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# Boxplots of error estimates for the two approaches with Gumbel distribution used for design flood level estimation at cross-sections (A,B). The red lines represent the median (50th percentile), the lower and upper ends of the blue box represent the 25th and 75th percentile, respectively. Outliers are represented by red dots.

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The design of flood defence structures requires the estimation of flood water levels corresponding to a given probability of exceedance, or return period. In river flood management, this estimation is often done by statistically analysing the frequency of flood discharge peaks. This typically requires three main steps. First, direct measurements of...

## Contexts in source publication

**Context 1**

... application of the simulation framework described in Section 2 and depicted in Figure 1, to a 98-km reach of the Po River (Section 3), allows us to compare the accuracy and precision of the two approaches for the estimation of design flood levels. Figure 3 shows boxplots of errors when the estimation of design flood levels is based on the two methods that are considered in this study. Water 2019, 11, x ...

**Context 2**

... application of the simulation framework described in Section 2 and depicted in Figure 1, to a 98-km reach of the Po River (Section 3), allows us to compare the accuracy and precision of the two approaches for the estimation of design flood levels. Figure 3 shows boxplots of errors when the estimation of design flood levels is based on the two methods that are considered in this study. Figure 3. Boxplots of error estimates for the two approaches with Gumbel distribution used for design flood level estimation at cross-sections A and B. The red lines represent the median (50th percentile), the lower and upper ends of the blue box represent the 25th and 75th percentile, respectively. ...

**Context 3**

... 3 shows boxplots of errors when the estimation of design flood levels is based on the two methods that are considered in this study. Figure 3. Boxplots of error estimates for the two approaches with Gumbel distribution used for design flood level estimation at cross-sections A and B. The red lines represent the median (50th percentile), the lower and upper ends of the blue box represent the 25th and 75th percentile, respectively. ...

**Context 4**

... left panel of Figure 3 refers to the common approach (when the estimation is based on annual maximum flows), and it shows a substantial underestimation, which is larger in section A, where the rating curve has larger errors (Figure 2). Moreover, and there is no substantial improvement in accuracy and precision after increasing the sample size from 30 to 100 years. ...

**Context 5**

... right panel of Figure 3 refers to the alternative approach (when the estimation is based on annual maximum levels), and it shows an underestimation of the design flood by 0.8 m with a more The left panel of Figure 3 refers to the common approach (when the estimation is based on annual maximum flows), and it shows a substantial underestimation, which is larger in section A, where the rating curve has larger errors (Figure 2). Moreover, and there is no substantial improvement in accuracy and precision after increasing the sample size from 30 to 100 years. ...

**Context 6**

... right panel of Figure 3 refers to the alternative approach (when the estimation is based on annual maximum levels), and it shows an underestimation of the design flood by 0.8 m with a more The left panel of Figure 3 refers to the common approach (when the estimation is based on annual maximum flows), and it shows a substantial underestimation, which is larger in section A, where the rating curve has larger errors (Figure 2). Moreover, and there is no substantial improvement in accuracy and precision after increasing the sample size from 30 to 100 years. ...

**Context 7**

... right panel of Figure 3 refers to the alternative approach (when the estimation is based on annual maximum levels), and it shows an underestimation of the design flood by 0.8 m with a more significant reduction of uncertainty after increasing the sample size from 30 to 100 years. These differences are substantial when the estimation of design flood levels is used for flood defence design or risk assessment. ...

**Context 8**

... example application demonstrates that the alternative approach can work better than the common approach in terms of both accuracy and precision (Figure 3). These results were unavoidable as they were associated with the specific test site, as well as the arbitrary assumptions about the parent distribution and the hydraulic model. ...

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## Citations

... Therefore, the probabilistic methods based on hydrological observation data statistic analysis have been more popular in studying and predicting diverse design hydrological characteristics. In particular, it is in the case of forecasting maxima water discharges relating to riverine floods [1][2][3]. ...

... Usually, the time series of observed maxima water discharges are considered and analyzed in the frame of the stationary hypothesis. To forecast design discharges of low annual probability of exceedance, parametrical probability distributions are used as predictive models [2]. Returning to the problem of flood risk management, it should be reminded that Directive 2007/60/EC (the EU Flood Risk Directive) [5] defines flood risk quantitatively as "the combination of the probability of a flood event and the potential adverse consequences for human health, the environment, cultural heritage and economic activity associated with a flood event". ...

