## 1. Introduction

Estimates of variability in heavy and extreme daily precipitation are important for understanding climate variability and change. Such estimates are, however, much more uncertain compared to those derived for mean precipitation. First, estimates of heavy precipitation depend closely on the accuracy of the daily records; they are more sensitive to missing values (e.g., Zolina et al. 2005). Second, uncertainties in the estimates of heavy and extreme precipitation are boosted by statistical significance problems owing to the low occurrence of such events. As a consequence, continental-scale estimates of the variability and trends in heavy precipitation might generally agree qualitatively but may exhibit significant quantitative differences.

For the European continent, most results hint at a growing intensity of heavy precipitation over the last five decades. Klein Tank and Können (2003, hereafter KTK03) reported primarily positive linear trends in extreme precipitation indices up to 5% decade^{−1} from their analysis of daily station data for the period 1945–95. Similar conclusions were derived in the continental-scale and regional studies discussed in the Intergovernmental Panel on Climate Change (IPCC) Fourth Assessment Report (AR4; see Trenberth et al. 2007) and published by Easterling et al. (2000), Frei and Schär (2001), Groisman et al. (2005, hereafter G05), Zolina et al. (2005, 2008), Brunetti et al. (2004, 2006), Moberg et al. (2006), and Alexander et al. (2006). These results do, however, reveal considerable spatial inhomogeneities in the variability patterns of heavy precipitation. For example, KTK03 and Frei and Schär (2001) noticed strong spatial noise in their estimates of linear trends for Europe and the Swiss Alps, respectively. Kunkel et al. (1999, 2003) and G05 found similar results for the North American continent. Hundecha and Bardossy (2005), Zveryaev (2006), and Zolina et al. (2008) showed a pronounced seasonality in the long-term trends of precipitation characteristics over Europe. However, the accurate estimation of heavy precipitation for seasons and individual months is even more difficult (compared to annual estimates) because of the lower number of events in the records (e.g., Moberg et al. 2006). Climate model simulations (Zwiers and Kharin 1998; Yonetani and Gordon 2001; Semenov and Bengtsson 2002; Watterson and Dix 2003; Kiktev et al. 2003) and reanalyses (Zolina et al. 2004) show, in general, more consistent spatial patterns of climate variability in heavy precipitation than station data. However, different methods for estimating heavy precipitation applied to the model output may also give very different results.

The methodology used to estimate heavy and extreme precipitation is guided by the definition of what is termed as a heavy or extreme precipitation event. We should distinguish between the extreme event in an absolute sense and the so-called relative extremeness. Changes over time in the absolute value of precipitation extremes (quantified, e.g., as precipitation exceeding a reasonably chosen threshold) are not necessarily related to changes in extremeness quantified, for example, though the fraction of annual/seasonal total due to most wet days, and vice versa. Absolute values of heavy precipitation can be estimated using the empirical thresholds (G05) or upper percentiles of the precipitation distribution (e.g., Groisman et al. 1999; Zolina et al. 2005, 2008). Absolute values of heavy rainfalls do not allow us to quantify the extent to which the variations are influenced by changes in precipitation totals or by the variability in the occurrence of strong precipitation events. The increase of heavy precipitation may be connected with changes in precipitation totals or with an increasing precipitation intensity in combination with a decreasing number of wet days (e.g., Semenov and Bengtsson 2002; Brunetti et al. 2004; Zolina et al. 2004, 2008).

Approaches to quantify extremeness are so far based on empirical extreme precipitation indices (KTK03; Moberg et al. 2006; Alexander et al. 2006). These indices quantify the precipitation contribution to the total, which is caused by days with precipitation exceeding some percentile (e.g., 75% or 95%). Such indices (R75tot or R95tot, respectively) allow us to account for the fractional contribution of the most wet days to the precipitation totals and help us to distinguish between changing mean precipitation and the changing occurrence of the wettest days in forming precipitation extremes. The accurate estimation of these indices for seasonal or monthly time series is, however, severely hampered by the limited number of wet days per season or month.

