flux are not reliable enough, and a forecasting tool that is based only on the moisture fluxes would be better. However, it seems that T^*_d accumulated in the area of extra high upward vertical velocity shows a better direct relationship to EPEs than T_d independent of this condition.

AUTHOR CONTRIBUTIONS

Blanka Gvoždíková: Conceptualization; data curation; formal analysis; methodology; visualization; writing - original draft. Miloslav Müller: Conceptualization; supervision; writing-review & editing.