Uncertainty based design floods

: Many statistical distributions approximate well the frequent values of the maximum discharges, but have a very large spread for medium or rare probabilities of exceedance. This scattering defines a range of uncertainty of maximum discharges outside the measured values. Based on the upper (U) and lower (L) values of the uncertainty interval, a maximum discharge flood (MDF) and a maximum volume flood (MVF) are defined for each probability of exceedance. This approach is in agreement with the bivariate analysis, the contour lines for a certain probability of exceedance putting into evidence an infinity of couples (maximum discharge, flood volume). If the most critical combinations are selected, the MDF and MVF are derived. Apart from maximum discharge and flood volume, the shape of the design flood, characterized by the time to peak and the total duration, is also important. The easiest way to obtain the design flood is to use an analytical curve that passes through the characteristic points of the flood. Another possibility, which was largely developed in this paper, is to normalize the floods, then to define clusters of floods with similar shapes, and to obtain an average dimensionless flood for each class. Finally, a family of design floods with different shapes, but characterized by the same parameters for each probability of exceedance, are derived. The use of these floods in the design or operation of hydrotechnical works or in the delineation of flooded areas is presented at the end.


Introduction
The design flood represents the maximum flood that a hydraulic structure can safely pass [1] or that can guarantee the safety of hydraulic structures [2,3]. The magnitude of the design flood is usually associated with the probability of exceedance of the maximum discharge [4]. The probability of exceedance represents the complementary value of the security degree of the objective to be protected.
In the current practice, the flood hydrograph for the probability of exceedance % is constructed considering the maximum discharge % and the output volume of a hydrological model, previously calibrated [3].
The correct estimation of the design flood is necessary for proper planning, design and operation of major hydraulic structures (dams and reservoirs, dikes, diversions) or structures associated with water (bridges, culverts, drainage systems, pumping stations), as well as for the delineation of the flooded areas and flood risk management.
-Envelope curves of observed floods -Methods based on the frequency analyses of observed floods. The analysis can be at-site or at regional level.
-Joint peak-volume design flood methodology, which derives the exceedance probability of the design flood volume given a design peak discharge based on a regional approach.
-Rainfall-based methods: a) The design event-based models consider that the probability of exceedance of the design flood is equal to the probability of exceedance of the input rainfall, (assumption which except for small catchments is not valid); b) In the joint probability approach, a deterministic rainfall-runoff model receives probability-distributed inputs to obtain probability-distributed outputs; c) The continuous simulation uses the registered or generated rainfall for a long period to compute the flood hydrographs, subjected then to statistical processing.
In the past and even quite recently, only the maximum discharge corresponding to different probabilities of exceedance was considered to be the main parameter of the design flood [5,[8][9][10][11][12].
This approach is leading in steady state hydraulic simulation to the worst extension of the flooded areas. The result is an over-evaluation of the flood extent, since the flood attenuation in the floodplain is not taking into account the storage in the depression areas and the corresponding decrease of the maximum discharge in the downstream sections.
The volume of the design flood is also of maximum interest especially in the case of large reservoirs, which storing partially the flood attenuate the peak discharge, as it is the case of the Three Gorges Reservoir [13].
In bivariate flood frequency analysis both the maximum discharges and the corresponding flood volume are considered [14][15][16][17][18][19][20][21][22]. This approach is necessary if the purpose is to derive the storage for flood control of the reservoirs and the spillways characteristics. The maximum discharges and the flood volumes are not independent variables, and can be studied using either the bivariate frequency analysis [23], or copulas which overcome the drawbacks of the bivariate distributions [14,24].
Chuntian et al [25] considered not only the peak discharge and the flood volume, but also the peak time. Other authors [14,26,27] characterize a flood by peak, volume, duration, which are correlated random variables.
Mediero et al. [3] generated a set of synthetic hydrographs that preserved the statistical characteristics of the observed peaks, volumes and durations, while the hydrograph shape was defined by two variables: the time of the peak and the location of the hydrograph centroid.