To overcome this limitation, we develop an estimate of the fractional contribution from the wettest days to the total, which is less hampered by the limited number of wet days. To this purpose, we make the assumption that the probability density function (PDF) of daily precipitation follows a gamma distribution and determine the parameters of the gamma distribution from the observations. Then we derive a theoretical distribution of the fractional contribution of any percentage of wet days to the total from the gamma cumulative distribution function (CDF). Of course other types of statistical distributions can be used if more appropriate. In the following section, this measure will be used for the analysis of long-term changes in precipitation extremes over Europe from rain gauge data.

## 2. Station data

We base our analysis on long-term daily precipitation rain gauge observations over Europe obtained from merging the following three datasets: the Royal Netherlands Meteorological Institute (KNMI) collection, known as the European Climate Assessment (ECA) daily dataset (Klein Tank et al. 2002); the regional German Weather Service (DWD) collection; and the collection of the Hydrometeorological Center of Russia (Zolina et al. 2004). The merged dataset consists of 306 synoptic stations, which cover periods from one or two decades up to more than 100 yr. We selected from this collection those time series that cover the period from 1951 to 2000. In addition, we require that the sampling density of the selected time series is sufficient for estimating the statistical characteristics of daily precipitation according to the guidelines put forward by Zolina et al. (2005). According to Zolina et al. (2005), linear trends in heavy precipitation estimated from seasonal time series are practically insensitive to gaps in daily time series smaller than 10%. To ensure the robustness of the results, we only use station time series with gaps below 1% in this study. This leaves us with 150 stations for our analysis. The omission of time series for which the gamma PDF was rejected as a suitable fit (see section 3) resulted in the elimination of an additional 34 stations; therefore, 116 stations were finally used for our analysis. The spatial distribution of these stations over Europe (see, e.g., Fig. 2) is quite inhomogeneous, with the highest density in central Europe and a considerable drop in the number of sites in southeastern and eastern Europe. Recently, KNMI made available the so-called blended daily precipitation time series in which many daily gaps in records were filled with data from neighboring SYNOP stations. Although this updated collection is practically free of gaps, we decided not to use it in this study to avoid unpredictable effects of mixing two types of precipitation records.

## 3. Methods

As we mentioned in section 1, absolute values of heavy or extreme precipitation (either estimated using regional thresholds or from the upper percentiles of the gamma distribution) can be largely influenced by variations in precipitation totals. According to Groisman et al. (1999), a 5% increase of the mean (with no change in the number of wet days and the shape of the distribution) results in a 20% increase of the probability of daily precipitation exceeding 1 inch (25.4 mm). Using this concept, Easterling et al. (2000) and G05 have shown that a considerable part of Europe is characterized by disproportional changes in heavy precipitation and in precipitation totals. Alternatively, extreme precipitation may increase as a result of a decreasing number of wet days and the changing shape of the probability distribution, while the precipitation total remains the same or even decreases.

*R*is the precipitation total for period

_{j}*j*,

*R*is the daily precipitation at wet day

_{wj}*w*(

*R*≥ 1 mm) of period

*j*, and

*R*95 is the 95th percentile of precipitation during the long-term period (1961–90 in KTK03). When estimating R95tot, we experience an uncertainty for areas or seasons with a limited number of wet days. Fifty percent of European stations in winter and about 80% in summer have fewer than 40 wet days on average. Interannual standard deviations (std) of the number of wet days (7–13 days for winter and 4–11 days in summer) imply that estimates of interannual variability in the R95tot index will be even more uncertain. For the monthly estimates, or the higher (e.g., 99th) percentiles, the uncertainty will grow drastically, making any conclusion about changes in extremeness doubtful.

_{wn}*F*(

*x*) is given by where Γ(

*α*) is the Gamma function,

*α*is the nondimensional shape parameter determining the skewness of the PDF, and

*β*is the scale parameter that holds the dimension of the variable analyzed and steers the stretch and squeeze of the PDF. The mean intensity of precipitation is equal to

*α*×

*β*, implying an interdependence of the shape and scale parameters. We fitted the gamma distribution to daily time series for individual seasons from 1951 to 2000 for every station by estimating the parameters using the maximum likelihood method (Greenwood and Durand 1960; Wilks 1995). The goodness of fit was evaluated using a Kolmogorov–Smirnov (K–S) test with the null hypothesis, that the empirical data of daily precipitation are drawn from the gamma distribution. This test was performed for every seasonal time series at each station with more than five days with precipitation per season according to the criterion of Semenov and Bengtsson (2002). The significance test resulted in the rejection of several locations, primarily in Southern Europe, which led to 116 remaining stations of the 150 stations for further analysis. The chi-square test rejected only 25 stations based on the 95% significance level. To minimize the uncertainty associated with fitting gamma PDF to the data, we performed our census on the basis of the K–S test.

