Developing the IsoProbability Curves of Combination of Rainfall and Water Level for Urban Drainage Infrastructure Planning - A Case Study in Ho Chi Minh City
Vietnam National University, Vietnam
HCMC University of Natural Resources and Environment, Vietnam

ABSTRACT
To ensure efficiency and cost-effectiveness, drainage systems often need the capability to handle drainage such that the overload phenomenon only occurs with a return period greater than a certain value specific to the system. For urban areas situated in low-lying regions and influenced by tides, the flow in the drainage system depends on two natural factors: rainstorm rainfall in the area (R), and water level at the outlet (H) in the time of rainstorm. Consequently, the return period of the overload phenomenon in the drainage system will be determined by the combination of R, H causing the overload. Before conducting hydraulic calculations, it is essential to construct probability curves for the R, H combinations. This article aims to introduce a method for constructing iso-value curves representing the return period of R, H combinations. The method is applied to the case of Ho Chi Minh City as an example.
1. INTRODUCTION
Throughout history, urban drainage systems have played a crucial role in urban infrastructure, serving to gather and transport stormwater and wastewater away from populated regions (Larsen and Gujer 1997). Despite continuous advancements, the task of creating a well-functioning and efficient drainage system remains a formidable undertaking (Birkmann et al. 2010).
Urban drainage systems are most often designed with a specific return period (Tuyls et al. 2018). It is common for the return period of the designed rainfall to be predetermined depending on the characteristics of the drainage area (PUB 2018; Welsh Government 2018; CPHEEO 2019). Intensity-duration-frequency (IDF) curves have played an important role in the design of urban drainage systems (Ombadi et al. 2018; Sarhadi and Soulis 2017; Veneziano and Yoon 2013). These curves offer valuable insights into rainfall intensity for specific durations and occurrence frequencies. Sherman (1931) and Bernard (1932) were the first to construct an IDF curve based on historical rainfall data. Later studies have improved the method to increase the accuracy of the IDF curves (Zhao 2021; Cheng and Aghakouchak 2014; Kim et al. 2008; Shrestha et al. 2017; Guo 2006). With the wider application of hydrodynamic mathematical models, a new method of drainage system design has begun to be developed in which the rainfall return period is determined to optimize the benefits based on the analysis of flood risks (Fortunato et al. 2014; Yang et al. 2019). This method is based on the principle of minimizing costs including the costs of construction, maintenance, and operation of the system, and maximizing benefits including social-environmental and economic losses due to possible floods.
Due to the influence of factors other than the rain, the return period of the flooding is not of a design rainfall (Tuyls et al. 2018). Only for drainage systems where flow depends only on rainfall (such as in the case where the sewer mouth is higher than the water level at the outlet or the flow in the sewer is supercritical), the return period of the flooding is identical to the one for rainfall. In cities located in low-lying areas and affected by tides, flooding occurs due to a combination of factors, including rainstorm rainfall, high tides, and occasional river currents overflowing (Tingsanchali 2012; Shi et al. 2022; Qiang et al. 2021; Silva et al. 2017). From a hydraulic point of view, the tide is the water level, and the overflow of the river is also caused by water level, so it can be said that urban flooding depends on the combination of two factors: rainstorm rainfall in the area and water level at outlet. The return period for flooding is identical to one of these combinations. Considering this fact, instead of the return period for rainfall, the return period for flooding was used (Drainage Services Department 2018; British Standard Insitute 2013). In a rare study conducted by Lian et al. (2013), the joint impact of rainfall and tides on flooding was analyzed using the copula presented by Sklar (1959) which established the joint distribution function of water level and rainfall. With the help of the joint distribution function, the joint probability of any combination of rainfall and tidal level can be assessed.
The aim of this paper is to introduce another approach to construct the isofrequency curves of the combination of rainfall and water level suitable for urban drainage design, as well as for analysis of flood risk. The proposed method has been applied to construct isofrequency curves by combining rainfall and water level data in specific areas of Ho Chi Minh City, Vietnam. This approach enhances the flood risk assessment by integrating both rainfall and water level variables, offering a significant improvement over traditional single-variable methods. The integration of these factors not only increases predictive accuracy but also supports the optimization of urban drainage system design, particularly in coastal cities influenced by tidal patterns. This methodological advancement provides a more comprehensive framework for addressing flood-related challenges in urban environments.
