Acta hydrotechnica 32/57 (2019), Ljubljana ISSN 1581-0267 Open Access Journal Odprtodostopna revija UDK/UDC: 519.2:556.166(282.243.7) Izvirni znanstveni članek - Original scientific paper Prejeto/Received: 30.10.2019 Sprejeto/Accepted: 25.12.2019 Characterisation of the floods in the Danube River basin through FLOOD FREQUENCY AND SEASONALITY ANALYSIS Analiza značilnosti poplav v povodju reke Donave s pomočjo VERJETNOSTNE ANALIZE IN ANALIZE SEZONSKOSTI Martin Morlot12, Mitja Brilly1, Mojca Šraj1* 1 Faculty of Civil and Geodetic Engineering, University of Ljubljana, Jamova 2, Ljubljana, Slovenia 2 IHE Delft, Westvest 7, Delft, Netherlands Floods are natural disasters that cause extreme economic damage and therefore have a significant impact on society. Understanding the spatial and temporal characteristics exhibited by floods is one of the crucial parts of effective flood management. The Danube River with its basin is an important region in Europe and floods have occurred in the Danube River basin throughout history. Flood frequency analysis (FFA) and seasonality analysis were performed in this study using the annual maximum discharge series data from 86 gauging stations in order to form a comprehensive characterisation of floods in the Danube River basin. The results of the study demonstrate that some noticeable clusters of stations can be identified based on the best-fitting distribution regarding FFA. Furthermore, the best-fitting distributions regarding FFA for the stations in the Danube River basin are generalized extreme values (GEV) and log Pearson type 3 (LP3) distributions as among 86 considered gauging stations, 76 stations have one of these two distributions among their two best fits. Moreover, seasonality analysis demonstrates that large floods in the Danube River basin mainly occur in the spring, and flood seasonality in the basin is highly clustered. Keywords: Danube River basin, Floods, Flood Frequency Analysis (FFA), seasonality. Poplave so ena izmed naravnih nesreč, ki povzročajo veliko gospodarsko škodo in zato močno vplivajo na družbo. Razumevanje prostorskih in časovnih značilnosti poplav je eden od ključnih dejavnikov učinkovitega upravljanja voda. Povodje reke Donave je pomembna regija v Evropi, poplave v povodju pa se pojavljajo že skozi celo zgodovino. V raziskavi smo izdelali celovito analizo poplav v povodju reke Donave, in sicer verjetnostne analize visokih vod in analize sezonskosti na podlagi maksimalnih letnih pretokov s 87 vodomernih postaj. Rezultati verjetnostnih analiz kažejo opazno grupiranje vodomernih postaj glede na *Stik / Correspondence: mojca.sraj@fgg.uni-lj.si © Morlot M. et al.; This is an open-access article distributed under the terms of the Creative Commons Attribution - Noncommercial - ShareAlike 4.0 Licence. © Morlot M. et al.; Vsebina tega članka se sme uporabljati v skladu s pogoji licence Creative Commons Priznanje avtorstva -Nekomercialno - Deljenje pod enakimi pogoji 4.0. https://doi.org/10.15292/acta.hydro.2019.06 Abstract Izvleček 73 Morlot M. et al.: Characterisation of the floods in the Danube River basin through flood frequency and seasonality analysis - Analiza značilnosti poplav v povodju reke Donave s pomočjo verjetnostne analize in analize sezonskosti _Acta hydrotechnica 32/57 (2019), 73-89, Ljubljana_ najustreznejšo porazdelitveno funkcijo. Kot najustreznejši porazdelitveni funkciji za analize visokih vod na vodomernih postajah v povodju reke Donave sta se izkazali generalizirana porazdelitev ekstremnih vrednosti (GEV) in logaritemska Pearsonova 3 porazdelitev (LP3). Izmed 86 obravnavanih vodomernih postaj je bila kar za 76 postaj ena od omenjenih dveh funkcij med prvima dvema najustreznejšima. Analiza sezonskosti je pokazala, da se velike poplave v povodju reke Donave pojavljajo večinoma spomladi, sezonskost pojavljanja poplav pa je močno regijsko pogojena. Ključne besede: povodje Donave, poplave, verjetnostna analiza visokih vod, sezonskost. 