Monthly extreme rainfall risk envelope graph method development and application in Algeria

Water resources are among the most precious commodities for the socio-economical sustainability of any country, and therefore all over the world effective assessment of their protection against the climate change impacts are advised by scientific and technological means (Cook 2017). Anthropogenic activities in modern life trigger not only air and water pollution, but more alarmingly atmospheric pollution especially due to the greenhouse gas (GHG) emissions, which initiated the discussion on global warming and consequent climate change impacts on energy, economy and environment, in general, and societal sustainability. In this context, the Intergovernmental Panel on Climate Change reports (IPCC 2007, 2013, 2014) are for general guidance all over the world. It is indicated that an increase in the extreme precipitation risk during the 21st century is likely in the Mediterranean areas (Christensen et al. 2013) and the same result is emphasized further for the southern Mediterranean countries by Lionello & Scarascia (2020) using the average rainfall intensity and its fraction during intense events.

Over the Mediterranean countries, the extreme precipitation events have caused significant damage in different sectors of the economy, health and the environment. During 1990–2006, the damage and deaths caused by floods in the Mediterranean region totaled 29.14 billion Euros with the largest number of affected persons in Algeria (Llasat et al. 2010; Khodayar et al. 2015). According to Gaume et al. (2016), the primary natural cause of flooding in this region is the short intense rainfall bursts inducing the convection flows in the Mediterranean Sea. Advance knowledge on rainfall extremes would contribute to better flood planning as well as accurate protection designs by means of hydraulic structures such as flood protection channels and storm sewers. For this purpose, it is important to express the magnitude of extreme events by their occurrence frequency using probability distribution functions (PDFs; Alam et al. 2018). The application of statistical theory to model the extreme rainfall is required for different regions since the precipitation events are more dependent on topography, coastal area closeness and general circulation air movement.

In Algeria, several studies have been devoted to analyze and identify regional and local changes in the extreme climatic events (Taibi et al. 2015; Djerbouai & Souag-Gamane 2016; Korichi et al. 2016; Achour et al. 2020). Benkhaled (2007) found that the generalized extreme value (GEV) and Gumbel PDFs provide the best fit with almost similar results based on a return period of 100 years for four annual maximum daily rainfall records in the Cheliff-Zahrez basin, northwest Algeria, where Benhattab et al. (2014) identified the GEV PDF as the most appropriate regional distribution for annual maximum daily rainfall records, which are characterized by a Mediterranean climate. A comparison between the GEV and Gumbel PDFs on annual maximum daily rainfall was established by Boucefiane et al. (2014) at some rain gauge stations located in the steppe region of western Algeria. The authors proved that the Gumbel PDF suggested by the National Hydraulic Resources Agency (ANRH 2007) was not suitable for this region. The GEV and the Generalized Logistics (GLO) PDFs were identified as the most appropriate for modeling annual maximum daily rainfall in northern Algeria (Zeggane & Boutoutaou (2017). Many authors have been interested in frequency distributions of monthly maximum daily rainfall records in several different regions around the world (Şen & Eljadid 1999; Eslamian & Feizi 2007; Bhakar et al. 2008; Mandal & Choudhury 2015; Alam et al. 2018). Şen & Eljadid (1999) found that in Libyan arid regions (border country of Algeria), the Gamma PDF provided the best fit. In the same way, Amiri & Eslamian (2010) suggested that the GEV and Pearson type-III were suitable for Iranian arid region monthly precipitation records. Currently, most Algerian engineers who applied the annual extreme precipitation data for design purposes used the Gumbel PDF (ANRH 2007), despite the fact that it produces small design rainfall values for large return periods (Koutsoyiannis 2004, 2007; Zahar & Laborde 2007; Shabri & Mohd Arif 2009; Cavicchia et al. 2018). Consequently, any engineering water structure design based on such calculations showed improper performances.

Based on the above insights, the overall goal of this study is to identify the PDF for monthly maximum daily rainfall records at a set of three meteorology stations in the Hodna basin, Algeria. A new approach is presented under the name of envelope curves for monthly extreme rainfall assessments. The envelope curve or broken line in a plane subsumes all the alternatives under its umbrella, i.e., it is the maximum boundary of the events depending on the hydro-meteorological variable, which is the monthly extreme precipitation value in this paper.

In case of intensity–duration–frequency (IDF) curve absence in a region, envelope curves provide rainfall amounts based on a set of risk levels and return periods. The envelope curves appear as a straight line on double-logarithmic Cartesian coordinate systems between the extreme monthly rainfall amounts and return periods. The application of the proposed methodology has been performed for three meteorology stations, monthly extreme rainfall records.

In general, the study area and the meteorology station locations that are considered in this study are given in Figure 1. The study area is within the Hodna basin as shown in Figure 2.

