Abstract

Floods are among the most destructive natural hazards, with climate change expected to increase their frequency and intensity, particularly in arid regions where runoff is characterized by high variability. Accurate assessment of these events is critical for the safety and management of hydraulic structures. This paper addresses the design flood revaluation over dam spillways via frequency analysis of the instantaneous peak discharges recorded at the level of the Foum El Gherza dam (Algeria). Six distributions widely used in flood frequency analysis were considered and compared using three model selection criteria. The results showed that the Pearson Type 3 distribution is more appropriate to fit the annual maximum data, while the generalized Pareto distribution is used to fit the peaks-over-threshold. Thus, the largest instantaneous peak discharge observed is an event with a return period of 35 years, and the instantaneous peak flow value has a yearly probability of 1/200 is twice as great than the design flood. These results confirm the significant flood risk on the population living downstream that must be given great interest and caution.

1 Introduction

Flood-related studies are attracting more attention in the field of water sciences, with the observation that economic losses caused by extreme floods have increased significantly. With the effects of climate change, it is expected that global warming will lead to an increased frequency of droughts and floods, with Africa being severely affected (Trenberth et al. 2003; IPCC 2022; Bates et al. 2008). Therefore, dam construction is one of the structural measures largely used for flood risk mitigation. However, it can also be a source of devastating floods linked to huge under-evaluations of the design flood. In fact, with the creation of hydrological databases over the years, several dams around the world have been found to have insufficient discharge capacity (You et al. 2012; Lempérière 2017).

Two common approaches are used to estimate a design flood: rainfall-runoff modeling (Moretti and Montanari 2008; Beven 2010; Breinl 2016) and the fitting of a probability distribution function to a flow series of either annual maxima (AM) or peaks-over threshold (POT) (Hosking and Wallis 1987; Madsen et al. 1997; Viglione et al. 2013; Yan and Moradkhani 2016). The latter approach, generally referred to as Flood Frequency Analysis (FFA), has remained relevant to applications as well as to hydrology research (Gaume 2018; Kobierska et al. 2018). In general, increasing the amount of explanatory information incorporated into the model and the amount of information contained in the model output also increases the complexity of the model and the uncertainty associated with the model output. According to Masson (1992), in hydrology, the only realistic approach is a statistical approach. Nevertheless, the FFA is affected by various sources of uncertainty, including: the choice of the sample technique (peak-over-threshold or annual maximum flows), a limited sample size, the selection of a suitable probability distribution function, the method of parameter estimation for the chosen distribution function, and errors in the observed annual peak flows derived from a rating curve (Sonuga 1972; Laio et al. 2009; Di Baldassarre et al. 2012).

Since the introduction of the probability theory in flood estimation procedures by Fuller (1914) who calculated floods of different return periods for catchments in the U.S, several frequency distributions and methods have been considered for describing the flood data around the world. The most commonly used are:

• Normal (NOR),
• Two parameter log-normal (LN2),
• Exponential (EXP),
• Gamma (GAM),
• Pearson Type-3 (P3),
• Log Pearson Type-3 (LP3),
• Generalized extreme value (GEV),
• Gumbel (Gumbel),
• Fréchet (EV2) and
• Generalized Pareto distribution (GPD) (Cunnane 1989; Laio et al. 2009; WMO 2009; Vivekanandan and Shukla 2015).

This last method is suitable to model exceedances beyond a high threshold (Grimshaw 1993). The model was introduced for the first time by Pickands (1975) and has been presented by many authors such as Davison (1984), Van Montfort and Witter (1985), Hosking and Wallis (1987), and Coles (2001). Traditionally, the threshold was chosen before fitting by balancing bias and variance, giving the so-called fixed threshold approach. The threshold must be sufficiently high to ensure that the asymptotics underlying the GPD approximation are reliable, thus reducing the bias. However, the reduced sample size for high thresholds increases the variance of the parameter estimates (Scarrott and MacDonald 2012). In fact, many threshold selection methods are available in the literature, but there is no single technique that has been shown to work well in all applications. Nevertheless, graphical diagnosis methods and approaches based on goodness-of-fit tests (GoF tests) are the most popular (Coles 2001; Choulakian and Stephens 2012; Caeiro and Gomes 2016; Zoglat et al. 2014; Solari et al. 2017; Bader et al. 2018; G’Sell et al. 2016; Durocher et al. 2018; Silva Lomba et al. 2020).

Selection of an appropriate probability distribution and associated parameter estimation procedure is a classic problem facing hydrologists and has been widely researched (Vogel et al. 1993a; 1993b; ÖnÖz and Bayazit 1995; Laio et al. 2009; Merz and Thieken 2009; Haddad and Rahman 2011). This issue is of practical concern because different distributions may produce significantly different estimates for the same return period (Rahman et al. 2013). Furthermore, there is no theoretical basis for the use of a single distribution over others (Saf 2009). This can be addressed by using formal statistical procedures involving GOF tests and diagnostic tests (Zhang 2002; Vivekanandan and Shukla 2015). Of these, the Bayesian Information Criterion (BIC) and Akaike Information Criterion (AIC) are the two most commonly-used criteria in model selection (Fabozzi et al. 2014). Laio et al (2009) proposed another selection criterion based on the Anderson-Darling test (ADC). The studies were carried out by a Monte Carlo simulation to investigate the robustness of the model selection criteria in recognizing the real parent distributions and concluded that the ability of these criteria to recognize the correct parent distribution from the available data varies from case to case, and these are rather more effective for two-parameter distributions. Haddad and Rahman (2011) stated that ADC is more successful in correctly recognizing the parent distribution than AIC and BIC when the parent is a three-parameter distribution. On the other hand, AIC and BIC are better at correctly recognizing the parent distribution when the parent is a two-parameter distribution. Di Baldassarre et al. (2009) demonstrated that model selection criteria such as AIC, BIC, and ADC, which are seldom used in hydrological applications, can help to identify the best probability model.

In fact, discrepancies in flood frequency estimation may also result from the use of different parameter estimation methods (Saf 2009). Several statistical methods are available for evaluating distribution parameters by sample data, but the commonly used ones are method of moments (MOM), method of maximum likelihood (MLE), and the L-moments approach. The MOM method is one of the simplest techniques conventionally used in hydrology. In most cases, the first three central moments are required to estimate the parameters, i.e., mean, variance, and skewness. From a statistical point of view, the MLE is a more robust method as compared to the MOM, and in most cases provides estimators with good statistical properties, while the L-moments approach is less affected by extremes in the data series (Hosking 1990; Hosking 1997) and can be used to model a wide range of theoretical distributions (Rahman et al. 2013). Today, the L-moments method is widely used for estimating various hydrometeorological variables and has become popular for FFA (Gubareva and Gartsman 2010; Gaume 2018).

Surface runoff in arid watercourses is generally characterized by distinct periods of runoff separated by no flow. However, some wadis present permanent or seasonal flow with a very low base flow (Kallel 2002). Indeed, in some years, even the highest discharges are very low and/or zero and may not be representative of the physical processes driving large floods and can unduly influence the fit of the distribution to AM (Cohn et al. 2013; Benameur et al. 2017). Unusually small values can result in relatively poor estimates of the large flood quantiles. Therefore, the identification and removal of Potentially Influential Low Flows (PILFs) is considered an important step in the AM approach. The impact of PILFs on the upper tail of the distribution can be avoided with the POT series.