2. METHODOLOGY
2.1 Empirical isofrequency curves of rainfall–water level combinations
The Weibull formula for estimating the probability of occurrence of an event in which its X parameter has the value larger than or equal to x is presented as follows (Chow et al. 1988).
![]() |
(1) |
Where:
n | = | total number of samples, |
m | = | rank of sample values sorted in ascending order, and |
x | = | value of ranking m. |
At the same time, the return period also can be calculated (Kumar and Bhardwaj 2015):
![]() |
(2) |
Where:
N | = | number of years of observation. |
Note that the exceedance probability and the return period are interchangeable.
Plot the points with coordinates X and P on a graph with the horizontal axis as P and the vertical axis as X. Then, draw a smooth curve passing through the center of these points to obtain the empirical probability curve.
We can extend Equation 1 and Equation 2 to the case that the events are defined by two parameters such as rainfall of a rainstorm (R) and water level at drainage system outlet in the time of rainstorm (H). The restriction now is that instead of satisfying just one criterion, X ≥ x, two criteria R ≥ r and H ≥ h must be satisfied simultaneously. The probability of occurrence of an event described by the combination of two parameters (R, H) is then:
![]() |
(3) |
Where:
m | = | number of events satisfying simultaneously R ≥ r and H ≥ h. |
The return period in this case can also be calculated by an equation similar to Equation 2.
In the case of events represented by one parameter, parameter X as in Equation 1, the probability of an event is represented by a curve in the coordinate system (P, X). However, in the case where events are represented by two parameters, namely rainfall R, and water level H, as in this study, if plotted, the probability of the event would be a curved surface in a 3-dimensional space with the vertical axis being probability P, and the two horizontal axes being R and H.
Figure 1 shows 64 rain events observed over 14 years at the Ly Thuong Kiet rain gauge station along with the water level at the Phu An hydrological station on the Saigon River at the time of the shower plotted on the plane (R, H). These rain events have a duration of 90 ± 15min. The locations of the combinations are depicted by circular points (Figure 1).
Determining the number of combinations exceeding the threshold can be easily described by Figure 1. For example, consider point A as a rain event with r = 49.7 mm. This rain event occurred when the water level at Phu An station was h = -56 cm. We can count 12 rain events that occurred and simultaneously satisfied 2 conditions: R ≥ r = 49.7 mm and H ≥ h = 56 cm (including point A). Thus, according to Equation 3, the probability of a combination satisfying the condition (R ≥ r = 49.7 mm, H ≥ h = 56 cm) is P = 0.18. The return period of the event exceeding the threshold r, h can also be calculated according to Equation 2 as T = 1.25 years.
Figure 1 Distribution of combinations of rainstorm rainfall at Ly Thuong Kiet station and water level at Phu An station occurring over 14 years.
Determining the number of combinations exceeding the threshold can be easily described by Figure 1. For example, consider point A as a rain event with r = 49.7 mm. This rain event occurred when the water level at Phu An station was h = -56 cm. We can count 12 rain events that occurred that simultaneously satisfied 2 conditions: R ≥ r = 49.7 mm and H ≥ h = 56 cm (including point A). Thus, according to Equation 3, the probability of a combination satisfying the condition R ≥ r = 49.7 mm, H ≥ h = 56 cm, is P = 0.18. The return period of the event exceeding the threshold r, h can also be calculated according to Equation 2 as T = 1.25 years.
After that, the exceedance probability for all observed combinations R, H is determined, and the isovalue curves of return periods for the combination R, H are plotted. One of the ways is to use bi-linear interpolation. A mesh of triangular elements is constructed from the points of rain events using Delaunay triangulation. The isovalue curves of the return period will be linearly interpolated within each triangle. Figure 2 shows the curves plotted in this way. The isovalue curves can then also be smoothed out by hand.
Figure 2 Isovalue curves of the return period of the combination R, H plotted using bi-linear interpolation
2.2 Smooth the empirical isofrequency curves of rainfall–water level combinations
The isofrequency curve generated by bi-linear interpolation from the calculated return period values at each R, H combination may not be smooth and does not guarantee accuracy when determining combinations at small frequencies. To address this limitation, mathematical methods will be employed.