In Algeria, the Hodna basin ranks as the fifth biggest and is situated in the center Isser, Soummam, Hauts plateaux, Chott Merlhir, Zahrez and Chelif surrounding basins as shown in Figure 2. The surface area is about 26,000 km², and it is between two topographic ensembles of Atlas and African Sahara. It also lies within two mountainous regions with 1,800–1,900 m elevations in the north and 600–900 m in the south.

The surface slope encourages rapid surface flow leading to occasional floods and flash floods. The Honda basin has four different feature geomorphological units as mountainous region, Hodna valley, Chott sabkha and R'mel piedmont regions. Table 1 indicates the geographic locations and basic statistical parameters for the three stations that are considered in this study.

The average precipitation in the Hodna basin at about 400 m elevations is 200 mm, but at higher elevations (about 1,800 m), it reaches 600–700 mm. In winter, snow falls on the Hodna Mountains and remains on the surface for 20–30 days at the maximum.

The temperature degrees vary according to elevation and local conditions. Evaporation is rather high, and due to the effect of wind, the surface water ponds lose water rather rapidly. In the Hodna basin, average maximum temperature changes according to the geographic features. The average annual temperatures are 18°C in the northwest, 18.5 °C in the center and northeast and 17.9 °C in the south of the basin (Hasbaia et al. 2012; Salhi et al. 2013).

Geographically, the watershed region is between 3°1′-6°12′E and 34°15′-36°15′N within three different climatic regions (Figure 3), according to Köppen's climate classification, namely the Mediterranean hot summer climates (Csa) in the northern part of the watershed, a belt of cold semi-arid climate (BSk) in the center and desert (BWh) with semi-arid (BSh) climate in the southernmost portion of the watershed (Zeroual et al. 2019).

The rainfall map of the Hodna Basin is prepared based on CHIRPS satellite-gauge data at 0.05° resolution over the 2000–2014 period (Funk et al. 2015; Figure 3).

The spatial distribution of rainfall is characterized by a strong gradient from the northwest to Chotelhodna, with a less marked South Chotelhodna gradient. The average annual rainfall drops from around 500–600 mm on mountain ridges of the Hodna (region of Bordj Bou Arreridj) to less than 200 mm at Chottelhodna. The average annual rainfall in the southern region ranges between 300 and 390 mm (Boussaada and Ain Rich locations). This high variability is due to the north-south contrast of the basin (Hasbaia et al. 2017). The low precipitation values found at the center and south of the basin are due to the remoteness of the region from the Mediterranean Sea and to the mountainous obstacle constituted by the El-Hodna Mountain, which prevent the progression of the moisture-bearing northerly winds. Summer drought is general, but there is also a frequent dry period in winter.

In general, individual storm rainfall amounts assessment toward IDF curves determination provides flood risk assessment foundations. In semi-arid and arid regions, such storm rainfall records are not available, and therefore, it is necessary to base the flood assessments on daily total rainfall amounts, which are in most cases representative of individual rainfall events. Extreme rainfall amount predictions are among the most significant feature identifications because their consequent results may cause floods and flash floods leading to both human property and loss of life.

The PDF matching the available data at each meteorology station is achieved through the execution of the following steps. Let the given annual monthly extreme rainfall records be representative as X₁, X₂, X₃, … , X_n, where n is the number of data.

• (1)

  Arrange the given data into ascending order, hence a new non-decreasing sequence is obtained as Y₁ < Y₂ < Y₃< . .<. . <Y_n with ranks, m, as 1, 2, 3, … , m = n, respectively.

• (2)
  Attach to each value in the ordered sequence a non-exceedence probability value, P_m, according to the following empirical formulation:

  

  (4)
• (3)

  Plot the scatter diagram of ordered sequence versus corresponding probability values (Y_m versus P_m). Hence, a non-decreasing systematic scatter diagram appears. Furthermore, if one wants to work with exceedence probabilities, then the scatter diagram takes the form of non-increasing form after plot of Y_m versus (1 − P_m). Hence, empirical systematic probability points scatter appear on the normal paper.

• (4)

  These points are fitted with the best PDF among many alternatives such as normal (Gauss), log-normal, Weibull, two- and three-parameter Gamma (Pearson III) PDFs which is achieved through the MATLAB program software written by Şen (2020).

• (5)

  Finally, the plot of the best fit PDF on the scatter diagram leads to figures with a set of risk levels and also the type and parameters of the best theoretical PDF as explained above in Equations (1)–(3).

One must keep in mind that there is a difference between the probability and risk. The probability is given by Equation (4), whereas the risk depends on decision makers such as 5 and 10%, for design purpose after the identification of the theoretical PDF.

In this study, three meteorology stations are taken into consideration and detailed explanation about the methodological application is written for one of them because the same procedure is used in three of them. Figure 4 presents the results obtained for each month at Ain Nssissa (Station 1) after the application of the previous steps to the monthly extreme rainfall data. It is obvious that in each month, only two- and three-parameter Gamma PDFs are the most suitable theoretical PDFs.

