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Analysis of extreme hydrological Events on THE danube using the Peak Over Threshold method

Analysis of extreme hydrological Events on THE danube using the Peak Over Threshold method J. Hydrol. Hydromech., 58, 2010, 2, 88­101 DOI: 10.2478/v10098-010-0009-x VERONIKA BACOVÁ-MITKOVÁ, MILAN ONDERKA Institute of Hydrology SAS, Racianska 75, 831 02 Bratislava 3, Slovakia; Mailto: mitkova@uh.savba.sk The Peak Over Threshold Method (POT) was used as an alternative technique to the traditional analysis of annual discharge maxima of the Danube River. The POT method was applied to a time-series of daily discharge values covering a period of 60 s (1931­1990) at the following gauge stations: , , Wien, Bratislava and . The first part of the paper presents the use of the POT method and how it was applied to daily discharges. All mean daily discharges exceeding a defined threshold were considered in the POT analysis. Based on the POT waves independence criteria the maximum daily discharge data were selected. Two theoretical log-normal (LN) and Log-Pearson III (LP3) distributions were used to calculate the probability of exceeding annual maximum discharges. Performance of the POT method was compared to the theoretical distributions (LN, LP3). The influence of the data series length on the estimation of the N- discharges by POT method was carried out too. Therefore, with regard to later regulations along the Danube channel bank the 40, 20 and 10- time data series were chosen in early of the 60- period and second analysed time data series were selected from the end of the 60- period. Our results suggest that the POT method can provide adequate and comparable estimates of N- discharges for more stations with short temporal coverage. KEY WORDS: Danube River, Extreme Hydrological Events, Flood Frequency Analysis, Peaks Over Threshold (POT) Method, Daily Runoff, Return Period. Veronika Bacová-Mitková, Milan Onderka: ANALÝZA EXTRÉMNYCH HYDROLOGICKÝCH SITUÁCIÍ NA DUNAJI VYUZITÍM METÓDY POT. J. Hydrol. Hydromech., 58, 2010, 2; 41 lit., 7 obr., 5 tab. Príspevok sa zaoberá analýzou extrémnych hydrologických udalostí na Dunaji metódou Peak Over Threshold (POT). Metóda POT sa pouzíva ako alternatíva urcovania N-rocných prietokov k metóde rocných maxím pri analýzach extrémnych hydrologických udalostí. Pre výskyt vrcholových prietokov sa zvycajne predpokladá Poissonova distribúcia. Základnými vstupnými údajmi pre statistickú analýzu sú 60-rocné casové rady priemerných denných prietokov a 60-rocné rady maximálnych rocných prietokov v nami zvolených staniciach: , , Viede, Bratislava a ­ za obdobie 1931­1990. Extrémne hydrologické udalosti na Dunaji boli analyzované metódou POT, ktorá zaha vsetky maximálne denné prietoky povodní za dané obdobie, presahujúce zvolenú prahovú hodnotu. Na zostavenie teoretickej ciary prekrocenia boli vybrané dve teoretické rozdelenia pravdepodobnosti: logaritmicko-normálne rozdelenie (LN) a Pearsonovo rozdelenie III. typu (LP III). Druhým cieom príspevku bolo analyzova vplyv zmeny dzky casového radu na odhad N-rocných prietokov. V práci boli 60-rocné casové rady údajov skrátené na 40, 20 a 10-rocné rady. V závere sme porovnali a zhodnotili získané výsledky statistických odhadov N-rocných prietokov vo zvolených staniciach. Z výsledkov analýzy vyplýva, ze metóda POT dáva pomerne dobré odhady N-rocných prietokov aj pre krátke casové rady údajov. KÚCOVÉ SLOVÁ: tok Dunaja, extrémne hydrologické udalosti, frekvencia výskytu povodní, metóda POT, denný prietok, doba opakovania prietokov. Introduction The annual maximum series approach is the most frequent method used in probabilistic hydrology. 88 However, this approach considers only one value per , which may result in loss of information. For example, some peaks within a may be greater than the maximum discharge in other s hence they can be ignored (Kite, 1997; Chow et al., 1988). In principle, annual discharge maxima contain critical information on the peak flow, however their use is limited by two factors: 1. the length of the series of annual maxima can be very short, 2. the annual maxima time-series may be interrupted and thus they may not allow us to infer the antecedent conditions in the basin preceding a given peak. The first limiting factor produces uncertainties in interpreting statistical analyses, while the latter constrain implies that statistical models built on a phenomenological basis must rely on ancillary data in order to validate the underling hypotheses on the antecedent state of soil moisture (Claps and Laio, 2003). This situation is avoided in the Peaks Over Threshold method (POT). Data series of the POT method considers all values exceeding a certain predefined threshold (Bayliss, 1999; Rao and Hamed, 2000). The POT method has been proposed as an alternative analytical tool to the method of annual discharge maxima for analysis of extreme hydrological events. This method was discussed in a number of papers (Langbein, 1949; Todorovic, 1970; Cunnane, 1973; Rosbjerg, 1977; Madsen at al. 1997 and Lang at al. 1999). In practice, however, it seems to be meaningful to consider not only the annual discharge maxima but also flood events that exceed safety limits. The idea is to derive the distribution and magnitude of annual floods from assumed distributions of the annual occurrence of events and the magnitudes of the POT. Shane and Lynn (1964) assumed that the Poisson distribution is valid for the occurrence of flood peaks, and flood magnitudes exhibit an exponential distribution. Zelenhasic (1970) investigated the distribution of annual maximum floods assuming the Poisson distribution for annual event occurrence and an exponential distribution for their magnitudes. Önöz and Bayazit (2001) showed that for flood estimation, negative binomical (or binomical) models in combination with the exponential distribution of peak heights are almost identical in performance as the Poisson model. This result is in agreement with the findings of Kirby (1969) and Cunnane (1979), and makes it unnecessary to prefer the binomical or negative binomical models even when the Poisson process hypothesis is rejected by statistical tests. It is easier to use the Poisson model because it leads to much simpler expressions for the N- flood and its sampling variance. In the 1980s and early 1990s the statistical method was generalized in different ways, including time-dependent parameters (North, 1980); correlated peak values (Rosbjerg, 1985), risk estimation (Konecny and Nachtnebel, 1985; Rasmussen and Rosbjerg, 1989), Bayesian approaches (Roussele and Hindie, 1976; Rasmussen and Rosbjerg, 1991 a)), a fixed number of peaks (Buishand, 1989), seasonality (Rasmussen and Rosbjerg, 1991b), and other alternatives to the exponential distribution of threshold exceedances such as Weibull (Ekanayake and Cruise, 1993), Lognormal (Rosbjerg, 1987 b); Rosbjerg et al., 1991), and the generalised Pareto (Davison and Smith, 1990; Wang, 1991; Rosbjerg et al., 1992). An extensive analysis of the methods for calculation of N- discharges is given in the paper of Szolgay et al. 2003; Kohnová and Szolgay, 2000. Baca and Mitková, 2007 reported changes in the occurrence frequency of extreme hydrological events in a small agricultural basin using the POT method. The pros and cons of using either the POT method or series of annual discharge maxima for a statistical estimation of design values for the Danube-Bratislava station was investigated by Mitková et al. (2003). Recently Lang at al. (1999) discussed the issue of threshold selection, models suitable for the occurrence processes of the peaks and for the distribution of their magnitudes, and the correspondence between the POT and annual maximum flood distributions. They concluded that the main difficulties of the POT approach concern the selection of the threshold level and of the occurrence process. Several applications of the POT method for estimates of design discharge have been shown in worldwide; however in Slovakia, this method has been rarely used. For example in the UK, the POT flood database of the Summary statistics and seasonality was carried out for POT method. This report describes the growth of the POT database, the data extraction procedures