*x*,

_{i}*i*= 1, …….

*n*is the daily precipitation,

*n*is the number of wet days (the size of the sample), and

*y*in (3) is bounded by [0, 1] and depends on

*α*and

*n*being independent on

*β*. Simple mathematical transformation (see appendix) leads to the following explicit form for the distribution (3) of the variable

*y*: where

*F*

_{2}

^{1}(

*a*,

*b*,

*c*,

*y*) is the Gaussian hypergeometric function (details are given in the appendix). The PDF of this distribution has the following form (see also the appendix): Similarly to (4) and (5), it is possible to consider the ratio of the sum of precipitation during the arbitrary taken

*k*wettest days to the precipitation total for 1 ≤

*k*≤

*n*: Simple transformations (see appendix) lead to the following formulas for the CDF: and for the PDF where the Gaussian hypergeometric function

*F*

_{2}

^{1}(

*a*,

*b*,

*c*,

*y*) is determined by the Eq. (A10) (see appendix). We will use the distribution of fractional contribution (DFC) for the distribution of the ratio (3) for the gamma distribution. In a practical application of the proposed distribution, the goodness of the fit and the sensitivity of the distribution to perturbations of the parameters should be evaluated. Since the suggested distribution is the precise mathematical consequence from the gamma distribution of a random variable, its goodness of fit is determined by the fit of the original distribution.

Figure 1 demonstrates the application of the DFC to the summer [June–August (JJA)] daily precipitation time series in Potsdam, Germany (52.383°N, 13.067°W), for 1959 and 1985 and in Elista, Russia (46.320°N, 44.300°E), for 1959. Table 1 presents the statistical characteristics of precipitation for these time series. Figure 1a shows several marked extremes, while Fig. 1b exemplifies a uniformly distributed precipitation intensity. Figure 1c gives an example of one extreme event for a relatively small number of wet days. The PDFs (Fig. 1d) and the CDFs (Fig. 1e) are very different for these three cases. Thus, the PDF and the CDF of DFC for the Potsdam (1985) time series are characterized by much smaller probabilities of high fractional contribution to the total compared to, for example, the Elista (1959) time series, characterized by the highest probabilities of high fractional contribution. Using Eqs. (4) and (5) and Eqs. (7) and (8), we can compute an analog of the R95tot index, derived from the theoretical distribution (R95tt herein) and compare it with the R95tot values. Practically, we first estimated the 95th percentile for the period 1950–2000 from the 50-yr time series and then derived the CDFs of the DFC according to Eq. (7) for each individual season or year from 1950 to 2000. Using these CDFs, we computed the fraction of precipitation total yielded by the wet days with precipitation exceeding the long-term 95th percentile—that is, simulating the procedure of KTK03 for the computation of the R95tot index.

The first time series (Potsdam, 1959, JJA) is characterized by the highest R95tot and R95tt indices. The smallest values of extreme precipitation indices are obtained for the 1985 summer time series of Potsdam. The R95tt index is 50%–60% smaller than the R95tot index for the first two time series, but it is very close to the estimate of R95tot for the third time series (Elista, 1959, JJA). Thus, the two indices may be different from each other. The new index is based on the theoretically derived fraction of wet days, which is not limited by the finite number of wet days. The DFC implies that changes in the character of heavy precipitation are controlled not only by the shape parameter of the gamma PDF but also by depending on the actual number of wet days. The change of heavy precipitation with the shape parameter assuming a fixed scale parameter may be associated with differences in the fraction of the precipitation total provided by a given number of the wettest days. Thus, the new index provides additional information about the character of extreme precipitation, and its more robust estimation allows for a better understanding of the nature of the observed changes.