1. Introduction Water resource management, planning, flood mapping, hydrologic/hydraulic designs, etc. all require a detailed knowledge about past flood events. Floods are defined in the Oxford dictionary as "an overflow of a large amount of water beyond its normal limits" (Stevenson, 2010). In more hydrologically sound language, Jarvis (1936) defined flood as "a relatively high flow as measured by either gauge height or discharge quantity", which is the definition used in this study. Naturally occurring flood events vary in their frequency. In order to determine characteristics of the flood events and more specifically the design discharge rates and their frequency of occurrence (i.e. return period), probability theory methods can be applied. Return period is defined as "the average interval of time within which the given flood will be equalled or exceeded once" (ASCE, 1953, p. 1221). Flood frequency analysis (FFA) is one of the most commonly applied hydrologic procedures used to analyse the relationship between discharge and return period of the floods, which is unique for each individual gauging station (e.g. WMO, 1989; Bezak et al. 2014; Poduje et al., 2014; Vittal et al., 2015). FFA can be performed using two types of sample definitions, namely annual maximum (AM) series and peaks-over-threshold (POT) series (e.g. Mitková and Onderka, 2010; Bezak et al., 2014; Sraj et al., 2015). The AM method relies on finding the maximum discharge rates for each year, which are then used as the annual maximum sample. This method's limitation is that individual large events, while still quite intense, may not actually be the yearly maximum and thus are not included in the sample. In order to compensate for this issue, the POT method can be applied; however, the POT method also comes with its own limitations, such as setting the threshold, which could be very subjective and may not always help distinguish between different events (Bezak et al., 2014). In most practical cases the annual maximum discharge series (AM) is used to perform FFA (e.g. Sraj et al. 2012; Bezak et al. 2014; 2016; Bezak and Mikos, 2014). FFA is usually performed using statistical analysis in combination with different probability distribution functions. Several theoretical distribution functions are available for modelling exceedance magnitudes (e.g. WMO, 1989; Bezak et al. 2014). Generalized extreme values (GEV), Gumbel (G), Log normal (LN), Pearson type 3 (P3), Log Pearson type 3 (LP3), Generalized logistics (GL) are among the most commonly used ones. Some countries have guidelines suggesting which distribution to use for FFA. The four distributions that are most commonly used for FFA of AM series in individual countries are the GEV distribution (Australia, Austria, Cyprus, Germany, France, Italy, Lithuania, Slovakia, Spain), the Gumbel distribution (Finland, Greece), the GL (UK), and the log-Pearson III (USA, Australia, Lithuania, Poland, Slovenia) (Kobierska et al., 2018). In order to fit the distribution parameters to the data, several methods can be applied, namely methods of moments, the method of L-moments, and maximum likelihood method (e.g. Hosking, 1990; Mahdi and Cenac, 2005; Grimaldi et al., 2011; Bezak et al., 2014). How the methods perform is influenced by the sample size and skewness of the data (Sankarasubramanian and Srinivasan, 1999); however, the method of L-moments has been found to be one of the most reliable methods in many researches because it is unbiased and more efficient than other methods (Lin and Vogel, 1993; Hosking 74 Morlot M. et al.: Characterisation of the floods in the Danube River basin through flood frequency and seasonality analysis - Analiza značilnosti poplav v povodju reke Donave s pomočjo verjetnostne analize in analize sezonskosti _Acta hydrotechnica 32/57 (2019), 73-89, Ljubljana_ and Wallis, 2005; Shahzad and Asghar, 2013; Bezak et al., 2014; Simkova and Picek, 2017). The selection of the best-fitting distribution can be a challenge since more than one distribution may fit the data well (Salas