Brunner et al [17] used for the characterization of a flood event the following parameters: peak discharge, time to peak, duration and flood volume. However, for the bivariate return periods they used only two design variables, peak discharge and flood volume; although the flood duration was mentioned as a potential variable, a window of 72 hours, which was kept constant, was considered for the flood volume estimation.
Brunner et al [28] obtained the Synthetic Design Hydrograph based on peak magnitude and flood volume, using frequency analysis, and a selected probability density function to model the shape of a normalized registered hydrograph.
Other authors included under the name of loading system variables the following characteristics describing the flow: peak discharge, flood hydrograph shape and duration, sediment transport rate, volume and concentration and rate of driftwood transport [29,30].
In the present paper the synthetic flood, which represents a statistical footprint of a river location, is characterized by 4 main parameters [31,32]: -% -maximum discharges corresponding to the probability of exceedance P% --the time to peak of the flood hydrograph --total duration of the flood hydrograph --compactness coefficient, defined as follows: where is the flood volume. The parameters , , are computed as the average values of the 4-5 most significant registered floods ranked according to the maximum discharge [31]. In the statistical processing of the hydrological data, the complete data series is rarely used.
Two methods are employed to select data of interest from the complete series [9, [33][34][35][36][37]: -Annual maximum series or BM (Block maxima), obtaining the extreme value series; in this case, the annual maximum discharges is the data to be processed [5]; -Partial-duration series or POT (Peak Over Threshold), all events which exceed a pre-defined value are included in the analysis [5,[38][39][40][41].
Prior to any statistical analysis, the hydrological data have to be checked for mutual independence and identical distribution, the homogeneity and the lack of trend of the sample data.
The stationarity of data, although in most of the cases implicitly supposed, is questionable due to land cover and land use changes or of climate change [42]. Statistical tests like Wald-Wolfowitz, Turning point, Mann-Whitney-Wilcoxon and Mann-Kendall were used if the basic statistical assumptions are fulfilled [43][44][45].
The most commonly used distributions in hydrology can be divided into four groups [37]: the normal family (normal, log-normal, log-normal type 3), the general extreme-value (GEV) family (GEV, Gumbel, log-Gumbel, Weibull), the Pearson type 3 family (Pearson type 3, log-Pearson type 3), and the generalized Pareto distribution.
According to the selection procedure, the following distributions are preferred [49]: Following a comparison of different statistical distributions, one of them is selected and is considered to be adequate for a country or a region. Thus, in the United States, comparing Lognormal, Gamma, Gumbel, Log-Gumbel, Hazen and Log-Pearson3 (LP3) the recommended distribution for flood statistical analyses is LP3 [5]. Australia and Slovenia also opted for LP3 distribution, UK prefers GEV and Generalized logistics, while China uses Lognormal distribution [41,48]. Pearson III distribution was found to be the best distribution for the extrapolation of the regional curves in Danube basin [32]. LN distribution and LP3 distribution were used for the AM analyses on the Danube River [50,51].
If more distributions are analysed, the most suitable distribution is selected using goodness-of-fit tests, like Kolmogorov-Smirnov, Anderson-Darling or Chi-Squared tests [45,52].
Other authors [27,42] utilized the root mean square error, Akaike information criterion and Bayesian information criteria to select the most appropriate statistical model.
The true value of the statistical estimates is expected to lie within a confidence interval, whose size depends on the confidence level of the test [34].
The confidence level is expressed as = 1 − , where is the significance level and represents the probability of incorrectly (type I error) rejecting the null hypothesis when it is true [53]. Usually, is chosen to be 95%. The confidence interval of an estimate indicates the uncertainty associated with the mean of the estimated value. The upper and lower limits of the confidence interval are the confidence limits, representing the margin of error around the estimated value.
The best distribution to fit the empirical data is however unknown [6] or could even change in time as a consequence of aleatory uncertainty, due to the time variability and the length of the maximum discharge's series [52,54]. Out of this, the registered data are subjected to epistemic uncertainty due to incomplete knowledge of the hydrological system [54][55][56]. The uncertainty is also present as a result of climate change scenarios [57]. Due to aleatory and epistemic uncertainty, the statistical estimates belong to an uncertainty interval, which is not similar to the confidence interval.
The former depends on the spreading of the analysed statistical distributions, while the latter is associated to a given distribution.
This paper is organized as follows: Section 2 presents the Method used in this study; Section 3 provides 2 Case studies; Section 4 contains Discussions, and Section 5 Conclusions.

