Keywords

1 Introduction

The drainage systems play a crucial role in urban areas since they mostly convey the runoff during rainfall events. The capacity of a drainage system is mainly based on the rainfall characteristics of the local urban areas in which intensity–duration–frequency (IDF) curves are normally selected for design purpose as a convenient form of rainfall information. In urban areas, natural land surfaces have commonly been replaced by artificial surface. As a result of this, the direct runoff increases, which may put more pressure on the drainage systems. As such, the performance of drainage systems may no longer meet the requirements and thereafter results in water-loggings or inundations. Besides, rainfall characteristics have been changed, especially increasing in extreme rainfall events, due to global warming [1,2,3], which are now significantly contributing in exacerbating inundation in urban areas.

Ho Chi Minh City (HCMC) is known as the biggest economic city of Vietnam. This coastal city is also at the highest risk of floods and inundation [4, 5]. Located in low-lying lands of the Sai Gon–Dong Nai River basin with a complex river network, HCMC is easily affected by different flood sources [6,7,8]. During rainy season, the inundation in HCMC is more frequently due to a long rainy period from May to November. The average cumulative rainfall is about 2000 mm per year with approximately 160 rainfall events. Obsolete and accomplished sewer systems, together with a heavy rainfall, could make the city highly on inundation, traffic jam and paralysis. This situation may become worse and effect more citizens in the future because of the ongoing rapid population and rising in extreme rainfall [7]. Viet [9] investigated the change in rainfall intensity due to the global climate and urbanization by using Wenzel formula. The author reported that HCMC witnessed a significant increase in rainfall intensity during the period of 1955–2007. In another study on HCMC’s rainfall, Nguyen et al. [10] developed the rainfall IDF curve for HCMC using rainfall data from 1980 to 2015. The result showed that the generalized extreme value (GEV) distribution is appropriate for modeling extreme rainfall corresponding to different durations. Similarly, there are several studies on the rainfall IDF curve for HCMC [11, 12]. However, in the authors’ knowledge, the suggestion of a reasonable rainfall intensity formula with local parameters for HCMC could not be found in previous papers so far. Hence, the aim of this research is to provide the updated information on the rainfall IDF curve for HCMC. Besides, the reasonable empirical formula-based IDF curves for frequencies of 2, 5, 10, 50 and 100 years are also proposed to assist the local government in planning or upgrading the drainage systems in urban areas.

2 Data and Methodology

The methodology adopted for this research is presented in Fig. 1. Verified sub-hourly rainfall data from Tan Son Hoa station is used in the present analysis. The data is provided by the National Hydro–Meteorological Service (NHMS) of Vietnam. The time series span through 64 years from 1952 to 2015 and was checked carefully for quality.

Fig. 1
A flow chart defines how the best empirical formula with local parameters based on R S S is chosen from the collection of maximum rainfall data for various short-duration in minutes. It undergoes frequency analysis using log-normal, Gumbel, G E V, probability distribution selection, and developing I D F curves.

Employed research methodology

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There are several probability distributions in the literature that can be used for frequency analysis of extreme value. For comparison purpose, the Normal, Log-Normal (LN), Pearson III (P3), Log-Pearson III (LP3), Gumbel, GEV distributions are used for frequency analysis in this study. The probability distribution functions employed in this paper are shown in Table 1.

Table 1 Description of probability distribution functions
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The best-fit probability distribution is selected by the Akaike information criterion (AIC) [13] and Bayesian information criterion (BIC) [14]. The distribution with the lowest AIC and BIC values is supposed to be the most appropriate. The forms of AIC and BIC are shown by Eqs. 1 and 2:

$$ {\text{AIC}} = - \,2\log L\left( {\theta |X} \right) + 2k $$
(1)
$$ {\text{BIC}} = - \,2\log L\left( {\theta |X} \right) + k\ln \left( n \right) $$
(2)

where n is the sample size, k is the number of parameters in a given model, −log L(θ|X) is the minimized negative log-likelihood function.

