ABSTRACT
Rainfall intensity is considered one of the basic factors in designing hydrological models based on rainstorm data. The objective of this research is to employ novel intensity–duration–frequency curves and develop empirical equations for rainfall intensity in the city of Kirkuk. The reduction formula adopted by the Indian Meteorological Department was used to divide the maximum daily rainfall for short periods of 0.5, 1, 2, 4, 6, 12, and 24 h. Three key methods of frequency analysis (lognormal distribution, log Pearson type III distribution, and Gumbel extreme value distribution) were utilized to formulate a statistical relationship based on rainfall intensity data from between the years 1981 and 2023 for a gauging station upstream of the Kirkuk city basin to provide the best data set for all periods of 100, 50, 25, 10, 5, and 2 years. Research has shown that the logarithmic distribution is the best fit for modelling the relationship between the annual maximum rainfall at Kirkuk station and duration of the rainfall. The goodness-of-fit results indicate that the lognormal distribution statistically outperforms other distribution models. Hence, the generated rainfall intensity, duration, and frequency curves that were developed led to estimating the intensity of precipitation to build forecasting and hydrological behaviour of the Kirkuk city basin.
HIGHLIGHTS
The paper created rainfall intensity–duration–frequency (IDF) curves using statistical distribution tests adapted to Kirkuk city, Iraq.
Historical rainfall data were analysed.
The generated IDF curves were compared to current methodologies or standards.
The IDF curves for Kirkuk city's rainfall durations and return periods were derived from the statistical analysis.
Data constraints or model selection uncertainties were identified and advice for future research is presented.
INTRODUCTION
It is important to have a good idea of both the infrastructure's intended purpose and the physical conditions in which it will work. The rainfall intensity–duration–frequency (IDF) curves are used in several hydrological studies of hydraulic facilities and infrastructure that require safe hydraulic design. The IDF curve establishes a relationship between the rainfall intensity, duration, and frequency at which it occurs (Tuama Al-Awadi & Affairs 2016; Ewea et al. 2018). Thus, in the process of the management of stormwater, the design criteria of infrastructure system components are applied to the events of rainfall to the heavy return period (Mohammed et al. 2021). The data are typically presented as IDF curves that are obtained from analysis of the statistics regarding events that are highly unusual or exceptional based on the intended objective of the application; it can be generated employing various time intervals ranging from immediate daily maximum intensity to annual, monthly, and weekly intensities (Nyamathi & Yashas Kumar (2021)).
Different studies indicate the use of different methodologies to develop IDF curves for different regions worldwide. Table 1 summarizes these studies' most important findings and methods.
A literature review on rainfall intensity–duration–frequency (IDF) interval curves
Reference . | Location . | Method/Approach . | Key findings . |
---|---|---|---|
Ewea et al. (2018) | Saudi Arabia | Gumbel type-I distribution | Developed empirical equations for rainfall intensity |
Tuama Al-Awadi & Affairs (2016) | Baghdad, Iraq | Log Pearson type III distribution | Established the relationship between rainfall intensity, duration, and frequency |
Mohammed et al. (2021) | Nigeria | Lognormal probability distribution | Suggested using IDF curves for designing infrastructure systems |
Nyamathi & Yashas Kumar (2021) | India | Gumbel's extreme value distribution | Presented IDF curves obtained from statistical analysis |
Qarani Aziz et al. (2023) | North Iraq | Gumbel method | Developed empirical rainfall intensity equations for each station |
Hussein et al. (2023) | North Iraq (Mosul) | Log Pearson type III distribution | Found log Pearson type III suitable; rainfall intensity increases with return periods and decreases with duration |
Mahdi & Mohamedmeki (2020) | Baghdad, Iraq | Gumbel, log Pearson type III, lognormal | Found no significant difference among methods; log Pearson type III is slightly preferred |
Hussein (2014) | Karbala, Kut, Iraq | Log Pearson type III distribution | Found log Pearson type III to be the best fit |