If f(x) is the probability density function, the exceedance probability curve will be described by the cumulative distribution function F(x) as follows:
![]() |
(3) |
In meteorology and hydrology, the probability density functions for flow, water level, and rainfall often take the form of a bell-shaped curve. In the case where x is a single variable, one of the most commonly used functions to approximate the distribution of these variables is the Pearson III function (Singh and Singh 1988):
![]() |
(4) |
Where:
aP | = | location parameter, |
bP | = | scale parameter, |
cP | = | shape parameter, and |
Γ(cP) | = | Gamma function. |
![]() |
(5) |
When cP is a positive integer, the integral (3) leads to the result:
![]() |
(6) |
Another distribution function that is commonly used in the fields of meteorology and hydrology is the Weibull function (Singh 1987):
![]() |
(7) |
Its cumulative density function has the form:
![]() |
(8) |
Where:
aW | = | location parameter, |
bW | = | scale parameter, and |
cW | = | shape parameter. |
The important difference between Pearson III distribution and Weibull is that the Weibull distribution is more symmetric. Also, the Weibull cumulative density function is steeper. For the combination of R, H, we propose that the exceedance probability function be constructed as follows:
![]() |
(9) |
The rainfall distribution will be approximated by the Pearson III function, while the water level distribution will be approximated by the Weibull function. In addition, the parameter bP of the Pearson III function is assumed to depend on the water level (H), and the bw and cw of the Weibull function depend on the rainfall (R). The expressions of these parameters are suggested as follows:
![]() |
(10) |
![]() |
(11) |
![]() |
(12) |
The coefficients aP, bP0, bP1, bP2, cP, aW, bW0, bW1, bW2, cW0, cW1, and cW2 will be determined such that the error of F (R, H) compared to the empirical exceedance probability P is minimized:
![]() |
(13) |
Where:
S | = | total area of the mesh of triangular elements and is constructed from the points of rain events using Delaunay triangulation. |
Equation 9 ensures the basic characteristics of the cumulative density function. It has the value 1 at (R = aP, H = aW) and vanishes to zero when R and H approach infinity.
Once the coefficients are determined, using the exceedance probability calculated by Equation 9, the isovalue curves of return period of combination R, H will be plotted. Figure 3 presents the isovalue curves of return period of combination R, H plotted from the results calculated by Equation 9 for the case of 90-minute rainfall at Ly Thuong Kiet station.
Figure 3 Isofrequency curve of the R, H combination of Ly Thuong Kiet rain gauge station with a rain duration of 90 minutes.
3. APPLICATION FOR HO CHI MINH CITY AND DISCUSSION
3.1 Study area
Ho Chi Minh City is a largest city in southern Vietnam. The city is in downstream of the Dong Nai River system and influenced by tides from the East Sea (South China Sea). The Saigon River, a branch of the Dong Nai River system, flows through the city. The Phu An National Hydrological Station on the Saigon River is located right in the center (see Figure 4). The water level of this station is often used to calculate the design water level of the city's drainage system. The highest peak water level ever observed at Phu An station is 1.75 m. Meanwhile, 75.6% of the city's area is no higher than 2.0 m. Due to the low terrain, drainage of rainwater is quite difficult. During high tides, many areas of the city are flooded, even when it is not raining.
Ho Chi Minh City is rarely hit by storms but often flooded over large areas during heavy rain, especially if heavy rain occurs at high tide. The rainy season in Ho Chi Minh City lasts from May to November. The duration of a rainstorm usually lasts about 60–120 minutes. The amount of rain in each rainstorm rarely exceeds 120 mm.
Every year the city has about 10 rainstorms with rainfall greater than 100 mm. Recently, due to climate change, heavy rainstorms have occurred more frequently. Rainfalls also usually occur only in part of the city. It is very rare for rain to occur simultaneously in the entire city.
3.2 IsoProbability curves of combination of R, H for Ho Chi Minh City
Several rain gauge stations have been installed in Ho Chi Minh City. However, observation data for only 5 stations is available. At these stations, rainfall observation data spans more than 10 years, which is sufficient to serve the frequency analysis of the combination R, H. Information about these stations is presented in Table 1 and shown in Figure 4. The isofrequency curves of the combination of rainstorm rainfall of these stations and water level during the rain event at Phu An station at 4 recurrence periods of 2 years, 3 years, 5 years, and 10 years, and 3 rain events of 60 minutes, 90 minutes, and 120 minutes are established.
Table 1 Rain gauge stations.
No. | Rain gauge stations | Data | Number of years of data | ||
From | To | ||||
1 | Nguyen Huu Canh | 2010 | 2023 | 14 | |
2 | Duong Quang Ham | 2010 | 2023 | 14 | |
3 | Ly Thuong Kiet | 2010 | 2023 | 14 | |
4 | Phuoc Long A | 2010 | 2023 | 14 | |
5 | Thanh Da | 2013 | 2023 | 11 |
Table 2 also introduces the error of the approximation function, revealing that the error is relatively small, not exceeding 3.9%.