The federal guidelines for detection of low outliers in flood flow frequency computation in the United States recommend the Grubbs and Beck test (Grubbs 1969; Grubbs and Beck 1972). This test uses the data to calculate a one-sided, 10-percent significance-level critical value based on a log-normal distribution of the data. The major criticism of the Grubbs-Beck test is that it rarely identifies more than one outlier in flood samples, even though multiple outliers are common in actual flood records (Gotvald et al. 2012; Lamontagne et al. 2012). Rosner (1975; 1983) developed a sequential two-sided outlier test based on a generalization of the Grubbs test (1969) which usually detected outliers either too small or large. Cohn et al (2013) developed the generalized test for detecting multiple potentially influential low outliers in flood series, based on the Grubbs-Beck test statistic, and like Rosner’s (1975) method but without an a priori defined number of outliers.

The frequency analysis was applied in the arid catchment of the Abiod wadi, near Biskra, on the suspended sediment concentrations at McHounech gauge station, 18 km upstream of the Foum El Gherza Dam, by Benkhaled et al. (2014) where the LN2 distribution was selected. In the same station, Benameur et al. (2017) performed a series of observed daily average discharges which revealed that the POT procedure is more relevant than the AM, due to the high inter-annual variability of the data as well as the relatively short record length. Indeed, the only information about floods provided by this station are the daily average discharges measured over the period 1971–1998. Nevertheless, the design of hydraulic structures to control floods is often based on the instantaneous peak flow (Fill and Steiner 2003). Furthermore, the station was damaged during the strong floods of 1999/2000, and it is since non-operational. In addition, it is lacking in records.

In the arid to semi-arid areas, since it has been widely stated that hydrologic data are severely limited (Wheater and Al-Weshah 2002; Wheater 2008), the dams design flood is supposed to be doubtable. For instance, Bretas et al. (2010) investigated three collapsed masonry dams in Algeria; Gran Cheurfas, 39.9 m in height, finished in 1884 and failed in 1885, Hamiz, 38.1 m in height, finished in 1885, and the Habra dam, 38.1 m in height, finished in 1871 and failed in 1881. Among which, this last is more documented since it cost 209 human lives. According to Wegmann (1911) and Chanson (2002), the Habra dam breach was caused by inadequate spillway capacity after an unusually severe storm. More recently, since the construction of the Djorf-Torba dam in 1968 on the wadi Guir, in southwestern Algeria, the records acquired have led to an upward increase in the flood probability estimation. Hence, it would take a 50-100-year flood, instead of 1000, for the structure to discharge 1000 m³/s; thus, it was necessary to review the discharge capacity of the spillway (Belouni et al. 1983). Furthermore, several hydrologic events observed at the level of Foum El Gherza dam spillway, constructed in the late 1940s in southeastern Algeria, extremely exceeded the discharge capacity of the spillway, which was built to safely discharge 730 m³/s at a design flood of 1200 m³/s. One of these was the October 2011 flood, where the water overflowed the dyke for the first time since the dam’s construction (Benameur et al. 2017). Moreover, currently, 2/3 of its capacity is silted and it has serious water leakage problems (Toumi and Remini 2004; Tebbi et al. 2009; Boualem et al. 2015; Benfetta et al. 2017). Thus, floods are a significant risk for the population living downstream that must be given great interest. Nevertheless, to the best of our knowledge, no paper has been yet published addressing FFA via the AM and POT series for the evaluation of the dam design flood using direct flood data observation after more than half century of exploitation. Hence, the aim of this paper is to highlight the necessity for the re-evaluation of the design flood estimates in the Algerian arid area dams using instantaneous peak flow data from the Foum El Gherza dam station.

2 Methodology

The objective of FA is to associate the magnitude of events to their frequency of occurrence through probability distribution using the AM and/or POT series.

For FA studies, it is necessary to check data for outliers and consistency. One of the most frequently used methods to detect outliers for a univariate dataset is based on the concept of the boxplot procedure suggested by Tukey (1977) for normally distributed data while the flood data is assumed to follow other distributions (El Adlouni and Ouarda 2013; Zhao and Yang 2019). So, in this case, the adjusted boxplot for skewed data proposed by Hubert and Vandervieren (2008) is used. The adjusted boxplot includes a robust measure of skewness in the determination of the whiskers.

In arid regions, the surface runoff is generally characterized by distinct periods of runoff separated by no flow, in some years even the highest discharges are very low or zero and may not be representative of the physical processes driving large floods and can unduly influence the fit of the distribution using the AM method (Cohn et al. 2013; Benameur et al. 2017). Hence, the Multiple Grubbs-Beck (MGB) test developed by Cohn et al. (2013) is used to identify PILFs. The MGB test checks if the k-th smallest flow is unusually low and, if it is, uses this discharge to define a threshold for censoring discharges below the threshold (Ball et al. 2019).

Before performing a frequency analysis for any variable, the data series must meet certain statistical criteria such as independence, homogeneity, and stationarity. Therefore, the Wald and Wolfowitz (1943), Wilcoxon (1945), and Mann-Kendall (Mann 1945; Kendall 1975) tests were systematically used to test respectively data independency, homogeneity, and stationarity. These tests are nonparametric and widely applied in FFA (Bobée and Ashkar 1991; Yue et al. 2002; Yue and Pilon 2004; Kundzewicz et al. 2005; Khaliq et al. 2009; Karahacane et al. 2020).

AM flood data are often found to be skewed, which has led to the development and use of many skewed distributions in FFA. In this study, six probability distributions widely used in flood frequency analysis were considered. These were the GEV, Gumbel, LN2, P3, LP3, and EV2 distributions. They are two- or three-parameter distributions offering different levels of flexibility. The Extreme Value distributions family (Gumbel, EV2, and GEV) are designed specifically for maxima, while the Pearson families (P3, LP3) are highly flexible in modeling the right-skewed nature of extreme floods (WMO 2009). The reason behind the selection of the above probability distributions is that most of these have been recommended in various countries for at-site flood frequency (Cunnane 1989; Laio et al. 2009; WMO 2009; Vivekanandan and Shukla 2015). The selection of the appropriate distribution was then done using the following goodness-of-fit tests:

1. Anderson–Darling criterion (ADC) (Laio et al. 2009),
2. Akaike information criterion (AIC) (Akaike 1974), and
3. Bayesian information criterion (BIC) (Schwarz 1978).

The best model is the one which achieves the minimum test values of the comparing models. The proposed procedure is then performed over simulated series by carrying out Monte Carlo simulations from the original sample. Accordingly, a base distribution should be chosen, rather than the contending distributions. Cunnane (1989) proposed the five-parameter Wakeby distribution as a parent distribution for its flexibility. Hence, we adopted it as a base distribution to generate 2000 simulations to examine the performance of the probability distributions.

Peak-over-threshold flood data were fitted using the GPD distribution. Selecting an appropriate threshold value, denoted as u, is important because the estimated extreme events by the POT model are sensitive to the changes in u. In our study, a hybrid procedure, consisting of graphical methods and GOF tests, was proposed to specify a suitable threshold value for the POT model. First, the mean residual life plot (MRL) and the parameter stability plot recommended by Coles (2001) were used to specify a range of suitable threshold value for the POT model. However, an apparent limitation of the graphical methods is that threshold detection from a range of potential optimum threshold values is conducted based on subjective criteria. Further insight can be provided by an objective method based on the goodness of fit tests of the GPD to identify the lowest threshold above which the data provides adequate fit to the GPD distribution model. In this line, the LRT (Likelihood Ratio Test) and the score tests outperformed the various threshold selection techniques compared by Zoglat et al. (2014). They were adopted and the lowest threshold above which the null hypothesis is not rejected, and at all the higher thresholds considered was selected.

