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A new method for estimating flood peak discharge and extreme rainfall: Case study of Firat basin

The commonly-used design parameter for hydraulic structures is the annual maximum instantaneous streamflow recorded by conventional gauging stations. Increased hydroclimatic variability in recent years and the resultant flooding raise questions concerning the flood risk estimations from the short flow records in Turkey. The method described in this study has been selected according to the likely estimates for the peak flow values at different return periods for the gauged basins. Hence, estimation of the peak flow values for regions with poor or rich discharge datasets could be implemented. In theory, this developed method may be used to estimate the peak flow values at any point on a river network, and not only at basin outlets. In this research, a case study has been conducted on the Firat basin, on which the largest dams in Turkey have been built, by employing a novel approach for developing a new method that calculates the peak flood flows and extreme rainfall. The results demonstrate that the approach is sound and can be employed in the prediction of peak rainfall and flow parameters in river basins.


Introduction
From time immemorial, investigations have been made into the probability of occurrence of river floods. Based on that knowledge, river-engineering works have been designed and flood protection measures have been taken. Yet, the data available are insufficient to draw firm conclusions on the future effectiveness of these interventions. The more reliable the discharge data from the past, the smaller the risk of failure of the design conditions for flood protection measures. The estimation of the probability of occurrence of peak floods is open to improvement. To that end, other estimation methods will be used, the data series will be extended and different methods of data processing will be used.
Peak discharge information is required to determine the dam design and appropriate size water conveyance systems such as natural channels, diversion canals, bridge openings, etc. The accurate prediction of stream flows is essential to the planning of our water resource systems. This paper addresses the practical state of the art of techniques to predict flood peaks and their associated frequency of occurrence. Statistical relationships will be investigated as means of predicting the peak discharges. The statistical graphical or analytical methods of flood flow estimation seem to be well established in the literature, Gumbel [1] , Chow [2] , Benson [3] , Yevjevich [4] , Haan [5] . Generally, a graphical method by plotting annual peak flow on a log-normal probability paper using the Weibull plotting position formula, or an analytical method using the log Pearson type distributions is recommended.
Rossi et al. [6] describe the theoretical considerations to obtain a parent flood distribution that closely represents the real flood experience, the existence of the annual flood series of Italian river basins.
Keim and Faiers [7] explored heavy rainfall distributions by season and the associated differences in seasonal quantile estimates for selected recurrence intervals in Louisiana, as a result of the findings of other investigators.
Adamowski [8] considers the currently used parametric analysis of the " annual maximum" flood series. They reveal unimodal and multi-modal probability density functions for floods in two Canadian Provinces Ontario and Quebec.
Adamowski [8] considers the currently used parametric analysis of the " annual maximum" flood series. They reveal unimodal and multi-modal probability density functions for floods in two Canadian Provinces Ontario and Quebec.
Nonparametric frequency analysis has been introduced as an alternative method. This method also revealed unimodal and multimodal "annual maximum" and "peak over threshold" flood probability density function shapes in both Provinces.
Luxemburg et al. [9] analyzed the statistical properties of flood runoff of North Asian rivers under conditions of climate change.
Bakker and Luxemburg [10] consider the problem of heterogeneous distributions of floods, as research in the area of frequency analysis has been rather limited on this item, although several investigators confess that the assumption of homogeneity of flood distributions may not be valid. Therefore, the estimates of probabilities of exceedance are often very unreliable. The heterogeneity of the series of annual maximum runoffs can be explained by the fact that different extreme floods are caused by different mechanisms (ice melt, rains, cyclones, etc.).
Mantje et al. [11] try to identify the different homogeneous subsets in a heterogeneous distribution (although the latter is often regarded as homogeneous in flood frequency analysis).

