1 Introduction

There are plenty of water resources planning and management projects, which require intensity–duration–frequency (IDF) curves for proper water-related designs. These curves express the rainfall duration, return period, and frequency triple relationship on a double logarithmic paper, which provides practical and simple calculations procedure to design rainfall intensity in mm/hour provided that the return period in years, and the design rainfall duration in minutes is decided on.

IDF curves are suggested by different researchers who suggested mathematical or empirical approaches (Chow 1964; Aron et al. 1987; Koutsoyiannis et al. 1998; Hanson 1995; Garcia-Bartual and Schneider 2001). Proper and accurate determination of IDF curves is possible if storm rainfall accumulation records are available. Unfortunately, IDF curves are not available for many areas in the world even in developed countries let along the others.

In the past, different researchers have proposed quite different IDF curve generation procedures for different parts of the world by various methodologies depending on the available data type in the region. Such curves are suggested for the United States by Bell (1969) and Chen (1983); for Vietnam monsoonal region by Nhat et al. (2006) and for the arid and semi-arid regions by different researchers (Venkata Ramana et al. 2008; AlHassoun 2011; Elsebaie 2012; El-Sayed 2011; Wayal and Menon 2014). These authors took into consideration the available recording rain-gauge records. For instance, Bell (1969) explained IDF curves for certain areas of Russia. There are researches from the Kingdom of Saudi Arabia for IDF curve generation by different authors (Al-Amri and Subyani 2017; Al-Shaikh 1985; Al-Khalaf 1997; Elsebaie 2012). Convenient IDF curves are given for Nigeria by Ogarekpe (2014). Das et al. (2016) have developed IDF curves on the basis of Gumbel extreme value probability distribution functions (PDFs). Although classically one needs recording rainfall gauge measurements, but IDF curve construction is suggested by Koutsoyiannis et al. (1998) even from non-recording stations. In cases of recording rain gauges unavailability, several different methodologies are used for IDF curve generations. For instance, Raiford et al. (2007) obtained IDF curves from isohyet map frequency analysis. Based on similar maps El-Sayed (2011) obtained a set of regional IDF curves for the Sinai Peninsula, Egypt. In Malaysia, Liew et al. (2014) adjusted IDF curve at remote locations by information transfer from the nearest stations’ meteorological records.

The main purpose of this paper is to propose a new methodology for IDF curve construction by processing the annual daily maximum rainfall (ADMR) records. The basis of the methodology is first to obtain a dimensionless–intensity–duration (DID) curve, which is then converted to IDF curve sets through the ADMR theoretical cumulative distribution functions (CDFs). The application of the suggested method is presented for four ADMR records from Turkey, Ceylanpinar region meteorology stations next to Syrian border.

2 Theoretical Intensity–Duration (ID) Curves

ADMR statistical frequency analysis is achievable by a suitable theoretical probability distribution function (PDF). Among the most frequently used such PDFs are Gamma, logarithmic normal, Gumbel and Pearson type III functions, which are explained in various books and papers (Pearson 1930; Gumbel 1958; Chow 1964; Kite 1977).

In the development of IDF curves, the first step is to find the relationship between the rainfall duration and intensity, i.e., intensity–duration (ID) curves. Provided that a single storm rainfall record is available then one can numerically obtain ID curve. In many regions, recording rain gauges are not available, but one should look for other means for ID curve generation. There are two alternatives for this purpose:

  1. (1)

    information transfer by means of a convenient method from nearby stations;

  2. (2)

    adaptation of a hypothetical ID curve;

In this paper, an innovative methodology is developed by means of dimensionless–ID curve (DID) concept. Determination of IDF curves is possible through a convenient PDF and logical–rational mathematical relationship among the three variables, rainfall intensity, I, rainfall duration, t and return period, R. The most commonly used of these theoretical expressions has the following general form (Chow 1964):

$$I = \frac{{aR^{b} }}{{t^{c} }}.$$
(1)

Herein, a, b, and c are model parameters, which can be estimated through the least squares technique provided that the measurement set of I, t and R is available. Equation (1) can be written succinctly also as

$$I = \frac{d}{{t^{c} }}.$$
(2)

In this expression d = aRb represents another constant value, since in any IDF work the return period is a pre-fixed value.

3 Dimensionless–Intensity–Duration (DID) Curve

Provided that the parameters in Eq. (1) are known, it is then possible to obtain the DID curve by execution of the following steps.

  1. (1)

    Intensity values can be calculated for a set of rainfall durations, t, which might be considered for humid areas as 1440 min (24 h) or for semi-arid and arid regions as 720 min or 360 min and 180 min, respectively (Şen 2011),

  2. (2)

    In this paper, the maximum intensity value is denoted by I1, and the maximum rainfall duration as t1440 = 1440,

  3. (3)

    Divide the intensities (rainfall durations) by the maximum intensity (duration) value, and hence, a DID curve is obtained for the site.