Materials and Methods
For the beginning, two statistical variables were separately processed: -Maximum annual discharges (Block Maxima approach); -Flood volumes, selecting those floods whose discharges exceed a specified threshold value (Peak Over Threshold approach).
Then, the compactness coefficient, the shape and the flood duration of the design flood were derived.

Maximum annual discharges
For the annual maximum discharges, first the aleatory and then the epistemic uncertainty was investigated.
The aleatory uncertainty is highlighted by analysing time series of different lengths of the registered or generated maximum discharges. The set of registered maximum annual discharges was split into two series: the first 30 values were statistically processed and then new values were added followed by new statistical processing. In each case, the graphical representations were plotted on the normal probability paper, allowing a better visualization of the cumulative distribution function especially for small probabilities of exceedance. It was noticed that for the same value of the probability of exceedance P% the maximum discharges % increase during wet periods, while after a dry period the same characteristics decrease, putting into evidence the aleatory uncertainty. The tendency of dry years and wet years to cluster together into longer dry and wet periods is known as the Hurst phenomenon [58].
A similar approach was used for the generated values of maximum discharges. Based on the sample data of maximum annual discharges, a set of 1000 values following the log Pearson 3 distribution and keeping the same statistical parameters was randomly generated. The discharges corresponding to different probabilities of exceedance were computed for successive series of generated values, by increasing step by step the number of processed discharges from 80 to 500. One noticed that the cumulative distribution curves cover a quite large interval of values for medium and rare probabilities of exceedance, putting into evidence the aleatory uncertainty. Thus, for the probability of exceedance 0.1% the difference between the maximum and the minimum theoretical discharge is about 10%. The aleatory uncertainty interval decreases with the increase of the probability of exceedance, being minimum in the central part of the distribution.
To investigate the epistemic uncertainty, using EasyFit software, a number of about 60 statistical distributions were analysed and then ordered according to Kolmogorov-Smirnov, Anderson-Darling and Chi-Squared tests. Different probability distributions fit well the empirical values, but the extrapolation outside the domain of measurements leads to a large spreading of the estimates as signalled by many authors [47,48,52,55,59].
During statistical processing, it was found that minimum 8-10 ranked distributions should be kept to define the lower and the upper limits of the epistemic uncertainty interval. That means that, for a given probability of exceedance %, the peak discharge % belongs to an interval denoted by ( , ) % where the symbols and refers to the lower and the upper limit respectively of the uncertainty interval. Consequently, % is not a unique value as usually accepted, but it has an infinite number of values for the same probability of exceedance %.
To avoid the arbitrary selection of the number of retained distributions, one proceeds as follows: -The discharges 0.1% computed with the analysed statistical distributions for the probability of exceedance 0.1% is statistically processed.
-The distributions inside, but close to the interval limits ( , ) % are then chosen as the lower and upper margins of the uncertainty interval.