## 4. Estimates of long-term changes in heavy precipitation over Europe

### a. Climatology of heavy precipitation characteristics

Figure 2 shows the climatological annual and seasonal distributions of the R95tt index derived from the DFC (4) and (7) and of the R95tot index (1) derived from the raw time series. For the annual values, the DFC-derived R95tt index leads to a qualitatively and very similar spatial pattern compared to the R95tot index, with the largest differences exceeding 20% in southern Europe and the smallest values of less than 14% in northeastern Europe. The pattern correlation between R95tt and R95tot derived from the annual time series is 0.58, being 0.68 and 0.48 for winter [December–February (DJF)] and summer (JJA), respectively. Figure 2 shows that the estimates of R95tot are on average systematically somewhat higher (about 8% of the mean values) compared to those of R95tt, when derived from the annual time series. The seasonal estimates of R95tot are somewhat lower than R95tt, especially in winter. The largest winter differences of 4%–5% (25% of mean values) are observed in central western Europe. During summer, the differences between R95tt and R95tot are smaller compared to winter, with even few locations in eastern Europe, where R95tot is slightly higher than R95tt.

The observed differences can likely be explained by sampling effects. Annual time series typically provide at least a few wet days, with precipitation exceeding a long-term 95th percentile for the computation of R95tot. According to Albert Klein Tank (2008, personal communication), R95tot falls to zero only in a few cases when computed from annual time series. When computed from seasonal time series, R95tot can frequently result in zero values affecting estimates of means and linear trends. For the annual time series, R95tot falls to zero just once during the 51-yr period for 25 of 116 locations. When R95tot is derived from the seasonal time series, the index drops to zero at least 10 times during 1950–2000 at 90 and 77 locations in winter and summer, respectively.

The uncertainty in the computation of the R95tot index is not directly related to the small number of wet days in the season. Even for a moderate *n* (e.g., 40–50 per season), there still could be a situation when none of the wet days exhibits precipitation exceeding the long-term 95th percentile. Figure 3 shows the occurrence histograms of the number of wet days for the cases when the R95tot index showed zero values over 51 yr for all stations (roughly 25% of all cases). The distributions are peaked at 30–40 days in summer and at 40–45 days in winter. Thus, even when the R95tot index is not equal to zero, its estimate from the raw data can be uncertain. In other words, the uncertainty of the computation of the R95tot index results from the finite number of wet days available rather than from the small number of wet days. This is demonstrated in Fig. 4 showing an example of the seasonal time series of R95tt and R95tot for the location Sodankyla, Finland (67.37°N, 26.65°E) for the winter season. The 50-yr mean R95tt index is larger by about 6% compared to the R95tot index. Interannual standard deviations are also different with 10.6% and 15.4% for R95tt and R95tot indices, respectively. Even when the R95tot index does not drop to zero, it may significantly deviate from the R95tt index as the result of estimation uncertainties. Running means of time series in Fig. 4 reveal differences in the decadal variability and trends between the two indices. The linear trend of about 5.42% decade^{−1} in R95tot is largely influenced by a substantial number of zero values in the first part of the record. The estimate of the linear trend in the R95tt index reveals 1.77% decade^{−1}. These effects will be further analyzed in sections 4b,c.

Figure 5 shows the interannual std of the annual and seasonal estimates of R95tt and R95tot. The strongest interannual variability of R95tt is observed in eastern and southeastern Europe, where std (R95tt) is higher than those in central western and northern Europe by approximately 3%–5%. Compared to the absolute mean values of R95tt (Fig. 2a), std increases relatively uniformly from 25% to 30% in western Europe to 40% in southern European Russia. For individual seasons, the interannual std of R95tt ranges from 40% to 70% of the mean values and also increases from western central Europe to the southeast. Compared to R95tt, the interannual std of R95tot is 20%–25% higher, ranging from 30% to 50% of the long-term mean for the index computed from annual time series (Fig. 5b) and from 50% to 90% for the index derived from seasonal time series (Figs. 5d,f). In summary, R95tot is applicable when derived from year-round time series, but it becomes quite uncertain for seasonal time series. The R95tt index, however, can be computed from any time series, which can be described by a gamma PDF. The R95tt index is to a lesser extent influenced by the noise resulting from the uncertainties of estimation of the number of days, with precipitation exceeding a given threshold, especially for seasonal time series. We will further discuss the limitations of R95tot index in section 5.

