Research Article Volume 1 Issue 3
Department of Civil and Environmental Engineering, University of Por1t Harcourt, Choba, Nigeria
Correspondence: Ify L Nwaogazie, Department of Civil and Environmental Engineering, University of Por1t Harcourt, Choba, Nigeria
Received: September 01, 2017  Published: September 15, 2017
Citation: Nwaogazie IL, Ekwueme MC. Rainfall intensitydurationfrequency (IDF) models for Uyo city, Nigeria. Int J Hydro. 2017;1(3):6366. DOI: 10.15406/ijh.2017.01.00012
Suitable rainfallIntensityDurationFrequency (IDF) models for Uyo City were developed using the statistical method of leastsquares. A total of 18 rainfall models were developed based on 10 years rainfall data. These models are categorically grouped into two sets. The first set contains 10 rainfall models which represent inverse relationship between rainfall intensities and duration for specified frequencies called IntensityDurationModels. The second set contains 8 rainfall models which represent rainfall intensities and frequencies for specified duration called IntensityFrequency Model. The two sets of models developed in this research recorded high and positive values of coefficient of correlation, an indication of good curve fitting with values of IntensityDuration Models ranging from 0.948 to 0.977 and intensity frequency model ranging from 0.641 to 0.938. These set of models developed will serve as valuable tools for predicting rainfall events in Uyo City as well as for use in the design of hydraulic structures, urban drainage and flood control.
Keywords: Rainfall modeling, Intensity, Duration, Frequency, Least squares, Uyo City
Rainfall models can be expressed mathematically to represent a relationship between Intensity, Duration & Frequency (IDF). An IDF model provides the basic probabilistic rainfall information for runoff estimation and establishes a statistical basis for design, using the return period as the measure of frequency of failure.^{1−4} The IDF curves are standard tools of hydrologic risk analysis and design. Rainfall of a place can be completely defined if the intensities, durations and frequencies of the various storms occurring at that place are known. A major problem in rainfall analysis is the interpretation of past record of rainfall events in terms of future probabilities of occurrence.^{5,6} Whenever an intense rain occurs, its magnitude and duration are generally known from the meteorological readings. This available data can be used to determine the frequencies of the various rains usually referred to as frequency analysis. The frequencydata for storms of various durations so obtained can be represented by IntensityDurationfrequency Curves. Due to unavailability of rainfall data in most developing countries like Nigeria, rainfall frequency studies are very limited with only few meteorological stations usually at airport across the country. All stations where records exist, because of equipment failures continuous records for upwards of 30 years is very rare.^{5} Thus, the need for engineers to analyze available data more critically for proper planning and design of hydraulic structures is very necessary.^{7,8 }The aim of this study is to develop and calibrate two sets of rainfall models using observed rainfall amounts and durations. The first is the intensity duration model of quotient formula while the second is the intensityreturn period model of power law. The developed models are to find great application in design of stormwater drains, culverts, erosion protection structures, etc.
Study area: Uyo city is the state capital of Akwa Ibom, strategically located within the oil rich Niger Delta Southern of Nigeria and bounded to the South by Atlantic Ocean. It has coordinates of latitude 5^{0}2’N and longitude 7^{0}55’E and it is situated at elevation 45 meters above mean sea level (Figure 1).
Data collection: The rainfall data for Uyo City were obtained from the archives of the meteorological observatory, Department of Oceanography and Regional Planning, University of Uyo. The rainfall data obtained covered the period between 2002 and 2011.
Data analysis: Selected rainfall durations of 15, 30, 45, 60, 90, 120, 180 and 300 minutes were extracted from the rainfall data and the maximum intensities computed from Equation (1).
$i\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{A}{t}\text{}$ (1)
Where i: Intensity in mm/hr; A: Amount of rainfall in mm; t: Duration of rainfall in hours
The resulting intensity values evaluated from Equation (1) are ranked in decreasing order of magnitude such that the largest value is assigned 1. Return period is then computed using the Weilbull’s plotting position formula^{9} as:
$P\text{\hspace{0.17em}}\left({x}_{m}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{m}{n\text{\hspace{0.17em}}+\text{\hspace{0.17em}}1}$ (2)
Where: $P\text{\hspace{0.17em}}\left({x}_{m}\right)\text{\hspace{0.17em}}\text{\hspace{0.17em}}$ Probability of exceedence of variate x_{m}_{}; m: Rank of descending values with largest equal to 1; n: Number of years of record
Rainfall intensities and durations are tabulated against selected frequencies of 1.1, 1.22, 1.38, 1.57, 1.83, 2.2, 2.75, 3.7, 5.5 and 11 years. Next, the inverses of intensities were obtained to establish the relationship between rainfall intensity and duration for specified frequency.
Intensityduration quotient models: Frequencies of rainfall events calculated as the reciprocal of their corresponding probabilities are as presented in Table 1. Many formulas have been proposed in the past to fit rainfallIntensityDurationFrequency Curves, but most of them are in a form with intensity inversely proportional to duration.^{5‒10} The quotient model was used for the development of relationship between intensity of rainfall and its duration at specific return period. The quotient model is given by^{9 }as;
$i\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{a}{t\text{\hspace{0.17em}}+\text{\hspace{0.17em}}b}$ (3)
Where i: rainfall intensity (mm/hr); t: rainfall duration (hr); a, b: regional constants
Equation (3) is linearized by taking its reciprocal to become Equation (4):
$y={a}_{0}+{a}_{1}x$ (4)
Where: $y=\text{\hspace{0.17em}}1/i\text{\hspace{0.17em}};$
${a}_{0}=\text{\hspace{0.17em}}b/a;\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{a}_{1}=\text{\hspace{0.17em}}1/a.\text{\hspace{0.17em}}$
Equation (4) was solved using regression analysis via XLSTAT to produce the intensity duration models shown in Table 2. The rainfall intensity duration curves are shown in Figure 2. Apparently, the quotient model was calibrated by using the rainfall data on intensities and durations (Table 1) to obtain the regional constants of and in Equation (3) as presented in Table 3.
Intensityreturn period power models: The power model was employed in developing relationship between rainfall intensity and return period at specified duration. The power model^{5 }is given by:
$i=a{R}^{b}$ (5)
Where I, rainfall intensity (mm/hr); R, return period (years); &, regional constants
Equation (5) is linearized by logarithmic approach, viz;
$\mathrm{log}i=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{log}\text{\hspace{0.17em}}\text{\hspace{0.17em}}a\text{\hspace{0.17em}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}b\mathrm{log}\text{\hspace{0.17em}}\text{\hspace{0.17em}}R$ (6)
Where: ${a}_{1}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}b\text{\hspace{0.17em}}\text{\hspace{0.17em}}and\text{\hspace{0.17em}}\text{\hspace{0.17em}}{a}_{0}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\mathrm{log}\text{\hspace{0.17em}}a$
Regression analysis was equally employed to solve and calibrate Equation (6) using data from (Table 1) to produce the rainfall intensityfrequency models shown in Table 2. The Intensity Frequency Curves are shown in Figure 3.
S/No 
Rainfall intensity, mm/hr 