Zeri et al. (2023) | Major cities in Iraq | Global Satellite Mapping of Precipitation (GSMaP GC) | Developed IDF curves using satellite data |
Sun et al. (2019) | Singapore | GSMaP GC | Improved reliability and accuracy of IDF curves, especially for short durations |
Ashok Kumar et al. (2023) | India | Gumbel distribution, empirical reduction method | Found IDF curves useful for deriving design storms |
AlHassoun (2011) | Riyadh, Saudi Arabia | Gumbel, log Pearson type III, lognormal | Developed an empirical formula for estimating design rainfall intensity |
Odiong & Agunwamba (2023) | Nigeria | Non-stationary rainfall model | Developed a high-accuracy model and compared it with other models for calibration and validation |
Anand et al. (2023) | India | Tool development | Developed a tool for engineers to calculate rainfall IDF curves |
Onen & Bagatur (2017) | Saudi Arabia | Simplified Gumbel's method | Estimated flood discharge using gene expression programming and regression analysis models |
Akpen et al. (2019) | Lokoja, Nigeria | Log Pearson type III distribution | Recommended using IDF curves for designing hydraulic structures |
Hidayati et al. (2023) | Pontianak City, Indonesia | Log Pearson type III distribution | Derived IDF curve using rainfall data from four stations |
Reference . | Location . | Method/Approach . | Key findings . |
---|---|---|---|
Ewea et al. (2018) | Saudi Arabia | Gumbel type-I distribution | Developed empirical equations for rainfall intensity |
Tuama Al-Awadi & Affairs (2016) | Baghdad, Iraq | Log Pearson type III distribution | Established the relationship between rainfall intensity, duration, and frequency |
Mohammed et al. (2021) | Nigeria | Lognormal probability distribution | Suggested using IDF curves for designing infrastructure systems |
Nyamathi & Yashas Kumar (2021) | India | Gumbel's extreme value distribution | Presented IDF curves obtained from statistical analysis |
Qarani Aziz et al. (2023) | North Iraq | Gumbel method | Developed empirical rainfall intensity equations for each station |
Hussein et al. (2023) | North Iraq (Mosul) | Log Pearson type III distribution | Found log Pearson type III suitable; rainfall intensity increases with return periods and decreases with duration |
Mahdi & Mohamedmeki (2020) | Baghdad, Iraq | Gumbel, log Pearson type III, lognormal | Found no significant difference among methods; log Pearson type III is slightly preferred |
Hussein (2014) | Karbala, Kut, Iraq | Log Pearson type III distribution | Found log Pearson type III to be the best fit |
Zeri et al. (2023) | Major cities in Iraq | Global Satellite Mapping of Precipitation (GSMaP GC) | Developed IDF curves using satellite data |
Sun et al. (2019) | Singapore | GSMaP GC | Improved reliability and accuracy of IDF curves, especially for short durations |
Ashok Kumar et al. (2023) | India | Gumbel distribution, empirical reduction method | Found IDF curves useful for deriving design storms |
AlHassoun (2011) | Riyadh, Saudi Arabia | Gumbel, log Pearson type III, lognormal | Developed an empirical formula for estimating design rainfall intensity |
Odiong & Agunwamba (2023) | Nigeria | Non-stationary rainfall model | Developed a high-accuracy model and compared it with other models for calibration and validation |
Anand et al. (2023) | India | Tool development | Developed a tool for engineers to calculate rainfall IDF curves |
Onen & Bagatur (2017) | Saudi Arabia | Simplified Gumbel's method | Estimated flood discharge using gene expression programming and regression analysis models |
Akpen et al. (2019) | Lokoja, Nigeria | Log Pearson type III distribution | Recommended using IDF curves for designing hydraulic structures |
Hidayati et al. (2023) | Pontianak City, Indonesia | Log Pearson type III distribution | Derived IDF curve using rainfall data from four stations |
The main objective of this work is to formulate IDF curves for Kirkuk city for varying return periods, where three different methods of statistical distributions are utilized to achieve the objectives. The software EasyFit 5.6 is applied to verify the probability distribution function for the data of the maximum daily precipitation. These curves and equations are very important for enhancing hydrological analysis that can be used for the safe and economical design of hydraulic components represented by designing systems of drainage in urban areas, such as storm sewers, canals, and any required hydraulic structures in the city of Kirkuk.