An important assumption for the application of the POT method is that the extracted peaks are independent. A commonly applied method is to consider a minimum separation interval between successive peaks (Cunnane 1979; Lang and Bobe 1999; Silva et al. 2012). However, since two observed floods in the study area are generally distinct by no flow, maximum peaks of each flood hydrograph were considered, and their independence was tested using the Wald and Wolfowitz test.

The commonly used methods for evaluating distribution parameters by sample data are the MOM, MLE and L-moments approach. Nowadays, the L-moments method is widely used for FFA (Gubareva and Gartsman 2010; Gaume 2018). Thus, the L-moments approach was adopted in this research paper. More information about the used probability distributions, parameter estimation methods, and the GOF tests can be found in Cunnane (1989), Laio et al. (2009), and WMO (2009).

All the simulations in this study were carried out within the R environment, which is an open-source software for statistical computing and visualization (http:// www.r-project.org/), and using the free libraries nsRFA (Viglione 2020) and POT (Ribatet 2007).

3 Study Area and Data

3.1 Study area

The Foum El Gherza dam, located 18 km east of Biskra in Algeria, is a concrete arch (126 m) dam with abutment on the left bank (60.3 m) and overall height of 65 m above the streambed for an initial capacity of 47 million m³. It was built in the late 1940s on the Abiod wadi for the irrigation of the palms of Sidi Okba, Garta, Seriana, and Thouda. It covers an area of 3 km² for a current capacity of 16.5 million m³, which means that it has lost nearly 2/3 of its volume due to silting. The dam was designed to release 730 m³/s with two free surface spillways, each 18 m wide and 4.35 m high, and 110 m³/s with the bottom outlet for a design flood of 1200 m³/s. The stored water volume within the dam along the height of the spillway is 14.07 hm³.

The Abiod wadi watershed covers an area of 1300 km² with a perimeter of 200 km. It originates in the Chelia (2326 m high) and Ichemoul (2100 m high) mountains in the Aures, and crosses Tighanimine, and then the wadi gradually flows into the canyons of the Ghoufi and M’chouneche gorges before flowing into the Saharian gorge Foum El Gherza (295 m high) crossing 85 km from its source (Figure 1). The climate ranges between semi-arid to arid. Average annual precipitation is strongly modified by the relief and varies from 450 mm upstream to 100–150 mm in the Sahara plain.

Figure 1 Location of the Foum El Gherza dam.

The wadi valley is mainly composed of sedimentary rocks, comprising alternating limestone, marl, soft sediments (sandstones, conglomerates) and some evaporites (gypsum) dating from the Paleogene. The relief is rugged, characterized by slopes ranging between 12.5% and 25% for half of the area, and from 3% to 12.5% for another 40% of the area. Land cover is a mix of rocky outcrops, highly eroded soil, sparse vegetation, and a few forests, crops, gardens, and pastures (Hamel 2009).

The SCET monography (1971) (French acronym of “The Central Society for the Territory Equipment”) discusses a flood which occurred on September 15th and 16th, 1902, i.e., before the construction of the Foum El Gherza dam, which estimates the peak discharge between 800 m³/s and 1500 m³/s. Dalloni (1939) reports that the only flood relatively well-known is one that occurred on June 1st ,1920, when the peak discharge reached 330 m³/s at Foum El Gherza. According to Duquesnoy (1949) the discharges of exceptional floods should not exceed 500 m³/s.

3.2 Data description

The main gauging station used in this study is one that is managed by the National Agency for Dams and Transfers (NADT) of Biskra and installed at the level of Foum El Gherza Dam spillway. Another gauging station managed by the NAWR (National Agency of Water Resources) is located at M'chouneche, 18 km upstream from the dam, where only the average daily flows are monitored. However, this station is no longer operational since it was damaged during the 1994–1995 floods. The data extends over the period 1951–2011. The total number of recorded events is about 405, averaging seven floods per year. It is the only station to produce instantaneous flood discharge in the Abiod Wadi catchment. The water level was recorded by a limnimeter and transformed into flood discharge from the variation of the reservoir water level at scale time. The flood data are generally recorded at hourly scale time. The details reported are the mean flow, the maximum flow, the volume, the total duration, and the maximum water depths discharged by the spillway.

4 Results

The instantaneous peak flows and the resulted annual maxima (AM) and peak over threshold (POT) time series are reported in Figure 2. The event of October 2011 is not included since the instantaneous peak discharge of the flood is under discussion, despite that the released discharges and volume are almost known, and exceed 1500 m³/s and 58 hm³, respectively. It appears from the figure that the instantaneous peak flows of the observed events vary from a minimum value of 1.1 m³/s to a maximum value of 2152.00 m³/s. The mean daily discharge, calculated over the hydrologic year (365 days), is about 1.40 m³/s and the “wet discharge” denoted Q_w, i.e., the yearly average discharge calculated over the period of wadi flow (and not calculated over the full year) is about 68 m³/s.

Figure 2 Time series of the annual maximum instantaneous peak flows and peak over threshold flows observed at Foum El Gherza dam (1951–2011).

Since the objective is to study floods, only significant events will be considered. Thus, the Multiple Grubbs-Beck test is used to identify the PILFs. Hence, three possible PILFs have been detected and censored, which are 1.1 m³/s, 30.87 m³/s, and 51.03 m³/s, corresponding respectively to the hydrologic years 1974, 1971, and 1975.  

Unlike the AM method, in the case of the POT method, we consider all events above a determined threshold. The plot of mean excess against threshold (MRL plot), featured in Figure 3, shows that the graph looks somewhat linear from u = 50 m³/s until u = 220 m³/s. Therefore, for thresholds that vary on this range, the parameter stability plots are shown in Figure 4. In practice we want to find a threshold for which the scale and modified shape estimators are approximately stable. Hence, the figure shows that the fairly linear stability of the parameters can be identified starting at 169 m³/s. Next, the information is gained from the MRL plot, and the parameters stability plot is analysed using the LRT and the score tests (Figure 5) with the aim of the objective determination of the threshold. In Figure 5, the lowest threshold above which the p-value of the LRT and score tests are larger than 0.05, and remain larger than 0.05 at all higher thresholds is u = 206.5 m³/s.

Figure 3 Mean residual life plot for Foum El Gherza instantaneous peak flows.

Figure 4 Stability plot for shape and modified scale parameters for Foum El Gherza instantaneous peak flows.

Figure 5 LRT and score test results for Foum El Gherza instantaneous peak flows.

It is noted that the instantaneous peak flow of 1000 m³/s is exceeded only four times in the first part of the observation period (1951–1980), whereas it is exceeded eleven times in the second part (1980–2011). Moreover, more than 50% of the instantaneous peak flows exceeded the flow rate of 750 m³/s in the second part of the period.

The basic statistical parameters corresponding to the entire chronicles are presented in Table 1, while histograms and the skewed boxplots are shown in Figure 6. The latter shows no existence of outliers, and that the mean flow is greater than the median, which means that the distributions are right-skewed.

