Uncertainty Analysis of Intensity-Duration-Frequency Curves of Babylon City

Rainfall is an important aspect of water resources that needs to be accurately measured. Hydrologic models require accurate estimates of mean rainfall [1]. Extreme rainfall events pose challenges to society in terms of human and economic impact and require a multidisciplinary approach to be analyzed. To construct hydraulic structures that manage stormwater runoff, it is essential to gather data on the amount of intense rainfall over different time periods. This information is typically represented through an Intensity-Duration-Frequency relationship [2]. Babylon City is situated in central Iraq, along the Babylon branch of the Euphrates River, approximately 100km south of Baghdad. The population of Babylon province is around 3,000,000 residents, with an annual growth rate of about 3%. Serving as the provincial capital, Babylon City is located adjacent to the ancient city of Babylon and in close proximity to the historic cities of Borsippa and Kish. Covering an area of 5116 km², Babylon City is primarily characterized by agricultural activities. The region benefits from extensive irrigation facilitated by the Babylon canal, enabling the cultivation of diverse crops, fruits, and textiles. The river flows through the city center, flanked by date palm trees and other arid vegetation, which effectively mitigate the adverse effects of dust and desert winds.

IDF curve is a tool used in the design of hydraulic structures like drainage networks, bridges, and road culverts. It helps engineers estimate extreme rainfall levels by analyzing the frequency of occurrence for different durations. This is done through a technique called frequency analysis (FA), which involves fitting a probability distribution to the recorded rainfall intensities. By using this approach, engineers can determine the appropriate design parameters to ensure the structures can withstand extreme weather conditions [3]. The study estimated the likelihood of an event occurring every 100 years for combinations of annual maximum rainfall measurements collected from rain gauges in Southeastern Arizona using different frequency distributions and methods. The Log Normal distribution can be successfully used to fit a straight line through the upper data points for durations ranging from 5 to 120 minutes. Fitting thunderstorm rainfall data through mathematical or visual methods may lead to errors in estimating 100-year rainfall depths, with mathematical fitting more likely to overestimate. These graphs continue to be analyzed by researchers interested in climate and hydrological risk [4]. Used the magnitude of annual flood peak maxima at 152 representative sites totaling 6728 years of record were obtained from department of water affairs and forestry. The value of 50 year design flood by using three distribution obtained (Log Pearson type III, Generalized extreme value, and Extreme value type I distributions), the statistical analysis confirm that the widely used Log Pearson type III distribution using conventional moments estimators. The Gumbel distribution is commonly utilized to represent various time intervals of yearly maximum precipitation in a singular FA framework. This method presumes that the various durations are autonomous and frequently incorporates simplifications, which have been utilized by numerous authors and services to establish IDF curves for extreme precipitation [5]. Estimated one day and two to five consecutive days annual maximum rainfall of various return periods from 2 to 100 years at In Banswara, Rajasthan, India, researchers used three probability distributions (Normal, Log Normal, and Gamma) to analyze the data and determine the best fit. They compared the distributions using the Chi-square value and conducted frequency analysis using a frequency factor. The findings revealed that the Normal distribution function was the most suitable for predicting rainfall patterns for one, two, and three consecutive days. On the other hand, the Gamma distribution function provided the best fit for four and five consecutive days [6].

In another study conducted in the city of London, researchers developed IDF curves for three different weather scenarios: historic, wet, and dry. They utilized rainfall data collected by the Meteorological Service of Canada from 1943 to 2001. The rainfall data was gathered for various durations, ranging from as short as 5 minutes to as long as 24 hours. The researchers extracted extreme values for each duration from the time series data and fitted them to Gumbel's extreme value distribution. This analysis is crucial for assessing water availability for agriculture, industry, and other human activities, as rainfall serves as a vital factor in these sectors.

Understanding how rainfall is distributed in time and space is vital for a country's economy. Instead of relying solely on summary statistics, having knowledge of the actual distribution of precipitation greatly enhances various applications of rainfall data. Researchers have conducted numerous studies to explore different methods of representing real rainfall patterns for specific purposes [7].