et al., 2013). Various tests can be applied in order to check the adequacy of the tested distribution functions and find the best-fitting distribution. It is therefore recommended that several of them should be applied, since the goodness-of-fit tests are not necessarily unbiased and may favour one distribution over another (Kidson and Richards, 2005). NIST/SEMATECH (2010) provides several possibilities for goodness-of-fit tests such as the point plot correlation coefficient (PPCC) test, the Anderson-Darling (AD) test, and the Kolmogorov-Smirnov (K-S) test. Other statistical tests, such as root-mean-square error (RMSE), mean absolute error (MAE), and the Akaike information criterion (AIC), are commonly used in statistics and have proven to be reliable methods of findings the best fits against given data (Iacobellis et al., 2010). Critical values for different tests can be find in the literature (e.g. (Chowdhury et al., 1991; Zeng et al., 2015; Bezak and Mikos, 2014). Seasonality analysis is an important part of flood characterisation. Several studies on the seasonal patterns of discharge data in Europe or in individual parts of the Danube River basin can be found in the literature (e.g. Parajka et al., 2009; 2010; Barbalic and Petras, 2012; Hall and Bloschl, 2017; Bloschl et al., 2017; 2019). Parajka et al. (2009) analysed the precipitation and discharge data from stations in Slovakia and Austria. They argued that seasonality depended primarily on local characteristics. The study of Hall and Bloschl (2017) suggests that the geographical location of a station in Europe is a good indicator of its seasonal flood characteristics. The frequency of floods in the Danube River basin increased in the last decades (e.g. major floods in 2002, 2005, 2006, 2009, 2010, 2013, and 2014), increasing the need for a more effective and harmonized regional and cross-border cooperation on flood protection (Sraj et al., 2019). Furthermore, reliable design discharge estimation is still a challenge for engineers and essential part of flood protection measures. Understanding the spatial and temporal characteristics of floods is one of the crucial parts of their effective management. Therefore, the main aims of the study are as follows: (i) a comprehensive analysis of discharge data series of the gauging stations in the Danube River basin, (ii) finding the most appropriate distribution functions for FFA in the Danube River basin and (iii) characterisation of the floods in the Danube River basin as regards seasonality. 2. Study area and data 2.1 The Danube River basin The Danube is the second longest river in Europe after the Volga, with a length of 2850 km (Jones, 2007). It is a truly pan-European river, as it flows through 10 countries and its basin extends into 9 more. It is recognized as the world's most international basin. The countries in its basin are Germany, Austria, Slovakia, Hungary, Croatia, Serbia, Bulgaria, Romania, Moldova, Ukraine, Poland, Switzerland, Italy, Slovenia, Bosnia and Herzegovina, Montenegro, Macedonia, and Albania (Fig. 1). The Danube River basin extends over an area of 817,000 km2 and is home to a population of 83 million people. As such, it is very important to local economies that water be provided for industry, agriculture, and municipalities, and, in some countries, it is important also for transportation. However, it is as a result subject to heavy impacts from such activities, with pollution being a growing concern (Jones, 2007; ICPDR, 2011). 