Ranking 
Frequency 


duration, minute 









15 
30 
45 
60 
90 
120 
180 
300 

1 
172.573 
124.663 
86.427 
121.83 
87.0773 
61.1585 
56.211 
41.9269 
1 
11 
2 
163.409 
119.978 
82.8496 
112.5281 
85.0476 
59.8105 
53.0503 
38.80302 
2 
5.5 
3 
151.708 
104.28 
78.864 
100.342 
83.4075 
57.7255 
51.777 
38.2766 
3 
3.7 
4 
139.422 
98.4352 
73.3573 
91.2015 
80.5089 
54.0191 
51.1037 
35.9584 
4 
2.75 
5 
128.458 
92.7854 
68.7327 
85.167 
79.2747 
52.0555 
49.9207 
35.4334 
5 
2.2 
6 
119.173 
82.7512 
64.6732 
79.2391 
74.2752 
49.7316 
49.1423 
30.06 
6 
1.83 
7 
108.857 
74.9842 
59.8449 
70.5582 
65.5146 
48.1305 
47.1733 
27.9932 
7 
1.57 
8 
98.6088 
67.3948 
56.9279 
62.77 
60.8361 
44.1305 
46.2993 
26.2766 
8 
1.38 
9 
51.1012 
40.5122 
34.7056 
30.1114 
34.9794 
24.611 
24.4176 
17.9222 
9 
1.22 
10 
33.258 
22.2692 
19.1185 
18.3389 
19.113 
10.76755 
9.7309 
4.9919 
10 
1.1 
Table 1 Rainfall ranking and corresponding intensity
S/No 
Duration 
Intensity frequency model* 
Parameters ± 
1 
15 
$i\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{}89.4829{R}^{0.3374}$ 
CC: 0.920 
GF: 0.846 