Table 1 Statistical parameters corresponding to AM and POT time series.

Statistics	AM	POT
	1951-2011	1951-2011
Effective	58.00	112
Min	91.81	266.60
Mean	683.50	617.50
Median	558.30	473.90
Max	2152.00	2152.00
Standard deviation	520.91	441.17
Cv	0.76	0.714
Cs	0.90	1.51
Ck	-0.11	1.62

Figure 6 Histogramm and skewed boxplot of the AM and POT series.

The results of the required hypothesis testing on the considered data (AM and POT) are given in Table 2. After applying the Wald-Wolfowitz, Wilcoxon, and Mann-Kendall tests respectively, we conclude that the independence, homogeneity, and stationarity are accepted at any of the standard significance levels (1%, 5%, and 10%).

Table 2 Hypothesis tests results.

Tests	Statistic	AM	POT
		1951–2011	1951–2011
Wald–Wolfowitz	p-value	0.720	0.237
	a-value	-5.281	6.354
Wilcoxon	p-value	0.199	0.125
	W-value	337	1282
Mann-Kendal	p-value	0.440	0.227
	τ-value	0.070	0.077

While the GPD distribution is used for the POT data, the results of the application of the three models’ selection criteria to the simulated series from the original sample using the five-parameter Wakeby distribution as a parent distribution are summarized in Figure 7. It is clear from the figure that the P3 and LP3 distributions outperformed the other distributions. The AIC and BIC criteria tests select the P3 distribution for the highest number of series (16 stations out of 44, i.e., 59% and 53.45 % cases respectively) followed by the LP3 distribution which was selected for 39.7 % of cases by the AIC, and 38.9 % cases by the BIC. The ADC test selects the LP3 and P3 distributions for 57.2 % and 33.55% of cases, respectively. The AIC, BIC, and ADC tests were then applied to the original data and the results presented in Table 3 confirm that the P3 distribution is the best choice.

Figure 7 Percentage of cases in which each candidate distribution is selected by each criterion.

Table 3 Tests for the selection of the appropriate distribution.

Law of probability	AM (1951-2011)
	AIC	BIC	ADC
LN	870.3	874.4	0.3219
GUMBEL	878.9	883.0	0.6897
EV2	879.7	883.8	1.0509
GEV	876.6	882.8	0.5878
P3	864.4	870.6	0.1230
LP3	869.6	875.8	0.2068

The correspondent L-moment distribution parameters are obtained and shown in Table 4. The resulting models are then fitted and presented in Figure 8. 

Table 4 L-moment parameters estimation.

	Parameters
	Location (β)	Scale (λ)	Shape (γ)
AM	-58.51	399.93	1.855
POT	206.5	340.65	0.17

Figure 8 Model fitted to Foum El Gherza dam data.

One of the objectives at-site FFA is to estimate a different return period associated with a given flood magnitude. To gain this purpose, Table 5 resumes the values resulting from the fitted models.

Table 5 Estimated quantiles for different return period from the fitted models.

	Return periods (y)
	2	5	10	20	50	100	200
AM (P3-L-moments)	592.39	1132.25	1499.04	1848.70	2294.67	2623.66	2947.39
POT (GPD-L-moments)	457.20	837.73	1167.96	1539.80	2104.20	2594.00	3145.52

5 Discussion

This study gives us clear insight regarding the necessity for design flood reviews in the arid to semi-arid area dams. The study is carried out using FFA of the AM and POT series. Due to the high inter-annual variability characterizing surface runoff in the arid area watercourses, the multiple Grubbs-Beck test is used to identify a threshold below which data are considered PILFs and removed. So, three values are censored, and all the values are less than the mean wet discharge of the wadi. This approach is consistent with the methodology adopted by Lamontagne et al. (2016) and Rima et al. (2024), which emphasizes the importance of censoring low-magnitude outliers to prevent bias in extreme quantile estimation.

One of the primary objectives of this paper is to investigate the selection of an appropriate probability distribution for AM flood data. Three selection criteria tests are adopted: the AIC, the BIC, and the ADC, to identify the best-fit probability distribution model. It is found that the Pearson type III (P3) distribution is the best choice for the description of probability behaviors of AM events, while the GPD distribution is adopted for fitting POT series. Although regional hydrological studies across northern Algeria favor Gamma and Weibull distributions (Hebal and Remini, as cited in Benameur et al. (2017), our finding aligns with the work of Bouanani (2004) in the Tafna catchments and is further validated by recent literature (Fitriana et al. 2025; Shah et al. 2025). These studies confirm that asymmetric distributions like P3 and its variants often outperform other distributions for capturing the behavior of extreme flows in arid and semi-arid environments. Crucially, although the GPD distribution is the standard for the POT approach (Pan et al. 2022), the convergence of both AM and POT models at high return periods (50–200 years) justifies the use of partial duration series to bolster hydrological reliability in data-scarce regions.  

The revision of the hydrologic estimation related to instantaneous peak discharge shows that a flood with a twenty-year return period is large enough to release 730 m³/s instead of a millennial flood, which is generally adopted for concrete dams. Furthermore, the instantaneous peak flow value that has a yearly probability of 1/200, which is twice as great as the project flood. This large gap between the resulting flood magnitude and the builders’ estimates cannot be pushed to floods of exceptional frequencies, because of lack of sufficient data. But the development of knowledge related to the hydrology of the area makes plausible "project floods" whose instantaneous peak discharge would significantly exceed 1500 m³/s, i.e., the largest quoted peak discharge in the Abiod wadi before the dam construction (SCET 1971). This alarming gap is not an isolated case; similar underestimations in older dam designs have been reported in recent re-evaluations (You et al. 2012; Lempérière 2017; Arganis-Juárez et al. 2019; Hussain et al. 2022; Azha et al. 2026).

In fact, the information gained from the FFA of instantaneous peak flows under stationary conditions are essential but insufficient for many purposes. Flood severity is not only defined by its peak value, but also by its volume and duration (Javelle et al. 2003; Chen et al. 2006). Moreover, a significant number of theories and methods have been devoted to extreme and rare flood evaluation for big dam security. Hence, further works should focus on the revision of the hydrologic study for the determination of project flood hydrographs for high-return periods under stationarity and non-stationarity conditions.

6 Conclusion

The aim of this study was to evaluate the design flood using FFA performed on instantaneous peak flow data observed at the Foum El Gherza dam after more than a half-century of exploitation. These goals were achieved by assessing the performance of several probability distributions. A Monte Carlo simulation from the original records was used and fit the distributions to these samples. Subsequently, the fits were evaluated according to several goodness-of-fit criteria.

Based on the results presented herein we conclude the following:

• The P3 and LP3 distributions are the most appropriate to fit the AM data.
• The AM and POT approaches give similar flood estimates for return periods varying from 50 to 200 years.
• A twenty-year return period flood is large enough to release 730 m³/s instead of the millennial flood, which is generally adopted for concrete dams.
• 1/200-year probability for instantaneous peak flow values is twice as great as the project flood of the builders’ estimates.

The results herein confirm the presence of hydrological risk at the study area that must be given great interest and caution. A variety of flood mitigation and dam rehabilitation techniques for the Foum El Gherza dam should be deeply studied to ensure environmental and social compliance. This includes raising the height of a dam, constructing one or more additional spillways or fuse plugs, and modifying catchment flood characteristics by building flood detention devices or even an upstream dam.