75 Morlot M. et al.: Characterisation of the floods in the Danube River basin through flood frequency and seasonality analysis - Analiza značilnosti poplav v povodju reke Donave s pomočjo verjetnostne analize in analize sezonskosti _Acta hydrotechnica 32/57 (2019), 73-89, Ljubljana_ Figure 1: The Danube basin area. Slika 1: Povodje reke Donave. The Danube has many tributary rivers (327) and the four most important that contribute at least 10% of the Danube final average flow are the Drava, the Tisza, the Sava, and the Inn (ICPDR, 2011). Other important tributaries are also the Moravia, the Iskar, the Siret, and the Prut River. The Danube River basin area has diverse geographic and climatic characteristics. An array of different climates is found in the basin area with continental climates influencing the majority of the basin and a minority being influenced by the Atlantic and Mediterranean climates (UNDP/GEF, 2010). Several mountain chains are also part of the basin. The Alps are located at the basin's northwest, the Carpathian Mountains in the east/centre, and the Dinarides constitute some of its south-west border. These are generally the wettest parts of the basin, with areas in the Alps that see rainfall as high as 3200 mm/year and between 750 and 2000 mm/year in the Carpathian Mountains. The mountains influence a large part of the basin with their rain shadows (UNDP/GEF, 2010). Several low-lying areas can be found as well. The lowlands of Moravia in the Czech Republic and the Tisza valley in Romania are fairly dry (up to 750 mm/year) and the Danube Delta is the driest (400 mm/year of precipitation) (UNDP/GEF, 2010). The Danube River basin has experienced a number of significant floods over recent centuries with a total of 78 in the past 900 years, of which 23 occurred in the 18th century, before any significant flood protection was designed (ICPDR, 2011). Water marks have set record levels three times since 2002 and 5 significant floods have occurred in the past decade (ICPDR, 2011). 2.2 Data The daily discharge time series used in the study has been obtained from the Institute of Hydrology at the Slovak Academy of Sciences as part of the UNESCO International Hydrological Program projects (Ninov and Brilly, 2017). The database consists of daily discharges from 86 measuring stations (21 of them being directly on the Danube River) in 11 countries in the Danube River basin (Table 1, Fig. 2). The number of available years for data varies from station to station with the maximum being for the station of Bratislava in Slovakia (D10) with 131 years of available data and the minimum being for a station Kozluk Jajce 76 Morlot M. et al.: Characterisation of the floods in the Danube River basin through flood frequency and seasonality analysis - Analiza značilnosti poplav v povodju reke Donave s pomočjo verjetnostne analize in analize sezonskosti _Acta hydrotechnica 32/57 (2019), 73-89, Ljubljana_ located in Bosnia (76) with 20 years of available data. The list of considered stations and their main characteristics is presented in Table 1. All data series were manually and visually examined (plotting values against time) for data errors (e.g. jumps in timing, large differences to surrounding stations), stability across time, and potential outliers to check their homogeneity. Additionally, it should be mentioned that there are several gaps in the data; a few sets of data have gaps during the Second World War (1939-1945), others have some years missing during the Balkan Wars (in the 1990s), and some stations have random years missing inexplicably. It should also be noted that possible human influences on discharges were not investigated in this study; therefore, no station was excluded from analysis for this reason. It is also important to notice that the considered stations and the rivers they are located on have different characteristics regarding their elevation, basin characteristics, climate etc. This means that the range of flow-rates and their averages vary greatly from station to station. Figure 2: The considered water-gauging stations. Station codes are presented in Table 1. Slika 2: Obravnavane vodomerne postaje. Številke postaj so podane v preglednici 1. Table 1: List of the considered gauging stations and their basic characteristics. Preglednica 1: Seznam upoštevanih vodomernih postaj in njihovih lastnosti. No. Code Country Station Name Lat Lon Basin Area Height above # of years fm3l see level [ml 1 1 GE Inn-Oberaudorf 47.65 12.20 9712 464 107 2 2 