### b. Estimates of linear trends in heavy precipitation characteristics computed from annual time series

We computed the linear trends of the R95tt and R95tot indices as well as trends of the 95th percentile of daily precipitation derived from gamma distribution (*p*_{95} herein) and trends of the precipitation totals. The Student’s *t* test, together with the nonparametric Mann–Kendall test, was used to estimate the statistical significance of the trends. The trend estimates were further analyzed using the Hayashi (1982) reliability ratio, from which the confidence intervals for the statistical significance were established. Finally, we applied the field significance test to the individual trend estimates (Livezey and Chen 1983) to quantify from the binominal distribution the overall significance of the trend patterns. An extended discussion on the estimation of trend significance is presented in Zolina et al. (2008).

Figure 6 shows linear trends in R95tot (Fig. 6a) and R95tt (Fig. 6b) for the period 1950–2000. The spatial trend pattern (Fig. 6a) is qualitatively very close to that reported by KTK03 for the period 1946–99. Statistically significant upward tendencies of up to 3% decade^{−1} in central Europe and in eastern European Russia agree well with earlier estimates by KTK03. Significantly negative trends are observed in the northern part of western continental Europe (−2.2% decade^{−1}) and in southeastern Europe (−1.7% decade^{−1}). The spatial pattern of linear trends in R95tt (Fig. 6b) is more homogeneous compared to that for the R95tot index. Besides central western Europe, significantly positive trends (up to 2% decade^{−1}) are identified over Scandinavia. In eastern European Russia, the number of stations showing positive trends doubles, showing a higher trend (up to 3% decade^{−1}). The field significance of the positive upward tendency in R95tt holds over all of Europe at the 99% level and separately for eastern (east of 25°E) and western (west of 25°E) Europe at the 99% and 95% levels, respectively. R95tot, however, exhibits field significance only at the 95% and 90% levels, respectively.

Linear trends in the absolute values of precipitation extremes (*p*_{95}) computed from annual time series and in annual precipitation totals (Figs. 6c,d) along with the trend estimates in R95tt (Fig. 6b) show that over eastern Europe, the trend pattern in *p*_{95} is more consistent with the trend pattern in totals, while in the central Europe, it is more similar to that for the R95tt index. This indirectly implies that the increase in the occurrence of extreme precipitation (*p*_{95}) in central Europe is primarily caused by a growing contribution from wet days, while in eastern Europe, the growing intensity of precipitation extremes was likely driven by an increase of precipitation totals.

### c. Changes in heavy precipitation indices for individual seasons

For seasonal time series, the uncertainties of estimation for R95tot are higher compared to the annual time series. Thus, they may significantly affect the characteristics of long-term variability in heavy precipitation. For this reason, researchers are forced to use lower percentiles for the threshold when working with seasonal time series. For instance, Scaife et al. (2008) recently analyzed North Atlantic Oscillation (NAO) projections onto European precipitation extremes and used R90tot. We expect that the extended R95tt index will demonstrate better skills in capturing the variability in seasonal heavy precipitation.

Figure 7 shows the trends in R95tt derived for the four seasons. In winter (Fig. 7a), the strongest linear trends of 3%–5% decade^{−1} are observed in central Europe and in eastern European Russia. The spring pattern (Fig. 7b) shows a clear upward tendency (1.5%–3% decade^{−1}) in western Europe and in eastern European Russia, while central Europe and northern European Russia are characterized by a downward tendency from −1.5 to −2.5% decade^{−1}. The summer pattern (Fig. 7c) shows negative trends (up to −3% decade^{−1}) in western and central Europe and upward tendencies in eastern European Russia. In autumn (Fig. 7d), positive trends over western Europe are superimposed with the locations of downward tendencies, while in eastern European Russia the changes are positive (more than 4% decade^{−1}). Overall, the trends in R95tt in central Europe are likely seasonally dependent with upward tendencies during winter, spring, and autumn, and negative trends in summer. Over eastern and southeastern European Russia, however, trends in R95tt show primarily upward changes for all seasons. These conclusions are confirmed by the field significance test. The field significance of the positive trend pattern in R95tt in winter holds over all of Europe at the 99% level. In summer, the pattern of negative trends for western Europe is confirmed by a field significance test at the 95% level.