Goodness-of-fit test
It is the work of determining the magnitudes of hydrological variables corresponding to given frequencies or recurrence intervals. Procedures involved in frequency analysis include: (1) collecting a random sample of the interested hydrological variable; (2) finding the best-fit distribution for the sample by a goodness-of-fit (GOF) test or other appropriate methods; and (3) determining the magnitude of the hydrological variable corresponding to a given probability of exceedance using the best-fit distribution. Two GOF tests, namely the chi-square test and the Kolmogorov-Smirnov test, are often used for the selection of probability distributions for hydrological variables [12] . Another method of goodness-of-fit test is the method based on ordinary moment-ratio diagrams [13] .

The scope of the study
This study was developed, to yield a satisfactory first estimate of the discharge and corresponding water level at a certain point along the river starting from rainfall forecasts. An improved estimation method for the probability of occurrence of flood peak discharges. Improved accuracy of the probability of exceedance estimates of flood peak as a result of this determination. Identification of the flood properties at rivers that determine further downstream. A method to determine these downstream water levels, their probability of exceedance, and accuracy. Operational flood discharge prediction, especially early forecasting, enhances operational decision-making. As the flood event proceeds, the availability of more elaborate data and the use of more sophisticated flood forecasting models may enable more accurate predictions.
In this study, a new methodology, different from the distribution analyses, was developed to estimate the annual maximum instantaneous stream flows and the precipitation depths measured by weather stations in 100, 200, 500, 1,000, and 10,000 years, and a correlation was obtained. Then the values determined by the assistance of this correlation have been compared with the GOF test results.

Methodology
If the annual maximum precipitation values measured for river basins are arranged in ascending order as Q 1 < Q 2 < Q 3 … < Q N , a Q-T variation curve would be obtained as seen in Figure 1. For a time period of T years, the T years-recurrence peak-flow Q-T is defined as a value of discharge, which statistically occurs every T year. For any Q i in Figure 1, the equation below could be asserted: The variables employed in the equation above stand for the following: Q i = value of the measured streamflow, precipitation depth or similar variable at the i th year; a i = the coefficient of i th year; T i = i th year. Consequently, Q 2 = a 2 × T 2 × ln(T 2 ) or in short, Q 2 = a 2 × 2ln (2).
The a i values are as shown in Figure 2.
The values of a 50 , a 100 , a 200, a 500 , a 1,000 , and a 1,0000 are required for determining the values such as Q 50 , Q 100, Q 200 , Q 500 , Q 1,000 , and Q 10,000 that are beyond the scope of the N-year observation period for which measurements have been taken, could be determined in turn by the assistance of the chart in Figure 2 as well as the main values determined from the a i calculations. The equation below holds between the values of a i , and a i + 1 : Simplifying, (i + 1) × a (i + 1) = a i − 1 is obtained. For instance, concerning the relationship between the 16 th and 17 th years; 17 × a 17 = 16 × a 16 , hence, N × a min = 100 × a 100 = 1000 × a 1,000 = 10,000 × a 10,000 . Therefore, the equation below could be derived since the product of N × a min must be constant: The T-year streamflow as well as any time-dependent variable could be determined by equation 3. With N denoting the measurement year, the following streamflow values have been obtained; Q 50 = a min × N × ln(50) yields 50-year streamflow Q 100 = a min × N × ln(100) yields 100-year streamflow Q 200 = a min × N × ln(200) yields 200-year streamflow Q 500 = a min × N × ln(500) yields 500-year streamflow Q 1,000 = a min × N × ln(1,000) yields 1,000-year streamflow Q 10,000 = a min × N × ln(10,000) yields 10,000-year streamflow In the equations given above: a min = The minimum value obtained from the graph of measured values in Figure 2 or from equation 2 using the a i calculations. If no observation in excess of the N-year value has been made within the year of measurement, the value of a min is reached at year N as seen in Figure 2. However, this situation is encountered rarely. Generally, amin is reached before the N th year since some measurements greater than N-year values are observed within a particular observation year. In this case, if we denote the year where amin has been reached as N amin , the product of (a min × N amin ) stays constant. Therefore, the following equations hold: (a min × N amin ) = 100 × a 100 = 1,000 × a 1,000 = 10,000 × a 10,000 . Consequently, equation 3 could be stated in the following form: Q T = a min × N amin × ln(T)