GE Inn-Passau-Ingling 48.65 13.45 26084 289 87 3 3 GE Lech-Landsberg 48.04 10.88 2295 582 107 4 4 GE Regen-Regenstauf -> replaced 49.22 12.17 2658 337 107 5 5 GE Salzach-Burghausen 48.16 12.83 6649 352 107 6 6 GE Issar-Plattling 48.77 12.88 8839 316 82 7 7 AT Enns - Steyr, Ortskai 48.04 14.43 5915 284 55 8 8 AT Traun - Ebensee 47.80 13.76 1257.6 422 55 9 9 CZ Morava-kromeriz 49.30 17.40 7014 184 93 10 10 CZ Morava - Straznice 48.93 17.30 9147 163 88 77 Morlot M. et al.: Characterisation of the floods in the Danube River basin through flood frequency and seasonality analysis - Analiza značilnosti poplav v povodju reke Donave s pomočjo verjetnostne analize in analize sezonskosti _Acta hydrotechnica 32/57 (2019), 73-89, Ljubljana_ 11 11 CZ Jihlava - Ivanice 12 12 CZ Svratka - Zidlochovice 13 13 SK Morava-Moravsky Jan 14 14 SK Bela-Podbanske 15 15 SK Vah-Liptovsky Mikulas 16 16 SK Vah- (Trnovec - 1921-1962) Sala 17 17 SK Hron-Banska Bystrica 18 18 SK Hron-Brehy 19 19 SK Kysuca- Kysucke Nove Mesto 20 20 SK Topla-Hanusovce nad Toplov 21 21 SK Krupinica-Plastovce 22 22 SK Ipel-Holisa 23 23 SK NITRA-NITRIANSKA STREDA 24 24 HU Raba-Arpas 25 25 HU Tisza-Vasarosnameny 26 26 HU Tisza-Szolnok 27 27 HU Tisza-Szeged 28 28 HU SZAMOS-C SENGER 29 29 HU MAROS-MAKO 30 30 HU SAJO-FELSOEZSOLCA 31 35 SR Tisza-Senta 32 36 SR Lim-Prijepolje 33 37 SR Drina-Bajina Basta 34 38 SR Sava-Sremska Mitrovica 35 39 SR Moravica-Arilje 36 40 SR Ibar-Lopatnica Lakat 37 41 SR Zapadna Morava - Jasika 38 42 SR Juzna Morava-Mojsinje 39 43 SR Velika Morava-Ljubicevski most 40 44 HR Drava-Donji Miholjac 41 45 HR Kupa-Jamnicka kiselica 42 46 HR Sava-Zagreb 43 47 HR Orljava-Pleternica most 44 48 HR Una-Kostajnica 45 49 SL Sava-Catez 46 50 SL Krka-Podboéje 47 51 SL Savinja Laško 48 52 SL Sava-Litija 49 57 RO Szamos-Satu Mare 50 60 RO Crisul Negru-Zerind 51 62 RO Maros-Arad 52 64 UK Siret-Storozhinec 53 65 UK Prut-Chernivcy 54 66 UK Tisza-Rakhiv 55 67 UK Tisza-Vylok 56 68 UK Teresva-Ust-Chorna 57 69 UK Rika-Mizhhirya 58 70 UK Latorycya-Mucacheve 59 71 UK Latorycya-Chop 60 72 UK Uzh-Uzhhorod 61 73 UK Prut-Jaremcha 62 74 BA Una-Kralje 49.08 16.41 2681 194 85 49.04 16.62 3939 178 93 48.60 16.94 24129.3 146 85 49.14 19.90 93.49 923 79 49.09 19.61 1107.21 548 86 48.16 17.88 11217.61 109 86 48.73 19.13 1766.48 334 76 48.41 18.65 3821.38 195 76 49.30 18.79 955.09 346 76 49.03 21.50 1050.05 160 76 48.16 18.96 302.79 139 76 48.30 19.74 685.67 172 76 48.30 18.10 2093.71 158 78 47.51 17.40 6610 NA 54 48.12 22.34 25100 102 126 47.17 20.19 73113 NA 88 46.25 20.17 138408 74 87 47.83 22.68 15283 113 78 46.22 20.48 30149 80 78 48.11 20.84 6440 107 117 45.93 20.10 141715 73 77 43.38 19.63 3160 442 83 43.97 19.55 14797 211 82 44.98 19.62 87966 72 82 43.75 20.12 832 322 45 43.63 20.57 7818 225 60 43.62 21.30 14721 139 49 43.63 21.48 15390 136 60 44.58 21.12 37320 73 77 45.78 18.20 37142 89 80 45.55 15.86 6895 101 53 45.79 15.96 12450 112 81 45.29 17.81 745 114 63 45.22 16.55 8876 103 73 45.89 15.61 10186.45 137 51 45.87 15.47 2238.12 146 73 46.15 15.23 1663.6 215 93 46.06 14.82 4821.43 230 79 47.80 22.88 15388 118 59 46.63 21.52 3702 87 58 46.18 21.32 27280 118 57 48.15 25.72 672 356 52 48.32 25.92 6890 165 93 48.07 24.22 1070 435 59 48.10 22.83 9140 118 52 48.33 23.93 572 524 58 48.53 23.50 550 439 59 48.45 22.72 1360 123 59 48.45 22.20 2870 105 49 48.62 22.30 1970 114 59 48.45 24.55 597 507 56 44.84 15.85 NA 209 25 78 Morlot M. et al.: Characterisation of the floods in the Danube River basin through flood frequency and seasonality analysis - Analiza značilnosti poplav v povodju reke Donave s pomočjo verjetnostne analize in analize sezonskosti _Acta hydrotechnica 32/57 (2019), 73-89, Ljubljana_ 63 75 BA Sana-Sanski Most 44.77 16.68 2008 156 27 64 76 BA Vrbas-Kozluk Jajce 44.37 17.29 3161 342 20 65 77 BA Bosna-Maglaj 44.54 18.09 6619 150 26 66 D01 GE Danube-Berg 48.27 9.73 4047 490 79 67 D02 GE Danube-Ingolstadt 48.75 11.42 20001 360 84 Danube-Regensburg- 68 D03 GE Schwabelweis 49.02 12.14 35399 324 84 69 D04 GE