Flood volumes
As previously mentioned, the partial series of the floods' volume is processed using a POT approach. There are different recommendations concerning the selection of the threshold: to be chosen low enough that at least one event in each year is included [9], to be equal with 85% of all daily discharges, to be close to long-term mean discharge or to include in average 4 maximum values per year [50]. Another possibility is to choose the threshold equal with the bankfull discharge, when the floodplain begins to be flooded. The bankfull discharge corresponds to a sharp change in the slope of the rating curve [60]. Finally, in case of a dam regulating the river flow, the threshold can be considered equal with the discharge capacity of the bottom gates of the dam.
In this paper, the threshold is chosen in such a way the total number n of selected floods is equal with the number N of the years with daily or sub-daily discharges. Thus, the empirical exceedance probability associated to each measured value can be interpreted as an annual probability [9].
If ≠ , the average duration of the sampling interval d is less than or greater than one year if more or fewer floods respectively than the number of years are selected. The theoretical probabilities of exceedance = ( ) corresponding to the maximum discharges over a period d other than the year must be converted into annual probabilities of exceedance = 1 = (1), which correspond to the same maximum discharges ( Figure 2).
whence it results: Therefore, the annual probability of exceedance corresponding to the same flow rate Q is: A similar relationship is obtained if working with the cumulative distribution function (the probability of not exceedance) instead of the exceedance probability: However, the calculation of the annual probability of exceedance in this way encounters some difficulties, because many of the usual distributions do not admit the inverse; of course, numerical inversion can be used if necessary.
An approximate solution can be obtained relatively easily, starting from the relation [34]: where: 1 is the annual probability of exceedance of the discharge % , and is the probability of exceedance of the same event for a period of N years.
Suppose that in N years with available data, a number of M floods above a certain threshold were selected, where M is greater than the number of years ( > ). In this case, the probability of exceedance of the same event for a period of M intervals of average duration = / (years) is: As shown before, the probability of exceedance must be put in correspondence with the annual probability of exceedance = 1 for the same calculated discharge, where d is the size of the average sampling interval expressed in years. Let be: Equating the values of the two discharges is obtained: Applying the inverse function results: or: It follows from here: The previous relation allows the calculation of the exceedance probability corresponding to the new average time interval = / years as a function of the annual probability of exceedance: where 1 is the annual probability of exceedance (corresponding to a period of 1 year), and = / is the probability of exceedance of the new interval.
Let be for example N = 85 years and M = 130 selected floods. According to the above formula, to obtain the discharge with an annual probability of exceedance 1%, the discharge corresponding to the probability of exceedance 0.65% based on the 130 selected values must be calculated. Thus, if If the number of m selected floods is less than the number of years (i. e. < ), the relation for the calculation of the exceedance probability corresponding to a time interval = / years is similar to previous one, but the average duration is now = / : Considering a threshold discharge ℎ 1 , all floods whose discharges exceeds this threshold are selected. The excess discharges are assumed to be independent with an arbitrary distribution [38].
However, the fulfilment of these basic assumptions can be checked.
In a certain year 2-3 floods can be selected, while in dry years no flood is retained for future analysis. For the selected floods, a second threshold ℎ 2 = ℎ 1 , where < 0.9, is introduced in order to derive the floods duration. It can be considered that ℎ 2 represents a discharge corresponding to the warning level, which announces the beginning of a flood. The threshold ℎ 2 is chosen in such a way to obtain distinct floods. That means that two consecutive floods have to be separated by at least three-times the average time to rise and the discharge between two consecutive flood has to drop below the value of the two thirds of the smaller of the two peaks [61,62].
For a selected flood, the flood duration (or the Total duration ) corresponds to all discharges ( ) > ℎ 2 ( Figure 3). The volume of interest is the integral of the discharges higher than the threshold [16].
The flood volumes over ℎ 2 are then statistically processed in the same way as used for the maximum discharges, resulting the interval of uncertainty for the volumes % denoted by ( , ) % . That means an infinite number of volumes % corresponds to the probability of exceedance %.
where ( , ) is a random vector whose components can be expressed in terms of their marginals: In this case, if represent maximum registered discharges, V is the total volume of the flood, adding both volumes above and under the threshold. If ℎ 2 was subtracted from the registered discharges, then V represents only the volume above the threshold.
As in the case of univariates, in hydrology the probability of exceedance of the same couple of statistical variables (maximum discharge and the corresponding flood volume) is of interest: It can be shown that: Thus, ( , ) % and ̃( , ) % meaning the probability of non-exceedance and the probability of exceedance respectively are not strictly complementary as in the case of univariates.
The design flood, characterized by the probability of exceedance % cannot have simultaneously both the maximum discharge % and the volume % , which belong to a hydrograph whose probability of exceedance is unknown being defined by the common probability of the marginal probabilities of peak and volume [3]. High peak discharge does not necessarily mean high hydrograph volume [63].
That means the point ( % , % ) is not situated on the isoline of exceedance probability ̃( , ) % , as it can be seen from Figure 4a. where the indexes U and L mean the upper and the lower limit respectively of the interval of uncertainty. This approach is similar to that recommended by Volpi and Fiori [16] in the context of bivariate flood frequency analysis. Another critical couple [16] is the equivalent of the vertex in the plane (u,v) situated at the intersection of the first bisector with the isoline of exceedance probability ( Figure 4b).
Prohaska and Ilić [22] recommended for flood management issues 4 characteristic points, from which 3 are on the isoline ̃( , ) % corresponding to the probability of exceedance %, while the 4 th is the "maximum possible hydrograph" whose parameters are ( % , % ), although the probability of exceedance of this couple is lower than %.