Table 2 indicates the extreme rainfall events corresponding to a set of risk levels, which also appear in each graph in Figure 4, only for Ain Nssissa meteorology station.

The summary of all that can be inferred from the graphs in Figure 4 is presented in Table 3.

In the absence of IDF curves, the relationship between the return periods or risk levels and monthly extreme rainfall values provides a scientific basis for extreme value calculation opportunity leading to graphs, which can be referred to as the risk extreme value diagrams and can be drawn from the values in Table 3 for Ain Nssissa station. The resulting graphs are given in Figure 5 on the double-logarithmic scales.

The combination of the information from the two previous sections provides an integrated monthly extreme precipitation and return period relationship as indicated in Figure 6. The first striking fact is an upper envelope boundary for extreme rainfall occurrence possibilities as described by means of ‘low’, ‘medium’ and ‘high’ risk levels. The quantitative rainfall values are presented in Table 5 corresponding to three return periods and risk levels. One can deduce the following significant points from these graphs:

• (1)

  The month with the least precipitation risk is July because its return period values remain almost below each monthly values.

• (2)

  The upper envelope of extreme precipitation occurrence possibility takes place along different months as June, August and September.

• (3)

  The ‘high’ risk cases are bound to appear in June, where the 100-year return period value corresponds to 380 mm in June. The next ‘high’ risk extreme rainfall amounts are bound to appear in May and then in November.

• (4)

  The ‘medium’ risk occurs in August corresponding to a 50-year return period with 110 mm rainfall expectation. The next month in this risk category is transition from June–August to September.

• (5)

  The ‘low risk’ flood occurrence risk is confined to September for return periods up to 50 years, and for example, the 10-year return period extreme rainfall amount is about 50 mm.

• (6)

  For any given return period in a year, one can classify the months according to their respective values from the top down.

• (7)

  Any given monthly rainfall value can be categorized according to the return period values from the smallest to the biggest.

In Figure 6(a), the Ain Nssissa meteorology station in the west presents extreme rainfall amount change by return period on double-logarithmic paper for each month. It is obvious that the lowest rainfall amounts appear in July, which could be regarded as drought impact occurrence in this month. On the other hand, the envelope broken line has three classes in sequence as ‘low’ in September, ‘medium’ in August and ‘high’ rainfall occurrence expectations in June. The extreme rainfall variations along the upper envelope broken line are less than 110 mm for ‘low’, between 100 and 130 mm for ‘medium’ and more than 130 mm for the ‘high’ extreme rainfall occurrences.

The envelope straight lines on double-logarithmic paper are given for Ngaous meteorology station in Figure 6(b), where the bunch of monthly lines are rather close to each other compared with the previous station, but there appears an off line in June, which has both lower rainfall and along the envelope line high-risk component for extreme rainfall occurrences. This point indicates that this month is for rather ‘low’ extreme rainfall occurrences in the domain of the dry period. However, it also has the ‘high’ extreme rainfall occurrence in the upper envelope boundary with more than 400 mm, which has a very occasional chance of occurring. September plays a dominant role in the ‘low’ extreme rainfall occurrences with less than 215 mm, whereas a short duration ‘medium’ extreme rainfall domain has extreme rainfall amounts that may vary from 215 to 400 mm.

Finally, Bourdj Bou Arreridj meteorology station in the south has an overwhelmingly lower extreme rainfall line in July with expected extreme rainfall amounts that vary from 0.1 to almost 200 mm. Along the envelope line, there are three distinctive monthly parts. The ‘low’ extreme rainfall values are along the August month for about a 15-year return period. September appears for ‘medium’ extreme rainfall expectations from 16 to 100 mm, and the ‘high’ part is during April with more than 100 mm extreme rainfall (Table 5).

Finally, Table 6 is prepared for monthly extreme precipitation values for 10-year, 50-year and 100-year return periods with corresponding risk levels of 0.10, 0.20 and 0.01, respectively.

The main purpose of the paper was to identify first the PDF for each monthly extreme rainfall value to find the return periods (inverse of risk levels), which play a significant role in extreme rainfall frequency analysis. The return period and monthly extreme rainfall value relationship is obtained for each month, which appeared in the form of straight lines on double-logarithmic paper. These double-logarithmic plots expose which months are for extreme rainfall events and the length of the return period. The application of the proposed methodology indicated that the monthly extreme rainfall PDFs have either two- or three-parameter (Pearson III) Gamma mathematical formulations. However, return period and monthly extreme rainfall values are in the form of power function for which parameters are also obtained. A new concept of extreme rainfall envelope is developed and applied for three stations, which provide useful information in the absence of IDF curve absences. The applications of these methodologies are presented for the Hodna drainage basin in Algeria through three meteorology stations' monthly daily maximum rainfall amounts.