Acknowledgments

The authors would like to express their deepest gratitude to Mr. Salah Sakhraoui from the National Agency for Dams and Transfers to provide us with the necessary data to accomplish this research.

Funding

This work was financially supported by the Directorate-General for Scientific Research and Technological Development (DGRSDT, its French acronym).

List of Abbreviations

AM	Annual maxima
POT	Peaks-over threshold
FFA	Flood Frequency Analysis
NOR	Normal
LN2	Two parameter log-normal
EXP	Exponential
GAM	Gamma
P3	Pearson Type-3
LP3	Log Pearson Type-3
GEV	GEV
EV2	Fréchet
GPD	Generalized Pareto
GOF	Goodness-of-Fit
AIC	Akaike information criterion
BIC	Bayesian information criterion
ADC	Anderson-Darling criterion
MOM	Method of moments
MLE	Maximum likelihood
LRT	Likelihood ratio test
SCET	Central Society for the Territory Equipment
NADT	National Agency for Dams and Transfers
NAWR	National Agency of Water Resources
Qw	Wet discharge
Min	Minimum
Max	Maximum
Cv	Coefficient of Variation
Cs	Coefficient of Skewness
Ck	Coefficient of Kurtosis

References

1. Akaike, H. 1974. “A new look at the statistical model identification.” IEEE Transactions on Automatic Control 19 (6): 716–723.
2. Arganis-Juárez, M.L., M. Preciado-Jiménez, Ó.A. Fuentes-Mariles, A. Mendoza-Reséndiz, R. Domínguez-Mora, F. De Luna-Cruz, and E.C. Elizondo. 2019. “Hybrid method to update the design flood shape for spillway operation in a dam: case study of the Huites Dam, Mexico.” Hydrological Sciences Journal 64 (8): 934–947. https://doi.org/10.1080/02626667.2019.1612067
3. Azha, S.F., L.M. Sidek, K. Kok, S.A. Ahmad, D.H. Saman, H. Basri, M.H. Zawawi, et al. 2026. “Enhanced Dam Safety and Resilience through Revised Probable Maximum Precipitation as Insights from Hydroelectric Scheme in Malaysia” Results in Engineering 29, 109090. https://doi.org/10.1016/j.rineng.2026.109090
4. Bader, B., J. Yan, J. and X. Zhang. 2018. “Automated Threshold Selection for Extreme Value Analysis via Ordered Goodness-of-Fit Tests With Adjustment for False Discovery Rate.” The Annals of Applied Statistics 12 (1): 310–329. https://doi.org/10.1214/17-AOAS1092
5. Ball, J., M. Babister, R. Nathan, W. Weeks, E. Weinmann, M. Retallick, I. Testoni (Editors). 2019. Australian Rainfall and Runoff: A Guide to Flood Estimation. Commonwealth of Australia. Available at: http://arr.ga.gov.au/
6. Bates, B.C., Z.W. Kundzewicz, S. Wu, and J.P. Palutikof (Eds.). 2008. Climate Change and Water. Technical Paper of the Intergovernmental Panel on Climate Change. Intergovernmental Panel on Climate Change (IPCC). Geneva, Switzerland.
7. Belouni, M.S., T. Sahraoui, J. Bouloc, and B. Goguel. 1983. “Restructuration du déversoir de Djorf-Torba. ” In French. La Houille Blanche 69 (3/4): 231–239. https://doi.org/10.1051/lhb/1983020
8. Benameur, S., A. Benkhaled, D. Meraghni, F. Chebana, and A. Necir. 2017. “Complete flood frequency analysis in Abiod watershed, Biskra (Algeria).” Natural Hazards 86 (2): 519–534. https://doi.org/10.1007/s11069-016-2703-4
9. Benfetta, H., A. Ouadja, and N. Hocini. 2017. “Enhancement of the study of water leaks in the Algerian dam of Foum El Gherza.” Arabian Journal of Geosciences 10, 482. https://doi.org/10.1007/s12517-017-3281-9
10. Benkhaled, A., H. Higgins, F. Chebana, and A. Necir. 2014. “Frequency analysis of annual maximum suspended sediment concentrations in Abiod wadi, Biskra (Algeria).” Hydrological Processes 28 (12): 3841–3854. https://doi.org/10.1002/hyp.9880
11. Beven, K. 2010. Rainfall-Runoff Modelling: The Primer, 2nd Edition. Wiley-Blackwell, New Jersey, USA, pp. 488.
12. Bobée, B. and F. Ashkar. 1991. The Gamma Family and Derived Distributions Applied in Hydrology. Water Resources Publications, University of California, pp. 203.
13. Boualem, R., B. Djillali, and M. Mohamed. 2015. “Silting of Foum El Gherza Reservoir.” GeoScience Engineering 61 (1): 1–9. https://geoscience.cz/ojs/index.php/GSE/article/view/72
14. Bouanani, A. 2004. “Hydrologie, transport solide et modélisation de l'écoulement de surface dans le bassin versant de la Tafna (NW Algérie).” In French. PhD thesis, Tlemcen University, Algeria, pp. 250.
15. Breinl, K. 2016. "Driving a lumped hydrological model with precipitation output from weather generators of different complexity." Hydrological Sciences Journal 61 (8): 1395–1414. https://doi.org/10.1080/02626667.2015.1036755
16. Bretas, E.M., J.V. Lemos, and P.B. Louranço. 2010. “Masonry gravity dams: A numerical application for stability analysis.” 8th International Masonry Conference. Dresden, pp. 1963–1972.
17. Caeiro, F. and M.I. Gomes. 2016. “Threshold Selection in Extreme Value Analysis.” In: Extreme Value Modeling and Risk Analysis. Chapman and Hall/CRC 2007, pp. 69–86.
18. Chanson, H. 2002. Hydraulics of Stepped Chutes and Spillways. CRC Press, pp. 424. https://doi.org/10.1201/9781003762300
19. Chen, Y.D., G. Huang, Q. Shao, and C.Y. Xu. 2006. “Regional analysis of low flow using L-moments for Dongjiang basin, South China.” Hydrological Sciences Journal 51 (6): 1051–1064. https://doi.org/10.1623/hysj.51.6.1051
20. Choulakian, V. and M.A. Stephens. 2012. “Goodness-of-Fit Tests for the Generalized Pareto Distribution.” Technometrics 43 (4): 478–484. https//doi.org./10.1198/00401700152672573
21. Cohn, T.A., J.F. England, C.E. Berebrock, R.R. Mason, J.R. Stedinger, and J.R. Lamontagne. 2013. “A generalized Grubbs-Beck test statistic for detecting multiple potentially influential low outliers in flood series.” Water Resources Research 49 (8): 5047–5058. https://doi.org/10.1002/wrcr.20392
22. Coles, S. 2001. An Introduction to Statistical Modeling of Extreme Values. Springer London (Springer Series in Statistics). https://doi.org/10.1007/978-1-4471-3675-0
23. Cunnane, C. 1979. “A Note on the Poisson Assumption in Partial Duration Series Models.” Water Resources Research 15 (2): 489–494. https://doi.org/10.1029/WR015I002P00489
24. Cunnane, C. 1989. Statistical Distributions for Flood Frequency Analysis. WMO Series no. 718, Operational Hydrology Report no. 33. Secretariat of the World Meteorological Organization, Geneva, Switzerland, pp. 63–73.