Compactness coefficient and shape of the design flood
There are the following cases to define the design flood (MDF or MVF): A. Reproducing the shape of floods that occurred in the past: In the case A1, the shape of the registered floods characterized by the maximum discharge and by the maximum volume is reproduced. In scenario A2, all the floods above the threshold are normalized and then, using a clusterization procedure, they are grouped into classes according to their shape. For instance, the floods can be divided into the following categories: 1 peak, 2 peaks or 3 peaks. The normalised floods with a single peak can be subdivided according to the position of the peak: peak below 0. Similar considerations apply to cases B1 and B2, with the difference that spline functions will be used to obtain the shape of synthetic floods.
The normalized floods, which are dimensionless, have the same shape as the original floods and are obtained as follows: where: - where: -weighting factor for the dimensionless flood [ -]. A class can contain one or more floods with similar shape. If a class contains only one flood, it will shape that class.
To obtain the compactness coefficient and the shape of the design flood, proceed as follows: a) The first K significant floods are retained in the descending order of the maximum flows. A recommended value is ≤ 5. A flood with a particular shape can be excluded, and the next flood will be chosen in descending order of peak flow. In addition, it is advisable to select floods with 2-3 peaks.
b) The selected floods from which the value of the threshold has been deducted are normalized, i.e., they are brought to a percentage scale on both abscissa and ordinate, the value of 100% corresponding to the flood duration, respectively to the maximum discharge of each flood.
c) The compactness coefficients =1, of the dimensionless floods are computed.
The compactness coefficient is obtained as follows for the dimensionless floods: d) The dimensionless time to peak , =1, of the normalized floods is also determined.
e) Maximum discharge flood. In the case A1 the shape of the maximum discharge flood will be similar to the shape of the first ranked of the registered flood in descending order of the maximum discharges. The compactness coefficient of this flood is denoted by 1 , while the dimensionless time to peak is denoted by ( ) 1  class. The compactness coefficient will be kept at the value 1 , while the time to peak will be different for each class. Similar considerations can be made for the cases B1 and B2: the compactness coefficient is kept at the value 1 in all cases, while the shape of the synthetic floods is given by analytical relations.
f) Maximum volume flood. In the case A1, all the dimensionless floods are taken into account, and the maximum value of the compactness coefficient is chosen: The index for which is maximum designates the sequence number of the flood in the descending ordered of the maximum discharges. The letter from the exponent has no meaning of power but of index to indicate volume. The dimensionless time to peak of the maximum volume flood is denoted by ( ) . The current time value is ( ) and the corresponding discharge is ( ) .
In Scenario A1, the shape of the maximum volume flood will be similar to the shape corresponding to the th ranked of the registered flood in descending order of the maximum discharges.
In the case A2, the compactness coefficient of the normalized floods will be kept at the value , while the time to peak will be different for each class.
Similar considerations can be made for the cases B1 and B2: the compactness coefficient is kept at the value in all cases, while the shape of the synthetic floods is given by analytical relations.

Duration of the synthetic flood
The flood volume above the threshold, without any reference to the probability of exceedance P%, is: where : is the flood volume above threshold ℎ 2 ;  -the compactness coefficient of the flood; -maximum discharge of the flood, The duration % of the synthetic flood for the probability of exceedance % is calculated with the relation: where : is the flood volume above threshold The duration % of the synthetic flood for the probability of exceedance % is calculated with the relation: where the parameters , % , % are replaced by 1 , ( ) % , ( ) % for the maximum discharge flood and by , ( ) % , ( ) % for the maximum volume flood respectively.
The synthetic flood duration % has a variable size, depending on the probability of exceedance % . For the same probability of exceedance, the duration % is shorter for the maximum discharge flood compared to the maximum volume flood.