adopted, and it considers briefly the seasonality of flooding. The database now holds over 77 000 peaks for 870 gauging stations throughout the UK with an average record length of nearly 20 s (Baylis and Jones, 1993). The paper consists of two parts. The first part deals with the estimation of N- discharges from the 1931­1990 period. The second part concerns the estimation of N- discharges from shorter data series. The purpose of this paper is to present alternative approaches for analysis of extreme hydrological events. Discharges with different return periods were estimated by the POT method and were finally compared with selected theoretical distributions of discharge maxima. The POT 89 method can give representative values of extreme flows from shorter time series and can be used to assess N- discharges. The POT method The basic idea is to extract from the daily discharges sequences a sample of peaks containing more than one flood peak per , in order to increase the available information with respect to the annual maximum analysis. The POT series includes all maximum discharges over the threshold. The number of the peaks in statistical series must be higher than N, where N is the number of s on record. The first threshold can be chosen near the longterm mean discharge. This value is rather low; POT series can have high diffusion and can include some insignificant maxims. Therefore, a threshold value is usually chosen so that POT data series includes in average 4 maximum values per . In order to provide independence of the POT data the following criterions were used (Bayliss, 1999): - Time period between two consequent peaks must be at least three times longer than the time of increasing of the first wave. - Minimum discharge between two peaks must be less than 2/3 of the peak height recoded during the first wave. 50 40 discharge 30 20 10 0 0 1 2 3 day 4 5 6 A B E D F peak C is time independent from E, and minimum discharge between E and C is less than 2/3 of the E, therefore the peak C can be included into the POT series. The peak of the wave A is less than the threshold and hence it cannot be included into the POT data series. The POT method is characterised by two main variables: - number of peaks in each ; - flow exceedances over threshold Z = x - xB . The occurrence of discharge maxima is a random process defined as: ( t ) = sup ; = x ­ x B , (1) where: ( t ) ­ occurrence of the discharge in time, ­ exceedance of the discharge, xB ­ threshold value (discharge), x ­ value of the current discharge (maxima discharge). The distribution function of annual maxima is F ( x ) = P { ( t ) x} . Number of peaks The number of peaks from the interval (0, t) ­ one in this analysis, is a random variable t that can take values 0, 1, 2, ... with probabilities p ( t ) = P { t = }. The occurrence of peaks in this time interval is described as a Markov process with the intensity function (2) ( t , ) = lim P { ( t + t ) - ( t )} t 0 (3) where ( t , ) ­ occurrence of peaks in time interval, ­ number of the peak, t ­ time, t ­ time difference of the occurrence of the number of peaks. The probability of occurrence of peak exceedances is p ' ( t ) = ( t , - 1) p -1 ( t ) - ( t , ) p ( t ) . p '0 ( t ) = - ( t ,0 ) p0 ( t ) . (4) The solution to this equation (Eq. (4)) represents the probability law of occurrence of peaks and depends on the form of intensity function (Vukmirovic, 1990). Fig. 1. Methodology of the QPOT selection. (Peaks E, C are included to POT data series). Obr. 1. Aplikácia výberu údajov do súboru QPOT. (Vrcholy E, C sú zahrnuté do POT radu). Application of the POT method for peak selection is presented in Fig. 1. The maximum discharge was obtained during the fourth day, therefore the peak E can be automatically included into the POT series. Increasing time of the wave E is about 15 hours. The peak D occurres less than 15 hours before peak E. Wave D is dependent, hence we cannot include this peak into the POT data series (the same situation is for peak wave F). The 90 ( t ) Poisson ( t , ) = ( t )(1 - a ) Bernoulli (binomical) ( t ) (1 + b ) Negative binomical Peaks over threshold Distribution function for peak exceedances flow is defined as: H ( z ) = P {Z z} . (6) (5) Distribution of annual maximum is obtained by combining the distributions of the number of peaks and distributions of peak exceedances over threshold value (Todorovic, 1970): F ( x ) = p0 + H ( x ) p ( t ) . =1 (8) If the number of peaks follows the Poisson distributions then F(x) has the form F ( x ) = exp {- [1 + H ( x)]} . (9) Distribution can be generalized as three-parameter gamma distribution with density function: k + 1 a k +1 k -1 a z a a z exp a h( z) = k k k +1 a a Return period Return period is defined by well-known equation (7) R ( x) = 1 . 1- F ( x) (10) Study area Distributions like two-parameter gamma, Weibull's, Erlang's or exponential are special cases of this general distribution. For estimation is recommended one-parameter distribution (exponential, Rayleigh's) or two-parameters distribution (Weibull or gamma). Exponential H(z) = 1 ­ exp(­z/), Weibull H(z) = 1 ­ exp(z/), where: z ­ variables (magnitude of exceedance discharge over threshold), µ, , ­ distribution parameters. Annual maximum The Danube is the second greatest river in Europe. The length of the Danube River is proximately 2 830 km and drains a basin covering 817 000 km2. Originating in the Black Forest in Germany at the confluence of the Brigach and the Breg streams, the Danube flows over some 2850 km before emptying into the Black Sea via the Danube Delta in Romania. The Slovak part of the Danube River is situated from rkm 1708.2 (river km) to rkm 1880.2. About 7.5 km of the river creates a natural border to Austria, 22.5 km is in Slovakia and the rest of 142 km is the state border to Hungary (Fig. 2). Between the Vienna basin and the Danube lowland, the Danube flows in concentrated channel with relatively high bed slope. After leaving the Small Morava SK Devín BRATISLAVA Ipe Hron Váh Linz I nn Tr au n Ybbs Korneurburg Wien WIEN En ns Danube Cuovo Ybb The Gabcíkovo water power station Medveov Komárno Fig. 2. Scheme of the selected profiles along the Danube River. (Circle points ­ selected stations.) Obr. 2. Schéma vybraného úseku povodia rieky Dunaj. (Vybrané stanice ­ kruhový bod.) max[m3s -1] 3a) max[m 3s -1] 3b) Wien 3c) Bratislava 3d) 3e) Fig. 3 a)­e) The annual maximum discharges during period 1931­1990 ( ­ long term mean annual maximum discharge). Obr. 3 a)­e) Maximálne rocné prietoky pocas rokov 1931­1990 ( ­ dlhodobý priemerný rocný maximálny prietok). Carpathians it keeps the slope and flows over its alluvial cone through a complicated network of branches and meanders downstream to the town of Medvedov. The different physical features of the river basin affect the amount of water runoff in its three sections. In the upper Danube, the runoff corresponds to that of the Alpine tributaries, where the maximum occurs in June when melting of snow and ice in the Alps is the most intensive. Runoff drops to its lowest point during the winter months. In the middle basin the phases last up to four months, with two runoff peaks in June and April. The April peak is local. It is caused by the addition of waters from the melting snow in the plains and from the early spring rains of the lowland and the low mountains of the area. Rainfall is important; the period of low water begins in October and reflects the dry spells of summer and autumn that are characteristic of the low plains. Water discharge data and method River regime conditions of the Danube River are subject to temporal changes. These changes result from natural processes (erosion, sedimentation, vegetation cover) or anthropogenic activities (training works, construction of hydropower stations). Due to water flow changes on the Danube River it is impossible to determine the full range of hydrological characteristics (for example Q100) only from the range of historical discharges at given gauging stations. Differences between peak discharges and daily means of the Danube River at Bratislava are changed about 166 m3 s-1 (3.5%) during the period 1917­2002. This difference is not significant for a large river such as the Danube River. Therefore, the POT