A comparison of the seasonal trend estimates for the R95tot index (no figure shown) with those for R95tt reveals serious discrepancies as a result of the uncertainties in the estimation of R95tot. Seasonal trend patterns in R95tot are quite noisy. The wintertime upward tendency is confirmed by R95tot only in western Europe. Noticeable discrepancies were also observed in summer for western Europe and during spring and autumn for central Europe. The linear trends in the R95tt and R95tot indices may even have different signs in some locations. The R95tot index tends to underestimate positive tendencies in winter and negative tendencies in summer compared to R95tt, likely masking the seasonality signal in linear trends.

Figure 8 shows estimates of linear trends in *p*_{95} and seasonal precipitation totals for individual seasons. In winter the positive trends in *p*_{95} over southeastern European Russia (Fig. 8a) are consistent with the pronounced upward tendencies in precipitation totals, showing an increase up to 8% decade^{−1}. In central western Europe, however, seasonal winter totals (Fig. 8b) do not show significant upward tendencies, while *p*_{95} does. Trends in *p*_{95} in this area are likely driven by the change of the fractional contribution to the total as a result of very wet days. In the summer season, *p*_{95} does not show significant trends in central Europe (Fig. 8c), while a significant downward tendency is observed in precipitation totals (Fig. 8d). Nevertheless, in eastern Europe, the trends in precipitation totals and *p*_{95} are qualitatively consistent. Thus, changes in extreme precipitation in eastern Europe are likely driven by the change in totals, while in central western Europe, they are obviously imposed by changes in the fractional contribution from very wet days. This is consistent with the results of Easterling et al. (2000) and G05, who showed that in central Europe, changes in precipitation extremes occur disproportionally to the precipitation totals.

### d. Association of European extreme precipitation with the North Atlantic Oscillation

Discrepancies between R95tt and R95tot are evident not only in linear trends but also in characteristics of short-period interannual variability. Correlation coefficients between R95tt and R95tot derived from the annual time series exceed 0.7 for 89 locations of 116, with correlation above 0.8 in 27 locations. For the seasonal time series, correlations are considerably smaller. In winter the correlation coefficient between the two indices is above 0.7 for only 63 of 116 locations. Correlations computed for the detrended time series practically do not differ from those derived from the original time series.

Differences in the interannual variability of the two indices should have consequences for the analysis of the effect of large-scale circulation patterns onto heavy precipitation in Europe. Recently, Scaife et al. (2008) projected the winter NAO index onto R90tot for the period 1900–2000. They reported significantly positive correlations with NAO in a few locations over Scandinavia (up to 0.55) and significantly negative correlations over southern Europe. No significant correlations were found in eastern Europe. We computed correlations between the winter (DJF) NAO index (Hurrell 1995) and heavy precipitation characteristics (Fig. 9). For R95tot, Fig. 9a shows low positive correlations in central Europe and Scandinavia (the highest correlation coefficient is 0.39) and negative correlations lower than −0.40 over eastern European Russia and the Iberian Peninsula. With R95tt (Fig. 9b), the level of correlation is considerably higher and the spatial pattern is more pronounced. The highest significant positive correlation with the NAO index is 0.60, with correlations in central western Europe around 0.5. Negative correlations form pronounced patterns over southern Europe and south European Russia, with correlations varying from 0.3 to 0.5. The pattern of correlations between R95tt and NAO compared to that formed by R95tot is characterized by more points with significant correlations and less spatial noise. The pattern of the correlation of R95tt with NAO holds field significance at 95% level, while a similar pattern for R95tot shows field significance at the 90% level only. Figures 9c,d show the correlations of the NAO index with *p*_{95} and the precipitation total. The pattern of correlations for *p*_{95} is very close to that for R95tt and differs from that for R95tot, implying that the R95tt index is better at capturing the association of European extreme precipitation with the NAO. It is also interesting to note that although in central and western Europe the correlation pattern for *p*_{95} is quite consistent with that for the total, in eastern Europe correlations of *p*_{95} with NAO are qualitatively more consistent with those for R95tt.