Time to peak of the design flood
In the cases A1 and B1 (a single flood for MDF and another one for MVF), the time to peak for the design flood is: where is replaced by ( ) 1 for the maximum discharge flood and by ( ) for the maximum volume flood. The computed value for ( ) % corresponds to maximum discharge flood or to maximum volume flood respectively.
In the cases A2 and B2 (a family of design floods) the time to peak of the normalized floods in each class is computed with a similar relation, % being the same no matter the class.

Construction of synthetic flood waves
As previously mentioned, a synthetic flood is a discharge hydrograph defined by the following elements: peak discharge, flood volume, duration and shape.
The statistical processing of the annual maximum discharges is undertaken by EasyFit commercial software, while the other steps in defining the design floods are made using different in-house software IHS.
The algorithm for the construction of synthetic floods is the following: 1) Identify the period with available data for analysis.
2) Select the following data: 3) Choose the threshold ℎ 2 and determine the floods' volume above it.

4)
Test the independence, homogeneity and stationarity of the obtained series. 9) Establish the shape of the synthetic flood a) In the case A1 for the maximum discharge flood, the shape of the flood ranked first following the descending order of the maximum discharges is selected. The compactness coefficient of this flood is 1 , while the dimensionless time to peak is ( ) 1 . For the maximum volume flood, the flood shape corresponding to the maximum value of the compactness coefficient is chosen.

5)
The dimensionless time to peak of the maximum volume flood is denoted by ( ) and is different for MDF and MVF.
b) In the case A2, the normalized floods are grouped into classes according to their shape (see Figure 5) and then an average dimensionless flood is derived for each class ( Figure 6). The compactness coefficient will be kept as in Scenario A1 for MDF and MVF respectively, but the time to peak is different for each class.
c) Similar considerations are made for the cases B1 and B2.  respectively with the maximum discharge (from which the threshold ℎ 2 was substracted) corresponding to the probability of exceedance P%, and add to the ordinates thus obtained the threshold value ℎ 2 . The timing in days on the abscissa is obtained by multiplying the adimensional time with the flood duration. Thus for the maximum discharge flood one obtains:

10) Calculate the flood duration
For the maximum volume flood the coordinates of the synthetic flood are: The notations keep their original meaning. The indexes MDF and MVF refer to maximum discharge flood and maximum volume flood.
The time to peak ( ) % for MDF and ( ) % for MVF respectively are implicitly obtained with the above relations. In the case A1 there is only one design flood for the MDF and another one for MVF respectively, but with different compactness coefficients and times to peak.
They reproduce the shape of significant floods which occurred in the past. In the case A2 a family of design floods for MDF and MVF respectively, which reproduce shapes of registered floods in the past, is obtained. The design floods in each category or sub-category for maximum discharge flood are characterized by the same parameters: % , % , , but the shape and the peak time are different. The same rule applies to the maximum volume flood, but % and % are different from the similar parameters of the maximum discharge flood.
In cases B1 and B2, the synthetic flood is characterized by the same parameters,

Case studies
In the following, 2 case studies will be presented. They were initiated in international cooperation, but were developed after the projects' finalization. The first case study was analysed in  The time series were checked for potential trend, finding that the hypothesis of stationarity is justifiable. A future direction of research is to consider the time dependency of the distribution parameters [42] in order to derive the worst possible flood for a given probability of exceedance.
However, the interval of uncertainty, as it is defined, is able to include a possible non-stationarity of the hydrological data. The results of the statistical tests concerning the mutual independence and identical distribution, the homogeneity and the lack of trend of the annual maximum discharges are presented in Table 1. The explanation for the lack of trend of the annual maximum discharges is the large size of the Danube river basin, which is able to compensate at least in the lower Danube reaches the local effects of the climate changes. Thus, one can suppose that for the lower Danube the variation of the discharges % is due mainly to natural variability. The aleatory uncertainty for a LP3 distribution was put into evidence by analysing different lengths of registered or generated data at Turnu Măgurele on Danube River, while the epistemic uncertainty analysis was based on the selection of a number of statistical distributions which fit relatively well the selected discharges from 1931 till 2008 at the same gauge station. The graphical representations of the cumulative distribution function were plotted on the probability format, allowing the visualization of the cumulative curve. The dark solid line in Fig. 9 represents the curve of the probabilities of exceedance for the initial set of 78 annual maximum discharges, while the grey solid lines correspond to different volumes of the sample.  The results obtained using the first 9 ranked distributions according to Kolmogorov-Smirnov test are presented in Table 2   * Values from statistical processing to which the threshold value was added Figure 11. Uncertainty interval of the maximum discharges POT at Turnu Măgurele gauge station

