Estimating the Intensity Equations for Rain Intensity Frequency Curves (Mosul /Iraq)

: The relationship between the rainfall intensity, duration, and frequency (IDF) is widely used in water resources engineering in designing hydraulic structures, such as culverts and sewage systems, and for reducing and controlling floods. In this study, the curves of (IDF) were found for five rain stations registered in different regions in northern Iraq, i

relying on the (IMD) equation, applying two distribution methods, i.e., the Pearson type-III logarithmic distribution method and Gumbel Distribution, for different durations and return periods.All findings indicated that while rain intensity increased with more extended return periods, it reduced as rainstorm duration increased.The statistical tests found that the log Pearson distribution was the best distribution method in this city.Hamaamin [4] constructed IDF curves from daily rainfall data for different periods with different return periods to predict the rainfall intensity in Sulaymani.The study found a good agreement between the precipitation intensity using the general empirical formula and the IDF curves, with an excellent correlation coefficient between the two results.Mahdi and Mohamedmeki [5] developed intensity-sustainability-frequency (IDF) curves for the rain falling on the former city of Baghdad.Frequency analysis was performed by Log Pearson Type III, Gumbel Distribution Theory, and Log Normal Distribution to achieve rainfall intensities for different return periods based on the equation derived from Bernard.The Kolmogorov-Smirnov method was used to compare the three distributions' findings and test their fit.The results were satisfactory at the 10% probability level.Majeed et al. [6] found the intensity-duration-frequency IDF curves and identified their equations for the Iraqi city of Najaf.Also, the authors determined which distribution predicated the highest precipitation intensity of the three common distributions used in this area, i.e., Gumbel, Log-Normal, and Log Pearson Type III distributions.The results showed that the Gumbel distribution provided the best rainfall intensity for various return times and durations.The IDF rainfall intensity curves were drawn by Kareem et al. [7] to determine their equations for the city of Erbil.The empirical equations and IDF curves were generated using statistical distributions such as Gumbel and (LPT III) using daily precipitation data for various durations and return periods.The outcomes demonstrated a correlation coefficient (R 2 = 1) between the rainfall intensity obtained from the IDF curves and the experimental formula.The study aimed to obtain IDF curves in some regions in north Iraqi and derive empirical IDF equations for various return periods using three statistical distributions for maximum daily rainfall data.These curves and equations, necessary for building urban drainage works, such as culverts, storm sewers, and other hydraulic structures, have yet to be studied in these parts of Iraq.This study aims to obtain the IDF curves and develop empirical equations to find the rainfall intensity at five stations in north Iraq.Maximum daily rainfall was collected and disaggregated to short duration for different periods, i.e., 10, 20, 30, 60, 120, 180, 360, 720, and 1440 minutes.Three methods were used to examine the estimated rainfall in (mm) and its intensities in (mm/h) for various return intervals and durations (Gumbel, LPT III, and Log-Normal).

Study Area and Rainfall Collection Data
The study area is located in north Iraq and bordered by longitudes (41°-43°) East and latitudes (35°-36°) North.The region includes six selected stations, i.e., Mosul, Tal-Afar, Sinjar, Rabia, and Tal -Abta.The region is surrounded by a mountain range from the north and east and plains from the west and south inside Iraq.The average annual precipitation ranges between 350 and 1000 mm [8].Fig. 1 is a map showing the climate monitoring stations' locations included in the study.Table 1 shows a description of the statistical and hydrological characteristics of the data of those stations.To generate IDF curves, daily rainfall depth in mm was obtained from the Iraqi Meteorological Organization and Seismology (unpublished data) for different periods at five stations distributed in Mosul.

2.2.Generation Short-Duration Rainfall
The rainfall depth of short periods can be derived from the daily maximum rainfall data, given that it is unavailable at the meteorological station, depending on the Indian equation that was proposed by Indian Meteorological Department, IMD, which is known as the empirical reduction formula, as given in Eq. (1), from Ref's.[3,9].IMD applies to calculating the precipitation depth for different periods of less than 24 hours.
Where Pt: rainfall depth in mm for time t (hours).P24: maximum daily rainfall depth in (mm) t: the duration (hours) of rain for which rainfall depth is needed.

ANALYSIS OF FREQUENCY DISTRIBUTION AND DEVELOPMENT IDF CURVES
Obtaining the best probability distribution among the different theoretical distributions is necessary to analyze the rainfall intensity curves of rainfall amounts at constant duration.For this study, three distributions, i.e., the Gumbel distribution, the Log Pearson III distribution, and the Log Normal distribution, were used to assess the annual maximum values for all accessible periods statistically.

Gumbel Distribution (GD)
This method is one of the most used probability distribution functions in hydrological studies to predict flood peaks and maximum rain intensity.GD method is used to analyze the (IDF) curves due to its suitability for extreme data modeling.The Gumbel distribution calculates for various return periods (2, 5, 10, 25, 50, and 100 years) for each period, as shown in the following equations [10,11].PT = Pave + K×S (2) Where Pt is the frequency of rainfall in (mm) for each period and for a return period (T) in years.Pavg is the average of maximum rainfall values for specific periods, as in Eq. (3).
(3) K is the Gumbel frequency coefficient found in special tables [12] or from Eq. ( 4).)� (4) S is the standard deviation of (P) rainfall data, Eq. ( 5): Then rain intensity It in (mm/hour) for any return period T and duration t can be found by applying Eq. ( 6): Where Td is the duration in hours.