Rainfall curves, also known as IDF curves, provide insights into the amount of water that falls within a specific timeframe. These curves are valuable for predicting potential flooding in an area or calculating the duration and volume of rainfall. To analyze extreme rainfall events and make predictions about future water flow, it is essential to examine the relationship between different duration levels. Multivariate models offer a promising approach to investigate this significant question. Among these models, the copula theory has gained widespread application in hydrometeorology [8]. The research aims to analyze the uncertainty in IDF curves for Babylon City to determine the expected rainfall. With extreme rainfall events becoming more frequent, the risk of flooding is on the rise. It's crucial to upgrade stormwater drainage systems in urban areas to handle these increasing extreme rainfall events. Currently, historical IDF curves are vital for designing these drainage systems. Hence, there is an urgent call to review the existing IDF curves essential for urban rainwater drainage design. This study addresses flood risk by developing rainwater harvesting systems to manage rainfall effectively (Figure 1).

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Figure 1. The studied zone location

Appropriate test quality provides objective procedures to determine whether the presumed theoretical distribution provides a sufficient description of what has been observed insufficient model, such tests serve only to reject an insufficient model; Proving the accuracy of the model is a challenging task. In this paper, three types of tests, namely Chi-square, Kolmogorov-Smirnov and Anderson-Darling tests, are applied to a wide range of distributions [10]. These tests are conducted on all distributions examined in the study.

3.1 Kolmogorov-Smirnov index (K-S)

K-S test is based on statistical Measures cumulative scheme deviation from presumption Cumulative distribution function [11]. The tests showed that the results success in all tests (3).

3.2 Chi-square index

On the other hand, the Chi-squared statistic relies on determining the appropriate number of categories for data grouping in a histogram. However, there is no definitive rule to determine the correct number to utilize [12]. This test is calculated based on the following relationship:

$\chi^2=\sum_{i=1}^k \frac{\left(o_i-E_i\right)^2}{E_i}$               (1)

When conducting statistical analysis, we use the terms "O_i" to represent the observed number of observations in a particular class interval, and "E_i" to represent the expected number of observations based on the probability distribution being examined. To determine the Chi-square value, we refer to published $\chi^2$ tables that provide specific values corresponding to different degrees of freedom. Tables 3 and 4 are essential in statistical calculations and help us assess the significance of our findings. at the 5% level of significance [13]. Results of Chi-square index are shown in Table 4. The tests showed that the results success in all tests.

Table 3. The Kolmogorov-Smirnov index values for Babylon station and with 95% of confidence level

Table 4. The of index Chi-square

3.3 Anderson-Darling index

The Anderson-Darling normality test is a statistical method that checks if the range of tested data matches the expected range, confirming or refuting earlier hypotheses. Unlike the K-S test, it focuses more on the extremes, providing a more detailed analysis. Yet, it needs specific critical values for different distributions and evaluates how well the data align with a particular distribution. The closer the data distribution fits the expected one, the lower the test statistic [14]. The Anderson-Darling test statistic is computed from the relationship:

$\mathrm{AD}=-\mathrm{n}-\frac{1}{\mathrm{n}} \sum_{\mathrm{i}=1}^{\mathrm{n}}(2 \mathrm{i}-1)\left[\ln \left(\mathrm{F}_{\mathrm{i}}\right)+\ln \left(1-\mathrm{F}_{\mathrm{n}-\mathrm{i}+1}\right)\right]$        (2)

where, F is the cumulative distribution function of the specified distribution and n is the number of elements in the sample. Results of A-D index are shown in Table 5. The tests showed that the results success in all tests when significance α=0.05.

Table 5. Anderson-Darling index for the station that are used in the paper

At the Babylon station in Iraq, IDF curves are created using three different statistical distributions: the Gumbel extreme value distribution, the Normal distribution, and the Log Normal distribution. The findings reveal that, when considering durations of 15, 30, and 60 minutes, Normal distribution proves to be more accurate and reliable compared to Gumbel and Log Normal distributions. As illustrated in Figures 4 and 5, Normal distribution was successful through the proximity of curve periods with each other, the quality of the fit tests between the three methods of distribution (gamma, normal and Log Normal) is to test Anderson-Darling as shown in Table 5, significant value (α) is equal (0.05) and degrees of freedom are equal (1). The constant value of the three distributions was less than the critical value, which means that the data tracked the distribution Anderson-Darling That is, we accept the false premise. On the other hand, we note that the test results from the distribution of gamma in the Moments method Lower than Normal and log Normal distributions, indicating that the distribution of gamma in the Moments method is the best. So "the three distributions (Normal and Log Normal and Gamma) are successful and suitable for test Chi-square and Kolmogorov-Smirnov and Anderson-Darling. The impact of IDF curves on the design, operation and maintenance of engineering infrastructure such as urban drainage systems, channels and bridges. In standard hydrological practice, IDF curves are developed by analysing rainfall data at observation points. Further studies are recommended when more information and data are available to verify the results obtained and update IDF curves.