The application of these steps to Eq. (2) with different sets of c and d values results in the same DID curve, as presented in Fig. 1.

Fig. 1
figure 1

Dimensionless–intensity–duration curves

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It is important to notice that the DID curves depend only on one parameter, c; whatever the value of d they all come on the same DID curve. One can obtain similar DID curves for any given value of c. This figure shows that for high rainfall intensity regions, c is close to 0.9, whereas for low intensity has small c values. Most often in practical studies, its value is in the medium around 0.5–0.7 range.

The availability of DID curve provides opportunity for possible intensity–duration curves’ determination from the ADMR data PDF.

4 IDF Curves Generation and Application

As mentioned in the previous section, the determination of the AMDR PDF from a set of data opens way to calculate any AMDR value corresponding to a given risk level. It must be remembered at this stage that the risk, r, is the reverse of the return period, R, r = 1/R, which implies that an IDF curve provides numerical information about the intensity provided that return period is given.

The study area, Ceylanpınar, is near the Turkish–Syrian political boundary, as shown in Fig. 2. Four meteorology station record and location characteristics are given in Table 1. This area is one of the most significant zones in Turkey for surface and groundwater resources with transboundary problems. The ADMR records are given in Table 2 for each station for 1440-min duration.

Fig. 2
figure 2

Ceylanpınar study area location

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Table 1 Meteorology station locations
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Table 2 Annual daily maximum rainfall amounts (mm)
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IDF curve from AMDR and DID values can be calculated after the application of the following steps.

  1. (1)

    The most suitable theoretical PDF is identified from the AMDR data availability at each station. After necessary calculations, it appears that the logarithmic normal PDF is suitable for each station as in Fig. 3, where the point scatters and the theoretical PDF function are shown together. It is to be noticed that rather than the CDF, its corresponding CDF is used in the applications,

    Fig. 3
    figure 3

    Logarithmic-normal PDF fittings, a Ceylanpınar, b Sanlıurfa, c Mardin, d Diyarbakır

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  2. (2)

    The return period, R, (risk, r) set is considered as R = 2 years (r = 0.50), R = 5 years (r = 0.20), R = 10 years (r = 0.10), R = 25 years (r = 0.04), R = 50 years (r = 0.02), and R = 100 years (r = 0.01). Accordingly, calculated rainfall amounts for 1440-min duration are presented in Table 3 for each meteorology station.

    Table 3 Return periods and rainfall amounts (mm)
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  3. (3)

    The rainfall amount logarithmic normal PDF helps to calculate risk for each return period. Table 4 presents the rainfall amounts corresponding to exceedence probabilities at each station.

    Table 4 Rainfall amounts (mm)
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  4. (4)

    Available information at these four stations are transferred to DID curve, as shown in Fig. 4:

    Fig. 4
    figure 4

    DID curves

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$$i_{d} \, = \,\frac{0.0364}{{d_{d}^{0.683} }},$$
(3)

where dd and id are dimensionless duration and intensity, respectively.

  1. (5)

    After knowing the value of dd, one can calculate for any rainfall duration, t, from full duration, tr = 1440 min, according to the following expressions:

$$t\, = \,t_{r} t_{d} .$$
(4)

The rainfall intensity can be calculated by taking into account the rainfall amounts from Table 4, the intensity as

$$i = i_{d} r_{e} .$$
(5)
  1. (6)

    Fig. 5 exposes the application of all the previous steps at each station. The results of this research are plotted on the same IDF curves that have been obtained from the recording rain-gauge storm rainfall measurement charts. It is obvious that the methodology in this paper yields almost the same results, which indicate the validity of suggested procedure. In Fig. 5, broken lines are obtained from a set of individual storm rainfall charts at each station.

    Fig. 5
    figure 5

    IDF curves a Ceylanpınar, b Sanlıurfa, c Mardin, d Diyarbakır

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5 Conclusions

The rainfall intensity concept and its derivation from recording rainfall charts are among the basic requirements for any hydrological engineering water resources management and design studies. In general, intensity–duration–frequency (IDF) curves are not available, but few might exist for some specific areas. Although storm rainfall records are not available, but annual daily maximum rainfall (ADMR) data are registered at each meteorology station. The question is how to benefit from the ADMR records for IDF curve generation? In this paper, three steps are suggested for IDF curve establishment from the ADMR records. These are determination of the most suitable theoretical probability distribution function (PDF) from ADMR records, dimensionless–intensity–duration (DID) curve development and conversion of the ADMR records to IDF curves through the DID curve. The application of the methodology is presented for data from the four meteorology stations from the southeastern province of Turkey, Ceylanpınar near the Syrian border. For each station, there are IDF curves and ADMR records for many years. The IDF curves are obtained through the DID methodology and they are compared with the already existing IFD curves and the relative error between the two sets is less than 10%, which shows the validity of the proposed methodology.