Flood volumes
The HIS software was used for the separation of the significant floods, using all daily data for the discharges between 1931-2008. The threshold for floods selection is ℎ 1 = 9700 m 3 /s, while ℎ 2 = 8300 m 3 /s. All discharges greater than 8300 m 3 /s were taken into consideration for computing the flood volume, followed by statistical processing of the volumes undertaken with EasyFit software. The obtained results are presented in Figure 13 and Table 3.  The HIS software separates the floods according to the threshold values ( ℎ 1 = 9700 and ℎ 2 = 8300 m3/s). In the following table the first 5 floods in decreasing order of the maximum discharge, as well as their main characteristics (starting date, maximum discharge, total volume, volume above the threshold ℎ 2 , compactness (shape) coefficient, time to peak and total duration above ℎ 2 ) are presented. Taking into consideration the compactness coefficient, the floods that occurred in 2006 and 1981 ( Figure 14) were considered for the next steps, by skipping the flood clusterization. Based on the dimensionless floods (Figure 15), synthetic floods were obtained both for maximum discharge flood and for maximum volume flood. In the following only the synthetic floods for Qupper_Vlower (Class1) and Qlower_Vupper (Class 2) are presented (Figures 16 and 17; Tables 5 and 6).   Using the presented methodology, the floods above the threshold 2 (which was set to 90 m3/s, almost the average multiannual flow) were normalised and then grouped into classes of similar shapes ( Figure 22). In some classes only a flood with a very particular shape is represented (classes 6; 8; 9 and 10), while in other classes (1; 2; 3; 4; 5 and 7) more floods are grouped. The statistical processing of the maximum discharges (BM and OT approach) and floods volume above the threshold are presented in Figure 23. Based on these results (Tables 9-11), the design floods were derived. Only the design floods for classes 9 and 10 are presented in the following, due to the exceptional character of these floods. The flood in class no. 9 reproduces the shape of 2008 flood, which was very compact and raised a difficult situation for flood attenuation, while the flood in class no. 10 has a similar shape with 2010 flood, characterized by 3 peaks, an exceptional duration and a huge volume.
(a)   The design floods for classes 9 and 10 have the following characteristics (Table 12):  In the Figure 24, the maximum discharge flood and maximum volume flood of the classes 9 and 10 for the probabilities of exceedance 0.1%; 1% and 10% are shown. The design floods in the class 9 and 10, for the same type (MDF or MVF) differ only by time to peak and the total duration, while the maximum discharge and volume respectively are the same.
It can be seen that the differences between the maximum discharge flood MDF and maximum volume flood MVF diminish as the probability of exceedance increases.
In the Figure 25, the maximum discharge floods 0.1%, 1% and 10% are represented on the same graph for class 9 and class 10 respectively.  The statistical distribution can be different for maximum discharges and for flood volumes. Usually, the confidence level is 95%, but it can be reduced to avoid a large difference between the upper and lower limit, which normally should be less than 20%.
-Selecting a large number of statistical distributions as largely presented in the previous chapters.
-Establishing a correlation between the maximum discharges and flood volumes and defining the confidence intervals both for discharges and volumes based on the Student distribution.
-Based on bivariate flood frequency analysis.
No matter what approach is used, the main idea is the same: to define upper and lower limits of the uncertainty intervals and to make the appropriate combinations corresponding to the maximum discharge flood characterized by the pair ( , ) % , respectively to the maximum volume flood 5. The design floods that reproduce the shape of registered floods are more credible than the analytic floods that pass through the characteristic points (Figure 1), respecting the volume or the compactness coefficient. If the flood that reproduces the shape of the registered floods in classes 9 and 10 at Rădăuți-Prut station overlaps on the same graph with the analytic flood characterized by the same maximum discharge, volume, time to peak and duration, it is found that there is a fairly good agreement in the case of unimodal floods ( Figure 27), but very important differences occur for multimodal floods ( Figure 28). If the purpose is to delineate the flooding areas, then the analytic floods will lead to approximately the same results as the floods reflecting the shape of the registered floods. However, if the purpose is to determine the operational framework rules for the optimal attenuation in a reservoir during flood conditions, it is preferable to use the shape of real floods.