DATA COLLECTION AND AREA OF STUDY
Climatic elements (Najdat 2015)
Climatic element . | Maximum . | Minimum . | Average . |
---|---|---|---|
Precipitation (mm/month) | 768.9 | 202.5 | 368.4 |
Relative humidity (%) | 73 | 23 | 47 |
Evaporation (mm) | 387.9 | 45.6 | … |
Wind speed (m/s) | 28 | … | 2.9 |
Evaporation in free water surface/year (mm) | … | … | 1,642.9 |
Temperature (°C) | 49.53 | −6.7 | 22.4 |
Climatic element . | Maximum . | Minimum . | Average . |
---|---|---|---|
Precipitation (mm/month) | 768.9 | 202.5 | 368.4 |
Relative humidity (%) | 73 | 23 | 47 |
Evaporation (mm) | 387.9 | 45.6 | … |
Wind speed (m/s) | 28 | … | 2.9 |
Evaporation in free water surface/year (mm) | … | … | 1,642.9 |
Temperature (°C) | 49.53 | −6.7 | 22.4 |


The distribution of rainfall depths derived from maximum annual values for various durations throughout the year.
The distribution of rainfall depths derived from maximum annual values for various durations throughout the year.
METHODOLOGY
An established method to understand the relationship between precipitation and storm frequency is the IDF curve. It is generally adopted to demonstrate the design of storms for different systems of water projects of certain recurrence intervals (AlHassoun 2011). It is important to understand that IDF curves are not generated from specific storm events and as such do not represent the probability of a particular storm's occurrence. The initial phase in developing the IDF curves is to fit various distributions of the frequency of theoretical amounts to the extreme rainfall for some fixed intervals.
The next logical phase is to proceed and then describe the variation of the parameters of the distribution with intervals by a functional relation. Then any duration and return period can be generated and formulated from the fitted relationships of rainfall intensity. In this study, the values of availability to the maximum annual of the intervals are statistically analysed by using three methods of distribution keys (Gumbel extreme value, lognormal, and log Pearson type III). The optimal fit was identified using EasyFit 5.6 software.
Rainfall intensity (I)

Frequency factor
For each of the return periods that were chosen, the frequency factor calculation using the chosen distributions is now complete. Gumbel, lognormal, and log Pearson type III distributions were utilized in this investigation. Table 3 describes the frequency factors for various distributions.
The factors of frequency for the different distributions
N/S . | Distribution . | Frequency . |
---|---|---|
A | Gumbel generalized extreme distribution value type-I | ![]() |
B | Lognormal | In the case of a normal distribution, the only difference is the variables of the logarithm to be used, i.e., ![]() |
C | Log Pearson type III | With G = 0, a factor of frequency is equal to Z of the standardization of the normal variable. With KT, G is roughly by Chow et al. (1988) as KT = (z) + [(z 2–1) k] + [1/3*(z 3–6*z) * k2] – [(z2 – 1)*k] + (2*k4 + 1/3* k5) where the value of k = [G/6]. The value of KT can be obtained from the table of standardization statistics |
N/S . | Distribution . | Frequency . |
---|---|---|
A | Gumbel generalized extreme distribution value type-I | ![]() |
B | Lognormal | In the case of a normal distribution, the only difference is the variables of the logarithm to be used, i.e., ![]() |
C | Log Pearson type III | With G = 0, a factor of frequency is equal to Z of the standardization of the normal variable. With KT, G is roughly by Chow et al. (1988) as KT = (z) + [(z 2–1) k] + [1/3*(z 3–6*z) * k2] – [(z2 – 1)*k] + (2*k4 + 1/3* k5) where the value of k = [G/6]. The value of KT can be obtained from the table of standardization statistics |
Gumbel method
Log Pearson type III
Lognormal method
The lognormal way works through the parallel procedure as log Pearson type III, used in Gumbel's method with normal KT (Shamkhi et al. 2022).
Fitting of a probability distribution
Evaluating process for suitability of probability distribution through statistical tools.
Evaluating process for suitability of probability distribution through statistical tools.
RESULTS AND DISCUSSION
Descriptive statistics for duration series
The statistic parameters of the interval's series of rainfall for 24 h indicated that the means of the rainfall sample for 43 years of data in 24 h are 56.69 mm, and the standard deviation is 17.95 mm. The sample coefficient of variation and skewness is 0.32 and 0.70, respectively, the excess kurtosis is 0.0043, and the variance is 322.37. The statistics descriptive of 0.5, 1, 4, 6, 12, and 24 h are presented in Table 4.