Danube-Pfelling 48.88 12.75 37687 308 82 70 D05 GE Danube - Hofkirchen 48.68 13.12 47496 300 107 71 D06 GE Danube-Achleiten 48.58 13.50 76653 288 107 72 D07 AT Danube- Linz (ab 1979: Aschach) 48.31 14.30 79490 248 60 Danube - Stein-Krems / 73 D08 AT Kienstock 48.38 15.46 96028.4 194 104 74 D09 AT Danube - Wien-Nußdorf 48.25 16.38 101700 156 107 75 D10 SK Donau - Bratislava 48.14 17.11 131338 128 131 76 D11 HU Danube-Nagymaro s 47.78 18.95 183534 100 115 77 D12 HU Danube-Mohac s 46.00 18.67 209064 80 78 78 D13 SR Danube-Bezdan 45.85 18.87 210250 81 76 79 D14 SR Danube-Bogojevo 45.53 19.08 251593 77 77 80 D15 SR Danube-Pancevo 44.87 20.64 525009 67 77 81 D16 SR Danube-Veliko Gradiste 44.80 21.40 570375 62 77 82 D17 RO Donau - Orsova 44.70 22.40 576232 44 106 83 D18 RO Danube_Zimnicea 43.63 25.36 658400 16 107 84 D19 RO Donau - Vadu Oii-Hirsova 44.68 27.92 709100 3 69 85 D20 RO Danube Ceatal Izmail 45.22 28.72 807000 0.6 77 86 D21 RO Danube - Reni 45.47 28.22 805700 4 90 3. Methods 3.1 Flood frequency analysis In the study, the AM method was applied to define a sample as it is an objective method, independent of the subjective determination of a threshold, which would be different from station to station. Furthermore, some authors reported that the advantage of the POT method is noticeable mainly on smaller samples (Robson and Reed, 1999; Mitkova and Onderka, 2010), which is not the case for most of the considered stations in the study. In order to define the AM samples for all considered stations of the Danube River basin, a code written in the R programming language was applied (R Core Team, 2018) using the daily discharge data series. The maximum daily discharge for each year, as well as the corresponding date of this discharge being filtered out. This was done using the package "hydroTSM" (Zambrano-Bigiarini, 2017b). It should be noted that not each annual maximum necessarily results in a flood; however, AM sampling is nevertheless the most commonly used sample definition method in flood frequency analysis and flood characterisation (e.g. Bezak et al., 2014; Bezak et al., 2016; Blöschl et al., 2017; Blöschl et al., 2019). The parameters of the considered distributions were estimated using the method of L-moments, which has been found as one of the most reliable methods in many studies (e.g. Hosking and Wallis, 2005; Bezak et al., 2014; Simkova and Picek, 2017). Additionally, L-moments are more robust and less sensitive to outliers than other methods (Hosking, 1990). Moreover, it was found to yield the best fit to the AM discharge sample for the Litija station on the Sava River, which is also a part of the Danube basin (Sraj et al., 2012). A detailed description of L-moments calculation procedure can be found in Hosking (1990). In the study, L-moments were calculated using the "lmomco" package (Asquith, 2018) in R software (R Core Team, 2018). The package generates the L-moments up to an order of 79 Morlot M. et al.: Characterisation of the floods in the Danube River basin through flood frequency and seasonality analysis - Analiza značilnosti poplav v povodju reke Donave s pomočjo verjetnostne analize in analize sezonskosti _Acta hydrotechnica 32/57 (2019), 73-89, Ljubljana_ 5, along with their ratios. Log L-moments needed for certain distributions were also calculated. In the next step of the study, several of the most commonly used distributions in univariate flood frequency analysis (FFA) were applied in order to find the best-fitting distribution, namely the General extreme value (GEV), Gumbel (G), Log normal (LN), Pearson type 3 (P3), Log Pearson type 3 (LP3), and Generalized logistics (GL). These distributions have been previously applied for FFA in many European studies. The LP3 and LN were tested on four stations in the Danube River basin by (Mitkova and Onderka, 2010). The LP3 is also the distribution recommended by the USGS for the determination of flood frequencies (England et al., 2018). The Gumbel and GEV methods were applied for FFA on data