Log-Pearson Type III
The Log-Pearson Type III (LP III) distribution method is applied to calculate the rainfall intensity for different durations and return

Study Area
periods to draw IDF curves using the same Gumbel method, as follows [12].
where logx ������ and σlogx are the mean and standard deviation based on logarithmic transformation.Cs is the coefficient of skewness obtained from Ref. [12].

Log-Normal Distribution
To use the log-Normal distribution approach, the rainfall quantities must be transformed into logarithmic numbers, i.e., the logarithmic values of the statistical variables.The frequency coefficient K taken from Ref. [13] follows the same process as the (LPIII) distribution; however, it is equivalent to the standard natural variable Z.

Derivation of IDF Empirical Equation
The IDF relationship is a widely used tool in water resources engineering, i.e., planning, designing, and operating water facilities.This equation expresses the linkage between the maximum rainfall intensity as a dependent variable and other parameters, such as duration and frequency of rainfall, as independent variables.So, (Bernard Equation) was applied to find the rainfall intensity [1], which is: where IT is intensity (mm/hr), Tr is the return period (years), and d is the duration (hours).The coefficients c, m, and e are regional coefficients [10].A log-log graph was created using non-linear regression analysis between rainfall intensity and duration for each return period to determine the constant (e).Values derived from interception points for each obtained equation were plotted on a log-log scale with recurrence intervals to determine the c and m values [11].

GOODNESS OF FIT
The relationship between observed and expected frequencies was examined using the goodness of fit test to select the best suitable type of probability distribution function (PDF) for the rainfall data in the research area.Three different goodness of fit tests were used in this investigation.An explanation of each is reported below.

Chi-Square Test
The chi-square distribution is one of the continuous probability distributions that was first described by Karl Pearson in 1900.It is expressed as follows.
where χ 2 is a random variable for which the sampling distribution is very close to the chisquare distribution.The symbols Oi and Ei refer to the observed and expected frequencies for the interval of the i-th histogram class.The symbol k denotes the number of class intervals.If the observed frequencies are near the corresponding expected frequencies, the χ 2 value will be small, showing a good fit; if not, it is a bad fit.Acceptance of the zero hypothesis results from a good fit, while rejection results from a bad fit.The critical region will therefore be in the right tail of the χ 2 .The critical value is found by χ 2 tables [11,14].The chi-square test of fit was run using the software Simple Fit 5.6 [15].

Kolmogorov-Smirnov Test
The test determines whether there is a good agreement between the observed data's frequency of occurrence and the expected frequency inferred from the distribution.Easy Fit 5.6 was utilized.The Kolmogorov-Smirnov test result that is lower value is the best fit for the selected distribution, and vice versa.Some steps can be taken to pass this test: • The data is put in descending order.
• The observed data's cumulative probability P(xi) is found using the Weibull equation [11].
Where n is the number of historical data, and m is the descending order of values.
• For all observed data, the assumed distribution was used to determine the theoretical cumulative probability F(xi).The maximum absolute difference determined from the following equation is the Kolmogorov-Smirnov test statistic (∆), calculated from the following equation [16].
• The tables saved in the Easy Fit 5.6 program are used to get the tabular (∆o) value of the Kolmokorov-Smirnov statistic for a specific degree of probability.• If the statistic's value (∆) is less than the value of (∆o), the hypothesis that the distribution fits with the assumed probability level is accepted.

Anderson-Darling (AD) Test
This test found good agreement between the observed data's frequency of occurrence and the predicted frequency derived from the distributions.If the A2 statistic's value exceeds the test's critical value, the null hypothesis is rejected at level (α) of probability.It is calculated from the following equation: Page where the data are Y1, Y2, ..., Yn, and n is the number of data.

RESULTS FINDING AND DISCUSSION
Figures (2)-( 16) display the results from the IDF curves obtained from the three used techniques, i.e., Gumbel, LPT III, and Log Normal, in all stations.The results obtained by the three approaches were consistent.After analyzing the equations derived from the intensity-duration-frequency (IDF) curves for the three probabilistic distributions that were applied for all stations, the coefficients (c), (m), and (e) were found.These parameters' values with the rain intensity formula obtained are shown in Table 2 for all stations and the three probability distributions.The goodness-of-fit of the three probability distributions was used in the study for 24 hours to choose the best suitable type of probability distribution function (PDF) for the studied area's rainfall data.Table 3 shows the test results.It can be shown that the Log Person III distribution was the most reliable in the study area because it was the first, i.e., the lowest value of the statistic among most distributions.

Fig. 1
Fig.1 Location Map of the Selected Stations in the Study.Table 1 Statistical and Hydrological Features of Data at the Selected Stations in the Study Area.

Fig. 6
Fig. 6 Log Pearson III Method IDF Curves at Tal-Afar Station.

Fig. 12
Fig. 12 Log Pearson III Method IDF Curves at Rabia Station.

Fig. 13
Fig. 13 Log Normal Method IDF Curves at Rabia Station.

Table 1
Statistical and Hydrological Features of Data at the Selected Stations in the Study Area.

Table 2
Parameter Values and the Rainfall Intensity Equation (IDF) That were Derived for All Stations.

Table 3
Results of the 24-Hour Series' Goodness of Fit Test.