method has been applied to daily discharge time series at selected gauging stations: , , Wien, Bratislava and . These gauging stations were chosen based on the availability of long-term discharge records. The record covers a period of 60 s (1931­1990). The annual maxima discharges are shown in Figs. 3 a)­e). The work consists of two parts. The first part is aimed at the POT method application on all selected data period. The second part concerns the estimation of N- discharges from shorter data series and the impact of the shortening data time series on the estimation of the design discharges. The daily discharges series of the period of 60 s were divided into three periods: 1. period of 40 s (1931­1971) and (1950­ ­1990); 2. period of 20 s (1931­1951) and (1970­ ­1990); 3. period of 10 s (1931­1941) and (1980­ ­1990). Period selection and period dividing of the data series is a subjective process. Therefore, with regard to later regulations of the Danube channel bank the 40, 20 and 10 time data series were chosen in early of the 60- period and second analysed time data series were chosen from the end of the 60- period. The threshold value at level limit on 0.85 percentile from mean daily discharges was chosen. The next, three filters were used to provide independence of the POT data. Theoretical Weibull distribution was applied for estimation N- discharges over threshold and number of peaks has Poisson distribution. A probability of the empirical exceedance curves of the maximum annual discharges in this methodology was determined by Cunnane (1988) relationship. P= m - 0.4 , n + 0.2 (11) where n is number of the s and m ­ serial number of the sort values. Basic hydrological parameters for assessment of the multi annual runoff are: long term mean annual maximum discharge, Cv ­ coefficient of variation, Cs ­ coefficient of asymmetry (Tab. 1). Log-normal distribution and Log-Pearson type III distribution are one of the most frequently used distributions in hydrology. Log-normal distribution is statistical distribution for which the log of the random variable is distributed normally. The LogPearson type III distribution (LP3) is a very important model in statistical hydrology. It is a flexible three-parameter family capable of taking many different shapes and has been widely used in many countries for modelling original (untransformed) annual flood series. Kolmogorov-Smirnov test for the evaluation of the theoretical distribution functions was used. In this type of test, the values of a tested sequence are not divided into classes ­ hence the test is particularly suitable for continuous distributions where the empirical distribution function Fn(x) is tested for consistency with an anticipated theoretical distribu93 T a b l e 1. Main statistical characteristics of the annual maximum discharges. T a b u k a 1. Hlavné statistické charakteristiky rocných maximálnych prietokov. Period 1931­1990 Q a [m s ] Cs Cv Q a [m3s-1] Cs Cv Q a [m3s-1] Cs Cv Q a [m3s-1] Cs Cv Q a [m3s-1] Cs Cv Q a [m3s-1] Cs Cv Q a [m3s-1] Cs Cv ACH 4121.21 1.71 0.27 4116.05 1.93 0.29 3923.50 0.51 0.22 3756.00 1.76 0.24 4200.92 1.86 0.29 4086.67 0.62 0.22 4102.20 0.77 0.27 KIE 5597.87 0.92 0.27 5416.95 1.09 0.25 5038.90 ­0.27 0.20 4906.60 ­0.51 0.19 5906.58 0.76 0.28 6018.15 0.54 0.28 5845.70 0.87 0.32 WIE 5535.77 0.68 0.22 5499.58 0.81 0.22 5220.85 ­0.80 0.18 4978.40 ­0.39 0.18 5733.80 0.72 0.23 5641.10 0.43 0.24 5477.50 0.14 0.24 BA 5718.28 0.85 0.25 5745.18 0.91 0.25 5432.60 ­0.42 0.21 5226.10 ­0.18 0.21 5895.80 0.95 0.26 5649.38 0.71 0.24 5657.64 0.41 0.23 NGM 5365.74 0.63 0.25 5434.40 0.08 0.23 5302.50 ­0.26 0.22 5075.00 0.39 0.24 5428.00 0.80 0.36 5289.70 1.20 0.30 5006.30 0.49 0.26 1931­1970 1931­1960 1931­1940 1950­1990 1970­1970 1980­1960 tion function F(x). Due to results of the Kolmogorov-Smirnov test we cannot reject hypothesis that daily maxima discharges comes from these distributions with 95% confidence. Therefore, theoretical distribution probability functions were used for parameters estimating of the theoretical probability curve of the maximum annual discharges: Lognormal distribution (LN) and Log-Pearson distribution III type (LP3). In this work two approaches of the estimating N discharges were compared. The POT method (Vukmirovic, Petrovic, 1995) has been compared with selected theoretical distribution functions of the annual maximum discharges. Results The Peak Over Threshold (POT) method has been used to analyse the extreme hydrological events on the Danube River. The POT method was applied to daily discharge time series over the period of 60 s (1931­1990). Our approach is based on application of the Weibull distribution for estimating of the N- discharges over the POT threshold values (Figs. 4 a)­e)). Number of peaks has Poisson 94 distribution and average numbers of peaks per for the whole period are presented in Tab. 2 with the maximum value (203 peaks) for gauging station . Distribution of the annual maxima discharges is obtained by combination of two mentioned distributions: 1. distribution of peak over threshold values and 2. the number of peaks distribution. Computed values of the N- discharges over the period 1931­1990 are shown in Fig. 5. As noted above, the Log-normal distribution and LogPearson III type distribution were used for parameters estimation of the theoretical exceedance curves. The POT method gives higher Q100 and discharge values than other applied distributions for gauging stations Bratislava and Wien during the period 1931­1990 (Fig. 5). Beacká (1976), Zatkalík (1965), and Svoboda (2000) present higher values of the discharges with the return period 100 and 1000 s than the values obtained by our statistical analysis. It may be resulted from our data period selection and the fact that some of the highest maximum discharges that were recorded on the Danube River were not taken into account for our analysis (historical floods occurred in the s 1501 and 1899). 12000 9000 Q [m3s -1] 4 a) 4 b) Wien 4 c) Bratislava 4 d) 4 e) Fig. 4. Line of the return period of the discharges over threshold for the Danube during the period 1931­1990 (Weibull). Obr. 4. Ciara doby opakovania prietokov nad prahovou hodnotou na Dunaji pocas obdobia r. 1931­1990 (Weibull). T a b l e 2. The Poisson distribution of the number of the peaks (1931­1990). T a b u k a 2. Poissonova distribúcia poctu vrcholov (1931­1990). 3.2 190 3.4 203 Wien 3.3 200 Bratislava 3.1 186 2.2 133 Peak/ Peak No. Q100 11000 10000 9000 8000 7000 Q60 [m3s -1] 6000 5000 4000 3000 2000 1000 0 Achl Kie Wie Ba Ng POT Lognormal Log-Pearson III 13000 12000 11000 10000 9000 Q60 [m3s -1] 8000 7000 6000 5000 4000 3000 2000 1000 0 Achl Kie Wie Ba Ng POT Lognormal Log-Pearson III Fig. 5. Discharges of the Danube River with return period 100 and 1000 s according to POT method, Log-normal distribution and Log-Pearson III distribution (60- period). Obr. 5. Prietoky Dunaja s dobou opakovania 100 a 1000 rokov poda POT metódy, logaritmicko-normálnej distribúcie a Pearsonovho rozdelenia III (60-rocný rad). T a b l e 3. The Poisson distribution of the number of the peaks for periods 1931­1970, 1931­1950, 1931­1941 and periods 1950­ 1990, 1970­1990 and 1980­1990. T a b u k a 3. Poissonova distribúcia poctu vrcholov pre obdobie 1931­1970, 1931­1951, 1931­1941 a obdobie1950­1970, 1970­1990 a 1980­1990. 