The statistics of descriptive intervals series
Statistic . | 0.5 h . | 1 h . | 4 h . | 6 h . | 12 h . | 24 h . |
---|---|---|---|---|---|---|
Size of the sample | 43 | 43 | 43 | 43 | 43 | 43 |
Range | 19.71 | 24.84 | 39.43 | 45.13 | 56.86 | 71.64 |
Mean | 15.60 | 19.65 | 31.20 | 35.71 | 45.00 | 56.69 |
Variance | 24.41 | 38.74 | 97.63 | 127.93 | 203.08 | 322.37 |
Standard deviation | 4.94 | 6.22 | 9.88 | 11.31 | 14.25 | 17.95 |
Coefficient of variation | 0.32 | 0.32 | 0.32 | 0.32 | 0.32 | 0.32 |
Standard error | 0.75 | 0.95 | 1.51 | 1.72 | 2.17 | 2.74 |
Skewness | 0.70 | 0.70 | 0.70 | 0.70 | 0.70 | 0.70 |
Excess kurtosis | 0.0043 | 0.0043 | 0.0043 | 0.0043 | 0.0043 | 0.0043 |
Statistic . | 0.5 h . | 1 h . | 4 h . | 6 h . | 12 h . | 24 h . |
---|---|---|---|---|---|---|
Size of the sample | 43 | 43 | 43 | 43 | 43 | 43 |
Range | 19.71 | 24.84 | 39.43 | 45.13 | 56.86 | 71.64 |
Mean | 15.60 | 19.65 | 31.20 | 35.71 | 45.00 | 56.69 |
Variance | 24.41 | 38.74 | 97.63 | 127.93 | 203.08 | 322.37 |
Standard deviation | 4.94 | 6.22 | 9.88 | 11.31 | 14.25 | 17.95 |
Coefficient of variation | 0.32 | 0.32 | 0.32 | 0.32 | 0.32 | 0.32 |
Standard error | 0.75 | 0.95 | 1.51 | 1.72 | 2.17 | 2.74 |
Skewness | 0.70 | 0.70 | 0.70 | 0.70 | 0.70 | 0.70 |
Excess kurtosis | 0.0043 | 0.0043 | 0.0043 | 0.0043 | 0.0043 | 0.0043 |
From Table 4, it can be observed that all data durations have a skewness value of approximately 0.70. This indicates that the duration or interval of data is distributed symmetrically. Moreover, extra kurtosis of the entire duration of data is uniform equal to 0.0043, meaning that the distribution of each data interval does not exhibit substantial divergence from a normal distribution. Table 4 presents the statistical descriptors of the EasyFit 5.6 software to get the finding of interval series.
Graphical determination of the best modelling to 0.5 h
Goodness-of-fit tests
The goodness-of-fit tests display the probability distributions that have been obtained, with the least statistic value being considered the most suitable fitting distribution. Table 4 clears the statistical results of the Kolmogorov–Smirnov, Anderson–Darling, and Chi-square tests of goodness of fit for 0.5 h durations. From Table 5, the lognormal and log Pearson distributions are commonly found to be the most suitable fit. Applying the Chi-square test, the 6th ranked was at lognormal distribution by Kolmogorov–Smirnov and Anderson–Darling goodness-of-fit tests. The 12th ranked was at log Pearson type III by the test of Kolmogorov–Smirnov, 2nd by the test of Anderson–Darling, and 3rd by the test of Chi-square. The 5th ranked was at Gumbel distribution by the test of Kolmogorov–Smirnov, 21st by the test of Anderson–Darling, and 17th by the test of Chi-square.
Tests of fit for 0.5h
Distribution . | Kolmogorov–Smirnov . | Anderson–Darling . | Chi-square . | |||
---|---|---|---|---|---|---|
Statistic . | Rank . | Statistic . | Rank . | Statistic . | Rank . | |
Gumbel | 0.0918 | 5th | 0.4388 | 21st | 0.5850 | 17th |
Lognormal | 0.0919 | 6th | 0.3930 | 6th | 0.2494 | 4th |
Log Pearson type III | 0.0961 | 12th | 0.3808 | 2nd | 0.2359 | 3rd |
Distribution . | Kolmogorov–Smirnov . | Anderson–Darling . | Chi-square . | |||
---|---|---|---|---|---|---|
Statistic . | Rank . | Statistic . | Rank . | Statistic . | Rank . | |
Gumbel | 0.0918 | 5th | 0.4388 | 21st | 0.5850 | 17th |
Lognormal | 0.0919 | 6th | 0.3930 | 6th | 0.2494 | 4th |
Log Pearson type III | 0.0961 | 12th | 0.3808 | 2nd | 0.2359 | 3rd |
Least sum of statistic model identification criterion
Table 6 displays the most suitable model among the three distributions of probability that were taken into account. It is evident from Table 5 that the lognormal distribution has the smallest statistical value of 0.03366, and the Gumbel distribution has a value of 0.11566. In cases where the logarithmic distribution best fits the relationship between the annual maximum rainfall at Kirkuk station and a 0.5-h duration, it is employed for subsequent analyses. Likewise, rainfall data with durations of 0.5, 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5, 6, 12, and 24 h were fitted with a probability distribution.