2 
30 
$i\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}60.1672{R}^{0.3758}$ 
CC: 0.914 
GF: 0.836 

3 
45 
$i\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{}53.804{R}^{0.2396}$ 
CC: 0.641 
GF: 0.886 

4 
60 
$i\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{}56.1293{R}^{0.3884}$ 
CC: 0.929 
GF: 0.864 

5 
90 
$i\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{}58.7314{R}^{0.2173}$ 
CC: 0.850 
GF: 0.722 

6 
120 
$i\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{}42.1231{R}^{0.1952}$ 
CC: 0.902 
GF: 0.821 

7 
180 
$i\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=44.4672{R}^{0.1090}$ 
CC: 0.938 
GF: 0.881 

8 
300 
$i\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}17.9242{R}^{0.4824}$ 
CC: 0.703 



GF: 0.494 
Table 2 Summary of all intensityreturn period power models
*IntensityFrequency models same as IntensityReturn period power model.
Regression parameters: CC, Coefficient of Correlation; GF, Goodness of Fit
S/N 
Frequency 
Intensity duration model 
Parameters ± 
1 
11 
$i\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{16393.4}{t\text{\hspace{0.17em}}+\text{\hspace{0.17em}}106.213}$ 
CC: 0.948 
$i\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{15060.2}{t\text{\hspace{0.17em}}+\text{\hspace{0.17em}}99.247}$ 
GF: 0.898 

2 
5.5 
CC: 0.970 

GF: 0.942 

3 
3.7 
$i\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{15479.9}{t\text{\hspace{0.17em}}+\text{\hspace{0.17em}}114.721}$ 
CC: 0.969 
GF: 0.939 

4 
2.8 
$i\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{14881}{t\text{\hspace{0.17em}}+\text{\hspace{0.17em}}118.601}$ 
CC: 0.966 
GF: 0.934 

5 
2.2 
$i\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{15432.1}{t\text{\hspace{0.17em}}+\text{\hspace{0.17em}}135.185}$ 
CC: 0.959 
GF: 0.920 

6 
1.8 
$i\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{14513.8}{t\text{\hspace{0.17em}}+\text{\hspace{0.17em}}135.849}$ 
CC: 0.958 
$i\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{12285}{t\text{\hspace{0.17em}}+\text{\hspace{0.17em}}119.533}$ 
GF: 0.918 

7 
1.57 
CC: 0.963 

GF: 0.928 

8 
1.38 
$i\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{11890.6}{t\text{\hspace{0.17em}}+\text{\hspace{0.17em}}145.66}$ 
CC: 0.954 
GF: 0.910 

9 
1.22 
$i\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{8695.65}{t\text{\hspace{0.17em}}+\text{\hspace{0.17em}}190.87}$ 
CC: 0.955 
$i\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{1776.2}{t\text{\hspace{0.17em}}+\text{\hspace{0.17em}}34.94}$ 
GF: 0.913 

10 
1.1 

CC: 0.977 



GF: 0.955 
Table 3 Uyo City rainfall intensityduration models for specified frequencies
Rainfall intensity is given in mm/hr and duration in minutes
Regression parameters: CC, Coefficient of Correlation; GF, Goodness of Fit
The first set of rainfall models involved finding solution of the quotient formula of Equation (3) using available rainfall intensity duration data for specified frequencies of 1.111 years. A total of 10 rainfall models were developed as presented in Table 3 with high and positive values of coefficient of correlations of 0.948 to 0.977. The second set of rainfall models involved finding solution of the power model of Equation (5) using rainfall intensities frequency data for specified durations of 15300 minutes. A total of 8 models were obtained as presented in Table 2 with moderate to high positive values of coefficients of correlation of 0.641 to 0.938 indicating good curve fittings. The plots of intensities against duration for the specified frequencies in Figure 2 and intensities against frequencies for the specified durations in Figure 3 formed a good basis for the predicted results. On the basis of these plots, Uyo city rainfall pattern can be characterized by the following observations. a. The curves are parallel; they do not cross each other; b. High and positive values of coefficients of correlation indicate best curve fittings; c. Higher return periods correspond with higher rainfall intensities for a given duration.^{11}
Based on the results of this study the following conclusion can be drawn:
None.
None.
©2017 Nwaogazie, et al. This is an open access article distributed under the terms of the, which permits unrestricted use, distribution, and build upon your work noncommercially.