1931­1970 Peak/ Peak No. 1931­1950 Peak/ Peak No. 1931­1941 Peak/ Peak No. 1950­1990 Peak/ Peak No. 1970­1990 Peak/ Peak No. 1980­1990 Peak/ Peak No. 3.1 123 3.2 63 3 30 3.33 130 3.25 65 3.9 39 3 120 3.2 64 3.2 30 3.475 19 3.5 70 3.7 37 Wien 3.3 130 Wien 3.1 62 Wien 3.2 32 Wien 3.37 135 Wien 3.45 69 Wien 3.6 36 Bratislava 2.8 112 Bratislava 2.7 54 Bratislava 2.9 29 Bratislava 3.35 134 Bratislava 3.7 74 Bratislava 4.1 41 2.1 84 1.9 38 2.1 21 2.37 95 2.55 51 3 30 For example recommended discharge values with the return period of 100-s, and 1000-s, for Bratislava are about 11000 m3 s-1 and 13500 m3 s-1 respectively, and the values estimated by the POT method were 10400 m3 s-1 and 12900 m3 s-1 respectively, for the period of 60 s (Tab. 4). The same approach was used for the analysis of the following data time series, 1931­1940, 1931­ 1950, 1931­1970, 1950­1990, 1970­1990 and 1980­1990 (Tab. 3). As shown, the mean number 96 of the peaks over the threshold for all gauging stations is about 3 peaks per/. The estimated values of the N- discharges over the shorter periods 1931­1970, 1931­1950 and 1931­1940 are shown in Figs. 6 a)­e). This figure reveals that the statistical method selection and its application to data series collection have influence to results of the statistical analysis. The application of the POT method for the period of 20 s gives comparable or higher values of the Analysis of extreme hydrological events on the Danube using the Peak Over Threshold method T a b l e 4. Comparison of the discharges with return period 100 and 1000 s for period 1931­1990. T a b u k a. 4. Porovnanie 100- a 1000-rocných prietokov pre casový rad r. 1931­1990. Qn [m3 s-1] 100 Achl Kie Wei Ba Ng 1000 Achl Kie Wei Ba Ng 1931­1990 POT LN 6720 7000 9900 9845 9170 9008 10400 10018 9000 9328 POT LN 8120 8425 12150 11978 10970 10651 12900 12119 10850 11298 LP3 7595 10164 8842 10163 9194 LP3 9904 12755 10278 12645 10989 Q100 11000 10000 9000 8000 7000 Q40 [m3s -1] 6000 5000 4000 3000 2000 1000 0 Achl Kie Wie Ba Ng POT Lognormal Log-Pearson III 13000 12000 11000 10000 9000 Q40 [m3s -1] 8000 7000 6000 5000 4000 3000 2000 1000 0 Achl Kie Wie Ba Ng POT Lognormal Log-Pearson III a) Q100 11000 10000 9000 8000 7000 Q20 [m3s -1] 6000 5000 4000 3000 2000 1000 0 Achl Kie Wie Ba Ng POT Lognormal Log-Pearson III 13000 12000 11000 10000 9000 Q20 [m3s -1] 8000 7000 6000 5000 4000 3000 2000 1000 0 Achl Kie Wie Ba Ng POT Lognormal Log-Pearson III b) Q100 11000 10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 0 Achl Kie Wie Ba Ng POT Lognormal Log-Pearson III 13000 12000 11000 10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 0 Achl Kie POT Lognormal Log-Pearson III Wie Ba Ng c) Fig. 6. Discharges of the Danube River with return period 100 and 1000 s according to POT method, Log-normal distribution and Log-Pearson III distribution; a) 1931­1970, b) 1931­1960, c) 1931­1940. Obr. 6. Prietoky Dunaja s dobou opakovania 100 a 1000 rokov poda POT metódy, logaritmicko-normálnej distribúcie a Pearsonovho rozdelenia III; a) 1931­1970, b) 1931­1960, c) 1931­1940. Q100 12000 10500 9000 Q40 [m3s -1] 7500 POT Lognormal Log-Pearson III 13500 12000 10500 9000 Q40 [m3s -1] 7500 6000 4500 POT Lognormal Log-Pearson III 6000 4500 3000 1500 0 Achl Kie Wie Ba Ng 3000 1500 0 Achl Kie Wie Ba Ng a) Q100 POT Lognormal Log-Pearson III 12000 10500 9000 Q20 [m3s -1] 7500 6000 4500 3000 1500 0 Achl Kie Wie Ba Ng POT Lognormal Log-Pearson III 13500 12000 10500 9000 Q20 [m3s -1] 7500 6000 4500 3000 1500 0 Achl Kie Wie Ba Ng b) Q100 12000 10500 9000 7500 POT Lognormal Log-Pearson III POT Lognormal Log-Pearson III 6000 4500 3000 1500 0 Achl Kie Wie Ba Ng Achl Kie Wie Ba Ng c) Fig. 7. Discharges of the Danube River with return period 100 and 1000 s according to POT method, Log-normal distribution and Log-Pearson III distribution; a) 1950­1990, b) 1970­1990, c) 1980­1990). Obr. 7. Prietoky Dunaja s dobou opakovania 100 a 1000 rokov poda POT metódy, logaritmicko-normálnej distribúcie a Pearsonovho rozdelenia III; a) 1950­1990, b) 1970­1990, c) 1980­1990). N- discharge estimations for stations , , Wien and Bratislava than the traditional analysis of annual discharge maxima. However, lower discharge values were obtained by the POT method in comparison with the Log-normal distribution for the station. The highest values of the N- discharges during the periods 1950­1990, 1970­1990 and 1980­ ­1990 were obtained by the POT method for the most gauging stations (Figs 7 a)­c)). From a comparison of the results of the beginning and the end of our 60- data period it can be concluded that 98 the differences between the values of N- discharges increased. For instance the difference for Q100 at is about 2279 m3s-1 and for is about 3339 m3 s-1 (10- period, Tab. 5). The results show that the differences between the discharge estimations obtained by statistical methods was Q100 = 4.8% and = 7% respectively during the period of 60 s. The maximum difference was obtained for Q100 = 6.9% and = 9% respectively during the period of 10 s (1980­ ­1990). Analysis of extreme hydrological events on the Danube using the Peak Over Threshold method T a b l e 5. Comparison of the discharges with return period 100 and 1000 s for different length of the data series. T a b u k a 5. Porovnanie odhadnutých 100- a 1000-rocných prietokov pre rôzne dzky casových radov pozorovaní. [m3 s-1] 100 Achl Kie Wei Ba Ng 1000 Achl Kie Wei Ba Ng [m3 s-1] 100 Achl Kie Wei Ba Ng 1000 Achl Kie Wei Ba Ng POT 7120 9700 9270 10000 9050 POT 8620 12000 11170 12200 10760 POT 7020 10700 9850 10800 9200 POT 8520 13300 12050 13400 11100 1931­1971 LN 7271 9253 8882 9976 9202 LN 8866 11137 10473 12078 11035 1950­1990 LN 7381 10210 9427 10105 9052 LN 8981 12418 11190 12180 11649 LP3 7400 9453 8663 10017 8694 LP3 9201 11614 9984 12174 9912 LP3 7646 10188 9651 10813 9031 LP3 9628 13338 11716 13938 12498 POT 6220 8500 7970 9500 8572 POT 8320 10150 9200 11560 10075 POT 6660 10300 9570 10400 8500 POT 7960 12800 11570 12940 10200 1931­1951 LN 6342 8102 8226 9047 9108 LN 7480 9533 9608 10779 10973 1970­1990 LN 6659 10367 9509 9526 8638 LN 7877 12626 11386 11401 10395 LP3 6208 7890 7515 8737 8553 LP3 7178 9063 8132 10084 9749 LP3 6866 10259 9512 9981 9099 LP3 8328 12422 11390 12499 11520 POT 5900 7250 8300 8590 8300 POT 6670 8250 7300 10300 9800 POT 6760 10300 8988 10300 8300 POT 8060 1252 10878 12900 9600 1931­1941 LN 6085 7890 7668 8741 8813 LN 7183 9280 8882 10420 10658 1980­1990 LN 7363 9678 9528 9686 8770 LN 9018 11737 11521 11630 10650 LP3 6207 7148 7382 8910 8391 LP3 7467 7738 8264 10825 9711 LP3 7790 9154 9212 9578 9212 LP3 10106 10564 10812 11385 11725 Conclusion and discussion The paper presents the results of our analysis of extreme hydrological events on the Danube River by the Peak Over Threshold method (POT). The first part of the paper presents the methodology of POT approach for the estimation of N- discharges at 5 gauging stations in the period 1931­1990. We have described two methods for estimating N- discharges: the peak over threshold method and the annual maxima discharges method. To apply the POT method is more difficult because a user himself has to choose a threshold. In this paper we have chosen the threshold to 85% quintile of all daily discharges. For the theoretical exceedance curves of annual maxima discharges were chosen the Log-normal and Log-Pearson III distributions. To evaluate the appropriateness of the theoretical distribution functions we used the Kolmogorov-Smirnov test. The POT and other distributions of the maximum discharge maxima (LN, LP3) were finally compared. The quality and stability of estimated N- discharges is affected by many different properties of the studying data set. Second part analyzed the impact of record length on the estimation of discharge with different return periods. The result of our analysis indicates that the POT method can provide adequate or even comparable results to the N- discharge estimates when used for multiple stations with short data coverage. In summary, the primary advantage of using the POT method is that it enables estimation of N- discharge from shorter time-series, and that the set of analyzed values involves less significant flood events that reflect the hydrological conditions in a basin. Second, the seasonality may also affect the obtained estimates then is better the series split into more homogenous sets, e.g. seasons. Jarusková and Hanek (2006) compared POT method with blokmaxima method for estimating of high return period of discharges series and resulted that the blockmaxima method is much more sensitive to one large observation. Giesecke et al (2002) and Szolgay et al. (2003) concluded that the statistical manner of the data series and uncertainty of statistical estimation are in accordance with the internal structure of data series. They asked: "Is it necessary to use all observations or is it better to choose only some periods due to hydrological, climatic and water management requirements?" Acknowledgements. This research was supported by the MVTS project "Flood regime of rivers in the Danube River basin", the Science Granting Agency under the contract No. VEGA 0096. 99 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Hydrology and Hydromechanics de Gruyter