## 5. Summary and conclusions

To quantify the fractional contribution of very wet days to the precipitation total, we suggested an index, R95tt, that represents an extension of the R95tot index (KTK03). Our index was derived from the probability distribution of daily fractions of precipitation totals, assuming that precipitation is distributed according to the gamma PDF. The new index, although computationally expensive, allows for a more accurate (compared to the traditional R95tot) estimation of the fraction of precipitation total during specified wet days for seasonal time series when the actual number of wet days is limited. Linear trends in R95tt and R95tot derived from the annual time series are qualitatively consistent and imply a growing occurrence of extreme precipitation up to 3% decade^{−1} in central western Europe and in south European Russia, with a more evident trend pattern for R95tt. At the same time, linear trends in the indices estimated from seasonal time series exhibit pronounced differences. In winter R95tt clearly reveals an upward tendency in western European Russia (up to 4% decade^{−1}). The summer R95tt index has a downward tendency in central western Europe. The new index also allows for a more evident association of the variability in European heavy precipitation, with the NAO index showing a higher correlation level compared to R95tot.

The new index, R95tt, allows us to avoid the uncertainty associated with the finite number of wet days that is inherent in R95tot. This uncertainty limits an accurate application of this measure to annual time series and to regions with sufficient numbers of wet days. Nevertheless, there is a considerable demand for the estimation of trends and shorter-term variability patterns on seasonal and monthly basis. Up to now most studies, including those which formed the basis for the IPCC AR4 (KTK03; G05; Alexander et al. 2006; Trenberth et al. 2007) were performed for the annual time series. The few estimates performed for seasonal time series (Moberg et al. 2006; Scaife et al. 2008) report patterns of trends, which are highly influenced by spatial noise, likely associated with the uncertainties of estimation of heavy precipitation indices from the raw data.

A sparse network of 116 stations over Europe may raise questions concerning the representativeness of the results for European continent. On one hand, we tested our results with field significance test. Our trend patterns are also similar to those reported by KTK03 who used more stations for the analysis. Seasonality in trend estimates in central Europe is consistent with Zolina et al. (2008) who used very dense regional network. On the other hand, given considerable methodological focus on the limitations of the number of wet days, we avoided to use stations with many gaps to provide a denser network.

An important issue is the estimation of the uncertainty in R95tot and R95tt indices and limitations for their use. On one hand, the DFC is the precise consequence from the gamma distribution. Its goodness of fit to the initial series with the K–S or the chi-square test provides the same level of significance for DFC. On the other hand, the stability of the DFC with respect to perturbations of individual variables should be also tested. This test requires a more complicated mathematical analysis and, although desirable, is outside the scope of this paper. Thus, the computation of the R95tt index is limited by the goodness of fit of the gamma distribution. This index cannot be computed in the locations where gamma PDF does not fit to the data (about 25% of stations in our case). In this sense, R95tot is more universal and can be generally applied to all locations. To make the R95tt index more generally applicable, we should consider lower significance levels for the goodness of fit of the gamma PDF, thus reducing the accuracy of the estimation of R95tt. On the other hand, the universality of the R95tot index is conditioned by the limited number of wet days. The index falling to zero represents the most remarkable manifestation of the uncertainty but even when R95tot is not zero, the uncertainties can still exist. In Table 2, we show the percentages of cases (for all seasons during 1950–2000 and all 116 stations) when the R*X*tot index falls to zero for different percentiles *X*. The traditional index can be accurately applied for the 95th and lower percentiles for annual time series and for the 80th percentile for seasonal time series. Remarkably, for the 99th percentile, which is frequently associated with extreme precipitation, the traditional index becomes uncertain even for annual time series. For these cases, we can consider the estimation of the R99tot index for blocks of several adjacent years (e.g., pentads). However, this approach seriously limits the analysis of the interannual variability of extreme precipitation, especially for relatively short time series. We also computed R95tot using the long-term thresholds derived from gamma distribution (as was done for R95tt index). Spatial patterns of trends in this mixed index (no figure shown) are quite similar to those in the classical index and are different from the R95tt index for all seasons.

There are several lines of development in this study. First, the suggested approach to estimate heavy precipitation also allows for the analysis of changes of the structure of the precipitation probability distribution over the years. Estimating the partial contribution of precipitation of different occurrences to the total, we can better quantify trends and shorter period variability in the intensity of precipitation of different intensity classes and thus analyze the evolution of daily precipitation PDFs. Earlier attempts to perform such an analysis (e.g., Brunetti et al. 2004; Zolina et al. 2004, 2008) were always limited by the uncertainties associated with the estimation of contributions for some classes from a limited number of wet days. An important issue is the derivation of similar fractional distributions for alternative (to gamma) probability density functions used for daily precipitation analysis. As we have already mentioned, in many regions, especially in southern Europe, the gamma distribution does not fit well to the daily precipitation data and different PDFs should be used. Thus, to properly estimate the fractional contribution to the total from very wet days, it is necessary to derive relations similar to DFC [e.g., Eqs. (4), (5), (7), and (8)] for these distributions.