... In the study, the scientific methods of theoretical and empirical research, analysis and synthesis, expert evaluation and comparison, formalization and modeling were used, including (1) extrapolation methods [14]; (2) fundamental and practical methods of mathematical statistics [11][12][13]15]; (3) specific statistical methods in hydrology [1][2][3][4][6][7][8][9][10]; (4) utility theory methods [16,17] and decision making methods under risk and uncertainty [8,9,18]. ...

Plotting position formulas provide a non-parametric means to estimate the observed hydrological data probability distribution. In particular, using a plotting position formula, a plot of the estimated values from a theoretical parametric probability distribution can be compared with the observation data. It allows an examination of the adequacy of the fit provided by parametric probability distributions. However, results of calculating empirical annual probabilities of exceedance observed maxima water discharges show an increase in the divergence between the estimates obtained using the different plotting position formulas in case of more extreme events. Thereby, the choice of a relevant plotting position formula becomes a challenge. Different plotting position formulas may be admissible options. This article shows that the divergence between the plotting position estimates can be extrapolated to predict design maxima water discharges of low exceedance probabilities.

... For example, the design annual exceedance probabilities in terms of prediction of maximum water levels and possible inundation zones because of floods may be established at 0.005, year -1 (or 0.5%, year -1 ), 1%, 2%, 5%, and 10%, or something else; the corresponding return periods of the design floods -200, 100, 50, 20, and 10 years, etc. In hydrological investigations relating to river floods, these estimations are usually done by statistically analysing the frequency of flood peak discharges [29][30][31][32][33]. Practically, it is done in such a way. ...

... In hydrological investigations relating to river floods, these estimations are usually done by statistically analysing the frequency of flood peak discharges [29][30][31][32][33]. Practically, it is done in such a way. Direct annual maximum water levels' h (m) measurements at a near-located gauging (hydrological) station are converted into maxima annual discharges Q (m 3 /s) by using a rating curve ) (h f Q = [7,33]. As a result of longterm (not less than 30-40 years) uninterrupted hydrological measurements, time series of annual maximum water discharges of floods are formed. ...

... Gathered data are statistically analysed, and, in the frame of the stationary hypothesis, a relevant maxima annual discharges' probability distribution is chosen, which has to fit the observed data [29,34,35]. This probability distribution is used to derive a predicted peak discharge corresponding to a chosen design return period p r T , or a chosen design annual probability of exceedance P [33]. In the next step, the established peak discharge of a chosen design annual probability of exceedance may be used as the input value for the hydraulic modelling to derive the corresponding design flood level [33,36], taking into account the current conditions with hydromorphological characteristics of the river channel and floodplain [37]. ...

There are a lot of analytical probability distributions that might be used to predict peak discharges of floods. However, there is no proper theoretical or another similar justification for choosing an appropriate parametric probability distribution to predict peak discharges of floods by using observed data. As a permissible hypothesis, any of recommended probability distributions can be considered providing it meets the given statistical criteria and other considerations for the adequacy of simulation are taken into account. In turn, more than seventeen plotting position formulas have been proposed. They provide a non-parametric means to estimate the observed data probability distribution. Using a plotting position formula, a plot of the estimated values from a theoretical parametric probability distribution can be compared with the observed data.The choice of a better plotting position formula for fitting the different probability distributions has been discussed many times in hydrology and statistical literature. However, no specific criterion for choosing these formulas has been proposed yet. Perhaps there is no need for such a criterion. Maybe, the diversity of estimates that can be obtained due to these formulas matters more. Due to the diversity of the different plotting position estimates, from the point of view of informational entropy, different plotting position formulas enable revealing epistemic (non-stochastic or subjective) uncertainty in predictions of hydrological extremes.Results of calculating empirical annual probabilities of exceedance observed maxima discharge employing various plotting position formulas show that increasing the predicting horizon toward low probable and more extreme events increases the divergence between the estimates obtained using the different plotting position formulas. Therefore, it is reasonable to assume that this divergence may be extrapolated to predict design maxima discharges of floods based on empirical estimates of plotting position probabilities.This paper proposes a numerically-analytical method using such an extrapolation. It is based on using different plotting position formulas, numerical calculations of plotting position probabilities, and extrapolation of the divergence between the obtained estimates. The method is tested in predicting the maxima discharges of 0.5% and 1% annual probability of exceedance for the Uzh River flowing in the Transcarpathia region, the hydrological station “Uzhhorod” data.

... Design flood therefore do not have a specific calculation format; it is safe to say that the design flood is practically a selected parameter which is strongly guided by data availability, experience and economic factor. The steps in estimating of design flood levels described by [30] often consists of three main steps: ...