Analysis of extreme hydrological Events on THE danube using the Peak Over Threshold method

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J. Hydrol. Hydromech., 58, 2010, 2, 88­101 DOI: 10.2478/v10098-010-0009-x VERONIKA BACOVÁ-MITKOVÁ, MILAN ONDERKA Institute of Hydrology SAS, Racianska 75, 831 02 Bratislava 3, Slovakia; Mailto: mitkova@uh.savba.sk The Peak Over Threshold Method (POT) was used as an alternative technique to the traditional analysis of annual discharge maxima of the Danube River. The POT method was applied to a time-series of daily discharge values covering a period of 60 s (1931­1990) at the following gauge stations: , , Wien, Bratislava and . The first part of the paper presents the use of the POT method and how it was applied to daily discharges. All mean daily discharges exceeding a defined threshold were considered in the POT analysis. Based on the POT waves independence criteria the maximum daily discharge data were selected. Two theoretical log-normal (LN) and Log-Pearson III (LP3) distributions were used to calculate the probability of exceeding annual maximum discharges. Performance of the POT method was compared to the theoretical distributions (LN, LP3). The influence of the data series length on the estimation of the N- discharges by POT method was carried out too. Therefore, with regard to later regulations along the Danube channel bank the 40, 20 and 10- time data series were chosen in early of the 60- period and second analysed time data series were selected from the end of the 60- period. Our results suggest that the POT method can provide adequate and comparable estimates of N- discharges for more stations with short temporal coverage. KEY WORDS: Danube River, Extreme Hydrological Events, Flood Frequency Analysis, Peaks Over Threshold (POT) Method, Daily Runoff, Return Period. Veronika Bacová-Mitková, Milan Onderka: ANALÝZA EXTRÉMNYCH HYDROLOGICKÝCH SITUÁCIÍ NA DUNAJI VYUZITÍM METÓDY POT. J. Hydrol. Hydromech., 58, 2010, 2; 41 lit., 7 obr., 5 tab. Príspevok sa zaoberá analýzou extrémnych hydrologických udalostí na Dunaji metódou Peak Over Threshold (POT). Metóda POT sa pouzíva ako alternatíva urcovania N-rocných prietokov k metóde rocných maxím pri analýzach extrémnych hydrologických udalostí. Pre výskyt vrcholových prietokov sa zvycajne predpokladá Poissonova distribúcia. Základnými vstupnými údajmi pre statistickú analýzu sú 60-rocné casové rady priemerných denných prietokov a 60-rocné rady maximálnych rocných prietokov v nami zvolených staniciach: , , Viede, Bratislava a ­ za obdobie 1931­1990. Extrémne hydrologické udalosti na Dunaji boli analyzované metódou POT, ktorá zaha vsetky maximálne denné prietoky povodní za dané obdobie, presahujúce zvolenú prahovú hodnotu. Na zostavenie teoretickej ciary prekrocenia boli vybrané dve teoretické rozdelenia pravdepodobnosti: logaritmicko-normálne rozdelenie (LN) a Pearsonovo rozdelenie III. typu (LP III). Druhým cieom príspevku bolo analyzova vplyv zmeny dzky casového radu na odhad N-rocných prietokov. V práci boli 60-rocné casové rady údajov skrátené na 40, 20 a 10-rocné rady. V závere sme porovnali a zhodnotili získané výsledky statistických odhadov N-rocných prietokov vo zvolených staniciach. Z výsledkov analýzy vyplýva, ze metóda POT dáva pomerne dobré odhady N-rocných prietokov aj pre krátke casové rady údajov. KÚCOVÉ SLOVÁ: tok Dunaja, extrémne hydrologické udalosti, frekvencia výskytu povodní, metóda POT, denný prietok, doba opakovania prietokov. Introduction The annual maximum series approach is the most frequent method used in probabilistic hydrology. 88 However, this approach considers only one value per , which may result in loss of information. For example, some peaks within a may be greater than the maximum discharge in other s hence they can be ignored (Kite, 1997; Chow et al., 1988). In principle, annual discharge maxima contain critical information on the peak flow, however their use is limited by two factors: 1. the length of the series of annual maxima can be very short, 2. the annual maxima time-series may be interrupted and thus they may not allow us to infer the antecedent conditions in the basin preceding a given peak. The first limiting factor produces uncertainties in interpreting statistical analyses, while the latter constrain implies that statistical models built on a phenomenological basis must rely on ancillary data in order to validate the underling hypotheses on the antecedent state of soil moisture (Claps and Laio, 2003). This situation is avoided in the Peaks Over Threshold method (POT). Data series of the POT method considers all values exceeding a certain predefined threshold (Bayliss, 1999; Rao and Hamed, 2000). The POT method has been proposed as an alternative analytical tool to the method of annual discharge maxima for analysis of extreme hydrological events. This method was discussed in a number of papers (Langbein, 1949; Todorovic, 1970; Cunnane, 1973; Rosbjerg, 1977; Madsen at al. 1997 and Lang at al. 1999). In practice, however, it seems to be meaningful to consider not only the annual discharge maxima but also flood events that exceed safety limits. The idea is to derive the distribution and magnitude of annual floods from assumed distributions of the annual occurrence of events and the magnitudes of the POT. Shane and Lynn (1964) assumed that the Poisson distribution is valid for the occurrence of flood peaks, and flood magnitudes exhibit an exponential distribution. Zelenhasic (1970) investigated the distribution of annual maximum floods assuming the Poisson distribution for annual event occurrence and an exponential distribution for their magnitudes. Önöz and Bayazit (2001) showed that for flood estimation, negative binomical (or binomical) models in combination with the exponential distribution of peak heights are almost identical in performance as the Poisson model. This result is in agreement with the findings of Kirby (1969) and Cunnane (1979), and makes it unnecessary to prefer the binomical or negative binomical models even when the Poisson process hypothesis is rejected by statistical tests. It is easier to use the Poisson model because it leads to much simpler expressions for the N- flood and its sampling variance. In the 1980s and early 1990s the statistical method was generalized in different ways, including time-dependent parameters (North, 1980); correlated peak values (Rosbjerg, 1985), risk estimation (Konecny and Nachtnebel, 1985; Rasmussen and Rosbjerg, 1989), Bayesian approaches (Roussele and Hindie, 1976; Rasmussen and Rosbjerg, 1991 a)), a fixed number of peaks (Buishand, 1989), seasonality (Rasmussen and Rosbjerg, 1991b), and other alternatives to the exponential distribution of threshold exceedances such as Weibull (Ekanayake and Cruise, 1993), Lognormal (Rosbjerg, 1987 b); Rosbjerg et al., 1991), and the generalised Pareto (Davison and Smith, 1990; Wang, 1991; Rosbjerg et al., 1992). An extensive analysis of the methods for calculation of N- discharges is given in the paper of Szolgay et al. 2003; Kohnová and Szolgay, 2000. Baca and Mitková, 2007 reported changes in the occurrence frequency of extreme hydrological events in a small agricultural basin using the POT method. The pros and cons of using either the POT method or series of annual discharge maxima for a statistical estimation of design values for the Danube-Bratislava station was investigated by Mitková et al. (2003). Recently Lang at al. (1999) discussed the issue of threshold selection, models suitable for the occurrence processes of the peaks and for the distribution of their magnitudes, and the correspondence between the POT and annual maximum flood distributions. They concluded that the main difficulties of the POT approach concern the selection of the threshold level and of the occurrence process. Several applications of the POT method for estimates of design discharge have been shown in worldwide; however in Slovakia, this method has been rarely used. For example in the UK, the POT flood database of the Summary statistics and seasonality was carried out for POT method. This report describes the growth of the POT database, the data extraction procedures adopted, and it considers briefly the