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

Rainfall patterns are bound to change as a result of global warming and climate change impacts. Rainfall events are dependent on geographic location, geomorphology, coastal area closeness and general circulation air movements. Accordingly, there are increases and decreases at different meteorology station time-series records leading to extreme events such as droughts and floods. This paper suggests a methodology in terms of envelope curves for monthly extreme rainfall event occurrences at a set of risk levels or return periods that may trigger the extreme occurrences at meteorology station catchments. Generally, in many regions, individual storm rainfall records are not available for intensity–duration–frequency (IDF) curve construction. The main purpose of this paper is, in the absence of individual storm rainfall records, to suggest monthly envelope curves, which provide a relationship between return period and monthly extreme rainfall values. The first step is to identify each monthly extreme rainfall records probability distribution function (PDF) for risk level and return period calculations. Subsequently, the return period rainfall amount relationships are presented on double-logarithmic graphs with the best power model as a set of envelope curves. The applications of these methodologies are implemented for three Hodna drainage basin meteorology station rainfall records in northern Algeria. It is concluded that the most extreme rainfall risk months are June, August and September, which may lead to floods or flash floods in the study area. A new concept is presented for the possible extreme value triggering months through the envelope curves as ‘low’, ‘medium’ and ‘high’ class potentials.


INTRODUCTION
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 ). 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 , , ) 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. ) and the same result is emphasized further for the southern Mediterranean countries by Lionello & Scarascia () 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. In Algeria, several studies have been devoted to analyze and identify regional and local changes in the extreme climatic events (Taibi et  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.

STUDY AREA AND METHODOLOGY
In general, the study area and the meteorology station locations that are considered in this study are given in

METHODOLOGICAL APPROACH
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 individ- where a > 0, b > 0 and 0 < c < x are the PDF parameters.
These are referred to as the location, scale and shape parameters, respectively. It can be given simpler form by defining that y ¼ (x À c)=a, and hence, it takes the following form: On the other hand, two-parameter Gamma PDF has easier parameter calculations and the mathematical expression has the following form: 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 X 1 , X 2 , X 3 , … , X n , where n is the number of data.
(2) Attach to each value in the ordered sequence a nonexceedence probability value, P m , according to the  following empirical formulation: (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 À 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 Sȩn ().
(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.

EXTREME RAINFALL RETURN PERIOD GRAPH
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 mathematical model is a power function with two parameters, a and b, which can be written as: where X represents the return period variable on the horizontal axis and Y is for the extreme rainfall amount. Feb.

May
Jun. Jul. Aug.
Min.  In Figure 5, the red straight lines are valid with intercept, a, and slope, b, values, which are calculated through the application of regression methodology. In cases of outliers, the regression line does not present a completely representative straight line. In Figure 5, the months of July, September, October, November and December have outliers, and therefore, the regression lines without outliers are presented by blue straight lines for these months. Table 4 includes the monthly parameters of power low model parameters.

DISCUSSION
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 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   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.

CONCLUSIONS
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.