Rainfall is the Major contributor to most of the flooding cases witnessed across Nsukka metropolis. Floods are among the most devastating natural disasters in the world, claiming more lives and causing more property damage than any other natural phenomena [2]. When land is submerged, drainage is the only way to reclaim such land [24]. This is perhaps the main form of flood prevention/control technique used in this area. However, the problem becomes that studied areas in the Nsukka metropolis have either no drainage or inadequate drainages.
The study involved physical site visitation and measurement of existing drainages. Measured drainages were either trapezoidal or rectangular with different dimensions. The Gumbel Extreme value (type 1) technique was used to analyze a 36 years rainfall data to produce an IDF curve for different return periods. An advanced composite runoff coefficient method was used to obtain the runoff coefficient. While making use of Bentley Civil Storm software, the identified areas and drainages were adequately modeled and simulated to check drainage overflow for a 5year Return Period. Particularly, catchment delineation was done using combination of Esri ArcGIS, Autodesk Civil3D and Bentley Civil Storm. The Rational method and Manning's equations were selected as preferred methods for the GVF Solver in Civil Storm Software.
This study identified that over 70% of drainages in the study area are not adequate. Twelve out of Sixteen major roads have drainage that are inadequate, which is also seen with the results from simulation showing overflow of the drainage. The major cause of the inadequacy is due to heavily silted drains. It is recommended that proper maintenance be done regularly, to dredging drainages.

... Since the natural disaster as extreme floods is the basis for planning and design of various hydraulic structures, hydrological forecasting, flood risk reflection characteristics such as trends of extreme floods and its changes, and its formation conditions, the probable maximal flood and its characteristics have a great practical importance. The determining of the probable maximal flood is the practical importance, especially for the planning, design, and operation of hydrotechnical structures (Apel et al., 2004;Blöschl et al., 2013;Okoli et al., 2019). ...

... Hence, the important task is obtaining reliable flood estimates. This can be achieved by using appropriate methodological approaches (McKerchar and Macky, 2001;Kjeldsen, 2015;Okoli et al., 2019). ...

The river floods are among the most dangerous natural disasters in the world. Each year, the spring floods cause the significant material damage in the different countries, including Ukraine. Knowledge of trends in such floods, as well as their probabilistic forecast, is of great scientific and practical importance. In last decades, the decreasing phase of cyclical fluctuations of the maximum runoff of spring floods has been observed on the plain rivers of Ukraine, including the Southern Bug River. In addition, there is an increase in air temperature. So, the actual task is the determine the modern probable maximum discharges estimates of spring floods in the Southern Buh River Basin as well as their comparison with the estimates that were computed earlier. It gives an opportunity to reveal possible changes of the statistical characteristics and values of the probable maximum discharges, to analyze and to discuss the reasons for these changes. For the investigation, we used the time series of the maximum discharges of spring floods for 21 gauging stations in the Southern Buh River Basin since the beginning of the observations and till 2015. The method of the regression on the variable that is based on the data of analogues rivers was used to bringing up the duration of the time series and restoration of the gaps. In the study, the hydro-genetic methods for estimation of the homogeneity and stationarity of hydrological series, namely the mass curve, the residual mass curve and the combined graphs. The distributions of Kritskyi & Menkel and Pearson type III for the frequency analysis were used. It has been shown in this study that the maximum discharges of spring floods of time series are quasi-homogeneous and quasi-stationary. It is explained the presence in the observation series of only increasing and decreasing phases of cyclical fluctuations, their considerable duration, as well as the significant variability of the maximal flow. The series of maximal runoff of spring floods are very asymmetric, which significantly complicates the selection of analytical distribution curves. The updated current parameters of the maximal spring flood runoff have not changed significantly. It can be assumed that such characteristics have already become stable over time, as the series of maximal runoff of spring floods already have phases of increasing and decreasing of long-term cyclic fluctuations. Анотація. У світі весняні повені на річках-одне з найнебезпечніших стихійних явищ. Щорічно весняні повені завдають значних матеріальних збитків у різних країнах світу, у тому числі, і в Україні. Важливе наукове і практичне значення має знання тенденцій таких повеней, а також їхній імовірнісний прогноз. В останні десятиліття на рівнинних річках України, до яких відноситься і річка Південний Буг, спостерігається маловодна фаза циклічних коливань максимального стоку весняної повені. Окрім того, спостерігається підвищення температури повітря. Отже, актуальним завданням є визначення сучасних ймовірних характеристик максимальних витрат весняної повені в басейні річки Південний Буг, а також їхнє порівняння з оцінками, які було розраховано раніше. Це дозволить виявити можливі зміни статистичних характеристик максимальних витрат весняної повені, проаналізувати та обговорити причини цих змін. Для дослідження використано ряди спостережень максимальних витрат весняної повені для 21 гідрологічного поста в басейні річки Південний Буг з початку спостережень по 2015 р. Для отримання більш достовірних оцінок ряди спостережень було приведено до багаторічного періоду та по можливості відновлено пропуски методом парної регресії. Для оцінки однорідності і стаціонарності рядів спостережень використано гідролого-генетичні методи, а саме сумарну та інтегральну криву відхилень, суміщені хронологічні графіки. Для апроксимації емпіричних точок використано розподіл Крицького-Менкеля та розподіл ІІІ типу Пірсона. У дослідженні пока-251 зано, що ряди максимальних витрат води весняної повені є квазіоднорідними та квазістаціонарними, оскільки мають тільки незавершені фази (підйому та спаду) довготривалих циклічних коливань. Ряди максимального стоку весняної повені є дуже асиметричними, що суттєво ускладнює побудову аналітичних кривих розподілу. Уточнені сучасні параметри максимального стоку весняної повені суттєво не змінились. Можна припустити, що такі характеристики вже стали стабільними з часом, оскільки ряди спостережень мають фази збільшення і зменшення довгострокових циклічних коливань.