25. Dalloni, M. 1939. Géologie appliquée de l’Algérie: métallogénie, hydrogéologie, agrogéologie. In French. Masson et Cie, Editeurs Libraires de L’Academi de Medecine 120, Paris, France.
26. Davison, A.C. 1984. “Modelling Excesses over High Thresholds, with an Application.” In: de Oliveira, J.T. (eds) Statistical Extremes and Applications. NATO ASI Series, vol 131, pp. 461–482. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-3069-3_34
27. Di Baldassarre, G., F. Laio, and A. Montanari. 2009. “Design flood estimation using model selection criteria.” Physics and Chemistry of the Earth 34 (10–12): 606–611. https://doi.org/10.1016/j.pce.2008.10.066
28. Di Baldassarre, G., F. Laio, and A. Montanari. 2012. “Effect of observation errors on the uncertainty of design floods.” Physics and Chemistry of the Earth, Parts A/B/C  42–44, 85–90. https://doi.org/10.1016/J.PCE.2011.05.001
29. Duquesnoy, M. 1949. Terres et Eaux – Le Barrage de Foum el Gherza. pp. 40. Gouvernement Général de l'Algérie.
30. Durocher, M., S.M. Zadeh, D.H. Burn, and F. Ashkar. 2018. “Comparison of automatic procedures for selecting flood peaks over threshold based on goodness-of-fit tests.” Hydrological Processes 32 (18): 2874–2887. https://doi.org/10.1002/hyp.13223
31. El Adlouni, S. and T.B.M.J. Ouarda. 2013. “Frequency Analysis of Extreme Rainfall Events.” In: Rainfall: State of the Science, eds F.Y. Testik and M. Gebremichael. AGU Advancing Earth and Space Sciences, pp. 171–188. 
32. Fabozzi, F.J., S.M. Focardi, S.T. Rachev, and B.G. Arshanapalli. 2014. “Appendix E: Model Selection Criterion: AIC and BIC.” In: The Basics of Financial Econometrics, edited by F.J. Fabozzi, S.M. Focardi, S.T. Rachev, and B.G. Arshanapalli. John Wiley & Sons, Inc. https://doi.org/10.1002/9781118856406.app5
33. Fill, H.D. and A.A. Steiner. 2003. “Estimating Instantaneous Peak Flow from Mean Daily Flow Data.” Journal of Hydrologic Engineering 8 (6): 365–369. https://doi.org/10.1061/(ASCE)1084-0699(2003)8:6(365)
34. Fitriana, F.S., S. Fischer, S.K. Singh, and M. Pahlow. 2025. “Regional flood frequency analysis for hydrologically diverse regions in New Zealand.” Australasian Journal of Water Resources 29 (2): 135–150. https://doi.org/10.1080/13241583.2025.2530280
35. Fuller, W.E. 1914. “Flood Flows.” Transactions of the American Society of Civil Engineers 77 (1): 564–617.
36. Gaume, E. 2018. “Flood frequency analysis: The Bayesian choice.” Wiley Interdisciplinary Reviews: Water 5 (4): e1290. https://doi.org/10.1002/wat2.1290
37. Gotvald, A.J., N.A. Barth, A.G. Veilleux, and C. Parrett. 2012. “Methods for Determining Magnitude and Frequency of Floods in California, Based on Data Through Water Year 2006.” U.S. Geological Survey Scientific Investigations Report 2012–5113, U.S. Department of the Interior, U.S. Geological Survey, in cooperation with Federal Emergency Management Agency, Reston, Virginia, pp. 38. http://pubs.usgs.gov/sir/2012/5113/
38. Grimshaw, S.D. 1993. “Computing Maximum Likelihood Estimates For The Generalized Pareto Distribution.” Technometrics 35 (2):185–191. http://doi.org/10.1080/00401706.1993.10485040
39. Grubbs, F.E. 1969. “Procedures for Detecting Outlying Observations in Samples.” Technometrics 11 (1): 1–21. https://doi.org/10.1080/00401706.1969.10490657
40. Grubbs, F.E. and G. Beck. 1972. “Extension of Sample Sizes and Percentage Points for Significance Tests of Outlying Observations.” Technometrics 14 (4): 847–854. https://doi.org/10.1080/00401706.1972.10488981
41. G’Sell, M.G., S. Wager, A. Chouldechova, R. Tibshirani. 2016. “Sequential Selection Procedures and False Discovery Rate Control.” Journal of the Royal Statistical Society: Series B (Statistical Methodology) 78 (2): 423–444. https://doi.org/10.1111/RSSB.12122
42. Gubareva, T.S. and B.I. Gartsman. 2010. “Estimating distribution parameters of extreme hydrometeorological characteristics by L-moments method.” Water Resources 37, 437–445. https://doi.org/10.1134/S0097807810040020
43. Haddad, K. and A. Rahman. 2011. “Selection of the best fit flood frequency distribution and parameter estimation procedure: a case study for Tasmania in Australia.” Stochastic Environmental Research and Risk Assessment 25, 415–428. https://doi.org/10.1007/s00477-010-0412-1
44. Hamel, A. 2009. “Étude de l'impact de l'érosion sur l'envasement du barrage de Foum El Gherza (Biskra).” In French. Magister thesis, Mentouri University, Constantine, Algeria, pp. 180.
45. Hosking, J.R.M. 1990. “L-Moments: Analysis and Estimation of Distributions Using Linear Combinations of Order Statistics.” Journal of the Royal Statistical Society, Series B 52 (1): 105–124.
46. Hosking, J.R.M. 1997. Regional Frequency Analysis: An Approach Based on L-Moments. Cambridge University Press, New York, USA, pp. 224.
47. Hosking, J. and J.R. Wallis. 1987. “Parameter and Quantile Estimation for the Generalized Pareto Distribution.” Technometrics 29 (3): 339–349. https://doi.org/10.2307/1269343
48. Hubert, M. and E. Vandervieren. 2008. “An adjusted boxplot for skewed distributions.” Computational Statistics and Data Analysis 52 (12): 5186–5201. https://doi.org/10.1016/j.csda.2007.11.008
49. Hussain, H.H., A.I. Al Obaidy, A.H. Hommadi, H.T. Al Hudaib, A.T. Al Masoodi, F.H. Saeed, and N.N. Al Saeedi. 2022. “Modifying the Spillway of Adhaim Dam, Reducing Flood Impact, and Saving Water.” Journal of Water Management Modeling 30, C485. https://doi.org/10.14796/JWMM.C485
50. IPCC. 2022: Climate Change 2022: Impacts, Adaptation, and Vulnerability. Contribution of Working Group II to the IPCC Sixth Assessment Report [H.-O. Pörtner, D.C. Roberts, M. Tignor, E.S. Poloczanska, K. Mintenbeck, A. Alegría, M. Craig, S. Langsdorf, S. Löschke, V. Möller, A. Okem, B. Rama (eds.)]. Cambridge University Press. https://www.ipcc.ch/report/ar6/wg2/
51. Javelle, P., T.B.M.J. Ouarda, and B. Bobée. 2003. “Spring flood analysis using the flood-duration–frequency approach: application to the provinces of Quebec and Ontario, Canada.” Hydrological Processes 17 (18): 3717–3736. https://doi.org/10.1002/hyp.1349