seasonality of flooding. The database now holds over 77 000 peaks for 870 gauging stations throughout the UK with an average record length of nearly 20 s (Baylis and Jones, 1993). The paper consists of two parts. The first part deals with the estimation of N- discharges from the 1931­1990 period. The second part concerns the estimation of N- discharges from shorter data series. The purpose of this paper is to present alternative approaches for analysis of extreme hydrological events. Discharges with different return periods were estimated by the POT method and were finally compared with selected theoretical distributions of discharge maxima. The POT 89 method can give representative values of extreme flows from shorter time series and can be used to assess N- discharges. The POT method The basic idea is to extract from the daily discharges sequences a sample of peaks containing more than one flood peak per , in order to increase the available information with respect to the annual maximum analysis. The POT series includes all maximum discharges over the threshold. The number of the peaks in statistical series must be higher than N, where N is the number of s on record. The first threshold can be chosen near the longterm mean discharge. This value is rather low; POT series can have high diffusion and can include some insignificant maxims. Therefore, a threshold value is usually chosen so that POT data series includes in average 4 maximum values per . In order to provide independence of the POT data the following criterions were used (Bayliss, 1999): - Time period between two consequent peaks must be at least three times longer than the time of increasing of the first wave. - Minimum discharge between two peaks must be less than 2/3 of the peak height recoded during the first wave. 50 40 discharge 30 20 10 0 0 1 2 3 day 4 5 6 A B E D F peak C is time independent from E, and minimum discharge between E and C is less than 2/3 of the E, therefore the peak C can be included into the POT series. The peak of the wave A is less than the threshold and hence it cannot be included into the POT data series. The POT method is characterised by two main variables: - number of peaks in each ; - flow exceedances over threshold Z = x - xB . The occurrence of discharge maxima is a random process defined as: ( t ) = sup ; = x ­ x B , (1) where: ( t ) ­ occurrence of the discharge in time, ­ exceedance of the discharge, xB ­ threshold value (discharge), x ­ value of the current discharge (maxima discharge). The distribution function of annual maxima is F ( x ) = P { ( t ) x} . Number of peaks The number of peaks from the interval (0, t) ­ one in this analysis, is a random variable t that can take values 0, 1, 2, ... with probabilities p ( t ) = P { t = }. The occurrence of peaks in this time interval is described as a Markov process with the intensity function (2) ( t , ) = lim P { ( t + t ) - ( t )} t 0 (3) where ( t , ) ­ occurrence of peaks in time interval, ­ number of the peak, t ­ time, t ­ time difference of the occurrence of the number of peaks. The probability of occurrence of peak exceedances is p ' ( t ) = ( t , - 1) p -1 ( t ) - ( t , ) p ( t ) . p '0 ( t ) = - ( t ,0 ) p0 ( t ) . (4) The solution to this equation (Eq. (4)) represents the probability law of occurrence of peaks and depends on the form of intensity function (Vukmirovic, 1990). Fig. 1. Methodology of the QPOT selection. (Peaks E, C are included to POT data series). Obr. 1. Aplikácia výberu údajov do súboru QPOT. (Vrcholy E, C sú zahrnuté do POT radu). Application of the POT method for peak selection is presented in Fig. 1. The maximum discharge was obtained during the fourth day, therefore the peak E can be automatically included into the POT series. Increasing time of the wave E is about 15 hours. The peak D occurres less than 15 hours before peak E. Wave D is dependent, hence we cannot include this peak into the POT data series (the same situation is for peak wave F). The 90 ( t ) Poisson ( t , ) = ( t )(1 - a ) Bernoulli (binomical) ( t ) (1 + b ) Negative binomical Peaks over threshold Distribution function for peak exceedances flow is defined as: H ( z ) = P {Z z} . (6) (5) Distribution of annual maximum is obtained by combining the distributions of the number of peaks and distributions of peak exceedances over threshold value (Todorovic, 1970): F ( x ) = p0 + H ( x ) p ( t ) . =1 (8) If the number of peaks follows the Poisson distributions then F(x) has the form F ( x ) = exp {- [1 + H ( x)]} . (9) Distribution can be generalized as three-parameter gamma distribution with density function: k + 1 a k +1 k -1 a z a a z exp a h( z) = k k k +1 a a Return period Return period is defined by well-known equation (7) R ( x) = 1 . 1- F ( x) (10) Study area Distributions like two-parameter gamma, Weibull's, Erlang's or exponential are special cases of this general distribution. For estimation is recommended one-parameter distribution (exponential, Rayleigh's) or two-parameters distribution (Weibull or gamma). Exponential H(z) = 1 ­ exp(­z/), Weibull H(z) = 1 ­ exp(z/), where: z ­ variables (magnitude of exceedance discharge over threshold), µ, , ­ distribution parameters. Annual maximum The Danube is the second greatest river in Europe. The length of the Danube River is proximately 2 830 km and drains a basin covering 817 000 km2. Originating in the Black Forest in Germany at the confluence of the Brigach and the Breg streams, the Danube flows over some 2850 km before emptying into the Black Sea via the Danube Delta in Romania. The Slovak part of the Danube River is situated from rkm 1708.2 (river km) to rkm 1880.2. About 7.5 km of the river creates a natural border to Austria, 22.5 km is in Slovakia and the rest of 142 km is the state border to Hungary (Fig. 2). Between the Vienna basin and the Danube lowland, the Danube flows in concentrated channel with relatively high bed slope. After leaving the Small Morava SK Devín BRATISLAVA Ipe Hron Váh Linz I nn Tr au n Ybbs Korneurburg Wien WIEN En ns Danube Cuovo Ybb The Gabcíkovo water power station Medveov Komárno Fig. 2. Scheme of the selected profiles along the Danube River. (Circle points ­ selected stations.) Obr. 2. Schéma vybraného úseku povodia rieky Dunaj. (Vybrané stanice ­ kruhový bod.) max[m3s -1] 3a) max[m 3s -1] 3b) Wien 3c) Bratislava 3d) 3e) Fig. 3 a)­e) The annual maximum discharges during period 1931­1990 ( ­ long term mean annual maximum discharge). Obr. 3 a)­e) Maximálne rocné prietoky pocas rokov 1931­1990 ( ­ dlhodobý priemerný rocný maximálny prietok). Carpathians it keeps the slope and flows over its alluvial cone through a complicated network of branches and meanders downstream to the town of Medvedov. The different physical features of the river basin affect the amount of water runoff in its three sections. In the upper Danube, the runoff corresponds to that of the Alpine tributaries, where the maximum occurs in June when melting of snow and ice in the Alps is the most intensive. Runoff drops to its lowest point during the winter months. In the middle basin the phases last up to four months, with two runoff peaks in June and April. The April peak is local. It is caused by the addition of waters from the melting snow in the plains and from the early spring rains of the lowland and the low mountains of the area. Rainfall is important; the period of low water begins in October and reflects the dry spells of summer and autumn that are characteristic of the low plains. Water discharge data and method River regime conditions of the Danube River are subject to temporal changes. These changes result from natural processes (erosion, sedimentation, vegetation cover) or anthropogenic activities (training works, construction of hydropower stations). Due to water flow changes on the Danube River it is impossible to determine the full range of hydrological characteristics (for example Q100) only from the range of historical discharges at given gauging stations. Differences between peak discharges and daily means of the Danube River at Bratislava are changed about 166 m3 s-1 (3.5%) during the period 1917­2002. This difference is not significant for a large river such as the Danube River. Therefore, the POT method has been applied to daily discharge