Figure 4 Location of rain gauge stations and water level station in Ho Chi Minh City.
Figures 5–9, below, depicts a graph showing isofrequency curves of combinations of rainstorm rainfall-water level R, H. The graph includes data from 5 rain gauge stations representing rain events with durations of 60 minutes, 90 minutes, and 120 minutes. These curves are drawn using the approximation function (6) for return periods of 2 years, 3 years, 5 years, and 10 years.
Figure 5 Isovalue curves of return period of combinations R, H of Ngu-yen Huu Canh rain gauge station.
Figure 6 Isovalue curves of return period of combinations R, H of Duong Quang Ham rain gauge station.
Figure 7 Isovalue curves of return period of combinations R, H of Ly Thuong Kiet rain gauge station.
Figure 8 Isovalue curves of return period of combinations R, H of Phuoc Long A rain gauge station.
Figure 9 Isovalue curves of return period of combinations R, H of Thanh Da rain gauge station.
The study employs both spatial and temporal analyses to assess the impact of rainfall events on water level fluctuations. Spatially, the correlation between rainfall and water level return periods is evaluated by considering the distance between rain gauges and water level gauges. Results reveal that locations closer to drainage outlets exhibit a more direct relationship, while distant stations introduce variability due to catchment storage and flow delay effects, as visualized in Figures 5–9. Temporally, the analysis examines the lag between peak rainfall intensity and peak water level occurrence. Short-duration, high-intensity events cause rapid water level rises, whereas longer-duration rainfall events lead to gradual accumulation. The figures highlight how the return period of a rainfall event translates into varying water level responses, underscoring the importance of considering both rainfall intensity and duration in understanding these dynamics.
Table 2 below presents the error associated with the approximation function, demonstrating that the error is relatively small, not exceeding 3.9%. This indicates that the mathematical approach used to generate a smooth function representing the isofrequency curves is robust and provides a strong fit to the original data. Such accuracy is critical for the design of urban drainage systems, as it enables more reliable predictions of flooding risks by effectively capturing the interaction between rainfall and water level events.
Table 2 Parameters of approximation functions.
TT | Station name | Duration (m) |
Coefficients | RMSE | ||||||||||||
α | aP | bP0 | bP1 | bP2 | cP | aW | bW0 | bW1 | bW2 | cW0 | cW1 | cW2 | ||||
1 | Phuoc Long | 60 | 4.10E-01 | 0 | 2.07E-01 | -3.87E-05 | -7.68E-10 | 2 | -400 | 1.90E-03 | 1.10E-04 | -2.11E-06 | 19 | 6.40E-01 | 1 | 6.88E-02 |
90 | 5.90E-01 | 0 | 7.77E-02 | -1.64E-05 | -1.49E-10 | 2 | -400 | 1.97E-03 | 1.17E-04 | -1.19E-06 | 16 | 4.30E-01 | 1 | 7.39E-02 | ||
120 | 4.80E-01 | 0 | 6.48E-02 | -1.17E-05 | -8.09E-10 | 2 | -400 | 1.90E-03 | 6.83E-05 | -5.05E-07 | 22 | 4.10E-01 | 1 | 5.09E-02 | ||
2 | Thanh Da | 60 | 2.27E+00 | 0 | 2.03E-01 | -1.21E-04 | -6.42E-10 | 2 | -400 | 1.88E-03 | 8.63E-05 | -9.63E-07 | 21 | 5.70E-01 | 1 | 7.66E-01 |
90 | 6.60E-01 | 0 | 3.62E-02 | -2.01E-04 | -1.58E-07 | 2 | -400 | 2.13E-03 | 2.14E-04 | -9.60E-07 | 11 | 5.50E-01 | 1 | 9.53E-02 | ||
120 | 7.40E-01 | 0 | 6.32E-02 | -1.17E-04 | -5.05E-08 | 4 | -400 | 2.11E-03 | 9.80E-05 | -1.40E-07 | 13 | 2.00E-01 | 1 | 4.98E-02 | ||
3 | Duong Quang Ham | 60 | 3.60E-01 | 0 | 1.79E-01 | -1.35E-05 | -7.67E-08 | 2 | -400 | 1.91E-03 | 9.79E-05 | -1.52E-06 | 18 | 5.30E-01 | 1 | 6.53E-02 |
90 | 6.40E-01 | 0 | 8.08E-02 | -1.57E-05 | -1.55E-09 | 2 | -400 | 1.96E-03 | 1.13E-04 | -1.11E-06 | 17 | 3.90E-01 | 1 | 5.96E-02 | ||