52. Kallel, M.R. 2002. "Développement intégré de l’hydrologie des oueds." In: Hydrology of Wadi Systems, edited by H. Wheater and R. Al-Weshah, Technical Documents in Hydrology No. 55, 123–144. Paris: UNESCO. Available at: https://unesdoc.unesco.org/ark:/48223/pf0000127229
53. Karahacane, H., M. Meddi, F. Chebana, H.A. Saaed. 2020. “Complete multivariate flood frequency analysis, applied to northern Algeria.” Journal of Flood Risk Management 13 (4): 1–48. https://doi.org/10.1111/jfr3.12619
54. Kendall, M.G. 1975. Rank Correlation Methods, Fifth Edition. Oxford University Press, London, UK, pp. 260.
55. Khaliq, M.N., T.B.M.J. Ouarda, P. Gachon, L. Sushama, and A. St-Hilaire. 2009. “Identification of hydrological trends in the presence of serial and cross correlations: A review of selected methods and their application to annual flow regimes of Canadian rivers.” Journal of Hydrology 368 (1–4): 117–130. https://doi.org/10.1016/J.JHYDROL.2009.01.035
56. Kobierska, F., K. Engeland, and T. Thorarinsdottir. 2018. “Evaluation of design flood estimates – a case study for Norway.” Hydrology Research 49 (2): 450–465. https://doi.org/10.2166/nh.2017.068
57. Kundzewicz, Z.W., D. Graczyk, T. Maurer, I. Pinskwar, M. Radziejewski, S. Svensson, and M. Szwed. 2005. “Trend detection in river flow series: 1. Annual maximum flow / Détection de tendance dans des séries de débit fluvial: 1. Débit maximum annuel.” In French. Hydrological Sciences Journal 50 (5): 797–810. https://doi.org/10.1623/hysj.2005.50.5.797
58. Laio, F., G. Di Baldassarre, and A. Montanari. 2009. “Model selection techniques for the frequency analysis of hydrological extremes.” Water Resources Research 45 (7): 1–11. https://doi.org/10.1029/2007WR006666
59. Lamontagne, J.R., J.R. Stedinger, C. Berenbrock, A.G. Veilleux, J.C. Ferris, D.L. Knifong. 2012. “Development of regional skews for selected flood durations for the Central Valley Region, California, based on data through water year 2008.” Scientific Investigations Report 2012–5130. Available at: http://www.usgs.gov/pubprod
60. Lamontagne, J.R., J.R. Stedinger, X. Yu, C.A. Whealton, and Z. Xu. 2016. “Robust flood frequency analysis: Performance of EMA with multiple Grubbs-Beck outlier tests.” Water Resources Research 52 (4): 3068–3084. https://doi.org/10.1002/2015WR018093
61. Lang, M. and B. Bobe. 1999. “Towards operational guidelines for over-threshold modeling.” Journal of Hydrology 225 (3–4): 103–117. https://doi.org/10.1016/S0022-1694(99)00167-5
62. Lempérière, F. 2017. “Dams and Floods.” Engineering 3 (1): 144–149. https://doi.org/10.1016/J.ENG.2017.01.018
63. Madsen, H., P.F. Rasmussen, and D. Rosbjerg. 1997. “Comparison of annual maximum series and partial duration series methods for modeling extreme hydrologic events: 1. At-site modeling.” Water Resources Research 33 (4): 747–757. https://doi.org/10.1029/96WR03848
64. Mann, H.B. 1945. “Nonparametric Tests Against Trend.” Econometrica 13 (3): 245–259. https://doi.org/10.2307/1907187
65. Masson, J.M. 1992. “Hydrologie et Satistiques in Séminfor 5 : Cinquième Séminaire Informatique de l’ORSTOM, Montpellier (FRA),1991/09/2-4, pp. 461–464. In French. Available at: http://www.documentation.ird.fr/hor/fdi:36027
66. Merz, B. and A.H. Thieken. 2009. “Flood risk curves and uncertainty bounds.” Natural Hazards 51, 437–458. https://doi.org/10.1007/s11069-009-9452-6
67. Moretti, G. and A. Montanari. 2008. “Inferring the flood frequency distribution for an ungauged basin using a spatially distributed rainfall-runoff model.” Hydrology and Earth System Sciences 12 (4): 1141–1152. https://doi.org/10.5194/hess-12-1141-2008
68. ÖnÖz, B. and M. Bayazit, M. 1995. “Best-fit distributions of largest available flood samples.” Journal of Hydrology 167 (1–4): 195–208. https://doi.org/10.1016/0022-1694(94)02633-M
69. Pan, X., A. Rahman, K. Haddad, and T.B.M.J. Ouarda. 2022. “Peaks-over-threshold model in flood frequency analysis: a scoping review.” Stochastic Environmental Research and Risk Assessment 36, 2419–2435. https://doi.org/10.1007/s00477-022-02174-6
70. Pickands III, J. 1975. “Statistical Inference Using Extreme Order Statistics.” The Annals of Statistics 3 (1): 119–131. https://doi.org/10.1214/aos/1176343003
71. Rahman, A.S., A. Rahman, M.A. Zaman, K. Haddad, A. Ahsan, and M. Imteaz. 2013. “A study on selection of probability distributions for at-site flood frequency analysis in Australia.” Natural Hazards 69, 1803–1813. https://doi.org/10.1007/s11069-013-0775-y
72. Ribatet, M. 2007. “POT: Modelling Peaks Over a Threshold.” R package version 1.1-0. The R Journal 7 (1): 34–36. https://journal.r-project.org/articles/RN-2007-008/
73. Rima, L., K. Haddad, and A. Rahman. 2024. “Low-Flow Identification in Flood Frequency Analysis: A Case Study for Eastern Australia” Water 16 (4): 535. https://doi.org/10.3390/w16040535
74. Rosner, B. 1975. “On the detection of many outliers.” Technometrics 17 (2): 221–227.
75. Rosner, B. 1983. “Percentage Points for a Generalized ESD Many-Outlier Procedure.” Technometrics 25 (2): 165–172. https://doi.org/10.1080/00401706.1983.10487848
76. Saf, B. 2009. “Regional Flood Frequency Analysis Using L Moments for the Buyuk and Kucuk Menderes River Basins of Turkey.” Journal of Hydrologic Engineering 14 (8): 783–794. https://doi.org/10.1061/(ASCE)HE.1943-5584.0000055
77. Scarrott, C. and A. MacDonald. 2012. “A Review of Extreme Value Threshold Estimation and Uncertainty Quantification.” REVSTAT Statistical Journal 10 (1): 33–60. https://doi.org/10.57805/revstat.v10i1.110
78. SCET. 1971 Monographie des grands barrages. Barrage de Foum El Gherza. Société Centrale pour l’Equipement du Territoire. In French. Available at: https://www.scribd.com/document/447285820/Calcul-de-l-Envasement-Du-Barrage-de-Foum-El-Gherza
79. Schwarz, G. 1978. “Estimating the dimension of a model.” The Annals of Statistics 6 (2): 461–446. http://www.jstor.org/stable/2958889
80. Shah, A.I., K. Das, and N.D. Pan. 2025. “Flood frequency analysis and susceptibility zonation of the Haora River Basin, Northeast India.” River 4 (1): 116–133. https://doi.org/10.1002/rvr2.70001