time series at selected gauging stations: , , Wien, Bratislava and . These gauging stations were chosen based on the availability of long-term discharge records. The record covers a period of 60 s (1931­1990). The annual maxima discharges are shown in Figs. 3 a)­e). The work consists of two parts. The first part is aimed at the POT method application on all selected data period. The second part concerns the estimation of N- discharges from shorter data series and the impact of the shortening data time series on the estimation of the design discharges. The daily discharges series of the period of 60 s were divided into three periods: 1. period of 40 s (1931­1971) and (1950­ ­1990); 2. period of 20 s (1931­1951) and (1970­ ­1990); 3. period of 10 s (1931­1941) and (1980­ ­1990). Period selection and period dividing of the data series is a subjective process. Therefore, with regard to later regulations of the Danube channel bank the 40, 20 and 10 time data series were chosen in early of the 60- period and second analysed time data series were chosen from the end of the 60- period. The threshold value at level limit on 0.85 percentile from mean daily discharges was chosen. The next, three filters were used to provide independence of the POT data. Theoretical Weibull distribution was applied for estimation N- discharges over threshold and number of peaks has Poisson distribution. A probability of the empirical exceedance curves of the maximum annual discharges in this methodology was determined by Cunnane (1988) relationship. P= m - 0.4 , n + 0.2 (11) where n is number of the s and m ­ serial number of the sort values. Basic hydrological parameters for assessment of the multi annual runoff are: long term mean annual maximum discharge, Cv ­ coefficient of variation, Cs ­ coefficient of asymmetry (Tab. 1). Log-normal distribution and Log-Pearson type III distribution are one of the most frequently used distributions in hydrology. Log-normal distribution is statistical distribution for which the log of the random variable is distributed normally. The LogPearson type III distribution (LP3) is a very important model in statistical hydrology. It is a flexible three-parameter family capable of taking many different shapes and has been widely used in many countries for modelling original (untransformed) annual flood series. Kolmogorov-Smirnov test for the evaluation of the theoretical distribution functions was used. In this type of test, the values of a tested sequence are not divided into classes ­ hence the test is particularly suitable for continuous distributions where the empirical distribution function Fn(x) is tested for consistency with an anticipated theoretical distribu93 T a b l e 1. Main statistical characteristics of the annual maximum discharges. T a b u k a 1. Hlavné statistické charakteristiky rocných maximálnych prietokov. Period 1931­1990 Q a [m s ] Cs Cv Q a [m3s-1] Cs Cv Q a [m3s-1] Cs Cv Q a [m3s-1] Cs Cv Q a [m3s-1] Cs Cv Q a [m3s-1] Cs Cv Q a [m3s-1] Cs Cv ACH 4121.21 1.71 0.27 4116.05 1.93 0.29 3923.50 0.51 0.22 3756.00 1.76 0.24 4200.92 1.86 0.29 4086.67 0.62 0.22 4102.20 0.77 0.27 KIE 5597.87 0.92 0.27 5416.95 1.09 0.25 5038.90 ­0.27 0.20 4906.60 ­0.51 0.19 5906.58 0.76 0.28 6018.15 0.54 0.28 5845.70 0.87 0.32 WIE 5535.77 0.68 0.22 5499.58 0.81 0.22 5220.85 ­0.80 0.18 4978.40 ­0.39 0.18 5733.80 0.72 0.23 5641.10 0.43 0.24 5477.50 0.14 0.24 BA 5718.28 0.85 0.25 5745.18 0.91 0.25 5432.60 ­0.42 0.21 5226.10 ­0.18 0.21 5895.80 0.95 0.26 5649.38 0.71 0.24 5657.64 0.41 0.23 NGM 5365.74 0.63 0.25 5434.40 0.08 0.23 5302.50 ­0.26 0.22 5075.00 0.39 0.24 5428.00 0.80 0.36 5289.70 1.20 0.30 5006.30 0.49 0.26 1931­1970 1931­1960 1931­1940 1950­1990 1970­1970 1980­1960 tion function F(x). Due to results of the Kolmogorov-Smirnov test we cannot reject hypothesis that daily maxima discharges comes from these distributions with 95% confidence. Therefore, theoretical distribution probability functions were used for parameters estimating of the theoretical probability curve of the maximum annual discharges: Lognormal distribution (LN) and Log-Pearson distribution III type (LP3). In this work two approaches of the estimating N discharges were compared. The POT method (Vukmirovic, Petrovic, 1995) has been compared with selected theoretical distribution functions of the annual maximum discharges. Results The Peak Over Threshold (POT) method has been used to analyse the extreme hydrological events on the Danube River. The POT method was applied to daily discharge time series over the period of 60 s (1931­1990). Our approach is based on application of the Weibull distribution for estimating of the N- discharges over the POT threshold values (Figs. 4 a)­e)). Number of peaks has Poisson 94 distribution and average numbers of peaks per for the whole period are presented in Tab. 2 with the maximum value (203 peaks) for gauging station . Distribution of the annual maxima discharges is obtained by combination of two mentioned distributions: 1. distribution of peak over threshold values and 2. the number of peaks distribution. Computed values of the N- discharges over the period 1931­1990 are shown in Fig. 5. As noted above, the Log-normal distribution and LogPearson III type distribution were used for parameters estimation of the theoretical exceedance curves. The POT method gives higher Q100 and discharge values than other applied distributions for gauging stations Bratislava and Wien during the period 1931­1990 (Fig. 5). Beacká (1976), Zatkalík (1965), and Svoboda (2000) present higher values of the discharges with the return period 100 and 1000 s than the values obtained by our statistical analysis. It may be resulted from our data period selection and the fact that some of the highest maximum discharges that were recorded on the Danube River were not taken into account for our analysis (historical floods occurred in the s 1501 and 1899). 12000 9000 Q [m3s -1] 4 a) 4 b) Wien 4 c) Bratislava 4 d) 4 e) Fig. 4. Line of the return period of the discharges over threshold for the Danube during the period 1931­1990 (Weibull). Obr. 4. Ciara doby opakovania prietokov nad prahovou hodnotou na Dunaji pocas obdobia r. 1931­1990 (Weibull). T a b l e 2. The Poisson distribution of the number of the peaks (1931­1990). T a b u k a 2. Poissonova distribúcia poctu vrcholov (1931­1990). 3.2 190 3.4 203 Wien 3.3 200 Bratislava 3.1 186 2.2 133 Peak/ Peak No. Q100 11000 10000 9000 8000 7000 Q60 [m3s -1] 6000 5000 4000 3000 2000 1000 0 Achl Kie Wie Ba Ng POT Lognormal Log-Pearson III 13000 12000 11000 10000 9000 Q60 [m3s -1] 8000 7000 6000 5000 4000 3000 2000 1000 0 Achl Kie Wie Ba Ng POT Lognormal Log-Pearson III Fig. 5. Discharges of the Danube River with return period 100 and 1000 s according to POT method, Log-normal distribution and Log-Pearson III distribution (60- period). Obr. 5. Prietoky Dunaja s dobou opakovania 100 a 1000 rokov poda POT metódy, logaritmicko-normálnej distribúcie a Pearsonovho rozdelenia III (60-rocný rad). T a b l e 3. The Poisson distribution of the number of the peaks for periods 1931­1970, 1931­1950, 1931­1941 and periods 1950­ 1990, 1970­1990 and 1980­1990. T a b u k a 3. Poissonova distribúcia poctu vrcholov pre obdobie 1931­1970, 1931­1951, 1931­1941 a obdobie1950­1970, 1970­1990 a 1980­1990. 