Conclusions
The hydrological processing of flood waves can be performed at different degrees of complexity, depending on the future utilization of the results. The simplest way, used in case of delineation of the flooded areas or for dikes designing, based on a steady state simulation, is to consider the maximum discharge corresponding to a probability of exceedance P%, which is treated in the following as a deterministic value, neglecting the associated aleatory and epistemic uncertainty. In such a case, for each probability of exceedance a unique extension of the flooded area is obtained.
If the hydrologic uncertainty is taken into account, for any distribution a confidence interval can be defined. The maximum discharges belong to this interval with a confidence level . Considering the upper and lower limits of the discharges, the uncertainty limits of the flooded areas are derived.
The problem gains in complexity when more distributions are used to fit the registered discharges. All of them approximate well the frequent values of the maximum discharges, but have a very large spread for medium or rare probabilities of exceedance. This scattering defines a range of uncertainty for the extrapolation of maximum discharges outside the measured values. The choice of the upper (U) and the lower (L) limits of this interval is a problem of expert judgement, but normally, the difference between the minimum and maximum values for 0.1% probability of exceedance should not exceed 20%. Based on the extreme values of the uncertainty interval, the maximum discharge flood ( , ) % and the maximum volume flood ( , ) % are defined for each probability of exceedance %. This approach is in perfect agreement with the bivariate analysis, the contour lines for a certain probability of exceedance putting into evidence an infinity of couples (maximum discharge, flood volume), which are not independent variables. From these couples, only the most critical combinations are selected, leading to MDF and MVF.
Apart from these parameters the shape of the design flood, characterized by the time to peak and the total duration, is also important. The easiest way is to use an analytical curve that passes through the characteristic points of the flood. Other possibility, which was largely developed in this paper, is to obtain dimensionless floods, then to define clusters of floods with similar shapes and finally to obtain an average dimensionless flood for each class. Based on them, design floods corresponding to different classes of clusterization, but characterized by the same parameters for each probability of exceedance, are derived.
More than one design flood has also been addressed by other researchers. Quite recently, Mediero et al [3] obtained a set of synthetic hydrographs that preserved the statistical characteristics of the observed peaks, while Volpi and Fiori [16] adopted an ensemble approach choosing the most critical events in term of hydrological loads on the hydraulic structures.
All design floods can be used to examine the consequences of their occurrence upon the environment and hydrotechnical works. However, in a simplified approach only the MDF reproducing the shape of the registered flood characterized by the maximum discharge and the MVF having a similar shape to the flood with the maximum registered volume can be used for further simulations.
The floods corresponding to the maximum discharges will be used for the design of spillways, or for the dike design in order to prevent high levels leading to dike overtopping. The most dangerous floods for reservoirs operation are those that are very compact, corresponding to the maximum compactness coefficient. In these cases, pre-emptying the reservoir based on a reliable hydrological forecast in order to increase the flood protection volume is a necessary management strategy.
For the dike stability, both the MDF and MVF can be used. The MDF, although the high levels have a short duration, can activate preferential routes of infiltration through the galleries of rodents, endangering the stability of the dike. On the other hand, the long duration of MVF floods also makes them dangerous because the infiltration curve can reach the downstream face of the dike. Both floods represent boundary conditions for transient computation of the water flow through the dike and its foundation. The critical gradients will then be computed, putting into evidence the sensitive parts of these hydraulic structures.
In the frame of this paper, 2 case studies were presented : a) processing Turnu-Măgurele daily