The model of best fit for 0.5h
Distribution . | Kolmogorov–Smirnov . | Anderson–Darling . | Chi-square . | LSSMIC . | Fit . |
---|---|---|---|---|---|
statistic . | statistic . | statistic . | |||
Gumbel extreme value | 0.09188 | 0.43883 | 0.58495 | 0.11566 | |
Lognormal | 0.09188 | 0.39239 | 0.54939 | 0.03366 | Best |
Log Pearson type III | 0.09611 | 0.38004 | 0.23587 | 0.28798 |
Distribution . | Kolmogorov–Smirnov . | Anderson–Darling . | Chi-square . | LSSMIC . | Fit . |
---|---|---|---|---|---|
statistic . | statistic . | statistic . | |||
Gumbel extreme value | 0.09188 | 0.43883 | 0.58495 | 0.11566 | |
Lognormal | 0.09188 | 0.39239 | 0.54939 | 0.03366 | Best |
Log Pearson type III | 0.09611 | 0.38004 | 0.23587 | 0.28798 |
Standard deviation and mean to the intensity of rainfall
Table 7 shows the finding of the standard deviation and mean obtained from each duration series, according to Equations (2) and (3) for computing the standard deviation and the mean for each data duration of rainfall intensities.
The rainfall intensity standard deviation and mean for all durations
Duration . | 0.5 h . | 1 h . | 1.5 h . | 2 h . | 4 h . | 12 h . | 24 h . |
---|---|---|---|---|---|---|---|
Mean | 15.60 | 19.65 | 22.50 | 24.76 | 31.20 | 45.00 | 56.69 |
Standard deviation | 4.94 | 6.22 | 7.13 | 7.84 | 9.88 | 14.25 | 17.95 |
Duration . | 0.5 h . | 1 h . | 1.5 h . | 2 h . | 4 h . | 12 h . | 24 h . |
---|---|---|---|---|---|---|---|
Mean | 15.60 | 19.65 | 22.50 | 24.76 | 31.20 | 45.00 | 56.69 |
Standard deviation | 4.94 | 6.22 | 7.13 | 7.84 | 9.88 | 14.25 | 17.95 |
Developing IDF curves through the generated intensity of rainfall design
CONCLUSION AND RECOMMENDATION
The developed IDF curves for Kirkuk city should be applied for calibration and verification processes according to the observed actual data using hydrological modelling, especially in geospatially and kinetic routing-based models, to determine the general hydrological behaviour of watersheds that impact the infrastructure of Kirkuk city. Then, the developed IDF curve can be utilized in the design data for different infrastructure systems (such as drainage systems, and in management of urban water systems such as bridges, sewers, and culverts). Due to variability patterns of rainfall data and climate change, the developed IDF curve must be applied to the review and for updating because of the lack of data and long-term observed data; in addition, the city of Kirkuk classified most of its watersheds as ungagged watersheds owing to the fact that its basins are not measured. Research has shown the logarithmic distribution is the best fit for modelling the relationship between the annual maximum rainfall at the Kirkuk station and its duration. The goodness-of-fit results indicate that the lognormal distribution statistically outperforms other distribution models. Additionally, the rainfall intensity increases as return periods increase but decreases as rainfall duration increases. Hence, the generated rainfall intensity, duration, and frequency curves that were developed led to estimating the intensity of precipitation to build forecasting and hydrological behaviour of the Kirkuk city basin.
DATA AVAILABILITY STATEMENT
All relevant data are included in the paper or its Supplementary Information.
CONFLICT OF INTEREST
The authors declare there is no conflict.