Since the best-fit probability distribution is found out, the T-year return level ZT can be calculated. Then, the rainfall IDF curves can be represented as a graph, in which rainfall intensity is placed on the vertical axis, rainfall duration is placed on the horizontal axis and the curves display the return period.

In order to easily use the results of rainfall IDF curves in practical applications, the present study uses some empirical formulas to describe these IDF curves, including Sherman (1931), Bernard (1932) and Wenzel (1982) formulas.

Sherman: \(I = \frac{a}{{\left( {T_{{\text{d}}} + b} \right)^{c} }}\)

Bernard: \(I = \frac{a}{{T_{{\text{d}}}^{n} }}\)

Wenzel: \(I = \frac{c}{{T_{{\text{d}}}^{e} + f}}\)

where I is rainfall intensity, Td is rainfall duration and a, b, c, e, n and f are local parameters.

To select the most suitable empirical formula with its local parameters for representing the rainfall IDF curves, the residual sum of square (RSS) is used. The empirical with the lowest value of RSS is considered as the suitable formula to describe the rainfall IDF curves.

$$ {\text{RSS}} = \mathop \sum \limits_{i = 1}^{n} \left( {y_{i} - f\left( {x_{i} } \right)} \right)^{2} $$
(3)

where yi is the ith value of the variable to be estimated, xi is the ith value of the explanatory variable and f(xi) is the estimated value of yi.

3 Results and Discussions

The AIC and BIC values between various probability distributions are shown in Table 2. The lowest value of AIC and BIC for each duration is bold. It can be seen that LN distribution shows its superiority over others. Hence, LN distribution is used for calculating return levels and developing IDF curves.

Table 2 AIC and BIC values between various distributions
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Table 3 presents the values of rainfall intensity corresponding to different rainfall durations and return periods. The IDF curve developed by LN distribution is shown in Fig. 2. It is clear that, in the same rainfall duration, rainfall intensity increases corresponding to higher return period. In the same return period, the rainfall intensity decreases when the duration increases. For example, for the 100-year return period, the value of precipitation intensity is 139.60 mm/h corresponding to duration of 30 min, meanwhile this value is 48.00 mm/h for the duration of 180 min.

Table 3 Values of rainfall intensity (mm/h) corresponding to different rainfall duration and return period
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Fig. 2
A multiline graph of intensity in millimeters per height versus duration in minutes. It plots five decreasing trends for 2, 5, 10, 20, and 100 years. All the trends start at 15 minutes and end at 175 minutes with different intensities. Data are approximate.

Rainfall IDF curve

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In comparison with research of Nguyen et al. [10], our results show that the values of rainfall intensity are higher than those values for the duration of 15 min. For instance, the value of rainfall intensity is 228.05 mm/h corresponding to duration of 15 min, meanwhile this value in the study of Nguyen et al. [10] is about 190.00 mm/h for the 100-year return period. For longer durations (e.g., 60 and 180 min), the rainfall intensity values in our study are not significantly different from those values of the previous study [10].

Table 4 shows the most suitable empirical formula for describing IDF curves corresponding to different return periods. It can be noted that the selected empirical formula is chosen based on the RSS value. It can be seen from Table 4 that the Wenzel formula is suitable for simulating IDF curves of 2, 10 and 20-year return period. Meanwhile, the Sherman formula is considered as the most suitable empirical formula for describing IDF curves of 5 and 100-year return period. Besides, the values of local parameters corresponding to selected empirical formulas are also shown in Table 4.

Table 4 Values of local parameters of the selected empirical formula
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4 Conclusion

This study has built the IDF curve for Tan Son Hoa rainfall station in the period 1952–2015 by applying the LN probability distribution function which is considered as the most suitable distribution for frequency analysis of rainfall intensity. The results from this study provide the values of rainfall intensity corresponding to different periods and frequency. Especially, short-duration rainfall (under 180 min) is considered to be quite important information in finding solutions to cope with waterlogging due to extreme rainfall in HCMC.

In addition, the local parameters of the characteristic of the IDF curve are also determined. This data can be useful to build the design hyetograph from the IDF curve and can be easily use in engineering design applications, such as serving the planning and designing of urban infrastructure projects.