Study case
The study area is situated adjacent to Keban Township, approximately 40 km to the west of Elazig province in eastern Turkey (Figure 3). The drainage system is characterized primarily by ephemeral streams of limited widths. The Euphrates is the main river in the study area, flowing from north to south. The elevation of the region ranges from about 1,000 m to over 2,500 m. The vegetation comprises scarce scrub grass and stunted trees. The area has a semi-arid climate characterized by dry summers and cold winters. According to the meteorological data for the period 1923 to 2010 (Turkish State Meteorological Service, Elazig), the average annual rainfall is 399 mm, with snowfall accounting for more than half of this precipitation. The lowest mean precipitation has been recorded in the months of July and August (6 and 4 mm, respectively), whereas the highest mean precipitation has been recorded in the months of April and May (58 and 60 mm, respectively). The mean annual temperature is 14.8 ℃, with July and August being the warmest months, with average temperatures around 28 ℃, whereas January and February are the coldest months, with temperatures around 3 ℃.
Keban dam, which is one of the largest dams in Turkey and the world, was built on the Euphrates River in the Upper Euphrates region. Keban Dam Reservoir spans an area of 67,500 km 2 . The construction of the dam had been launched in 1964 and the dam had become operational by 1974. The maximum streamflow values of Euphrates River for a period of N = 43 years between 1932 and 1974 (prior to the completion of Keban Dam) as measured by AGI owned by EIE [14] located in Keban have been displayed in Figure 4 and the streamflow analysis conducted by the new method is given in Table 1. Since the construction of Keban Dam has been fully completed in 1975, no streamflow measurements have been carried out in 1975. The obtained values have been given in Table  2 in addition to the GOF test results of the measured stream flows for 43 years.   The frequency analysis of the maximum streamflows belonging to Euphrates River: N = 43, Q avg = 1,685.6 m 3 /s, σ = 230, C s = -0.126. In Table 3, the analysis of the total rainfall for the 79-year period between 1928 and 2006 has been given and a correlation has been empirically developed to predict the annual rainfall depth h t according to the results of this methodology. This correlation is given in equation 5. The values obtained from this method have been compared with the calculated values using equation 5.

Conclusion
The goal of this study is to improve the understanding of peak flood discharge and extreme rainfall processes in river catchments. Nevertheless, continued hydrologic, hydraulic, and paleohydrologic research on catchments areas is needed that would contribute to a broad range of hydrologic research projects and investigations. An improved understanding of basic hydrologic and hydraulic processes will improve the available methods for the assessment of peak floods and extreme rainfall phenomena. These related studies depend on accurate data and hydrologic methods. The improved hydraulic methods can be incorporated into numerical simulation models of surface-water systems and could be useful to improve the analyses of hydrologic processes. The results of this method are also applicable to other rivers. Moreover, empirical correlations predicting the annual rainfall depths in the gauged catchments have been developed.
In time-related analyses, since the product a i .N i remains approximately constant, the results of the distribution analyses obtained from this method and the GOF tests could be compared, and thereby the most convenient results could be determined. Additionally, these results are more practical and reliable than the analysis methods such as MOM and Moment-L. If an observation is conducted for a long enough time and the curves produce reasonable values, the Q T and the h t values of the catchments could be determined with the assistance of a curve. For example, the total rainfall for Elazig could be calculated by the following formula: h t = 127.79T -0.011 × ln(T) An extreme rainfall that leads to a flash flood can be approached by a variety of methods. Among others, such methodologies as meteorological analysis, hydrological modeling, hydraulic modeling and analysis, and post-event campaigns for data retrieving (flood marks, peak flow timing through intervals) can be used to provide additional information for reliable annual peak discharge estimations.