1931­1970 Peak/ Peak No. 1931­1950 Peak/ Peak No. 1931­1941 Peak/ Peak No. 1950­1990 Peak/ Peak No. 1970­1990 Peak/ Peak No. 1980­1990 Peak/ Peak No. 3.1 123 3.2 63 3 30 3.33 130 3.25 65 3.9 39 3 120 3.2 64 3.2 30 3.475 19 3.5 70 3.7 37 Wien 3.3 130 Wien 3.1 62 Wien 3.2 32 Wien 3.37 135 Wien 3.45 69 Wien 3.6 36 Bratislava 2.8 112 Bratislava 2.7 54 Bratislava 2.9 29 Bratislava 3.35 134 Bratislava 3.7 74 Bratislava 4.1 41 2.1 84 1.9 38 2.1 21 2.37 95 2.55 51 3 30 For example recommended discharge values with the return period of 100-s, and 1000-s, for Bratislava are about 11000 m3 s-1 and 13500 m3 s-1 respectively, and the values estimated by the POT method were 10400 m3 s-1 and 12900 m3 s-1 respectively, for the period of 60 s (Tab. 4). The same approach was used for the analysis of the following data time series, 1931­1940, 1931­ 1950, 1931­1970, 1950­1990, 1970­1990 and 1980­1990 (Tab. 3). As shown, the mean number 96 of the peaks over the threshold for all gauging stations is about 3 peaks per/. The estimated values of the N- discharges over the shorter periods 1931­1970, 1931­1950 and 1931­1940 are shown in Figs. 6 a)­e). This figure reveals that the statistical method selection and its application to data series collection have influence to results of the statistical analysis. The application of the POT method for the period of 20 s gives comparable or higher values of the Analysis of extreme hydrological events on the Danube using the Peak Over Threshold method T a b l e 4. Comparison of the discharges with return period 100 and 1000 s for period 1931­1990. T a b u k a. 4. Porovnanie 100- a 1000-rocných prietokov pre casový rad r. 1931­1990. Qn [m3 s-1] 100 Achl Kie Wei Ba Ng 1000 Achl Kie Wei Ba Ng 1931­1990 POT LN 6720 7000 9900 9845 9170 9008 10400 10018 9000 9328 POT LN 8120 8425 12150 11978 10970 10651 12900 12119 10850 11298 LP3 7595 10164 8842 10163 9194 LP3 9904 12755 10278 12645 10989 Q100 11000 10000 9000 8000 7000 Q40 [m3s -1] 6000 5000 4000 3000 2000 1000 0 Achl Kie Wie Ba Ng POT Lognormal Log-Pearson III 13000 12000 11000 10000 9000 Q40 [m3s -1] 8000 7000 6000 5000 4000 3000 2000 1000 0 Achl Kie Wie Ba Ng POT Lognormal Log-Pearson III a) Q100 11000 10000 9000 8000 7000 Q20 [m3s -1] 6000 5000 4000 3000 2000 1000 0 Achl Kie Wie Ba Ng POT Lognormal Log-Pearson III 13000 12000 11000 10000 9000 Q20 [m3s -1] 8000 7000 6000 5000 4000 3000 2000 1000 0 Achl Kie Wie Ba Ng POT Lognormal Log-Pearson III b) Q100 11000 10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 0 Achl Kie Wie Ba Ng POT Lognormal Log-Pearson III 13000 12000 11000 10000 9000 8000 7000 6000 5000 4000 3000 2000 1000 0 Achl Kie POT Lognormal Log-Pearson III Wie Ba Ng c) Fig. 6. Discharges of the Danube River with return period 100 and 1000 s according to POT method, Log-normal distribution and Log-Pearson III distribution; a) 1931­1970, b) 1931­1960, c) 1931­1940. Obr. 6. Prietoky Dunaja s dobou opakovania 100 a 1000 rokov poda POT metódy, logaritmicko-normálnej distribúcie a Pearsonovho rozdelenia III; a) 1931­1970, b) 1931­1960, c) 1931­1940. Q100 12000 10500 9000 Q40 [m3s -1] 7500 POT Lognormal Log-Pearson III 13500 12000 10500 9000 Q40 [m3s -1] 7500 6000 4500 POT Lognormal Log-Pearson III 6000 4500 3000 1500 0 Achl Kie Wie Ba Ng 3000 1500 0 Achl Kie Wie Ba Ng a) Q100 POT Lognormal Log-Pearson III 12000 10500 9000 Q20 [m3s -1] 7500 6000 4500 3000 1500 0 Achl Kie Wie Ba Ng POT Lognormal Log-Pearson III 13500 12000 10500 9000 Q20 [m3s -1] 7500 6000 4500 3000 1500 0 Achl Kie Wie Ba Ng b) Q100 12000 10500 9000 7500 POT Lognormal Log-Pearson III POT Lognormal Log-Pearson III 6000 4500 3000 1500 0 Achl Kie Wie Ba Ng Achl Kie Wie Ba Ng c) Fig. 7. Discharges of the Danube River with return period 100 and 1000 s according to POT method, Log-normal distribution and Log-Pearson III distribution; a) 1950­1990, b) 1970­1990, c) 1980­1990). Obr. 7. Prietoky Dunaja s dobou opakovania 100 a 1000 rokov poda POT metódy, logaritmicko-normálnej distribúcie a Pearsonovho rozdelenia III; a) 1950­1990, b) 1970­1990, c) 1980­1990). N- discharge estimations for stations , , Wien and Bratislava than the traditional analysis of annual discharge maxima. However, lower discharge values were obtained by the POT method in comparison with the Log-normal distribution for the station. The highest values of the N- discharges during the periods 1950­1990, 1970­1990 and 1980­ ­1990 were obtained by the POT method for the most gauging stations (Figs 7 a)­c)). From a comparison of the results of the beginning and the end of our 60- data period it can be concluded that 98 the differences between the values of N- discharges increased. For instance the difference for Q100 at is about 2279 m3s-1 and for is about 3339 m3 s-1 (10- period, Tab. 5). The results show that the differences between the discharge estimations obtained by statistical methods was Q100 = 4.8% and = 7% respectively during the period of 60 s. The maximum difference was obtained for Q100 = 6.9% and = 9% respectively during the period of 10 s (1980­ ­1990). Analysis of extreme hydrological events on the Danube using the Peak Over Threshold method T a b l e 5. Comparison of the discharges with return period 100 and 1000 s for different length of the data series. T a b u k a 5. Porovnanie odhadnutých 100- a 1000-rocných prietokov pre rôzne dzky casových radov pozorovaní. [m3 s-1] 100 Achl Kie Wei Ba Ng 1000 Achl Kie Wei Ba Ng [m3 s-1] 100 Achl Kie Wei Ba Ng 1000 Achl Kie Wei Ba Ng POT 7120 9700 9270 10000 9050 POT 8620 12000 11170 12200 10760 POT 7020 10700 9850 10800 9200 POT 8520 13300 12050 13400 11100 1931­1971 LN 7271 9253 8882 9976 9202 LN 8866 11137 10473 12078 11035 1950­1990 LN 7381 10210 9427 10105 9052 LN 8981 12418 11190 12180 11649 LP3 7400 9453 8663 10017 8694 LP3 9201 11614 9984 12174 9912 LP3 7646 10188 9651 10813 9031 LP3 9628 13338 11716 13938 12498 POT 6220 8500 7970 9500 8572 POT 8320 10150 9200 11560 10075 POT 6660 10300 9570 10400 8500 POT 7960 12800 11570 12940 10200 1931­1951 LN 6342 8102 8226 9047 9108 LN 7480 9533 9608 10779 10973 1970­1990 LN 6659 10367 9509 9526 8638 LN 7877 12626 11386 11401 10395 LP3 6208 7890 7515 8737 8553 LP3 7178 9063 8132 10084 9749 LP3 6866 10259 9512 9981 9099 LP3 8328 12422 11390 12499 11520 POT 5900 7250 8300 8590 8300 POT 6670 8250 7300 10300 9800 POT 6760 10300 8988 10300 8300 POT 8060 1252 10878 12900 9600 1931­1941 LN 6085 7890 7668 8741 8813 LN 7183 9280 8882 10420 10658 1980­1990 LN 7363 9678 9528 9686 8770 LN 9018 11737 11521 11630 10650 LP3 6207 7148 7382 8910 8391 LP3 7467 7738 8264 10825 9711 LP3 7790 9154 9212 9578 9212 LP3 10106 10564 10812 11385 11725 Conclusion and discussion The paper presents the results of our analysis of extreme hydrological events on the Danube River by the Peak Over Threshold method (POT). The first part of the paper presents the methodology of POT approach for the estimation of N- discharges at 5 gauging stations in the period 1931­1990. We have described two methods for estimating N- discharges: the peak over threshold method and the annual maxima discharges method. To apply the POT method is more difficult because a user himself has to choose a threshold. In this paper we have chosen the threshold to 85% quintile of all daily discharges. For the theoretical exceedance curves of annual maxima discharges were chosen the Log-normal and Log-Pearson III distributions. To evaluate the appropriateness of the theoretical distribution functions we used the Kolmogorov-Smirnov test. The POT and other distributions of the maximum discharge maxima (LN, LP3) were finally compared. The quality and stability of estimated N- discharges is affected by many different properties of the studying data set. Second part analyzed the impact of record length on the estimation of discharge with different return periods. The result of our analysis indicates that the POT method can provide adequate or even comparable results to the N- discharge estimates when used for multiple stations with short data coverage. In summary, the primary advantage of using the POT method is that it enables estimation of N- discharge from shorter time-series, and that the set of analyzed values involves less significant flood events that reflect the hydrological conditions in a basin. Second, the seasonality may also affect the obtained estimates then is better the series split into more homogenous sets, e.g. seasons. Jarusková and Hanek (2006) compared POT method with blokmaxima method for estimating of high return period of discharges series and resulted that the blockmaxima method is much more sensitive to one large observation. Giesecke et al (2002) and Szolgay et al. (2003) concluded that the statistical manner of the data series and uncertainty of statistical estimation are in accordance with the internal structure of data series. They asked: "Is it necessary to use all observations or is it better to choose only some periods due to hydrological, climatic and water management requirements?" Acknowledgements. This research was supported by the MVTS project "Flood regime of rivers in the Danube River basin", the Science Granting Agency under the contract No. VEGA 0096. 99

Journal

Journal of Hydrology and Hydromechanicsde Gruyter

Published: Jun 1, 2010

References