81. Silva, A.T., M.M. Portela, and M. Naghettini. 2012. “Nonstationarities in the occurrence rates of flood events in Portuguese watersheds.” Hydrology and Earth System Sciences 16 (1): 241–254. https://doi.org/10.5194/hess-16-241-2012
82. Silva Lomba, J. and M.I. Fraga Alves. 2020. “L-moments for automatic threshold selection in extreme value analysis.” Stochastic Environmental Research and Risk Assessment 34 (3–4): 465–491. https://doi.org/10.1007/s00477-020-01789-x
83. Solari, S., M.M. Egüen, M.J. Polo, and M.A. Losada. 2017. “Peaks Over Threshold (POT): A methodology for automatic threshold estimation using goodness of fit p-value.” Water Resources Research 53 (4): 2833–2849. https://doi.org/10.1002/2016WR019426
84. Sonuga, J.O. 1972. “Principle of maximum entropy in hydrologic frequency analysis.” Journal of Hydrology 17 (3): 177–191. https://doi.org/10.1016/0022-1694(72)90003-0
85. Tebbi, F.Z., H. Dridi, L. Houichi, and F. Krimil. 2009. “Essai de modélisation de la gestion de barrage en zone aride: cas du barrage de Foum El Gherza W. Biskra.” Le Journal de l’Eau et de l’Environnement 8 (15): 99–105. In French. Available at: https://asjp.cerist.dz/en/article/37478
86. Toumi, A. and B. Remini. 2004. “Barrage De Foum El Gherza Face Au Probleme De Fuites D’Eau.” Larhyss Journal 3, 25–38. In French. Available at : https://www.larhyss.net/ojs/index.php/larhyss/article/view/42
87. Trenberth, K.E., A. Dai, R.M. Rasmussen, and D.B. Parsons. 2003. “The Changing Character of Precipitation.” Bulletin of the American Meteorological Society 84 (9): 1205–1218. https://doi.org/10.1175/BAMS-84-9-1205
88. Tukey, J.W. 1977. Exploratory Data Analysis,1st edn. Addison Wesley, Pearson, Reading Massachusetts, USA. pp. 688.
89. Van Montfort, M.A.J. and J.V. Witter. 1985. “Testing exponentiality against generalised Pareto distribution.” Journal of Hydrology 78 (3–4): 305–315. https://doi.org/10.1016/0022-1694(85)90108-8
90. Viglione, A. 2020. “nsRFA: Non-supervised Regional Frequency Analysis.” R package version 0.7-15. Available at: https://cran.r-project.org/web/packages/nsRFA/index.html
91. Viglione, A., R. Merz, J.L. Salinas, and G. Blöschl. 2013. “Flood frequency hydrology: 3. A Bayesian analysis.” Water Resources Research 49 (2): 675–692. https://doi.org/10.1029/2011WR010782
92. Vivekanandan, N. and S. Shukla. 2015. “Flood frequency analysis using method of moments and L-moments of probability distributions.” Cogent Engineering 2 (1): 1018704. https://doi.org/10.1080/23311916.2015.1018704
93. Vogel, R.M., T.A. McMahon, and F.H.S. Chiew. 1993a. “Floodflow frequency model selection in Australia.” Journal of Hydrology 146, 421–449. https://doi.org/10.1016/0022-1694(93)90288-K
94. Vogel, R.M., W.O. Thomas Jr., and T.A. McMahon. 1993b. “Flood-Flow Frequency Model Selection in Southwestern United States.” Journal of Water Resources Planning and Management 119 (3): 353–366. https://doi.org/10.1061/(ASCE)0733-9496(1993)119:3(353)
95. Wald, A. and J. Wolfowitz. 1943. “An Exact Test for Randomness in the Non-Parametric Case Based on Serial Correlation.” The Annals of Mathematical Statistics14 (4): 378–388. https://doi.org/10.1214/aoms/1177731358
96. Wegmann, E. 1911. The Design and Construction of Dams: Including Masonry, Earth, Rock-fill, Timber, and Steel Structures, Also the Principal Types of Movable Dams. 6th edn. John Wiley & Sons, New York, USA.
97. Wheater, H.S. 2008. “Modelling Hydrological Processes in Arid and Semi-Arid Areas: An Introduction.”  In: Wheater, H., Sorooshian, S., Sharma, K.D., eds. Hydrological Modelling in Arid and Semi-Arid Areas. International Hydrology Series. Cambridge University Press 2007: 1-20. https://doi.org/10.1017/CBO9780511535734.002
98. Wheater, H. and R. Al-Weshah. 2002. "Rainfall-Runoff Analysis and Modeling in Wadi Systems." In: Hydrology of Wadi Systems, Technical Documents in Hydrology 55, 87–111. Paris: UNESCO. Available at: https://unesdoc.unesco.org/ark:/48223/pf0000127229
99. Wilcoxon, F. 1945. “Individual Comparisons by Ranking Methods.” Biometrics Bulletin 1 (6): 80–83. https://doi.org/10.2307/3001968
100. WMO. 2009. “Management of Water Resources and Application of Hydrological Practices, Volume 2, Sixth Edition.” In: Guide to Hydrological Practices. Geneva, Switzerland: World Meteorological Organization. https://doi.org/10.1080/02626667.2011.546602
101. Yan, H. and H. Moradkhani. 2016. “Toward more robust extreme flood prediction by Bayesian hierarchical and multimodeling.” Natural Hazards: Journal of the International Society for the Prevention and Mitigation of Natural Hazards 81, 203–225. https://doi.org/10.1007/S11069-015-2070-6
102. You, L., C. Li, X. Min, and T. Xiaolei. 2012. “Review of Dam-break Research of Earth-rock Dam Combining with Dam Safety Management.” Procedia Engineering 28, 382–388. https://doi.org/10.1016/j.proeng.2012.01.737
103. Yue, S., T.B.M.J. Ouarda, B. Bobée, P. Legendre, and P. Bruneau. 2002. “Approach for Describing Statistical Properties of Flood Hydrograph.” Journal of Hydrologic Engineering 7 (2): 147–153. https://doi.org/10.1061/(ASCE)1084-0699(2002)7:2(147)
104. Yue, S. and P. Pilon. 2004. “A comparison of the power of the t test, Mann-Kendall and bootstrap tests for trend detection / Une comparaison de la puissance des tests t de Student, de Mann-Kendall et du bootstrap pour la détection de tendance.” Hydrological Sciences Journal 49 (1): 21–37. https://doi.org/10.1623/hysj.49.1.21.53996
105. Zhang, J. 2002. “Powerful Goodness-of-fit Tests Based on the Likelihood Ratio.” Journal of the Royal Statistical Society. Series B: Statistical Methodology 64 (2): 281–294. https://doi.org/10.1111/1467-9868.00337
106. Zhao, C. and J. Yang. 2019. “A Robust Skewed Boxplot for Detecting Outliers in Rainfall Observations in Real-Time Flood Forecasting.” Advances in Meteorology 1795673, 1–8. https://doi.org/10.1155/2019/1795673
107. Zoglat, A., S. El Adlouni, F. Badaoui, A.Amar, and C.G. Okiou. 2014. “Managing Hydrological Risks with Extreme Modeling: Application of Peaks over Threshold Model to the Loukkos Watershed, Morocco.” Journal of Hydrologic Engineering 19 (9): 05014010. https://doi.org/10.1061/(ASCE)HE.1943-5584.0000996