from rivers in France (Kochanek et al., 2014), being a country right beside the basin area of the Danube River. Sraj et al. (2012) compared all mentioned distributions in their study using the data of the Litija station on the Sava River, which is also a part of the Danube River basin. Distribution functions and equations for distribution parameter estimation can be found in Sraj et al. (2012). Distributions were fit to the AM samples with a code in the R programming language (R Core Team, 2018) using several functions. Traditional statistics such as root-mean-square-error (RMSE), mean-absolute-error (MAE), Akaike information criteria (AIC), as well as specific goodness-of-fit tests, such as the probability plot correlation coefficient (PPCC), Kolmogorov-Smirnov test (K-S), and Anderson-Darling test (AD), were applied to find the best-fitting distribution in this study. Detailed descriptions and equations can be found in Sraj et al. (2012). Several of these tests can be applied using packages in R software, whereas others had to be programmed manually. MAE and RMSE tests could be found in the "hydroGOF" package (Zambrano-Bigiarini, 2017a) and the A-D test could be found in the package "ADGof" (Bellosta, 2011). The K-S test is part of the package "stats" (R Core Team, 2018). PPCC and AIC tests were programmed manually for purposes of this study (Morlot, 2018). The best-fitting distributions are then ranked accordingly for each test. This was done by assigning the highest value (6) to the best-fitting distribution and so on to the worst-fitting distribution assigned with the lowest value (1). Finding the best result depends on the test characteristics, since for MAE, RMSE, and AIC the lower the value of statistics means the better the fit. According to the PPCC test, better fits have values closest to 1. For the K-S and A-D tests, values closest to 0 are better. In the next step all values of ranks (from 1 to 6) were added together for each distribution of individual station, with the maximum being 36 (6 distributions * 6 tests). The highest score provides the best-fitting distribution. Using this ranking system, it was possible to find the best-fitting distribution for each station of the Danube River basin. Based on these results as well as the geographical coordinates of the stations, a map showing the best-fitting distribution for each individual station in the Danube River basin was created. 3.2 Seasonality Seasonality analysis gives us important information about the time of flooding. It provides the most common season and dates for the occurrence of past floods, and allows us to predict the same parameters for the future floods (Bayliss and Jones, 1993; Burn, 1997). In order to estimate the seasonality of the floods (AM) and their variability, several steps must be conducted. Seasonality can be graphically presented by a circular diagram (Burn, 1997; Bezak et al., 2016; Hall and Bloschl, 2017). The day of the year is transformed into an angle 0t (for this purpose called angular date or Julian date J, which is the number of the day in the year between 0 and 365 (or 366 in a leap years)) by equation (1) (Bayliss and Jones, 1993): 1 365 (1) An average flood event, its average date of occurrence, and its season are also found for the AM series, as well as the variability between dates. The methodology used was proposed by Burn (1997) and defined by equations (2): 80 Morlot M. et al.: Characterisation of the floods in the Danube River basin through flood frequency and seasonality analysis - Analiza značilnosti poplav v povodju reke Donave s pomočjo verjetnostne analize in analize sezonskosti _Acta hydrotechnica 32/57 (2019), 73-89, Ljubljana_ x = 1If=1cos(0i) (2) y i n = -) sin(9i) i = 1 6 = tan-1 ( = r = jx2 + y2 where r represents the strength of seasonality. If r is near 1, the seasonality is strong and most of the considered events occurred at the same time of year. On the other hand, if r is closer to 0, the seasonality is not significant. Graphical