Finally, our approach allows for a more accurate comparison of the simulations of precipitation extremes using regional and global climate models with data. Since models may exhibit very different skills in representing extreme precipitation for different seasons and months, indices computed from the raw data (like R95tot) may not necessarily be effective for their validation against observations. Our new index can be effectively computed for individual seasons, and it can help to better evaluate the models’ abilities in simulating precipitation extremes.

## Acknowledgments

This study was supported by the North Rhine–Westphalia Academy of Science under the project “Large Scale Climate Changes and their Environmental Relevance.” We thank three anonymous reviewers and the editor for their comments and suggestions, which helped to greatly improve the first version of the manuscript. We appreciate the advice of Albert Klein Tank of KNMI (DeBilt) and discussions with Pavel Groisman of NCDC (Asheville). We also benefited from the support of the Russian Ministry of Science and Education under Grants 02.515.11.5032, NS-1566.2008.5, and MK–3459.2007.5 and from the Russian Foundation for Basic Research (Grant 08-05-00878-a).

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## Appendix

### Derivation of the DFC Distribution

*X*

_{1},

*X*

_{2}, … ,

*X*be independent and identically distributed random variables with a two-parameter gamma distribution (2). We are interested in the distribution of the ratio for the arbitrary chosen integer

_{n}*k*, 1 ≤

*k*≤

*n*, and for all real

*x*, 0 ≤

*x*≤ 1. Obviously, since the distribution of the ratio (A1) satisfies the equality it does not depend on the scale parameter

*β*. Thus, for the following derivation of the distribution, it is sufficient for simplicity to consider the case

*β*= 1. Introducing the notations we can state that the chain of equalities is valid: Inequality (A4) is valid because all variable are positive and identically distributed with probability being 1. Then, the inequality (A4) can be extended to Since a sum of the gamma-distributed independent random variables is also gamma distributed with an aggregated parameter, according to the full probability formula, inequality (A5) can be rewritten as follows: where

*p*(

*s*) = {1/Γ[(

*n*−

*k*)

*α*]} ×

*s*

^{(n−k)α−1}

*e*

^{−s}is the probability density function of the gamma distribution with the shape parameter (

*n*−

*k*)

*α*. In (A6),

*P*(

*U*<

*x*|

*V*=

*s*) denotes the conditional probability of the event 〈

*U*<

*x*〉 under the condition 〈

*V*=

*s*〉 (see, e.g., Prokhorov and Rozanov 1973). Then the inner probability under the integral in (A6) can be transformed to where

*γ*(

*μ*,

*z*) is the incomplete gamma function with the argument

*z*and the parameter

*μ*. Combining equalities (A5)–(A7) we arrive at the formula According to (Gradshteyn and Ryzhik 1971, 6.455.2) the integral in (A8) equals where is the hypergeometric Gauss series with

*a*= 1,

*b*=

*nα*,

*c*=

*kα*+ 1 and

*l*running from 0 to infinity. Eventually, we come to the relation which is the distribution sought and is identical to (5) and (7). For the distribution (7), the Gaussian hypergeometric function

*F*

_{2}

^{1}(

*a*,

*b*,

*c*,

*y*) is determined by (A10) with

*a*= 1,

*b*=

*nα*, and

*c*=

*kα*+ 1. Since it is a probability distribution, it satisfies the following properties: The distribution (A11) has a density function

*p*(

*x*) for 0 ≤

*x*≤ 1: which is identical to (6) and (8). This distribution has a singularity at 0 when

*α*< 1/

*k*and at 1 when

*α*< 1/(

*n*−

*k*).

Precipitation characteristics for three seasonal time series (refer to Fig. 1).

The percentages of cases (for all seasons during 1950–2000 and all 116 stations) when R*X*tot index falls to zero for different percentiles *X*. The remaining seasons are defined as March–May (MAM) and September–November (SON).