... The first hydro-meteorological approach employs historical rainfall-runoff data to estimate design rainfall converted to design rainfall-excess for its convolution using unit hydrograph to estimate design runoff/flood (Subramanya 2013). The second statistical flood frequency analysis employs measurements of annual maximum water levels at a river cross-section for the derivation of annual maximum flows for fitting to a suitable probability distribution to derive the design peak flow corresponding to a particular return period (Okoli et al. 2019). However, the unavailability of discharge data limits their application to ungauged catchments. ...

Wide field application of curve number (CN) methodology in design flood estimation and forecasting studies is a well-accepted reality. The parameter CN is often regarded as a constant, but it is in fact a random variable, for it relies on a number of factors governing the process of rainfall-runoff. This paper suggests an alternative procedure to estimate design runoff (Q) using design storm (P) and design curve number (CN) values. To this end, design CN values for different durations and return periods have been derived from 25 years of daily annual maximum P-Q data of 10 Indian catchments. When compared, the computed design Q from design CN values was either very close to or slightly greater than the observed Q. The performance is also evaluated using coefficient of determination (R2), percentage of bias (PBIAS), and normalized Nash-Sutcliffe efficiency (NNSE). The magnitude of design CN is found to decrease for longer durations and smaller return periods, and vice versa, and such a behavior invoked the development of an empirical relation for estimation of design CN from given return period and duration for gauged as well as ungauged watersheds. The statistical analysis revealed the general extreme value (GEV) to fit best the P-CN series, whereas multiple probability distributions, viz., GEV, Gumbel max, log Pearson 3, and Gen Pareto fitted best the Q series. The 50-year, 100-year, and 200-year runoff estimated from the P-CN series exceeded only marginally those derived from the conventional Q series approach.

... et. al, 1999); Rainfall-runoff Model (Lamb, 2005); Weibull formula (Selaman et. al, 2007); Log-Pearson Type III distribution (Sathe, et. al. 2012); Probability plotting method (Connell and Mohssen, 2017); California Method (Ganamala, and Kumar, 2017); Hazens Method (Ganamala, and Kumar, 2017); Gumbel's method (Bhagat, 2017) and Probability Models (Okoli et. al., 2019). From this above all methods, Gumbel's Method is most commonly used. So, this study were focus, used the peak discharge data and selected the method of Gumbel's distribution which is widely used for predicting extreme hydrological events such as floods (Zelenhasic, 1970;Haan, 1977;Shaw, 1983). ...

Quantification of peak flood (extreme-hydrological event) discharge of a channel or
river saying for a desired return period is a pre-requisite for design, planning and
management of hydraulic configurations like, dam, bridges, spillways, barrages, etc. this
paper figure out the result of a study carried out at estimation the frequency of Lower
Godavari River division’s flood by using the methods of Gumbel distribution which is one of the probability distribution methods used to model stream flows. The method was used to model the annual maximum river discharge for a period of 25 years (1991-92 to 2015-16). The analysis of the regression equation (R2) gives a value, which shows that this distributional method (Gumbel) is suitable for prophesying the expected flow in the future on this river.