120 | 7.10E-01 | 0 | 4.53E-02 | -7.96E-05 | -6.09E-08 | 2 | -400 | 1.99E-03 | 1.05E-04 | -5.45E-07 | 17 | 3.40E-01 | 1 | 4.34E-02 | ||
4 | Nguyen Huu Canh | 60 | 7.10E-01 | 0 | 9.03E-02 | -3.60E-04 | -3.38E-07 | 4 | -400 | 2.08E-03 | 1.37E-04 | -5.50E-08 | 14 | 3.60E-01 | 1 | 7.25E-02 |
90 | 4.20E-01 | 0 | 2.09E-01 | -4.15E-05 | -1.46E-07 | 5 | -300 | 2.29E-03 | 5.64E-05 | -7.36E-07 | 20 | 2.90E-01 | 2 | 4.81E-02 | ||
120 | 6.40E-01 | 0 | 3.73E-02 | -2.41E-04 | -2.55E-07 | 3 | -400 | 1.98E-03 | 1.05E-04 | -1.72E-07 | 18 | 4.30E-01 | 1 | 5.02E-02 | ||
5 | Ly Thuong Kiet | 60 | 2.70E-01 | 0 | 1.72E-01 | -8.77E-06 | -9.36E-08 | 2 | -400 | 1.89E-03 | 1.89E-03 | -9.57E-07 | 20 | 5.50E-01 | 1 | 7.80E-02 |
90 | 3.10E-01 | 0 | 1.45E-01 | -1.60E-04 | -1.60E-04 | 5 | -400 | 1.86E-03 | 6.91E-05 | -4.09E-07 | 25 | 5.10E-01 | 1 | 3.75E-02 | ||
120 | 6.50E-01 | 0 | 8.84E-03 | -1.18E-04 | -2.35E-09 | 2 | -400 | 1.96E-03 | 1.07E-04 | -1.15E-07 | 19 | 4.80E-01 | 1 | 7.32E-02 |
3.3 Discussion
On the one isofrequency curve of combination R, H, any pair of values with R, H has the same return period. However, the flooding potential of these R, H combinations is not uniform. Even within the same R, H combination, the roles of R and H in causing flooding at different locations within the drainage system vary. Specifically, near the drainage outlet, the flooding impact is primarily influenced by the H level. Conversely, at points farther from the outlet, changes in H do not significantly alter the flooding, with R playing a more critical role.
Therefore, to design a drainage system with a specific return period, after selecting the rain gauge station graph for the area, one must calculate multiple different R, H combinations taken from the isofrequency curve with the same recurrence interval. The R, H combination resulting in the highest flooding will be used as the design combination.
Therefore, to design a drainage system with a specific return period, after selecting the rain gauge station diagram for the area, one must calculate a trial with several combinations R, H taken at different locations on the isofrequency curve. The combination R, H that produces the highest flood will be used as the design.
4. CONCLUSIONS
This study introduces a novel methodology for constructing isofrequency curves that characterize the return periods of rainfall-water level R, H combinations, with direct applications for urban drainage planning and flood risk assessment. By applying this approach to five rain gauge stations in the inner-city area of Ho Chi Minh City, we have developed a set of isofrequency curves that provide critical insights into the relationship between rainstorm intensity and water level variations at drainage system outlets. These curves serve as essential tools for defining boundary conditions in hydrodynamic models, thereby enhancing the accuracy of urban flood simulations and optimizing the design of surface water drainage infrastructure.
Furthermore, integrating these isofrequency curves into drainage planning ensures compliance with national technical regulations on sewer overflow and flood recurrence cycles, contributing to improved flood resilience in rapidly urbanizing environments. The ability to assess and predict the joint probability of extreme rainfall and high-water levels allows for more effective decision-making in urban infrastructure planning, ultimately reducing flood risk and ensuring the sustainability of Ho Chi Minh City's drainage system. Future research should explore the integration of climate change projections and real-time hydrological data to further refine the applicability of these curves in dynamic urban environments.
ACKNOWLEDGMENTS
We acknowledge Ho Chi Minh City University of Technology (HCMUT), VNU-HCM for supporting this study.
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