presentation of the seasonality analysis was conducted using the 'circlize' package of the R programming language (Gu et al., 2014). For the purpose of the study, the seasons were assumed to occur always at the same dates. Season start and end dates and corresponding Julian and angular dates are presented in Table 2. Table 2: Definition of the seasons in the study. Preglednica 2: Opredelitev časovnih obdobij v raziskavi. Season Start and end of the season Julian date Angular date Winter Dec 22 356 -0.15 Mar 20 79 1.36 Spring Mar 21 80 1.37 Jun 21 172 2.96 Summer Jun 22 173 2.97 Sep 22 265 4.56 Fall Sep 23 266 4.57 Dec 21 355 6.11 4. Results and discussion 4.1 Flood frequency analysis The main aim of FFA was a comprehensive analysis of discharge data series from 86 gauging stations in the Danube River basin and finding the most appropriate distribution function for FFA in the Danube River basin. The best-fitting distribution function for each individual gauging station was found according to methodology described in the Methods section. The results of the analysis are presented in Fig 3, Fig. 4 and in Table 3. The most common best-fitting distributions are as follows: GEV (22 stations), Pearson type 3 (20 stations), log Pearson type 3 (18), and GL (18 stations). On the other hand, Gumbel (3 stations) and LN (5 stations) were less commonly selected as the best-fitting distributions for the considered gauging stations in the Danube River basin (Table 3). Table 3: No. of stations as regards the best-fitting distribution. Preglednica 3: Stevilo postaj glede na najbolje prilegajoco se porazdelitev. The best-fitting Rank 1 Rank 2 distribution (No. of (No. of stations) stations) GL 18 8 GEV 22 20 Gumbel 3 3 LN 5 8 LP3 18 37 P3 20 10 81 Morlot M. et al.: Characterisation of the floods in the Danube River basin through flood frequency and seasonality analysis - Analiza značilnosti poplav v povodju reke Donave s pomočjo verjetnostne analize in analize sezonskosti _Acta hydrotechnica 32/57 (2019), 73-89, Ljubljana_ Figure 3: The best-fitting distribution for individual gauging station in the Danube River basin (Morlot, 2018). Slika 3: Najbolje prilegajoča se porazdelitev za posamezno vodomerno postajo v povodju reke Donave (Morlot, 2018). Figure 4: Gauging stations with Log Pearson type 3 or GEVamong their two best-fitting distributions (Morlot, 2018). Slika 4: Vodomerne postaje s porazdelitvijo Log Pearson 3 ali GEV, kot eno izmed dveh najbolje se prilegajočih (Morlot, 2018). 82 Morlot M. et al.: Characterisation of the floods in the Danube River basin through flood frequency and seasonality analysis - Analiza značilnosti poplav v povodju reke Donave s pomočjo verjetnostne analize in analize sezonskosti _Acta hydrotechnica 32/57 (2019), 73-89, Ljubljana_ Statistical test results and scores according to the methodology described in Methods section are presented in Fig. 5. Detailed results for each individual station can be found in Morlot (2018). It should be mentioned that 13 of the considered stations have ties in the first place (two best-fitting distributions), with a score difference of zero. Furthermore, for the large majority of the considered stations (57 out of 86), the score difference between the first- and the second-best-fitting distribution is less than or equal to 4 (out of 36), indicating good performance for the best-fitting distributions. Some noticeable station clusters can be identified in Fig. 3. However, no large grouping of stations is particularly obvious in general. For example, the downstream section of the Sava River and its tributaries seem to favour the Pearson type 3 distribution, whereas the upstream section of the Sava River favours GEV and GL distributions. Furthermore, five consecutive stations on the Danube River in Austria yield GL distribution as the best-fitting one, whereas five other stations on the Danube River between Serbia and Romania favour GEV as the best-fitting distribution. Histogram of scores CM O S" °° C ffl 3